Add files using upload-large-folder tool

.gitattributes

@@ -181,3 +181,6 @@ CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf filter=lfs diff=lfs merge=lfs -tex
 mtE3T4oBgHgl3EQf6gu4/content/2301.04791v1.pdf filter=lfs diff=lfs merge=lfs -text
 JdAyT4oBgHgl3EQffviy/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf filter=lfs diff=lfs merge=lfs -text
+V9AzT4oBgHgl3EQfJ_vP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+x9FJT4oBgHgl3EQfhSy6/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf filter=lfs diff=lfs merge=lfs -text

29AyT4oBgHgl3EQfb_di/content/tmp_files/2301.00273v1.pdf.txt

@@ -0,0 +1,1110 @@
+arXiv:2301.00273v1  [math.PR]  31 Dec 2022
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+PETER B¨URGISSER
+Abstract. Consider a system f1(x) = 0, . . . , fn(x) = 0 of n random real polynomials in n
+variables, where each fi has a prescribed set of exponent vectors described by a set Ai ⊆ Zn
+of cardinality ti, whose convex hull is denoted Pi. Assuming that the coefficients of the fi are
+independent standard Gaussian, we prove that the expected number of zeros of the random
+system in the positive orthant is at most (2π)− n
+2 f0 (t1 − 1) . . . (tn − 1). Here f0 denotes the
+number of vertices of the Minkowski sum P1 + . . . + Pn. We also derive a better bound in
+the unmixed case where all supports Ai are equal, improving upon [8]. All arguments equally
+work for real exponent vectors.
+1. Introduction
+In many applications, we want to understand or find the (positive) real solutions of a system
+of multivariate polynomial equations, e.g., see [11, 16, 29]. Bezout’s theorem, which bounds
+the number complex zeros in terms of degrees, usually highly overestimates the number of real
+zeros. This can be already seen from Descartes’ rule of signs [10, p. 42], which implies that
+a real univariate polynomial with t terms has at most t − 1 positive zeros. In 1980, Khovan-
+skii [19] obtained a far reaching generalization of Descartes’ rule. He showed that the number
+of nondegenerate1 positive solutions of a system f1(x) = 0, . . . , fn(x) = 0 of n real polynomial
+equations in n variables is bounded only in terms of n and the number t of distinct exponent
+vectors occurring in the system. This result in fact allows for any real exponents. Following
+Kushnirenko, one speaks of fewnomial systems, with the idea that the number t of terms is small,
+see [20].
+Understanding the complex zeros of fewnomial systems is much simpler: the famous BKK-
+Theorem [2, 24] states that for given finite supports A1, . . . , An ⊆ Zn and Laurent polynomials
+fi(x) = �
+a∈A ci(a)xa1
+1 · · · xan
+n
+with generic complex coefficients ci(a), the number of complex
+solutions in (C×)n of a corresponding system f1(x) = 0, . . . , fn(x) = 0 is given by n! times the
+mixed volume of the Newton polytopes P1, . . . , Pn, where Pi is defined as the convex hull of Ai.
+Note that the number of real zeros has little to do with the metric properties of Pi: indeed,
+replacing Ai by a multiple mAi amounts to substituting xi by xm
+i (m > 0). Clearly, this does not
+change the number of positive real zeros of a fewnomial system, however Pi has been replaced
+by mPi.
+The bound on the number of real zeros obtained by Khovanskii is exponential in the number t.
+It is widely conjectured that this bound is far from optimal: in fact it is conjectured [27] that for
+fixed n, the number of nondegenerate positive solutions of a fewnomial system with t exponent
+vectors is bounded by a polynomial in t. Quite surprisingly, this question is open even for n = 2!
+Date: January 3, 2023.
+2010 Mathematics Subject Classification. 60D05, 14P99.
+Key words and phrases. fewnomials, random polynomials, real algebraic geometry, sparsity.
+Supported by the ERC under the European Union’s Horizon 2020 research and innovation programme (grant
+agreement no. 787840).
+1i.e., the Jacobian of the system does not vanish at the zero.
+1
+
+2
+PETER B¨URGISSER
+For results in special cases, we refer to [4, 1, 29, 22, 23, 3]. Moreover, there is a very interesting
+connection to complexity theory [21, 5].
+Given this state of affairs of real fewnomial theory, a possible way to advance is to ask
+what happens in generic situations. This can be made formal by considering random fewnomial
+systems. Fix supports A1, . . . , An ⊆ Zn of cardinality t1, . . . , tn, respectively, and consider a
+system of n random polynomials fi(x) as above, but now the coefficients ci(a) are assumed
+to be independent standard Gaussian. Let us denote by N(A1, . . . , An) the expectation of the
+number of nondegenerate positive real zeros of such system. Actually, we work in more generality,
+allowing any subsets Ai of Rn; see Section 4. There is a large literature on the real zeros of
+random polynomials: we refer to [8] for references.
+In [8] it was proven that N(A, . . . , A) ≤ 21−n�t
+n
+�
+. The main result of the present paper is an
+extension of this to the mixed case, where the fewnomials may have have different supports Ai.
+Our bound depends on the combinatorial structure of the Minkowski sum P1 + . . .+ Pn through
+the number of its vertices. We remark that our proof is quite different from the one in [8], which
+was quite indirect.
+Theorem 1.1. If Pi denotes the convex hull of the finite nonempty set Ai ⊆ Rn of cardinality ti,
+for i = 1, . . . , n, then
+N(A1, . . . , An) ≤ (2π)− n
+2 f0 (t1 − 1) . . . (tn − 1).
+Here f0 denotes the number of vertices of the Minkowski sum P := P1 + . . . + Pn.
+The bound in this theorem looks similarly to the one in a conjecture attributed to Kush-
+nirenko, which states that the number of positive nondegenerate zeros is always bounded by
+(t1 − 1) · · · (tn − 1). However, this was disproved in [15].2
+In the unmixed situation, where all supports are equal to A, it is well known [12] that the
+expected number of positive zeros can be expressed by the volume of the image of the Veronese
+like map Rn
+>0 → P(RA) sending x to [xa]a∈A. This is a consequence of the kinematic formula
+for real projective spaces. In the mixed situation, there is no such simple characterization: one
+has to work with the more complicated kinematic formula for products of projective spaces
+(Theorem 3.2) that we derive from [17, 9]. After passing to exponential coordinates w = log x,
+we bound the resulting integral over Rn with a strategy inspired by the theory of toric varieties.
+The normal fan of the polytope P affords a decomposition of Rn into the normal cones C at
+the vertices of P. The resulting integral over C can be bounded in terms of the characteristic
+function of the dual cone of C. Finally, an explicit a priori bound on the characteristic function
+(Proposition 2.4) completes the argument.
+1.1. The univariate case and a conjecture. The univariate case (n = 1) was settled by
+Jindal et al. [18]. They showed that for any subset S ⊆ R of cardinality t, we have
+(1.1)
+N(S) ≤ 2
+π
+√t − 1.
+Moreover, they constructed a sequence St ⊆ Z of supports of cardinality t with N(St) ≥ c
+√
+t for
+some constant c > 0. Consider for t1, . . . , tn ≥ 1 the supports A1 := St1 × 0 . . . × 0, . . . , An :=
+0 × . . . × 0 × Stn. These supports describe a system of n equations, where the ith equation
+depends on xi only. Therefore, N(A1, . . . , An) = N(St1) · · · N(Stn), which with the above leads
+to the lower bound
+(1.2)
+N(A1, . . . , An) ≥ cn√t1 · · · tn.
+2This conjecture was never published by Kushnirenko and apparently, he did not believe in it.
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+3
+We complement this by showing that for any A = S1 ×. . .×Sn in product form, the expectation
+N(A, . . . , A) can be expressed in terms of the N(Si) as follows.
+Proposition 1.2. If A = S1 × . . . × Sn for finite Si ⊆ R, then
+N(A, . . . , A) =
+πn
+vol(Pn) N(S1) · · · N(Sn).
+We conjecture that the lower bound (1.2) is optimal in the following sense.
+Conjecture 1. Let Ai ⊆ Rn be finite nonempty sets of cardinality ti with convex hull Pi, for
+i = 1, . . . , n. We denote by f0 the number of vertices of P1 + . . . + Pn. Then
+N(A1, . . . , An) ≤ κ(f0)√t1 · · · tn
+for some function κ : N → N. In particular, N(A, . . . , A) ≤ κ(f0) t
+n
+2 for A ⊆ Rn of cardinality t.
+In the special case A = S1 × . . . × Sn, by combining (1.1) with Proposition 1.2, we obtain
+N(A, . . . , A)vol(Pn) ≤ 2n√
+t, which is much smaller than what Conjecture 1 predicts.
+1.2. Improvement in unmixed case. We can improve the dependence on n in the bound of
+Theorem 1.1 in the case where all supports are equal.
+Theorem 1.3. For A ⊆ Rn of cardinality t ≥ 1 with convex hull P and f0 vertices, we have
+N(A, . . . , A) ≤
+1
+vol(Pn) f0
+�t−1
+n
+�
+.
+To compare this with the bound N(A, . . . , A) ≤ 21−n�t
+n
+�
+from [8], note that vol(Pn)−1 =
+Γ( n+1
+2 )π− n+1
+2 . Hence, for n → ∞, the new bound goes asymptotically faster to zero than the
+old one. However, the bound in [8] also holds for nonstandard centered Gaussian coefficients
+ci(a) ∼ N(0, σi(a)2).
+1.3. Location of zeros. We finish with a result on the typical location of the zeros. It is well
+known that for random real polynomials, the positive reals zeros x tend to accumulate around 1:
+see [12] for the dense and [18] for the sparse case. This means that w = log x accumulates
+around 0. We generalize this to multivariate systems as follows.
+Theorem 1.4. Fix a finite supports A1 . . . , An ⊆ Rn and consider a random system (4.3) with
+independent standard Gaussian coefficients ci(a) for the stretched supports mAi, where m ∈ Z>0.
+Fix ε > 0. Then the probability that the system has a zero w ∈ Rn with ∥w∥ > ε goes to zero, as
+m → ∞.
+2. Preliminaries
+2.1. A metric property of charts of real projective space. Consider the real projective
+space Pm. We shall identify the tangent space T[y]Pm at a point [y] := [y0 : . . . : ym] with Ry⊥.
+The standard Riemannian metric on Pm is defined by ⟨v, w⟩[y] := ∥y∥−2⟨v, w⟩ for v, w ∈ Ry⊥.
+We denote by Py the orthogonal projection onto Ry⊥.
+Consider the affine chart (Pm)y0̸=0 → Rm, which maps [y0 : . . . : ym] to y−1
+0 (y1, . . . , ym). Its
+inverse is given by
+π: Rm → (Pm)y0̸=0, (y1, . . . , ym) �→ [1 : y1 : . . . : ym].
+By [6, Lemma 14.8], the derivative of π at y′ := (y1, . . . , yn) satisfies Dy′π = ∥π(y′)∥−1Py. From
+this we conclude for the spectral norm
+(2.1)
+∥Dy′π∥ ≤ ∥π(y′)∥−1 ≤ 1.
+
+4
+PETER B¨URGISSER
+2.2. On the quantity σ. The relative position of two subspaces of a Euclidean vector space E
+can be quantified by a volume like quantity, which is crucial in the study of integral geometry
+in homogeneous spaces; see [17] and [9, §3.3]. To define this quantity, note first that there is an
+induced inner product on the exterior algebra Λ(E) given by (see [9, (2.1)])
+⟨v1 ∧ · · · ∧ vk, w1 ∧ · · · ∧ wk⟩ = det(⟨vi, wj⟩)1≤i,j≤k.
+More concretely, ∥v1 ∧ . . . ∧ vn∥ = | det[v1, . . . , vn]|, where [v1, . . . , vn] denotes the matrix with
+columns vi ∈ E = Rn
+Let V, W be linear subspaces of E of complementary dimensions. We define (see [9, (3.3)])
+(2.2)
+σ(V, W) := ∥v1 ∧ . . . ∧ vk ∧ w1 ∧ . . . ∧ wm∥ ∈ [0, 1],
+where v1, . . . , vk and w1, . . . , wm are orthonormal bases of V and W, respectively.
+Clearly,
+σ(V, W) = σ(W, V ). The extreme cases are: σ(V, W) = 0 iff V ∩ W ̸= 0 and σ(V, W) = 1 iff v
+and W are orthogonal.
+Proposition 2.1. We have σ(V ⊥, W ⊥) = | det p |, if p: V ⊥ → W denotes the restriction of the
+orthogonal projection E → W to V ⊥. Moreover, σ(V, W) = σ(V ⊥, W ⊥).
+Proof. Let ν1, . . . , νm be an orthonormal basis of V ⊥. We decompose νi = ν′
+i + ν′′
+i according
+to E = W ⊕ W ⊥. Then p(νi) = ν′
+i and | det p | = ∥ν′
+1 ∧ . . . ∧ ν′
+m∥. If ω1, . . . , ωk denotes an
+orthonormal basis of W ⊥, we have
+σ(V ⊥, W ⊥) = ∥ν1 ∧ . . . ∧ νm ∧ ω1 ∧ . . . ∧ ωk∥ = ∥ν′
+1 ∧ . . . ∧ ν′
+m ∧ ω1 ∧ . . . ∧ ωk∥ = ∥ν′
+1 ∧ . . . ∧ ν′
+m∥,
+the last equality holding since the span of the ν′
+i equals W, which is orthogonal to the span of
+the wj, which is W ⊥. This proves σ(V ⊥, W ⊥) = | det p |.
+For the second assertion, we use that | det p | = | det q |, where q: W ⊥ → V denotes the
+restriction of the orthogonal projection E → V to W ⊥, see [7, Lemma 5.4].
+□
+Clearly, the definition (2.2) can be extended to more than two subspaces; see [9, (3.5)]. But if
+W = W1 ⊕. . .⊕Wn is an orthogonal decomposition, we can reduce to the case of two subspaces:
+(2.3)
+σ(V, W1, . . . , Wn) = σ(V, W1 + . . . + Wn).
+This is a consequence of [9, Lemma A.6].
+2.3. Characteristic function of convex cones. We prove here an priori upper bound on the
+characteristic function of a convex cone, which is a key ingredient in the proof of Theorem 1.1.
+A convex cone C ⊆ Rn is called proper if it is n-dimensional and pointed, i.e., contained in a
+halfspace. It is well known that a convex C ⊆ Rn is proper iff its dual cone
+C∗ := {x ∈ Rn | ∀y ∈ C ⟨x, y⟩ ≥ 0}
+is proper. Let g ∈ GL(n, R). Then K := g(C) is a proper cone and gT (K∗) = C∗. We denote
+by int(C) the interior of C.
+We assign to a proper cone C ⊆ Rn the function
+(2.4)
+vC : int(C∗) → R>0, vC(x) :=
+�
+C
+e−⟨x,y⟩ dy.
+One calls vC the characteristic function (or Koszul-Vinberg characteristic) of C∗. It is a useful
+analytic tool for investigating convex cones, e.g., see [13, I.3] and [14]. E.g., Rn
+>0 is self dual and
+vRn
+>0(x) = (x1 · . . . · xn)−1 for x ∈ Rn
+>0.
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+5
+The homogeneity property vC(tx) = t−nvC(x) for t > 0, x ∈ int(C∗) is immediate to check.
+Moreover, the transformation formula implies the following invariance property: if g ∈ GL(n, R)
+and K := g(C), then gT (K∗) = C∗ and
+(2.5)
+vK(z) = | det g| vC(gT z)
+for z ∈ int(K∗).
+Remark 2.2. The function log vC is strictly convex and essentially equals Nesterov and Ne-
+mirowski’s universal self-concordant barrier function [26, §2.5], see [14] for the proof.
+The following result is well known, e.g., see [14, Thm. 4.1]. We provide the simple proof for
+the sake of completeness.
+Lemma 2.3. For x ∈ int(C∗) we have
+vC(x) = n! vol
+�
+y ∈ C | ⟨x, y⟩ ≤ 1
+�
+.
+Proof. We fix x ∈ int(C∗). For t ≥ 0 we define the n − 1-dimensional slice
+Ct :=
+�
+y ∈ C | ⟨x, y⟩ = t∥x∥
+�
+.
+By Fubini, we get
+vC(x) =
+�
+C
+e−⟨x,y⟩ dy =
+� ∞
+0
+voln−1(Ct)e−t∥x∥dt = voln−1(C1)
+� ∞
+0
+tn−1e−t∥x∥dt.
+Note that
+� ∞
+0
+tn−1e−t∥x∥dt =
+1
+∥x∥n
+� ∞
+0
+sn−1e−sds =
+1
+∥x∥n Γ(n) = (n − 1)!
+∥x∥n .
+Moreover, we have
+voln−1(C1) = n voln
+�
+y ∈ C | ⟨x, y⟩ ≤ ∥x|
+�
+= n ∥x∥n voln
+�
+y ∈ C | ⟨x, y⟩ ≤ 1
+�
+.
+It follows that
+vC(x) = n ∥x∥n voln
+�
+y ∈ C | ⟨x, y⟩ ≤ 1
+� (n − 1)!
+∥x∥n
+= n! voln
+�
+y ∈ C | ⟨x, y⟩ ≤ 1
+�
+,
+completing the proof.
+□
+The following technical result is essential for the proof of Theorem 1.1.
+Proposition 2.4. Let C ⊆ Rn be a proper cone and b1, . . . , bn ∈ C∗. Then we have
+∥b1 ∧ . . . ∧ bn∥ · vC(b1 + . . . + bn) ≤ 1.
+This bound is optimal.
+Proof. We denote by cone(b1, . . . , bn) ⊆ C∗ the convex cone generated by b1, . . . , bn. Without
+loss of generality, we may assume that b1, . . . , bn ∈ C∗ is a basis of Rn. Let b∗
+1, . . . , b∗
+n denote its
+dual basis, that is ⟨b∗
+i , bj⟩ = δij. In matrix terminology, [b∗
+1, . . . , b∗
+n]T [b1, . . . , bn] = In, hence
+(2.6)
+det[b∗
+1, . . . , b∗
+n] det[b1, . . . , bn] = ±1.
+Moreover, the definition of the dual basis implies that cone(b∗
+1, . . . , b∗
+n) is the dual cone of
+cone(b1, . . . , bn). Therefore, by duality, we get
+C ⊆ cone(b1, . . . , bn)∗ = cone(b∗
+1, . . . , b∗
+n).
+Put d := b1 + . . . + bn and let y ∈ C such that ⟨d, y⟩ ≤ 1. Since C ⊆ cone(b∗
+1, . . . , b∗
+n), we can
+write y = �
+i tib∗
+i with ti ≥ 0. Moreover �
+i ti = ⟨d, y⟩ ≤ 1. Thus we have shown the inclusion
+{y ∈ C | ⟨d, y⟩ ≤ 1} ⊆ conv{0, b∗
+1, . . . , b∗
+n}.
+
+6
+PETER B¨URGISSER
+This implies the inequality of volumes
+voln{y ∈ C | ⟨d, y⟩ ≤ 1} ≤ volnconv{0, b∗
+1, . . . , b∗
+n} = 1
+n!| det[b∗
+1, . . . , b∗
+n]|.
+Multiplying with n! | det[b1, . . . , bn]|, using (2.6) and taking into account Lemma 2.3, the assertion
+follows.
+The optimality is attained for C = Rn
++ and bi = diei with di > 0. Indeed, we have
+| det[b1, . . . , bn] |vC(d) = d1 · . . . · dn (d1 · . . . · dn)−1 = 1.
+□
+2.4. Vertices and normal fan of sums of polytopes. We recall here some basic facts about
+polytopes and their normal fans; see [30, §7.1] for more details.
+Let P ⊆ Rn be a full-dimensional polytope and v be a vertex of P. The cone Pv of P at v is
+defined as the convex cone generated by P − v. It is a proper cone. The dual cone of Pv, also
+called the inner normal cone of P at v, is defined as
+P ∗
+v := {y ∈ Rn | ∀x ∈ P ⟨x, y⟩ ≥ 0}.
+The cone P ∗
+v is also proper. The union over all P ∗
+v equals Rn. Moreover, for v1 ̸= v2, we have
+dim(P ∗
+v1 ∩ P ∗
+v2) < n. In fact, the P ∗
+v are the n-dimensional cones of the normal fan of P.
+We will need the following result.
+Lemma 2.5. Let P1, . . . , Pn be polytopes in Rn. There is an injective map
+Vert(P1 + . . . + Pn) → Vert(P1) × . . . Vert(Pn), v �→ (v1, . . . , vn)
+satisfying v = v1 + . . . + vn. Moreover, if we denote by Πi the cone of Pi at the vertex vi, then
+Π := Π1 + . . . + Πn is the cone of P1 + . . . + Pn at the vertex v1 + . . . + vn. In particular,
+Π∗ = Π∗
+1 ∩ . . . ∩ Π∗
+n.
+Proof. For the following, see [28, §1.7]. To a nonzero weight ω ∈ Rn we assign the face of Pi,
+given by
+F(Pi, ω) :=
+�
+w ∈ Rn | ⟨w, ω⟩ = min
+w′∈Pi⟨w′, ω⟩
+�
+.
+We have (see [28, Thm. 1.7.5])
+F(P1 + . . . + Pn, ω) = F(P1, ω) + . . . + F(Pn, ω).
+Suppose that F(P1 + . . . + Pn, ω) = {v} is a vertex. Then all F(Pi, ω) = {vi} are vertices
+and v = v1 + . . . vn.
+It is easy to see that the vi are uniquely determined by v.
+The map
+v �→ (v1, . . . , vn) is as required. The remaining assertions are clear.
+□
+3. Random intersections in products of projective spaces
+3.1. The kinematic formula. We specialize here the general kinematic formula for homo-
+geneous spaces from [9, Thm. A.2] to the case of products of real projective spaces (Theo-
+rem 3.2). For this purpose, we define the average scaling factor and we explain how to bound it
+in Lemma 3.5.
+Consider the product Ω := Pm1 × · · · × Pmn of real projective spaces. The product G :=
+O(m1 +1)×· · ·×O(mn +1) of orthogonal groups acts transitively on Ω. So Ω is a homogeneous
+space and we have an induced transitive action of G on the tangent bundle of Ω. We focus on
+the special hypersurfaces H1, . . . , Hn of Ω of the following shape
+(3.1)
+H1 := Pm1−1 × Pm2 × · · · × Pmn, . . . , Hn := Pm1 × Pm2 · · · × Pmn−1.
+They are determined upon selecting hyperplanes Pmi−1 in Pmi. Our goal is to investigate the
+average cardinality of the intersection Z ∩H1 ∩. . .∩Hn of an n-dimensional smooth submanifold
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+7
+Z ⊆ Ω with random Hi corresponding to independently chosen uniform random hyperplanes
+in Pmi.
+Fix a distinguished point ω ∈ Ω and denote by K the stabilizer group of ω.
+E.g., take
+ωi = [1 : 0 . . . : 0] for all i. Notice that we have an induced action of K on the tangent space
+T := TωΩ, which we can identify with the standard action of K = O(m1) × · · · × O(mn) on
+T = Rm1 × · · · × Rmn. This induces an action of K on the Grassmann manifold Gr(d, T ) of
+linear subspaces of T with codimension d. Note that this action is transitive if n = 1, but not
+for n ≥ 2.
+We assign to an n-dimensional smooth submanifold Z ⊆ Ω a map
+(3.2)
+Z → Gr(n, T )/K, z �→ KgNpZ
+as follows. For given p ∈ Z choose any g ∈ G such that gp = ω. The induced action of g maps
+the tangent space TpΩ to TωΩ = T . This transports the normal subspace NpZ ⊆ TpΩ of Z at p
+to gNpZ ⊆ T . Note that the K-orbit of the subspace gNpZ does not depend on the choice of g,
+which shows that the map (3.2) is well defined.
+We call the submanifold Z cohomogeneous if the map (3.2) is constant; see [9, A.5.1] and [25].
+For instance, a product Z = L1 × . . . × Ln of lines Li in Pmi is cohomogeneous: indeed, the
+map (3.2) sends any point p ∈ Z to the K-orbit of R × . . . × R.
+Definition 3.1. The average scaling factor function of the n-dimensional submanifold Z of
+Pm1 × · · · × Pmn is the function ¯σZ : Z → [0, 1] defined at p ∈ Z by
+¯σZ(p) := E Liσ(gNpZ, L1 × . . . × Ln),
+where g ∈ G satisfies gp = ω0, and the expectation is taken over uniformly random lines Li in
+T = Rm1 × · · · × Rmn (see (2.2) for the definition of σ).
+Note that due to the averaging over the K-orbit, the choice of g is irrelevant. The above
+definition is consistent with the one in [9, Def. A.1], since
+(3.3)
+σ(gNpZ, L1 × . . . × Ln) = σ(gNpZ, L1 × 0 × · · · × 0, . . . , 0 × · · · × 0 × Ln)
+by (2.3); indeed note that the n lines L1 × 0 × · · · × 0, ... are pairwise orthogonal.
+We introduce the notation
+ρn := E ∥x∥ =
+√
+2 Γ( n+1
+2 )
+Γ( n
+2 )
+≤ √n
+for standard Gaussian x ∈ Rn. We note that by [6, Lemma 2.25],
+(3.4)
+vol(Pmi−1)
+vol(Pmi)
+=
+1
+√π
+Γ( mi+1
+2
+)
+Γ( mi
+2 )
+=
+1
+√
+2π ρmi,
+We can now explicitly state the kinematic formula for products of real projective spaces.
+Theorem 3.2. For any n-dimensional submanifold Z of Pm1 × · · · × Pmn, we have
+E g∈G#(Z ∩ g1H1 ∩ . . . ∩ gnHn) = (2π)− n
+2 ρm1 · · · ρmn
+�
+Z
+¯σZ dZ,
+where the hypersurfaces Hi are defined in (3.1).
+Proof. If σK : Z × H1 × . . . × Hn → [0, 1] denotes the average scaling function from [9, Def. A.1],
+then [9, Thm. A.2] states that
+E g∈G#(Z ∩ g1H1 ∩ . . . ∩ gnHn) =
+1
+vol(Ω)n
+�
+Z×H1×...×Hn
+σK d(Z × H1 × . . . × Hn).
+
+8
+PETER B¨URGISSER
+By K-invariance and (3.3), we have σK(z, y1, . . . , yn) = ¯σZ(z) for all z ∈ Z and yi ∈ Hi.
+Therefore,
+E g∈G#(Z ∩ g1H1 ∩ . . . ∩ gnHn) = vol(H1) · · · vol(Hn)
+vol(Ω)n
+�
+Z
+¯σZ dZ.
+Finally, (3.4) gives
+vol(H1) · · · vol(Hn)
+vol(Ω)n
+=
+n
+�
+i=1
+vol(Pmi−1)
+vol(Pmi)
+= (2π)− n
+2 ρm1 · · · ρmn,
+which completes the proof.
+□
+Example 3.3. A product Z = L1 × . . . × Ln of lines Li is cohomogeneous. Theorem 3.2 implies
+that ¯σZ = (2/π)n/2(ρm1 · · · ρmn)−1.
+We shall focus on submanifolds Z arising as the image of an injective smooth map
+(3.5)
+ψ: U → Pm1 × · · · × Pmn, ψ(x) := (ψ1(x), . . . , ψn(x)),
+where the ψi : U → Pmi are smooth maps defined on an open subset U ⊆ Rn. Let us denote by
+Jψ(x) :=
+�
+det((Dxψ)T Dxψ)
+the absolute Jacobian of ψ at x. The transformation formula implies that
+(3.6)
+�
+Z
+¯σZ dZ =
+�
+U
+¯σZ(ψ(x))Jψ(x) dx.
+We next analyze the integrand on the right-hand side more closely.
+Lemma 3.4. Let x ∈ U and put Ti := Tψi(x)Pmi.
+Let λ1, . . . , λn be independent standard
+Gaussian linear forms on Ti. This defines the random linear forms λi ◦ Dxψi on Rn. Then
+ρm1 · · · ρmn ¯σZ(ψ(x))Jψ(x) = E λ1,...,λn ∥(λ1 ◦ Dxψ1) ∧ . . . ∧ (λn ◦ Dxψn)∥ .
+Proof. To simplify notation, we assume w.l.o.g. that p = ψ(x) is the distinguished point ω.
+We also identify Ti with Rmi. For ui ∈ Ti with ∥ui∥ = 1 consider the line Li = Rui and the
+orthogonal projection pi : Ti → Li, which is is given by pi(w) = µi(w)ui with the linear form on Ti
+defined by µi(w) := ⟨w, ui⟩. Thus the orthogonal projection pL : T1 × · · · × Tn → L1 × · · · × Ln
+is described by µ1, . . . , µn. This implies that
+(3.7)
+| det(pL ◦ Dxψ)| = ∥(µ1 ◦ Dxψ1) ∧ . . . ∧ (µn ◦ Dxψn)∥ .
+On the other hand, according to Proposition 2.1, we have
+σ(L1 × · · · × Ln, NpZ) = | det p′
+L|,
+where p′
+L : TpZ → L1 × · · · × Ln denotes the restriction of pL to TpZ. Applying the determinant
+to the composition of Dxψ with p′
+L, we get
+Jψ(x) | det p′
+L| = | det(pL ◦ Dxψ)|.
+By averaging over random lines Li, we deduce from the definition of ¯σZ and the above that
+Jψ(x)¯σZ(p) = Jψ(x) E Liσ(NpZ, L1 × · · · × Ln) = Jψ(x) E Li| det p′
+L| = E Li| det(pL ◦ Dψ)|.
+Finally, a standard Gaussian linear form on Ti is obtained as λi = riµi with independent random
+variables ri and ui, where ui is uniformly random in the unit sphere of Ti and r2
+i is χ2-distributed
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+9
+with mi degrees of freedom. Thus E ri = ρmi. Altogether, we obtain, using (3.7),
+ρm1 · · · ρmnJψ(x)¯σZ(p) = ρm1 · · · ρmnE | det(pL ◦ Dψ)|
+= ρm1 · · · ρmnE ∥(µ1 ◦ Dxψ1) ∧ . . . ∧ (µn ◦ Dxψn)∥
+= E ∥(λ1 ◦ Dxψ1) ∧ . . . ∧ (λn ◦ Dxψn)∥ ,
+which completes the proof.
+□
+3.2. Bounding the average scaling factor. In order to bound the quantity in Lemma 3.4,
+we use affine charts for the product of projective spaces. Denote by yi0, . . . , yimi coordinates
+for Pmi. Fix 0 ≤ ri ≤ mi for i = 1, . . . , n, and consider the inverse of the affine chart πiri : Rmi →
+(Pmi)yiri̸=0, see Subsection 2.1. We describe the maps ψi from (3.5) in these charts by smooth
+functions defined on open subsets of Rn,
+(3.8)
+ϕiri : Rn ⊇ Uiri → Rmi,
+satisfying ψi := πiri ◦ ϕiri. In order to simplify notation, we assume w.l.og. ri = 0 and write
+πi := πi0, ϕi := ϕi0. In these charts, the combined map ψ of (3.5) is represented by a map
+ϕ: U → Rm1 × · · · × Rmn, ϕ(x) = (ϕ1(x), . . . , ϕn(x))
+defined on some open subset U ⊆ Rn. We view the derivative M(x) := Dxϕ as a matrix of
+format (m1 + . . . + mn) × n with blocks Mi(x) := Dxϕi ∈ Rmi×n. For 1 ≤ ji ≤ mi, i = 1, . . . , n,
+we denote by M(x)j1,...,jn the n × n submatrix of M(x) obtained by selecting in the ith block
+the jith row.
+Lemma 3.5. Let x ∈ U be such that [yi] := ψi(x) ∈ (Pmi)y0̸=0 for all i. Then
+ρm1 · · · ρmn ¯σZ(ψ(x))Jψ(x) ≤
+�
+j1,...,jn
+| det M(x)j1,...,jn|,
+where the sum is over the n-tuples (j1, . . . , jn) ∈ [m1] × . . . × [mn].
+Proof. From ψi = πi ◦ ϕi we get Dψi = Dπi ◦ Dϕi, where we drop arguments for notational
+simplicity. Let λi : Ti → R be a linear form on Ti = Tψi(x)Pmi. Then, defining wi := λi ◦ Dπi,
+λi ◦ Dψi = λi ◦ Dπi ◦ Dϕi = wi ◦ Dϕi.
+If we identify λi ◦ Dxψi with a vector in Rn and wi with a vector in Rmi, then we have the
+matrix product of formats n × �
+i mi and �
+i mi × n,
+(3.9)
+R(x) :=
+
+
+(λ1 ◦ Dxψ1)T
+...
+(λn ◦ Dxψ1)T
+
+ =
+
+
+wT
+1
+0
+. . . 0
+0
+wT
+2
+. . . 0
+...
+...
+...
+0
+0
+wT
+n
+
+ ·
+
+
+M1(x)
+...
+Mn(x)
+
+ .
+Lemma 3.4 tells us that
+ρm1 · · · ρmn ¯σZ(ψ(x))Jψ(x) = E λi| det R(x))|,
+where the expectation is over independent standard Gaussian λi. Note that the resuling random
+vector wi := λi ◦ Dπi is not standard Gaussian anymore. However ∥Dπi∥ ≤ 1 by (2.1) and
+Lemma 3.6 below implies that E w2
+ij ≤ 1 for the jth component wij of wi.
+From Binet-Cauchy, we obtain from (3.9)
+(det R(x))2 =
+�
+j1,...,jn
+w2
+1j1 · · · w2
+njn(det M(x)j1,...,jn)2,
+
+10
+PETER B¨URGISSER
+where the sum is over all (j1, . . . , jn) ∈ [m1] × . . . × [mn]. Taking expectations yields
+E w(det R(x))2 ≤
+�
+j1,...,jn
+(det M(x)j1,...,jn)2.
+We conclude that
+E w| det R(x))| ≤
+�
+E w(det R(x))2� 1
+2 ≤
+�
+j1,...,jn
+| det M(x)j1,...,jn|,
+which completes the proof.
+□
+Lemma 3.6. Let A ∈ Rp×m with ∥A∥ ≤ 1. If y ∈ Rp is standard Gaussian, then the random
+variable z := yA satisfies E |zj|2 ≤ 1 for all j.
+Proof. From zj = �
+i yiaij we get z2
+j = �
+i,k yiykaijakj. Hence E z2
+j = �
+i a2
+ij. Finally, �
+i a2
+ij =
+∥A(ej)∥2 ≤ ∥A∥2 ≤ 1.
+□
+4. Mixed random fewnomial systems
+The goal here is to provide the proofs of the assertions in the introduction.
+Let us first
+introduce some notation.
+We assign to a real valued function c: A → R on a finite nonempty subset A ⊆ Rn the real
+analytic function FA,c : Rn → R
+(4.1)
+FA,c(w) :=
+�
+a∈A
+c(a)e⟨a,w⟩.
+In the special case where A consists of integer vectors, FA,c arises from the Laurent polynomial
+fA,c(x) = �
+a∈A c(a)xa by a substitution: FA,c(w) = fA,c(ew). Generally, we have the following
+equivariance property: for g ∈ GL(n, R) and b ∈ Rn,
+(4.2)
+FA+b,b.c(w) = e⟨b,w⟩FA,c(w),
+Fg(A),g.c(w) = FA,c(gT w),
+where (g.c)(a) := c(g−1a) and b.c(a) := c(a − b).
+Suppose now we have n such analytic functions encoded by ci : Ai → R, for i = 1, . . . , n.
+Throughout, we denote by ti the cardinality of Ai and by Pi its convex hull. We are interested
+in the number N of nondegenerate zeros w ∈ Rn of the system
+(4.3)
+FA1,c1(w) = 0, . . . , FAn,cn(w) = 0.
+Our goal is to study the expected number of nondegenerate zeros for random coefficient func-
+tions. More specifically, we denote by N(A1, . . . , An) the expectation of N, when all the coeffi-
+cients ci(a), for i ∈ [n] and ai ∈ Ai, are independent standard Gaussians. Clearly, N(A1, . . . , An)
+is invariant under permutations of the Ai. Also, N(A1, . . . , An) = 0 if ti = 1 for some i. More-
+over, we have N(A1, . . . , An) = 0 if dim(P1 + . . . + Pn) < n, see Lemma 4.2.
+Equation (4.2) implies the following invariance properties
+(4.4)
+N(A1 + b1, . . . , An + bn) = N(A1, . . . , An),
+N(g(A1), . . . , g(An)) = N(A1, . . . , An),
+where b1, . . . , bn ∈ Rn and g ∈ GL(n, R).
+Our main result is Theorem 1.1 stated in the introduction. Note that it gives the correct
+answer N(A1, . . . , An) = 0 if ti = 1 for some i.
+Example 4.1. In the case t1 = . . . = tn = 2, the Pi are segments. If they are linearly independent,
+P1 + . . . + Pn is a parallelepiped with 2n vertices. Thus, Theorem 1.1 gives N(A1, . . . , An) ≤
+(2/π)
+n
+2 . This can be easily verified directly as follows. Suppose Ai = {ai, bi}. We claim that
+N(A1, . . . , An) = 2−n if b1 − a1, . . . , bn − an are linearly independent. For showing this, by the
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+11
+invariance properties (4.4), it suffices to consider the case where Ai = {0, ei}. Then (4.3) amounts
+to the system ci(0) + ci(ei)ewi = 0, for i = 1, . . . , n, which has a solution iff ci(0)ci(ei) < 0, for
+all i. This happens with probability 2−n, hence indeed N(A1, . . . , An) = 2−n.
+4.1. Proof of Theorem 1.1. Let us look at a special instance of (3.5). To the given finite
+nonempty subsets A1, . . . , An ⊆ Rn, we assign the maps
+ψi : Rn
+>0 → P(RAi) ≃ Pmi, ψi(x) := [xai]ai∈Ai,
+where mi := #Ai − 1. Recall that Pi denotes the convex hull of Ai and put P := P1 + . . . + Pn.
+We consider the combined map
+(4.5)
+ψ: Rn
+>0 → Pm1 × · · · × Pmn, ψ(x) := (ψ1(x), . . . , ψn(x)).
+Lemma 4.2. The map ψ is injective iff P is n-dimensional. Moreover, if P is not n-dimensional,
+then rankDxψ < n for all x ∈ Rn
+>0.
+Proof. Assume ψ(exp(w)) = ψ(exp(w′)) for w ̸= w′ ∈ Rn Then there are ci ∈ R such that for
+all ai ∈ Ai we have ⟨ai, w − w′⟩ = ci. Hence, ⟨x, w − w′⟩ = ci for all xi ∈ Pi. It follows that
+⟨x, w − w′⟩ = c1 + . . . + cn for all x ∈ P. Hence dim P < n.
+Conversely, assume there is a nonzero w ∈ Rn and c ∈ R such that ⟨x, w⟩ = c for all x ∈ P.
+Then there are ci ∈ R such that ⟨xi, w⟩ = ci for all xi ∈ Pi. It follows that for any x ∈ Rn
+>0 and
+any s ∈ R we have
+ψi(eswx) = [(esw)aixai]ai∈Ai = [es⟨ai,w⟩xai]a∈Ai = [escixai]a∈Ai = ψi(x).
+Hence ψ is not injective. Moreover, w is in the kernel of the derivative of ψi at x.
+□
+We denote by Z the image of ψ. Then we can write
+N(A1, . . . , An) = E g∈G#(Z ∩ g1H1 ∩ . . . ∩ gnHn),
+where the hypersurfaces Hi are defined in (3.1). By Theorem 3.2 and (3.6), this can be expressed
+as
+(4.6)
+N(A1, . . . , An) = (2π)− n
+2 ρm1 · · · ρmn
+�
+Rn
+>
+(¯σZ ◦ ψ)Jψ dx.
+We make the coordinate change Rn → Rn
+>0, (w1, . . . , wn) �→ x = (e−w1, . . . , e−wn), which has
+the absolute Jacobian x1 · · · xn, and obtain (slightly abusing notation)
+(4.7)
+�
+Rn
+>
+(¯σZ ◦ ψ)Jψ dx =
+�
+Rn x1 · · · xn(¯σZ ◦ ψ)Jψ dw.
+Recall from Subsection 2.4 that each vertex v of P defines the inner normal cone Cv := P ∗
+v .
+We can write
+(4.8)
+Rn =
+�
+v
+Cv
+as the union over the vertices v of P. Moreover, we know that dim(Cv ∩ Cv′) < n for different
+vertices v, v′. Therefore, we can rewrite (4.7) as the sum
+�
+v
+�
+Cv
+x1 · · · xn(¯σZ ◦ ψ)Jψ dw.
+over the f0 many vertices v of P.
+Fix now a vertex v of P. According to Lemma 2.5, there are vertices vi of Pi, for i = 1, . . . , n,
+satisfying v = v1 + . . . + vn. Note that ai ∈ Ai.
+
+12
+PETER B¨URGISSER
+We define the map ϕi : Rn
+>0 → RAi\{vi} by
+ϕi(x) = (xai−vi)ai∈Ai\{vi} ∈ RAi\{vi} ≃ Rmi.
+Note that ϕi expresses ψi in the affine chart given by P(RAi)yvi ̸=0 → Rai∈Ai\{vi}, which maps
+[yai]ai∈Ai to y−1
+vi (yai)ai∈Ai\{vi}. So we are in the setting of Subsection 3.2 and ϕi is an instance
+of (3.8). The rows of the matrix M(x) := Dxϕ are labeled by the disjoint union A1⊔. . .⊔An and
+M(x) has n columns. For any n-tuple (a1, . . . , an) with ai ∈ Ai \{vi}, we denote by M(x)a1,...,an
+the n×n submatrix of M(x), obtained by selecting from M(x) the rows numbered by a1, . . . , an.
+We apply Lemma 3.5 to bound
+ρm1 · · · ρmn
+�
+Cv
+x1 · · · xn(¯σZ ◦ ψ)Jψ dw ≤
+�
+a1,...,an
+�
+Cv
+x1 · · · xn| det M(x)a1,...,an| dw,
+where the sum runs over all tuples (a1, . . . , an) with ai ∈ Ai \ {vi}. So there are m1 · · · mn many
+summands. To prove Theorem 1.1, it is sufficient to show that
+(4.9)
+�
+Cv
+x1 · · · xn| det M(x)a1,...,an| dw ≤ 1
+for each vertex v and each selection (a1, . . . , an).
+The component (row) of the derivative Dxϕi corresponding to ai ∈ Ai \ {vi} is given by
+(Dxϕi)ai = xai−vi(ai − vi)diag(x−1
+1 , . . . , x−1
+n ).
+Hence the n × n-submatrix M(x)a1,...,an of M(x) is given by
+M(x)a1,...,an = diag(xa1−v1, . . . , xan−vn)
+
+
+a1 − v1
+...
+an − vn
+
+ diag(x−1
+1 , . . . , x−1
+n ).
+Therefore, setting bi := ai − vi, we get
+x1 · · · xn det(M(x)a1,...,an) = xb1+...+bn det[b1, . . . , bn].
+Let us write Πi for the cone of Pi at the vertex vi. By definition, bi ∈ Π∗
+i . By Lemma 2.5,
+Π := Π1+. . .+Πn equals the cone of the polytope P = P1+. . .+Pn at the vertex v = v1+. . .+vn.
+Hence bi ∈ Π∗
+i ⊆ Π∗
+1 ∩ . . . ∩ Π∗
+n = Π∗ = Cv.
+We can therefore rewrite the left-hand side of (4.9) as
+(4.10)
+�
+Cv
+x1 · · · xn| det M(x)a1,...,an| dw =
+�
+Cv
+e−⟨b1+...+bn,w⟩| det[b1, . . . , bn]| dw.
+By Proposition 2.4, this is at most 1. This shows claim (4.9) and finishes the proof of Theo-
+rem 1.1.
+□
+4.2. Proof of Proposition 1.2. For finite Si ⊆ R, put A := S1 × . . . × Sn, and consider the
+maps
+ψi : R>0 → P(RSi), xi �→ [xai
+i ]ai∈Si,
+ψ: Rn
+>0 → P(RA), x �→ [xa]a∈A
+with images Zi and Z, respectively. The kinematic formula for real projective space gives (see
+[9, Cor. A.3])
+N(Si) = vol(Zi)
+vol(P1),
+N(A, . . . , A) = vol(Z)
+vol(Pn).
+The key insight is that Z is obtained as the image of Z1 × . . . × Zn under the Segre embedding
+P(RS1) × . . . × P(RSn) → P(RS1 ⊗ . . . ⊗ RSn) ≃ P(RA),
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+13
+which is well known to be isometric. Therefore, vol(Z) = vol(Z1) · · · vol(Zn), and this completes
+the proof of Proposition 1.2.
+□
+4.3. Proof of Theorem 1.3. Given is a finite subset A ⊆ Rn with convex hull P. By Lemma 4.2
+we can can w.l.o.g. assume that dim P = n. Consider the injective map
+(4.11)
+ψ: Rn
+>0 → P(RA), ψ(x) := [xa]a∈A
+with image Z ⊆ P(RA). The kinematic formula for real projective space [9, Cor. A.3] is con-
+siderably simpler than the one in Theorem 3.2, since O(m) acts transitively on the Grassmann
+manifolds Gr(k, Rm): we have
+(4.12)
+N(A, . . . , A) = vol(Z)
+vol(Pn) =
+1
+vol(Pn)
+�
+Rn
+>0
+Jψ(x) dx.
+We now proceed as in the proof of Theorem 1.3. We make the coordinate change x = e−w
+and decompose the resulting integral according to the decomposition (4.8) of Rn into the full
+dimensional cones Cv corresponding to vertices v. Thus
+�
+Rn
+>0
+Jψ(x) dx =
+�
+Rn x1 · · · xnJψ(x) dw =
+�
+Cv
+�
+Cv
+x1 · · · xnJψ(x) dw
+For a fixed vertex v of P, we consider the map ϕ: Rn
+>0 → RA\{v} defined by
+(4.13)
+ϕ(x) = (xa−v)a∈A\{v}.
+Then we have ψ(x) = π(ϕ(x)), where π is the inverse of the chart P(RA)yv̸=0 → RA\{v}. It is
+easy to verify that Jψ(x) ≤ Jϕ(x) using ∥Dϕ(x)π∥ ≤ 1, see (2.1).
+Le us view M(x) := Dxϕ as a matrix whose rows are labelled by elements of A \ {v}. and
+denote by M(x)a1,...,an the submatrix of M(x) obtained by selecting the rows labelled by the ai.
+Binet-Cauchy implies that
+Jϕ(x)2 = det(M(x)T M(x)) =
+�
+a1,...,an
+(det M(x)a1,...,an)2,
+with the sum running over all n-element subsets {a1, . . . , an} of A\{v}, of which there are
+�t−1
+n
+�
+many. This implies Jϕ(x) ≤ �
+a1,...,an | det M(x)a1,...,an|. We have arrived at
+�
+Cv
+x1 · · · xnJψ(x) dw ≤
+�
+a1,...,an
+�
+Cv
+x1 · · · xn| det M(x)a1,...,an| dw ≤
+�t − 1
+n
+�
+,
+where the right-hand inequality follows from Proposition 2.4 as in (4.10).
+□
+4.4. Proof of Theorem 1.4. The key observation is the following. Define for ε > 0
+Dε := {x ∈ Rn | ∥x∥ ≥ ε}.
+Lemma 4.3. Let C ⊆ Rn be a proper cone, d ∈ int(C∗), and ε > 0. Then
+lim
+m→∞ mn
+�
+C∩Bε
+e−m⟨d,w⟩ dw = 0
+Proof. Since ∩m≥1Dmε = ∅, basic integration theory implies
+lim
+m→∞
+�
+C∩Dmε
+e−⟨d,u⟩ du = 0.
+Making the change of variables u = mw shows the assertion.
+□
+
+14
+PETER B¨URGISSER
+We now observe the following. Let U ⊆ Rn
+>0 be open. Analogously as for (4.6), one shows
+that
+(2π)− n
+2 ρm1 · · · ρmn
+�
+U
+x1 · · · xn(¯σZ ◦ ψ)Jψ dw.
+equals the expected number of nondegenerate zeros in U of the random system (4.3).
+We follow the proof of Theorem 1.1. Note that stretching the support does not change the
+Newton polytopes Pi and P = P1 + . . . + Pn. Fix a vertex v of P. According to Lemma 2.5,
+there are vertices vi of Pi, for i = 1, . . . , n, satisfying v = v1 + . . . + vn. Tracing the proof
+of Theorem 1.1, one sees that it is sufficient to show that (compare (4.10)) for any selection
+a1 ∈ A1 \ {v1}, . . . , an ∈ An \ {vn}, the vectors bi = ai − vi satisfy
+lim
+m→∞
+�
+Cv
+e−m⟨b1+...+bn,w⟩| det[mb1, . . . , mbn]| dw = 0.
+However, this is a consequence of Lemma 4.3.
+□
+4.5. Additional comment. It is instructive to see how (4.12) directly follows from the more
+general kinematic formula in Theorem 3.2. Consider the injective map ψ from (4.11) with image
+Z ⊆ P(RA). We use ψ to define the map
+(4.14)
+ψd : Rn
+>0 → (P(RA))n, x �→ (ψ(x), . . . , ψ(x)).
+The image Zd = {(y, . . . , y) | y ∈ Z} ⊆ (Pm)n of ψd is the diagonal embedding of Z in the
+product of projective spaces. By Theorem 3.2 and (3.6) we have
+N(A, . . . , A) = (2π)− n
+2 ρn
+m
+�
+Rn
+>0
+(¯σZd ◦ ψd)Jψd dx.
+Via Lemma 4.4 below, we indeed conclude that
+N(A, . . . , A) =
+1
+vol(Pn)
+�
+Rn
+>0
+Jψ dx = vol(Z)
+vol(Pn),
+which is (4.12).
+Lemma 4.4. For x ∈ Rn
+>0 we have
+ρn
+m ¯σZd(ψd(x))Jψd(x) =
+(2π)
+n
+2
+vol(Pn)Jψ(x).
+Proof. Lemma 3.4 applied to the map ψd from (4.14) gives
+(4.15)
+ρn
+m ¯σZd(ψd(x))Jψd(x) = E λ1,...,λn ∥(λ1 ◦ Dxψ) ∧ . . . ∧ (λn ◦ Dxψ)∥
+where the λi are standard Gaussian linear forms on Tψ(x)Pm. Take an isometry Tψ(x)Pm ≃ Rm,
+view λi ∈ Rm as a vector, and view ∆ := Dxψ as a matrix in Rm×n. We note that Jψ(x) =
+�
+det(∆T ∆). The right-hand side of (4.15) can be written as the expectation E λi| det R(x)|,
+with the matrix
+(4.16)
+R(x) :=
+
+
+λT
+1 ◦ Dxψ
+...
+λT
+n ◦ Dxψ
+
+ =
+
+
+λT
+1
+...
+λT
+n
+
+ · ∆.
+We thus need to prove that
+(4.17)
+E λi| det R(x)| =
+(2π)
+n
+2
+vol(Pn)
+�
+det(∆T ∆).
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+15
+In order to show this, by the singular value decomposition, we may assume that ∆ =
+�
+D
+0
+�
+,
+where D = diag(σ1, . . . , σn). Note that
+�
+det(∆T ∆) = σ1 · · · σn. Then (4.16) can be written as
+R(x) = ΛD, where Λ ∈ Rn×n is a standard Gaussian square matrix and we get E Λ| det(R(x))| =
+σ1 · · · σn E w| det Λ|. It is well known that E Λ| det Λ| = ρnρn−1 · · · ρ1, e.g., see [6, Cor. 4.11]. On
+the other hand (see [6, Lemma 2.25])
+ρm =
+√
+2π vol(Pm−1)
+vol(Pm) ,
+hence ρnρn−1 · · · ρ1 = (2π)
+n
+2
+vol(Pn). We have thus verified (4.17).
+□
+References
+[1] Mart´ın Avenda˜no. The number of roots of a lacunary bivariate polynomial on a line. Journal of Symbolic
+Computation, 44(9):1280–1284, 2009.
+[2] D. N. Bernstein. The number of roots of a system of equations. Funkcional. Anal. i Priloˇzen., 9(3):1–4, 1975.
+[3] Fr´ed´eric Bihan and Boulos El-Hilany. A sharp bound on the number of real intersection points of a sparse
+plane curve with a line. Journal of Symbolic Computation, 81:88–96, 2017.
+[4] Fr´ed´eric Bihan and Frank Sottile. New fewnomial upper bounds from Gale dual polynomial systems. Mosc.
+Math. J., 7(3):387–407, 573, 2007.
+[5] Ir´en´ee Briquel and Peter B¨urgisser. The real tau-conjecture is true on average. Random Structures Algo-
+rithms, 57(2):279–303, 2020.
+[6] Peter B¨urgisser and Felipe Cucker. Condition, volume 349 of Grundlehren der Mathematischen Wis-
+senschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2013. The geometry
+of numerical algorithms.
+[7] Peter B¨urgisser, Felipe Cucker, and Pierre Lairez. Rigid continuation paths ii. structured systems. Preprint
+arXiv:2010.10997, 2021.
+[8] Peter B¨urgisser, Alperen A. Erg¨ur, and Josu´e Tonelli-Cueto. On the number of real zeros of random fewno-
+mials. SIAM J. Appl. Algebra Geom., 3(4):721–732, 2019.
+[9] Peter B¨urgisser and Antonio Lerario. Probabilistic Schubert calculus. J. Reine Angew. Math., 760:1–58,
+2020.
+[10] Ren´e Descartes. La G´eom´etrie. Librairie Scientifique A. Hermann, 1886. Digital reproduction of 2008 by
+Project Gutenberg (Ebook number: 26400).
+[11] Mathias Drton, Bernd Sturmfels, and Seth Sullivant. Lectures on algebraic statistics, volume 39 of Oberwol-
+fach Seminars. Birkh¨auser Verlag, Basel, 2009.
+[12] Alan Edelman and Eric Kostlan. How many zeros of a random polynomial are real? Bull. Amer. Math. Soc.
+(N.S.), 32(1):1–37, 1995.
+[13] Jacques Faraut and Adam Kor´anyi. Analysis on symmetric cones. Oxford Mathematical Monographs. The
+Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications.
+[14] Osman G¨uler. Barrier functions in interior point methods. Math. Oper. Res., 21(4):860–885, 1996.
+[15] Bertrand Haas. A simple counterexample to Kouchnirenko’s conjecture. Beitr¨age Algebra Geom., 43(1):1–8,
+2002.
+[16] Fritz Horn and Roy Jackson. General mass action kinetics. Arch. Rational Mech. Anal., 47:81–116, 1972.
+[17] Ralph Howard. The kinematic formula in Riemannian homogeneous spaces. Mem. Amer. Math. Soc.,
+106(509):vi+69, 1993.
+[18] Gorav Jindal, Anurag Pandey, Himanshu Shukla, and Charilaos Zisopoulos. How many zeros of a random
+sparse polynomial are real? In ISSAC’20—Proceedings of the 45th International Symposium on Symbolic
+and Algebraic Computation, pages 273–280. ACM, New York, [2020] ©2020.
+[19] Askold G. Khovanski˘ı. A class of systems of transcendental equations. Dokl. Akad. Nauk SSSR, 255(4):804–
+807, 1980.
+[20] Askold G. Khovanski˘ı. Fewnomials, volume 88 of Translations of Mathematical Monographs. American Math-
+ematical Society, Providence, RI, 1991.
+[21] Pascal Koiran. Shallow circuits with high-powered inputs. Proc. Second Symposium on Innovations in Com-
+puter Science, ICS, 2011.
+[22] Pascal Koiran, Natacha Portier, and S´ebastien Tavenas. On the intersection of a sparse curve and a low-degree
+curve: a polynomial version of the lost theorem. Discrete Comput. Geom., 53(1):48–63, 2015.
+
+16
+PETER B¨URGISSER
+[23] Pascal Koiran, Natacha Portier, and S´ebastien Tavenas. A Wronskian approach to the real τ-conjecture. J.
+Symbolic Comput., 68(part 2):195–214, 2015.
+[24] Anatoli G. Kushnirenko. Poly`edres de Newton et nombres de Milnor. Invent. Math., 32(1):1–31, 1976.
+[25] L´eo Mathis. The Handbook of Zonoid Calculus. PhD thesis, SISSA, 2022.
+[26] Yurii Nesterov and Arkadii Nemirovskii. Interior-point polynomial algorithms in convex programming, vol-
+ume 13 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM),
+Philadelphia, PA, 1994.
+[27] Kaitlyn Phillipson and J. Maurice Rojas. Fewnomial systems with many roots, and an adelic tau conjecture.
+In Proceedings of Bellairs workshop on tropical and non-Archimedean geometry (May 6-13, 2011, Barbados),
+Contemporary Mathematics, volume 605, pages 45–71, 2014.
+[28] Rolf Schneider. Convex bodies: the Brunn-Minkowski theory, volume 151 of Encyclopedia of Mathematics
+and its Applications. Cambridge University Press, Cambridge, expanded edition, 2014.
+[29] Frank Sottile. Real solutions to equations from geometry, volume 57 of University Lecture Series. American
+Mathematical Society, Providence, RI, 2011.
+[30] G¨unter M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag,
+New York, 1995.
+Institute of Mathematics, Technische Universit¨at Berlin
+Email address: pbuerg@math.tu-berlin.de
+

2dFAT4oBgHgl3EQfkh3i/content/tmp_files/2301.08612v1.pdf.txt

@@ -0,0 +1,1544 @@
+ARcode: HPC Application Recognition Through
+Image-encoded Monitoring Data
+1st Jie Li
+Department of Computer Science
+Texas Tech University
+Lubbock, TX, USA
+jie.li@ttu.edu
+2nd Brandon Cook
+National Energy Research Scientific Computing Center
+Lawrence Berkeley National Laboratory
+Berkeley, CA, USA
+bgcook@lbl.gov
+3rd Yong Chen
+Department of Computer Science
+Texas Tech University
+Lubbock, TX, USA
+yong.chen@ttu.edu
+Abstract—Knowing HPC applications of jobs and analyzing
+their performance behavior play important roles in system
+management and optimizations. The existing approaches detect
+and identify HPC applications through machine learning models.
+However, these approaches rely heavily on the manually extracted
+features from resource utilization data to achieve high prediction
+accuracy. In this study, we propose an innovative application
+recognition method, ARcode, which encodes job monitoring
+data into images and leverages the automatic feature learning
+capability of convolutional neural networks to detect and identify
+applications. Our extensive evaluations based on the dataset
+collected from a large-scale production HPC system show that
+ARcode outperforms the state-of-the-art methodology by up
+to 18.87% in terms of accuracy at high confidence thresholds.
+For some specific applications (BerkeleyGW and e3sm), ARcode
+outperforms by over 20% at a confidence threshold of 0.8.
+Index Terms—High Performance Computing, Application De-
+tection, Deep Learning, Convolutional Neural Network
+I. INTRODUCTION
+As HPC systems are approaching the exaFLOP era, the scale
+and complexity of HPC systems have increased significantly
+over the past few years. Administrators need to understand
+not only the performance of the hardware system, but also the
+typical applications and their characteristics, such as how they
+use the computing resources and how they have been executed
+before [1]–[5]. With the increase of computation capability, the
+resource contention and energy consumption increase as well.
+To improve HPC system efficiency, it is imperative to under-
+stand the characteristics of applications and to guide better
+resource-aware scheduling policies based on the knowledge
+of resource requirements of applications [6]–[8]. Moreover,
+emergent misbehaviour is becoming more prevalent due to the
+large scale and high utilization [9]. For system administrators
+striving to guarantee optimal system performance, detecting
+anomalies and potential errors of applications is an essential
+task [10]–[12].
+An online detection system that is capable of identifying
+applications in real-time, with little or no human intervention,
+would be a boon to system management. However, this is a
+daunting task. Large-scale HPC systems are generally shared
+by a variety of users from different domains. In addition to
+traditional large-scale simulation applications (e.g., molecu-
+lar dynamics, quantum chemistry applications, and climate
+simulations), emerging Machine Learning (ML) and Artificial
+Intelligence (AI) applications have become an increasingly
+critical part of the workloads on HPC systems. Frequently
+used applications and libraries are usually pre-installed on the
+system by system administrators. However, administrators do
+not necessarily possess the knowledge about the executions
+and characteristics of each application. Users could build,
+compile and name their own applications that are not shared
+with others. In addition, if users do not provide application
+information in job submission scripts, it is difficult to know
+what applications these jobs belong to. These various use
+cases and the imprecise mapping of job names to actual
+applications make it difficult to identify applications using
+naive approaches. As an example, we examined the application
+names derived from the job submission script on Cori and
+found that about 42.4% of the names were incorrect.
+Advanced methods for detecting and recognizing applica-
+tions can be divided into static analysis of binaries and/or
+scripts, which can be performed without running a job, and dy-
+namic analysis of system logs and performance metrics, which
+implies analysis during or after job execution. Early works
+explored static analysis of binaries to determine the semantic
+similarity between two applications [13], [14]. However, The
+complexity of HPC systems has reached a point where static
+analysis of the binaries used to run and maintain the detection
+system is no longer feasible. This approach is invasive to
+users’ data and requires a dedicated binaries collection and
+management infrastructure, which is not always of interest for
+system administrators. Moreover, even though the binaries are
+available, it does not perform well if the same application is
+compiled by a different compiler toolchain or optimization
+level [15]–[17].
+Collecting and analyzing system logs and performance
+metrics are critical to combat performance crisis, and they
+are prevalent in HPC systems, although the approach may
+vary. Recently, there has been a growing research interest
+in automatic detection that relies on extensive performance
+metrics and employs ML techniques to identify applica-
+tions [18]–[22]. A representative approach proposed by Ates
+et al. explored building supervised ML models with statistic
+features of monitoring metrics to classify applications [19].
+The classification model relies on thousands of statistical
+arXiv:2301.08612v1  [cs.DC]  20 Jan 2023
+
+features extracted from hundreds of time-series monitoring
+metrics to achieve high prediction accuracy. A major weakness
+of this approach is the high-latency responses of the detection
+model, and the statistical features are only accurate for repre-
+senting the application after the job is finished. In addition, the
+performance of feature-based models is highly dependent on
+feature engineering; using different features has the potential
+to deviate the classification performance [23].
+In this study, we extend the line of performance metrics
+based approaches and propose an innovative method called
+ARcode (stands for Application Recognition code). It is
+an application recognition method utilizing images encoded
+from performance monitoring metrics. Specifically, we lever-
+age monitoring metrics collected from HPC systems (§II-A)
+and encode time-series data to two-dimensional images to
+represent the resource usage patterns of HPC executions (or
+job signatures for simplicity, discussed in §III-B). We then
+build a dataset labelled with application names and train a
+Convolutional Neural Network (CNN) to build the classifi-
+cation model (§III-C). The contributions of this study are
+summarized below:
+• Contrary to other studies where datasets are generated
+from benchmarks and proxy applications, our dataset is
+built from real applications with different input data,
+resource allocations and run times, which well reflects the
+complex real scenarios. Specifically, we collect monitor-
+ing data from a production system and build a dataset of
+performance metrics of twelve popular HPC applications
+where the application names are labelled.
+• Our innovative methodology encodes time-series mon-
+itoring data into two-dimensional images, where the
+performance metrics are creatively represented in a much
+smaller size compared to the original data without losing
+important metric variations. The encoded job signatures
+can be used not only for application classification and
+detection, but also to inspire methods for predicting and
+estimating the resource usage of applications.
+• We use the CNN techniques and train the CNN model
+with the job signatures. The job signatures are generated
+from the performance monitoring data, thus do not in-
+volve collecting and analyzing users’ private data. The
+CNN model, on the other hand, does not require manual
+features engineering, making it easier to be tuned and
+adopted by any HPC sites.
+Through extensive experiments, we find that ARcode
+achieved a competitive classification performance in most
+cases and outperformed by up to 18.87% at high confidence
+thresholds compared to the state-of-the-art methods. When
+detecting some specific applications (e.g., BerkeleyGW and
+e3sm) with a confidence threshold of 0.8, ARcode is better
+than the state-of-the-art methods by over 20% in terms of ac-
+curacy. Meanwhile, ARcode retains the temporal information
+of the monitoring data and is able to recognize running jobs.
+This capability is not available in any state-of-the-art methods.
+The details of all these experiment evaluations are discussed
+Figure. 1: Workflow of LDMS on Cori
+in
+§IV. The ARcode model and dataset used in this study
+can be found in a separate submission of artifacts.
+II. BACKGROUND
+In this section, we briefly introduce the Cori system and
+how we collect job monitoring data. Then, we describe the
+workflow of the monitoring infrastructure being used and
+present the available job-level monitoring metrics.
+A. The Cori System
+Cori1 is a Cray XC40 system at National Energy Research
+Scientific Computing Center (NERSC). It consists of 2,388
+Intel Xeon “Haswell” processor nodes and 9,688 Intel Xeon
+Phi “Knight’s Landing” (KNL) nodes interconnected on Cray
+Aries High-Speed Network, which provides a peak perfor-
+mance of about 30 petaflops. Additionally, Cori is equipped
+with a large scratch Luster File System that provides 432
+GB/s of performance with a capacity of 28.5 petabytes to
+the compute nodes. Cori also has the Cray DataWarp based
+Burst Buffer, offering a 1.8 petabytes burst buffer storage with
+1.7 TB/s in peak bandwidth performance [24], [25]. With the
+mission of accelerating the pace of scientific discovery through
+HPC and data analysis, workloads running on the Cori system
+cover a wide range of scientific disciplines, including lattice
+QCD, materials science, climate research, high energy physics,
+astrophysics, and more.
+B. Monitoring Workflow
+In this study, we utilize the monitoring metrics collected
+from the CPU nodes of the NERSC Cori system, where the in-
+1https://docs.nersc.gov/systems/cori/
+
+Compute Node
+Compute Node
+LDMS Sampler Daemon 
+LDMS Sampler Daemon 
+Haswel
+Aggregation Node
+LDMS Aggregator Daemon
+Parguet
+Large Memory Node
+2TB of MemoryTABLE I: Selected Monitoring Metrics
+Sampler
+Metrics
+Derived Metrics
+Description
+cray aries sampler
+power
+power
+Node Power Consumption
+syspapi
+PAPI TOT INS
+IPC (PAPI TOT INS/PAPI TOT TOT)
+Instruction Per Cycle
+PAPI TOT TOT
+meminfo
+MemTotal
+Mem (MemTotal - MemFree)
+Memory Used
+MemFree
+Figure. 2: Overview of ARcode Design. ARcode encodes the time-series monitoring data into job signatures. In the offline
+training phase, ARcode trains the CNN from a labelled job signature dataset. In the running recognition phase, the CNN
+model is used to detect the encoded job signature. The CNN model is modified so that it can identify novel applications.
+band monitoring tool, LDMS is used [26]. The detailed discus-
+sion of how LDMS works on Cori is out of scope of this paper.
+In Figure 1, we illustrate its workflow for generating job-level
+performance metrics. The workflow includes the following
+steps: 1 LDMS samplers on Haswell and KNL nodes (two
+partitions in Cori) collect in-band metrics at a pre-configured
+frequency; 2 the monitoring data are then sent to aggregation
+nodes, where metrics collected from the same sampler are
+stored in CSV files under the same folder. Each CSV contains
+metrics from multiple nodes and the corresponding metrics for
+a job may span multiple CSV files. To improve the usability of
+the monitoring data, the NERSC LDMS [27], an LDMS data
+processing tool, takes care of the post-processing of CSV files.
+3 NERSC LDMS gets job IDs from Slurm sacct and joins
+the job IDs with CSV files, and 4 submits these files to large
+memory nodes for post-processing, where the same metrics of
+the same job are extracted. 5 The post-processed job-level
+metrics are saved in parquet files by metric samplers for future
+analysis. Compared to raw CSV files, job-level parquet files
+significantly improve the availability of monitoring data and
+the efficiency of querying job performance metrics.
+C. Available Metrics
+The available metrics collected through LDMS on HPC
+systems depend on the configurations and available samplers
+on the hardware platform. As routine tasks of monitoring
+the health of the Cori system, 34 different samplers collect
+metrics related to I/O, network, CPU counters, memory usage,
+power consumption, etc. The total number of metrics collected
+through the sampler varies from 12 to 3,016, depending
+on the sampler. The granularity of collected metrics is one
+second, generating approximately 400MB of monitoring data
+per second on a system wide basis.
+From the perspective of application recognition, it is
+unattractive and impractical to include all of these extensive
+monitoring metrics in one model because processing large
+amounts of time-series data is compute-intensive and time-
+consuming. On the other hand, we envision that the proposed
+methodology should be easily adopted by other HPC systems
+and the selected metrics should be common even when using
+different monitoring infrastructures. Therefore, we select five
+of these metrics and derive three representative metrics for
+creating job signatures. These three metrics are the power
+consumption of the compute node, instruction per cycle (IPC),
+and memory used as shown in Table I. These metrics are
+expected to be available through the monitoring infrastructure
+on a variety of HPC architectures.
+III. METRICS ENCODING AND CLASSIFICATION
+In this section, we first discuss design considerations and
+provide an overview of ARcode design. We then present
+the details of encoding monitoring data, including resampling
+job metrics, converting 1D time-series data into 2D images,
+and encoding multiple traces into a single image. Lastly, we
+introduce the CNN architecture for classifying the encoded
+images.
+A. Design Consideration and Overview
+As discussed in §I, existing solutions either perform a static
+analysis of binaries and/or scripts or leverage machine learning
+methods to extract key features out of extensive monitoring
+data to build application prediction and classification models.
+
+Known App
+App1
+App2
+Known App 2
+Known App n
+App 3
+App 4
+Unknown Apps
+App n(a) Original Power Consumption Trace
+(b) Downsampled Trace by Aggregating
+(c) Original Power Consumption Trace
+(d) Upsampled Trace by Padding
+Figure. 3: Resampling traces by aggregating and padding based on the original trace length and the predefined length. Figure
+a is the trace longer than the predefined length 128, in which the aggregating function is applied to downsample the trace.
+Figure c is a trace shorter than 128, in which the padding is used to upsample the trace. Figure b and d show the corresponding
+resampled traces, respectively.
+Our approach, ARcode, is similar to the monitoring data
+based approaches but with two design considerations: features
+learned without human intervention and data retaining tempo-
+ral information.
+First, current state-of-art application detection models, such
+as Taxonomist [19], extract statistical summaries from raw
+monitoring data to create a feature vector for machine learning
+models, where the classification performance is highly de-
+pendent on the quality of the manually constructed features.
+To improve the usability, one of our considerations is that
+the features of the monitoring traces can be learned without
+human intervention. Second, although statistical features are
+spatially efficient and lightweight when building detection
+models, they lose the temporal information of time-series data,
+thus limiting their use case. To improve the extensibility,
+the other consideration of our model is to retain temporal
+information. So it is able to use part of the monitoring data in
+detection, classification, and prediction of the resource usage
+of applications.
+With these considerations in mind, our proposed approach,
+ARcode, encodes entire time-series monitoring data of jobs
+into unified-size images and leverages deep learning tech-
+niques to learn features. The encoded image, which we named
+as job signature, is a representation of monitoring traces that
+preserve temporal performance behavior of the job.
+As shown in Figure 2, ARcode has two main components:
+the monitoring data encoding and the CNN model. The
+first component, the monitoring data encoding component,
+performs a series of operations on the raw monitoring data.
+It creates job signatures encoding the time-series data and
+represents the jobs as images. The second component is a CNN
+model that is customized to learn features from job signatures
+and to classify these job signatures. ARcode operates in two
+phases. The first phase is the offline training phase, where
+the CNN model is trained from a labelled job signature
+dataset. The training phase can be enhanced with transfer
+learning [28], where the convolutional layers can be transferred
+from a trained job detection model and only the layers that
+make predictions need to be trained. The runtime recognition
+phase is where ARcode operates to detect and to predict the
+applications by job signatures.
+Classification model should not be limited to classify
+known applications already seen in the training phase. To
+make ARcode practically useful, we introduce confidence
+thresholds to help identify applications. When the prediction
+probability exceeds the defined threshold, ARcode labels
+the job with the application name; otherwise, it marks the
+observation as unknown, indicating the job is likely to be a
+new application.
+B. Construction of Job Signature
+1) Resampling Traces: To construct job signatures that can
+be measured for the similarity between pairs of signatures, it
+requires the time series traces to have equal length. Given a
+trace, we resample it with a predefined length l. For a trace T
+of length n, we create T ′ by sampling the data points from T
+
+275.00
+Power Consumption (W)
+250.00
+225.00
+200.00
+75.00
+150.00
+125.00
+14000
+0
+2000
+4000
+6000
+8000
+10000
+12000
+Timesteps220.00
+Power Consumption (W)
+210.00
+200.00
+190.00
+180.00
+170.00
+20
+60
+80
+100
+120
+0
+40
+Timesteps300.00
+Power Consumption (W)
+250.00
+200.00
+.50.00
+100.00
+50.00
+0
+10
+20
+30
+40
+50
+Timesteps300.00
+Power Consumption (W)
+250.00
+200.00
+.50.00
+100.00
+50.00
+0
+20
+40
+60
+80
+100
+120
+Timesteps(a) Polar Coordinate of the Normalized Trace
+(b) Gramian Angular Summation (Left) and Difference (Right) Fields
+Figure. 4: Steps of Gramian Angular Field Conversion. Each encoded image has a resolution of 128 × 128.
+so that n′ = l, where n′ is the length of the sampled trace T ′.
+Note that the predefined length is usually set to a relatively
+small value to reduce the compute time when calculating the
+similarities.
+Considering most jobs on HPC run for hours or even days,
+their corresponding monitoring traces are usually longer than
+the predefined length, i.e, n > l. In this case, we downsample
+the traces. For each set of ⌊n/l⌋ data points, we apply one of
+the functions, such as mean, max, min, median, to calculate the
+aggregated value. Assuming we set the predefined length to be
+120, to resample the power consumption trace of a 10-minute
+job (i.e., the trace has 600 timesteps in total), every 5 data
+points should be aggregated. Figure 3a and Figure 3b illustrate
+the procedure of resampling the power consumption traces of
+a job using the mean value. The original trace contains more
+than 14,000 timesteps, while after resampling, the trace length
+becomes 128 timesteps.
+In case that n < l, i.e., the length of a job metric trace is less
+than the predefined length l, we upsample the original traces.
+Specifically, we add ⌊l/n⌋ paddings between consecutive data
+points and fill with the previous value. Figure 3c and Figure 3d
+show the traces before and after upsampling. After resampling,
+all traces have the length of l, irrespective of the duration of
+the job. The corresponding resampled traces T ′ will be used
+for further processing.
+2) Converting 1D time series to 2D images: As shown
+in Figure 3, the monitoring traces and the corresponding
+resampled traces are univariate time series. To take advantage
+of the feature learning in deep learning architectures, we
+convert 1D time series to 2D images.
+We utilize the Gramian Angular Field (GAF) to transform
+time series into images [29]. Specifically, the time series data
+is first normalized or scaled into the range of [−1, 1]. The
+normalized time series data is then represented in a polar
+coordinate instead of the typical Cartesian coordinate. A Gram
+Matrix like operation is applied on the resulting angles to
+construct 2D images.
+Given a time series trace T = {t1, t2, ..., tn} of n times-
+tamps, we rescale T to have the interval [−1, 1] by the equation
+below:
+˜ti = (ti − max(T)) + (ti − min(T))
+max(T) − min(T)
+(1)
+The value of the time series and its corresponding timestamp
+need to be accounted for so that no information is lost. These
+two quantities are expressed with the angle and the radius
+in polar coordinates, respectively. Mathematically, the angle
+is computed by arccos(˜ti), which lies within [0, π], and the
+radius variable is calculated by i/n, which is in [0, 1]. The
+point can be expressed in polar coordinates (φi, ri), where:
+�
+φi = arccos(˜ti),
+−1 ≤ ˜ti ≤ 1
+ri = i
+n,
+0 ≤ i ≤ n
+(2)
+The encoding function is a composition of bijective func-
+tions, producing one and only one result in the polar coordinate
+system. In addition, as opposed to Cartesian coordinates,
+polar coordinates preserve temporal dependency through the
+r coordinate. An example of a trace represented in polar
+coordinates is shown in Figure 4a, which is transformed from
+the normalized trace of Figure 3d.
+The temporal correlations between each pair of data points
+(ti, tj) are computed by considering the trigonometric sum-
+mation (cos(φi + φj)) or subtraction (cos(φi − φj)), leading
+to the Gramian Matrix called Gramian Angular Summation
+Field (GASF) or Gramian Angular Difference Field (GADF),
+respectively. The GASF is defined as follows:
+GASF =
+�
+����
+cos(φ1 + φ1)
+. . .
+cos(φ1 + φn)
+cos(φ2 + φ1)
+. . .
+cos(φ2 + φn)
+...
+...
+...
+cos(φn + φ1)
+. . .
+cos(φn + φn)
+�
+����
+(3)
+Through the GASF or GADF conversion, the diagonal Gi,i
+contains the original value of the scaled time series, while
+Gi,j represents the relative correlation by superposition of
+directions with respect to time interval |i − j|. Other details
+on time series encoding can be found from [29]. In Figure 4b,
+
+90°
+135°
+45°
+20.00
+180°
+0°
+0.25
+0.50
+0.75
+L.00
+225°
+315°
+270°60
+0
+20
+40
+80
+100
+120
+0
+20
+40
+60
+80
+100
+120
+1.00
+0.00
+-1.00
+120
+0.75
+100
+0.50
+80
+0.25
+0.00
+60
+-0.25
+40
+-0.50
+20
+-0.75
+0
+1.00
+.00
+0
+1we illustrate the encoded 2D images with GASF and GADF
+for the trace in Figure 4a. To keep it concise, we use GASF
+as the 1D time series conversion algorithm and refer to GASF
+as GAF in the following discussion.
+3) Encoding Multiple Time Series Traces: The procedure
+presented in §III-B2 only converts one monitoring trace to
+a 2D image. Since we have multiple monitoring traces cor-
+responding to the same job and we do not want to lose the
+correlation between each pair of traces. The question becomes
+how we can encode multiple time series to a single image such
+that a deep learning architecture can understand. To solve this
+problem, we are inspired by the concept of RGB channels,
+where each RGB channel emphasizes different aspects of the
+original image. Similarly, the power consumption trace, IPC
+trace, and memory used trace can be considered as the R
+channel, G channel, and B channel of the encoded image,
+respectively.
+The output of GAF conversion is nothing but a 2D matrix
+of floating-point numbers that fall in [−1, 1]. To visualize the
+job signature, we rescale the GAFs of the above-mentioned
+three traces to be in [0, 255] by using the below equation:
+�
+GAF = ⌊GAF + 1
+2
+∗ 255⌋
+(4)
+The �
+GAF contains the pixel value of the gray scale image.
+When combining �
+GAF power, �
+GAF ipc, �
+GAF mem as three
+channels of RGB, we create a single color image. Figure 5
+shows the GAFs of the power, IPC and memory usage of a
+job and its corresponding encoded job signature. It is worth
+noting that rescaling GAF to the range [0, 255] is only for
+the purpose of visualizing the job signature while the original
+GAF falling in [−1, 1] can be directly fed in the Convelutional
+Neural Network.
+It is important to note that the channel-like encoding
+methodology is not limited to three channels. Each time series
+trace is converted to a l ×l matrix by the procedure presented
+in §III-B2. An encoded job signature of three channels is
+a l × l × 3 matrix. Encoding one more metric is simply
+adding another dimension in the matrix. More formally, a job
+signature of c metrics is a l × l × c matrix. Even though it
+is not straightforward to be visualized when c is larger than
+3, the CNN model can “understand” and analyze the high-
+dimensional matrix.
+C. Classification Model
+Using the methodology presented in §III-B, the monitoring
+traces of jobs can be represented in images (i.e., job sig-
+natures). Therefore, detecting and identifying applications of
+jobs become an image recognition problem. In this subsec-
+tion, we first introduce the CNN, a deep learning technique
+that has been widely used in image classification problems
+and achieved promising results in many domains. Then, we
+present the CNN architecture that is specifically customized
+for classifying job signatures.
+1) Deep learning using CNN: The performance of con-
+ventional machine learning techniques depends heavily on
+data representation, which requires lots of efforts to design
+preprocessing pipeline and feature engineering. Such feature
+engineering is labor intensive and lacks the ability to extract
+discriminative information from the data [30]. Deep learning,
+on the other hand, explores the possibility to feed raw data to
+the algorithm and automatically discover the features needed
+for detection or classification. The key concept of deep learn-
+ing is to transform the representation at a lower level into a
+representation at a higher and more abstract level; and with
+composition of such transformations, complex functions can
+be learned [31]. As a widely-used deep learning technique,
+CNNs have been successful in image classification problems.
+2) Customized CNN Architecture for Job Signatures: The
+CNN architecture for classifying job signatures is designed as
+shown in Figure 6, which contains the following customized
+layers and parameters:
+1. Input layer for job signature: The inputs are job
+signatures of encoded multiple time series traces, as presented
+in §III-B. The resolution of the job signature is 128 × 128
+and the number of channels equals to the number of traces
+encoded in the job signature (i.e., three in our dataset). Note
+that for other cases that job signatures encode more than three
+monitoring traces, the input shape should be set accordingly.
+2. Layers for feature extraction: In our model, we have
+three layers of CNN to learn features from the job signature,
+each of which contains one convolution layer to extract
+features and one pooling layer to reduce the spatial dimension
+of the convoluted image. The convolution layer extracts image
+features by convolving the job signature with a set of kernels
+and produces one feature map for each kernel. We apply
+the activation function ReLU on the convoluted features. It
+replaces negative values with zero and keeps the positive value.
+The output of ReLU will be the input for the pooling layer.
+The first convolution layer learns 32 kernels, and the sec-
+ond and third convolution layers learn 64 and 128 kernels,
+respectively. We set the kernels to be size of 3×3 and use the
+same padding in the convolution to maintain the dimension of
+output as input. All three pooling layers perform max pooling
+that returns the maximum value from the portion of the image
+covered by a window of size of 2 × 2.
+3. Layers for classification: After going through the feature
+extraction layers, the customized CNN model flattens the
+final output and feeds it to a regular neural network for
+classification. The flattened vector is connected to a fully
+connected layer, through which non-linear combinations of
+the features can be learned. After passing through the fully
+connected layer, we use the softmax activation function in the
+last layer to get the probabilities of the input job signature
+being in a particular class.
+To prevent the CNN model from overfitting and improve the
+generalization of the CNN model, we add several dropout lay-
+ers in the model to randomly disable neurons during training,
+as shown in the dashed red parallelograms in Figure 6. In our
+
+Figure. 5: Encoding Multiple Time Series Traces into a Job Signature with a Resolution of 128 × 128.
+Figure. 6: Customized CNN Architecture for Classifying Job Signatures.
+experiments, the dropout rate before the flatten layer and in
+the fully-connected layer is set as 30% and 70%, respectively.
+IV. EXPERIMENTAL EVALUATION
+In this section, we introduce the experimental evaluation
+of our proposed classification model on a real-world dataset
+collected from the Cori system. Particularly, we first introduce
+the dataset built based on the proposed encoding methodology
+and then, we compare the performance of our model with the
+baseline methods in terms of accuracy.
+A. Dataset
+To the best of our knowledge, there are no publicly released
+monitoring traces with labelled application information col-
+lected from production HPC systems. Ates el al. [32] published
+a dataset containing the monitoring data of benchmarks and
+proxy applications. Google published its cluster traces that do
+not have application information [33]. Therefore, in order to
+train the CNN model for detecting applications running in
+production HPC systems, we built our own dataset.
+On Cori system, some other research groups had imple-
+mented the job metadata management service, where the
+metadata of jobs running on Cori are stored and managed
+in a MySQL database. The job metadata has a field named
+‘application name’ which is derived from the job submission
+script. However, based on our analysis, about 42.4% of derived
+names do not reflect the application accurately. Many of them
+have names like ‘test’, ‘bugs’, ‘b.sh’, etc. Therefore, to build
+a reliable labelled dataset, we select jobs that have accurate
+application names through the job metadata service. In addi-
+tion, since Cori system has the KNL partition and the Haswell
+partition, the monitoring metrics of the same application (even
+with same configurations and same input files) running on
+different architectures could potentially have large variations.
+To avoid discrepancy caused by the architecture, we only focus
+on the jobs running on the same architecture and we select
+KNL jobs as our dataset. Besides, we discard short jobs that
+run for less than 60 seconds since they are likely for testing
+purposes.
+The
+job
+IDs
+of
+the
+selected
+jobs
+are
+used
+with
+NERSC LDMS [27] to obtain the corresponding monitoring
+traces. Considering that a job may have multiple steps and
+that job steps may correspond to different applications, we
+treat the job steps of a job separately. In addition, since a job
+may use multiple nodes (nodes are exclusively used by the
+job on Cori), we aggregate the monitoring metrics from all
+involved nodes and use the average time-series traces to build
+the job signature.
+We select 23,665 jobs from 12 different applications to build
+the job signature dataset after eliminating those jobs that do
+not have monitoring metrics (due to data collection errors or
+historical traces are purged) and balancing the number of jobs
+of each application as much as possible. Details are listed in
+Table II. The numbers of allocated nodes for a job vary from
+1 to a maximum of 2,048. The BerkeleyGW, Espresso and
+
+Footunee Lonin:
+CNIN Ieyr 
+ JAI NING
+CNIN IyEr
+128 × 129 x3
+FT X 
+84 X 04 × B4
+[Poad mps
+32 量 $2 x 128
+Potpd mp
+L
+口
+口
+口0
+20
+40
+60
+80
+100
+120
+20
+40
+60
+80
+100
+120
+20
+40
+60
+80
+100
+120
+0.90 卡
+0.13 
+185.00
+0.12
+182.50
+120 -
+120
+120
+20 
+100
+100
+100 
+40
+80 
+80
+80 -
+60
+09
+60
+80 -
+40 
+40
+40 -
+100
+20 -
+M
+120
+ol
+01
+0
+20
+40
+60
+80
+100
+120
+GAF of Power (Channel R)
+GAF of IPC (Channel G)
+GAF of Mem (Channel B)
+Job SignatureTABLE II: Dataset of Job Signatures
+Application
+# of Jobs
+# of Nodes*
+Runtime(s)*
+Description
+BerkeleyGW
+1899
+1 / 2048
+60 / 172867
+For quasiparticle excitations and optical properties of materials.
+Espresso
+2000
+1 / 2048
+61 / 172858
+For electronic-structure calculations and materials modeling at the nanoscale.
+Gromacs
+1937
+1 / 64
+61 / 172836
+For simulations of proteins, lipids and nucleic acids.
+LAMMPS
+1999
+1 / 64
+60 / 172866
+For molecular dynamics with a focus on materials modeling.
+NWChem
+1992
+1 / 384
+71 / 169209
+For computational chemistry.
+VASP
+2000
+1 / 128
+69 / 172864
+For “ab-initio” quantum-mechanical molecular dynamics (MD) simulations.
+WRF
+1860
+1 / 256
+72 / 174608
+for atmospheric research and operational forecasting applications.
+aims
+2000
+1 / 192
+71 / 172959
+For “ab-initio” molecular simulations.
+chroma
+1996
+2 / 137
+488 / 147447
+For lattice Quantum Chromodynamics calculations (LQCD).
+cp2k
+1993
+1 / 128
+60 / 174608
+For quantum chemistry and solid state physics.
+e3sm
+1989
+1 / 2048
+60 / 173402
+For earth system modeling, simulation and prediction.
+su3
+2000
+1 / 36
+65 / 20277
+For lattice Quantum Chromodynamics calculations (LQCD)
+*The values on the left and right side of ‘/’ indicate the minimum and maximum values of the corresponding characteristics of the jobs.
+s3sm have extremely large-scale jobs where all cores of 2,048
+nodes are used, corresponding to 131,072 physical cores. The
+selected jobs also cover a large range of runtime variations.
+The shortest job, as defined earlier, runs for only 60 seconds.
+The longest job, limited by the Quality of Service of the KNL
+partition, runs for 48 hours. The dataset can be found in a
+separate submission of artifacts.
+It is worth mentioning that, it is up to users to set the resam-
+ling length l and define the resolution based on their computing
+capability. Nevertheless, the CNN model for images of larger
+resolution usually requires more complex architecture and
+the time for fine tuning and training the parameters of the
+architecture becomes high. In our experiments, we set the
+resampling length to be 128 and use a resolution of 128×128
+for job signatures.
+B. Baseline Methods
+To compare the performance of our proposed CNN model
+with state-of-the-art methods for detecting HPC jobs, we
+examine several other performance metrics based approaches.
+The Power Signature [18] and Taxonomist [19] are two repre-
+sentative methods that use statistic features of the time series
+monitoring data to build the classification model. The statistics
+include the minimum, maximum, mean, standard deviation,
+skew, kurtosis and the 5th, 25th, 50th, 75th, 95th percentiles.
+Using the same monitoring metrics as ARcode, namely power
+consumption, IPC and memory usage of the selected jobs,
+we extract statistical features and build several classifiers as
+baselines. These classifiers are Random Forest (RF), linear
+Support Vector Classifier (Liner-SVC), and non-linear Support
+Vector Classifier (SVC). In addition, we include a Random
+Forest model that uses only statistical features extracted from
+the power consumption trace. All of these models use default
+hyperparameters.
+C. Experiment Setup
+The ARcode and baseline models are trained and evaluated
+on a Cori GPU node, where 2 Skylake CPUs, 8 NVIDIA Tesla
+V100 GPUs and 384 GB DDR4 memory are provided. The
+proposed CNN is implemented using the TensorFlow Keras
+library in a Python 3.9 environment. The baseline models are
+implemented in Python leveraging scikit-learn library.
+Figure. 7: Loss (top) and accuracy (bottom) of the training
+and validation data. The vertical dashed line indicates the
+determined epoch of 50.
+We divide the dataset into 60% for training, 20% for
+validation, and 20% for testing. To mitigate overfitting and to
+increase the generalization of ARcode model, we determine
+the optimal epoch number by examining the loss and accuracy
+trends on training and validation data. As shown in Figure 7,
+after epoch reaches to 50, the training loss gets lower than
+validation loss while the accuracy of validation does not
+improve. Therefore, we set epoch to 50. While evaluating the
+classification performance on each application, we use ten-fold
+stratified cross validation to divide the dataset into ten disjoint
+partitions and evaluate model performance with each partition.
+We test the classification performance on different con-
+fidence thresholds. For each baseline classifier, we use the
+one-vs.-rest version of that classifier such that it produces a
+set of real-valued prediction scores for its decision instead
+of a labeled class. In ARcode model, the softmax function
+assigns probabilities of classes in each prediction. We compare
+the prediction probabilities with the confidence thresholds to
+determine the predicted class. For the prediction score si of
+class ci, i ∈ {1, ..., k}, the classifier assigns the label of the
+class by the following expression:
+�
+ci, where si = max(s1, ..., sk),
+if si ≥ threshold
+unknown,
+otherwise
+(5)
+In other words, when the maximum value of the prediction
+
+Training
+4.00
+Validation
+3.00
+S
+LoSS
+2.00
+1.00
+0.80 -
+Accuracy
+0.60
+0.40
+-
+40
+60
+120
+0
+20
+80
+100
+EpochsFigure. 8: Accuracy of ARcode and baseline classifiers at
+different confidence thresholds. The vertical dashed line in-
+dicates the confidence threshold (0.4) above which ARcode
+outperforms all other classifiers.
+Figure. 9: Accuracy of classifiers on each application at a
+confidence threshold of 0.8.
+scores is larger than the threshold, the class with the largest
+score is the predicted class. Otherwise, the classifier labels the
+observation with unknown. When the confidence threshold is
+0, it is the same with the vanilla multi-class classifier.
+D. Experimental Results
+We evaluate the capability of ARcode in identifying appli-
+cations in this subsection. First, we evaluate the classification
+performance on different confidence thresholds and examine
+the performance on each application. Then, we assess the
+performance of identifying applications that have not been
+trained with the models. In addition, since the job signatures
+encode temporal information of the monitoring metrics, we use
+a subset of the monitoring data to construct partial signatures
+to evaluate the ARcode model.
+1) Classification Performance: Figure 8 depicts the accu-
+racy of ARcode and baseline models on different confidence
+thresholds. As we can see from the figure, the Linear-SVC
+model has the worst performance at all confidence thresholds.
+When the confidence threshold is below 0.4, Random Forest
+outperforms all other classifiers, with the highest accuracy
+of 93.81% at a confidence threshold of 0. ARcode follows
+closely with an accuracy of 89.67%. When the confidence
+threshold is above 0.4, the accuracy of Random Forest de-
+creases and ARcode has the best performance. At a threshold
+of 1.0, the SVC model performs a little better than Random
+Figure. 10: Accuracy of ARcode and baseline classifiers on
+detecting novel applications at different confidence thresholds.
+ARcode achieved the highest accuracy when the confidence
+threshold is greater than 0.8.
+Forest, but it is still 18.87% worse than ARcode. At a
+confidence threshold of 0, the accuracy of Random Forest
+on power features is close to ARcode , but it decreases
+significantly as the confidence threshold increases. In sum-
+mary, the accuracy of all classifier decreases with increasing
+confidence thresholds, but the slowest decrease rate is observed
+for ARcode.
+We further examine the classification performance of
+ARcode and Random Forest on each application, as shown
+in Figure 9, where we use a confidence threshold of 0.8
+in the experiment. From this figure, we have the following
+observations. First, the accuracy of ARcode and Random
+Forest are very close in most cases. When detecting Gromacs,
+LAMMPS and cp2k, Random Forest performs slightly bet-
+ter. Second, for some applications such as BerkeleyGW and
+e3sm, ARcode achieves a significantly better accuracy. In
+BerkeleyGW classification, the accuracy of Random Forest is
+62.06% while ARcode achieves 83.94% accuracy. ARcode
+is also 20.21% better than Random Forest in detecting e3sm.
+2) Classification on Novel Applications: Considering that
+in HPC environments, ‘novel’ applications are more prevalent
+than those trained with the classifier, it is appropriate to
+define an ‘unknown’ class for all these novel applications. To
+be practically useful, the classification system must classify
+both known and unknown novel applications. We evaluate
+this capability by removing one application from the training
+set while keeping the testing set untouched. If the removed
+application in the testing set is predicted to be unknown, we
+mark it as a correct prediction.
+The classification performance for novel applications is
+depicted in Figure 10. From this figure, we can observe
+that Random Forest has the best prediction accuracy when
+the confidence threshold is below 0.8, while ARcode beats
+Random Forest once the confidence threshold is greater than
+0.8. ARcode, similar to its performance in known application
+detection, has relatively stable prediction accuracy across all
+confidence thresholds. However, the accuracy of Random
+Forest drops significantly in large confidence thresholds.
+
+100%
+80%
+60%
+Accuracy
+40%
+ARcode (CNN)
+Random Forest
+20%
+SVC
+LinearisvC
+Random Forest on Poweri Features!
+0%
+0.2
+0.0
+0.1
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Confidence ThresholdARcode(CNN)
+Random Forest
+100%
+80%
+Accuracy
+60%
+40%
+20%
+0%
+aims
+WRF
+BerkeleyGW Espresso Gromacs LAMMPS NWCHEM
+VASP
+chroma
+cp2k
+e3sm
+su3
+Applications100%
+80%
+60%
+Accuracy
+40%
+ARcode (CNN)
+Random Forest
+20%
+SVC
+Linearisvc
+Random Forest on Poweri Features
+0%
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Confidence ThresholdFigure. 11: Accuracy of ARcode on each application using
+different channels of the job signature.
+Figure. 12: Accuracy of ARcode on each application using
+partial job signature. The partial job signatures are all built
+from the metrics collected since the startup of jobs.
+3) Classification using Single Channel: In our experiment,
+ARcode encodes three channels of monitoring data in the
+job signature. It is worth knowing whether each channel plays
+the same importance in classification and whether we can use
+one channel for detection while still achieving high accuracy.
+To answer these questions, we train CNN models with each
+of these channels individually and analyze their classification
+performance. The results are shown in Figure 11.
+From the figure we can see that using any of the chan-
+nels gives a competitive accuracy when detecting Espresso,
+LAMMPS, WRF, and chroma compared to using all channels.
+For NWCHEM, the classification accuracy using the memory
+channel is significantly better than using the power or IPC
+channel. We can also see that all channels are important for
+detecting both Gromacs and cp2k with an accuracy improve-
+ment of 26.36% and 19.66%, respectively, compared to using
+only one of the channels. The memory channel is the most
+representative of these channels. In most applications, using
+the memory channel in detection has the closest accuracy to
+using all channels.
+4) Classification using Partial Signatures: The job sig-
+nature retains the temporal information of the time-series
+monitoring data, and the job signature generated before the
+end of the job still holds some of the attributes of the full
+job signature. We use the first 25%, 50% and 75% of the
+time-series data to create partial job signatures and compare
+the detection performance with the full job signature (i.e,
+a job signature built from 100% of the monitoring data).
+This experiment is performed to evaluate the usability of the
+ARcode model for detecting running applications, which is
+not available in the statistics-based models.
+The results are shown in Figure 12. It is not surprising that
+the larger the percentage of encoded data, the better the detec-
+tion performance achieved by ARcode. When encoded with
+25% of the monitoring data, ARcode achieves a relatively
+high accuracy of 77.27% in detecting WRF, but only 17.27%
+and 12.88% in detecting cp2k and Espresso, respectively.
+When detecting NWCHEM, the full job signature brings
+89.79% accuracy; however, the 75% partial job signature
+gives only 46.73% accuracy. The improvement from encoding
+more data varies from application to application. For example,
+encoding 25% more data brings an average improvement of
+24.67% for Espresso, but only 4.65% for WRF.
+5) Sensitivity to the resampling length: The performance
+evaluation described above is based on job signatures with
+a resolution of 128 × 128, i.e., the performance metrics are
+resampled to a length of 128. It is valuable to have knowledge
+of the sensitivity of the classification accuracy in terms of the
+resampling length such that an appropriate resolution can be
+chosen to achieve a balance between accuracy and training
+overhead. To explore this, we further build two job signature
+datasets with resampling lengths of 32 and 64, respectively,
+and use the same CNN architecture to train these models.
+Figure 13 depicts the accuracy (solid lines) and training
+time (horizontal dashed lines) of ARcode using different
+resampling lengths. Note that the training time is constant
+across different thresholds for the same resampling length, as
+illustrated by horizontal dashed lines. This is because training
+time does not vary according to confidence thresholds, which
+are set during the prediction of applications. As illustrated
+from the figure, the training times vary significantly among
+these three resampling lengths. It is over 185 seconds for
+the resampling length of 128. For signatures with resampling
+lengths of 64 and 32, it is 80 and 65 seconds, respectively.
+Additionally, we can observe from the figure that the higher
+resolution it is, the higher prediction accuracy. While the
+accuracy improvement from the resampling length of 64 to
+128 is not significant (only by 0.5% on average), the accuracy
+improvement from the resampling length of 32 to 64 is 3.0%
+on average.
+This result suggests that 64 is the optimal resampling length
+for building job signatures in our experiment, considering the
+trade-off between accuracy and training time. Its prediction
+accuracy is close to that of the highest resolution signature, but
+only uses 43.2% of the training time of the latter. Furthermore,
+the data size of the job signature generated with a resolution
+of 64 × 64 is only 25% of that with 128 × 128.
+V. RELATED WORKS
+Significant amount of work has been reported in the lit-
+erature on detecting and classifying applications. Identifying
+similarities and differences among binary executables was
+explored in [13]–[16]. These approaches are limited by the
+
+ZZIPC
+Memory
+Power
+All
+100%
+80%
+Accuracy
+60%
+40%
+20%
+0%
+WRF
+BerkeleyGWEspresso Gromacs LAMMPS NWCHEM 
+VASP
+aims
+chroma
+cp2k
+e3sm
+su3
+Applications25%
+75%
+100%
+50%
+Z
+100%
+80%
+Accuracy
+60%
+40%
+20%
+0%
+WRF
+BerkeleyGWEspresso Gromacs LAMMPS NWCHEM 
+VASP
+aims
+chroma
+cp2k
+e3sm
+su3
+ApplicationsFigure. 13: Accuracy and training time of ARcode using dif-
+ferent resampling lengths. The horizontal dashed lines indicate
+the training time.
+fact that, they cannot differentiate the same source program
+compiled by different compiler toolchains or optimization
+levels. To overcome this limitation, Blanket Execution [17]
+presented a binary differencing algorithm that compares the
+side effects of functions during executation, which is based
+on the insight that similar codes have semantically similar
+execution behavior.
+However, the binary based approach cannot be applied in the
+HPC environment, where it is impractical to conduct binary
+differencing among hundreds of thousands of executables. In
+addition, obtaining and managing binaries of users’ applica-
+tions are not always possible for HPC researchers. Instead
+of using binaries, Yamamoto et al. [34] used job scripts as
+job information to classify applications with text classification
+techniques. This approach, however, also requires dedicated
+collection and management infrastructures, which are not as
+prevalent as monitoring infrastructures for performance metric.
+Another line of work aims at characterizing HPC appli-
+cations through system logs. Liu et al. extracted features
+from combined logs of multiple subsystems to represent
+applications and build a machine learning model based on
+the eXtreme Gradient Boosting (XGB) algorithm to identify
+HPC applications [35]. DeMasi et al. collected and extracted
+features from Integrated Performance Monitoring (IPM) per-
+formance logs to fingerprint HPC codes [36]. Log analysis is
+more common in characterizing subsystem and user behavior,
+related works can be found in [37]–[40].
+Monitoring data based application detection has been ex-
+plored in [18]–[22]. As a early quantitative study of power
+consumption of HPC workloads, Combs et al. [18] studied
+the applicability of classifying applications through power
+consumption traces. Ramos et al. [21] relied on performance
+counters to model, fingerprint and clustering applications. Zou
+et al. [20] explored detecting illicit applications in GPU-
+accelerated HPC workloads. They used performance counters,
+data movement behavior and resources utilization traces to
+train machine learning models. Taxonomist [19], proposed
+by Ates et al., used over 700 system metrics and a time
+window spanning the whole execution to extract statistical
+features. They enhanced the classification model such that
+unknown applications can be detected. EFD [22] created
+key-values pairs that link execution fingerprints of system
+metrics to application and input size information to implement
+application recognition. Except for using measurements of
+system metrics, they also used the metrics name, node ID and
+time interval to create fingerprints.
+These studies, however, are based on the datasets built from
+the monitoring data of benchmarks and proxy applications,
+where a relative small range of configurations and input size
+are shown in the applications. On the contrary, the model
+and experiment results of ARcode are developed and tested
+on datasets collected from real applications in a large-scale
+production system. In addition, the models of these studies rely
+on the manually defined features extracted from time series
+monitoring data and some studies utilize the input information
+to enhance the detection model. In this work, ARcode allevi-
+ates the effort of features engineering and takes advantages of
+the capability of features learning in CNN model to detect and
+classify encoded monitoring data. Moreover, ARcode retains
+the temporal information of time-series performance metrics,
+enabling detecting applications before jobs are finished. This
+capability bridges a major gap in related works.
+VI. CONCLUSION
+Existing application detection methodologies on HPC sys-
+tems either relies on the data that are not prevalent or require
+intensive effort of feature engineering to build high accuracy
+models. In this study, we aim to provide a solution that can
+be easily used and adopted by any HPC site. We have intro-
+duced ARcode, an application recognition framework which
+is effective and extensible with the following characteristics:
+1) ARcode uses three most common monitoring metrics (i.e.,
+the power consumption of the compute node, instruction per
+cycle, and memory usage) available through the monitoring
+infrastructure on diverse HPC architectures. 2) ARcode allevi-
+ates the effort of feature engineering by leveraging the feature
+learning capability of CNN models. 3) ARcode’s channel-like
+encoding method allows easy encoding of additional metrics
+in job signatures. 4) ARcode encodes job monitoring data into
+images, which translates the application recognition problem
+into an image classification problem. Unlike the statistics-
+based approaches where the temporal information of monitor-
+ing data are lost, the job signature encodes metric variations
+over the runtime. Therefore, the job signature generated from
+part of monitoring data can still be used in detection, but with
+the sacrifice of accuracy.
+Although our evaluation is performed on job signatures
+generated from three common monitoring metrics, HPC re-
+searchers and system administrators can select other represen-
+tative metrics and build job signatures to detect and classify
+applications with specific characteristics such as CPU intensive
+applications and I/O intensive applications. In addition, we
+have seen lots of successful cases of applying image recogni-
+tion in the medical and automobile industries. Their experience
+in training high-accuracy models can be used in our model to
+further improve the recognition accuracy. Moreover, encoding
+
+100%
+128
+32
+64
+32
+64
+128
+200
+(accuracy)
+(training time)
+90%
+180
+Time (
+80%
+Accuracy
+140
+ining
+70%
+120
+Trai
+-100
+60%
+-80
+50%
+60
+0.0
+0.1
+0.2
+0.3
+0.6
+0.7
+0.8
+0.9
+1.0
+0.4
+0.5
+Confidence Thresholdmonitoring data in job signatures offers a new perspective
+of exploring and analyzing the performance monitoring data.
+In our future work, we will further explore the use of job
+signatures to predict resource usage, detect anomalies, and
+identify malicious applications.
+REFERENCES
+[1] J. Brandt, A. Gentile, J. Mayo, P. Pebay, D. Roe, D. Thompson, and
+M. Wong, “Resource monitoring and management with ovis to enable
+hpc in cloud computing environments,” in 2009 IEEE International
+Symposium on Parallel & Distributed Processing.
+IEEE, 2009, pp.
+1–8.
+[2] W. Allcock, E. Felix, M. Lowe, R. Rheinheimer, and J. Fullop, “Chal-
+lenges of hpc monitoring,” in SC’11: Proceedings of 2011 International
+Conference for High Performance Computing, Networking, Storage and
+Analysis.
+IEEE, 2011, pp. 1–6.
+[3] W. M. Jones, J. T. Daly, and N. DeBardeleben, “Application monitoring
+and checkpointing in hpc: looking towards exascale systems,” in Pro-
+ceedings of the 50th Annual Southeast Regional Conference, 2012, pp.
+262–267.
+[4] T. Evans, W. L. Barth, J. C. Browne, R. L. DeLeon, T. R. Furlani,
+S. M. Gallo, M. D. Jones, and A. K. Patra, “Comprehensive resource use
+monitoring for hpc systems with tacc stats,” in 2014 First International
+Workshop on HPC User Support Tools.
+IEEE, 2014, pp. 13–21.
+[5] J. Li, G. Ali, N. Nguyen, J. Hass, A. Sill, T. Dang, and Y. Chen,
+“Monster: an out-of-the-box monitoring tool for high performance
+computing systems,” in 2020 IEEE International Conference on Cluster
+Computing (CLUSTER).
+IEEE, 2020, pp. 119–129.
+[6] H. Shoukourian, T. Wilde, A. Auweter, and A. Bode, “Predicting
+the energy and power consumption of strong and weak scaling hpc
+applications,” Supercomputing frontiers and innovations, vol. 1, no. 2,
+pp. 20–41, 2014.
+[7] P.-F. Dutot, Y. Georgiou, D. Glesser, L. Lefevre, M. Poquet, and I. Rais,
+“Towards energy budget control in hpc,” in 2017 17th IEEE/ACM Inter-
+national Symposium on Cluster, Cloud and Grid Computing (CCGRID).
+IEEE, 2017, pp. 381–390.
+[8] O. Tuncer, E. Ates, Y. Zhang, A. Turk, J. Brandt, V. J. Leung, M. Egele,
+and A. K. Coskun, “Online diagnosis of performance variation in hpc
+systems using machine learning,” IEEE Transactions on Parallel and
+Distributed Systems, vol. 30, no. 4, pp. 883–896, 2018.
+[9] J. Dean, “Large-scale distributed systems at google: Current systems and
+future directions,” in The 3rd ACM SIGOPS International Workshop on
+Large Scale Distributed Systems and Middleware (LADIS 2009) Tutorial,
+2009.
+[10] E. Baseman, S. Blanchard, N. DeBardeleben, A. Bonnie, and A. Morrow,
+“Interpretable anomaly detection for monitoring of high performance
+computing systems,” in Outlier Definition, Detection, and Description
+on Demand Workshop at ACM SIGKDD. San Francisco (Aug 2016),
+2016.
+[11] M. C. Dani, H. Doreau, and S. Alt, “K-means application for anomaly
+detection and log classification in hpc,” in International Conference on
+Industrial, Engineering and Other Applications of Applied Intelligent
+Systems.
+Springer, 2017, pp. 201–210.
+[12] A. Borghesi, A. Libri, L. Benini, and A. Bartolini, “Online anomaly
+detection in hpc systems,” in 2019 IEEE International Conference on
+Artificial Intelligence Circuits and Systems (AICAS).
+IEEE, 2019, pp.
+229–233.
+[13] H. Flake, “Structural comparison of executable objects,” in Detection
+of intrusions and malware & vulnerability assessment, GI SIG SIDAR
+workshop, DIMVA 2004.
+Gesellschaft f¨ur Informatik eV, 2004.
+[14] T. Dullien and R. Rolles, “Graph-based comparison of executable objects
+(english version),” Sstic, vol. 5, no. 1, p. 3, 2005.
+[15] D. Gao, M. K. Reiter, and D. Song, “Binhunt: Automatically finding
+semantic differences in binary programs,” in International Conference
+on Information and Communications Security.
+Springer, 2008, pp.
+238–255.
+[16] M. Bourquin, A. King, and E. Robbins, “Binslayer: accurate comparison
+of binary executables,” in Proceedings of the 2nd ACM SIGPLAN
+Program Protection and Reverse Engineering Workshop, 2013, pp. 1–10.
+[17] M. Egele, M. Woo, P. Chapman, and D. Brumley, “Blanket execution:
+Dynamic similarity testing for program binaries and components,” in
+23rd {USENIX} Security Symposium ({USENIX} Security 14), 2014,
+pp. 303–317.
+[18] J. Combs, J. Nazor, R. Thysell, F. Santiago, M. Hardwick, L. Olson,
+S. Rivoire, C.-H. Hsu, and S. W. Poole, “Power signatures of high-
+performance computing workloads,” in 2014 Energy Efficient Supercom-
+puting Workshop.
+IEEE, 2014, pp. 70–78.
+[19] E. Ates, O. Tuncer, A. Turk, V. J. Leung, J. Brandt, M. Egele, and A. K.
+Coskun, “Taxonomist: Application detection through rich monitoring
+data,” in European Conference on Parallel Processing.
+Springer, 2018,
+pp. 92–105.
+[20] P. Zou, A. Li, K. Barker, and R. Ge, “Fingerprinting anomalous
+computation with rnn for gpu-accelerated hpc machines,” in 2019
+IEEE International Symposium on Workload Characterization (IISWC).
+IEEE, 2019, pp. 253–256.
+[21] V. Ramos, C. Valderrama, S. X. de Souza, and P. Manneback, “An
+accurate tool for modeling, fingerprinting, comparison, and clustering
+of parallel applications based on performance counters,” in 2019 IEEE
+International Parallel and Distributed Processing Symposium Workshops
+(IPDPSW).
+IEEE, 2019, pp. 797–804.
+[22] T. Jakobsche, N. Lachiche, A. Cavelan, and F. M. Ciorba, “An execution
+fingerprint dictionary for hpc application recognition,” in 2021 IEEE
+International Conference on Cluster Computing (CLUSTER).
+IEEE,
+2021, pp. 604–608.
+[23] B. Zhao, H. Lu, S. Chen, J. Liu, and D. Wu, “Convolutional neural
+networks for time series classification,” Journal of Systems Engineering
+and Electronics, vol. 28, no. 1, pp. 162–169, 2017.
+[24] Y. He, B. Cook, J. Deslippe, B. Friesen, R. Gerber, R. Hartman-Baker,
+A. Koniges, T. Kurth, S. Leak, W.-S. Yang et al., “Preparing nersc
+users for cori, a cray xc40 system with intel many integrated cores,”
+Concurrency and Computation: Practice and Experience, vol. 30, no. 1,
+p. e4291, 2018.
+[25] K. Antypas, N. Wright, N. P. Cardo, A. Andrews, and M. Cordery, “Cori:
+a cray xc pre-exascale system for nersc,” Cray User Group Proceedings.
+Cray, vol. 1, 2014.
+[26] A. Agelastos, B. Allan, J. Brandt, P. Cassella, J. Enos, J. Fullop, A. Gen-
+tile, S. Monk, N. Naksinehaboon, J. Ogden et al., “The lightweight
+distributed metric service: a scalable infrastructure for continuous mon-
+itoring of large scale computing systems and applications,” in SC’14:
+Proceedings of the International Conference for High Performance
+Computing, Networking, Storage and Analysis.
+IEEE, 2014, pp. 154–
+165.
+[27] B.
+Cook.
+(2021)
+NERSC
+LDMS
+Pipeline.
+[Online].
+Available:
+https://software.nersc.gov/ldms/nersc-ldms
+[28] S. J. Pan and Q. Yang, “A survey on transfer learning,” IEEE Trans-
+actions on knowledge and data engineering, vol. 22, no. 10, pp. 1345–
+1359, 2009.
+[29] Z. Wang and T. Oates, “Encoding time series as images for visual in-
+spection and classification using tiled convolutional neural networks,” in
+Workshops at the twenty-ninth AAAI conference on artificial intelligence,
+2015.
+[30] Y. Bengio, A. Courville, and P. Vincent, “Representation learning: A
+review and new perspectives,” IEEE transactions on pattern analysis
+and machine intelligence, vol. 35, no. 8, pp. 1798–1828, 2013.
+[31] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” nature, vol. 521,
+no. 7553, pp. 436–444, 2015.
+[32] E. Ates, O. Tuncer, A. Turk, V. J. Leung, J. Brandt, M. Egele, and
+A. K. Coskun, “Artifact for taxonomist: Application detection through
+rich monitoring data,” 2018.
+[33] J. Wilkes, “Google cluster-usage traces v3,” Google Inc., Mountain
+View, CA, USA, Technical Report, Apr. 2020, posted at https://github.
+com/google/cluster-data/blob/master/ClusterData2019.md.
+[34] K. Yamamoto, Y. Tsujita, and A. Uno, “Classifying jobs and predicting
+applications in hpc systems,” in International Conference on High
+Performance Computing.
+Springer, 2018, pp. 81–99.
+[35] Z. Liu, R. Lewis, R. Kettimuthu, K. Harms, P. Carns, N. Rao, I. Foster,
+and M. E. Papka, “Characterization and identification of hpc applications
+at leadership computing facility,” in Proceedings of the 34th ACM
+International Conference on Supercomputing, 2020, pp. 1–12.
+[36] O. DeMasi, T. Samak, and D. H. Bailey, “Identifying hpc codes via
+performance logs and machine learning,” in Proceedings of the first
+workshop on Changing landscapes in HPC security, 2013, pp. 23–30.
+[37] S. Chunduri, S. Parker, P. Balaji, K. Harms, and K. Kumaran, “Char-
+acterization of mpi usage on a production supercomputer,” in SC18:
+International Conference for High Performance Computing, Networking,
+Storage and Analysis.
+IEEE, 2018, pp. 386–400.
+
+[38] S.-H. Lim, H. Sim, R. Gunasekaran, and S. S. Vazhkudai, “Scientific
+user behavior and data-sharing trends in a petascale file system,” in
+Proceedings of the International Conference for High Performance
+Computing, Networking, Storage and Analysis, 2017, pp. 1–12.
+[39] G. K. Lockwood, S. Snyder, T. Wang, S. Byna, P. Carns, and N. J.
+Wright, “A year in the life of a parallel file system,” in SC18: In-
+ternational Conference for High Performance Computing, Networking,
+Storage and Analysis.
+IEEE, 2018, pp. 931–943.
+[40] T. Patel, S. Byna, G. K. Lockwood, and D. Tiwari, “Revisiting i/o be-
+havior in large-scale storage systems: the expected and the unexpected,”
+in Proceedings of the International Conference for High Performance
+Computing, Networking, Storage and Analysis, 2019, pp. 1–13.
+

6dFJT4oBgHgl3EQflizf/content/tmp_files/2301.11584v1.pdf.txt

@@ -0,0 +1,1959 @@
+Robust variance-regularized risk minimization
+with concomitant scaling
+Matthew J. Holland
+Osaka University
+Abstract
+Under losses which are potentially heavy-tailed, we consider the task of minimizing sums
+of the loss mean and standard deviation, without trying to accurately estimate the variance.
+By modifying a technique for variance-free robust mean estimation to fit our problem
+setting, we derive a simple learning procedure which can be easily combined with standard
+gradient-based solvers to be used in traditional machine learning workflows. Empirically,
+we verify that our proposed approach, despite its simplicity, performs as well or better
+than even the best-performing candidates derived from alternative criteria such as CVaR
+or DRO risks on a variety of datasets.
+Contents
+1
+Introduction
+2
+2
+Background
+3
+2.1
+Robust mean estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+2.2
+Good-enough ancillary scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.3
+A bridge between two problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2.4
+Overview of contributions and limitations . . . . . . . . . . . . . . . . . . . . .
+6
+3
+Theory
+7
+3.1
+Links to the mean-SD objective . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+3.2
+Guiding the optimal threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.3
+Deriving an algorithm using finite-sample theory . . . . . . . . . . . . . . . . .
+8
+3.4
+Stationary points of mean-variance . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+3.5
+Comparison with dual form of DRO risk . . . . . . . . . . . . . . . . . . . . . .
+11
+4
+Empirical analysis
+11
+4.1
+Simulated noisy classification on the plane . . . . . . . . . . . . . . . . . . . . .
+12
+4.2
+Classification on real datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+A Technical appendix
+18
+A.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+A.2 Convexity and smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+B Additional proofs
+20
+C Empirical test appendix
+27
+1
+arXiv:2301.11584v1  [stat.ML]  27 Jan 2023
+
+1
+Introduction
+Traditionally, the “textbook definition” of a statistical machine learning problem is formulated
+in terms of making decisions which minimize the expected value of a random loss [9, 27, 35].
+More precisely, the traditional setup has us minimize Eµ L(h) with respect to a decision h,
+where we denote random losses as L(h) ..= ℓ(h; Z), with a random data point Z ∼ µ, and ℓ(·)
+is a loss function assigning real values to (decision, data) pairs. This problem class is very
+general in that it covers a wide range of learning problems both supervised and unsupervised,
+but it is limited in the sense that it only aspired to be optimal on average, with no guarantees
+for other aspects of performance such as loss deviations, resilience to worst-case examples and
+distribution shift, sub-population disparity, and class-balanced error. While it is sometimes
+possible to account for these issues by modifying the base loss function ℓ (e.g., logit-adjusted
+softmax cross-entropy for balanced error [26]), there is a growing literature looking at prin-
+cipled, systematic modifications to the “risk,” i.e., a non-random numerical property of the
+distribution of L(h) to be optimized in h, leaving the base loss ℓ(·) fixed. Some prominent
+examples are weighted sums of loss quantiles [25], distributionally robust optimization (DRO)
+risk [13], conditional value-at-risk (CVaR) [7], tilted risk [20], and more general optimized
+certainty equivalent (OCE) risks [19], among others. It is well-known that many risks can be
+expressed in terms of location-deviation sums, with the canonical example being a weighted
+sum of the loss mean and standard deviation (or variance) [31, §2]. We refer the reader to some
+recent surveys [14, 16, 32] for more general background on developments in learning criteria.
+In this work, the criterion of interest is the mean loss regularized by standard deviation
+(SD), when losses are allowed to be heavy-tailed. More formally, we allow for heavy tails in
+the sense that all we assume is that the second moment Eµ|L(h)|2 is finite, and the ultimate
+objective of interest is the mean-SD criterion
+MSµ(h; λ) ..= Eµ L(h) +
+�
+λ Vµ L(h)
+(1)
+with loss variance denoted by Vµ L ..= Eµ(L − Eµ L)2, and weighting parameter λ ≥ 0. This
+mean-SD objective (1) and its mean-variance counterpart have a long history in the literature
+on decision making under uncertainty, including the influential work of Markowitz [23] on
+optimal portfolio selection. In the context of machine learning, it is well-known that one can
+obtain “fast rate” bounds on the expected loss when variance is small (see [11, §1]), though the
+problem of actually ensuring that loss deviations are sufficiently small is an entirely separate
+matter. In this direction, Maurer and Pontil [24] bound the (population) expected loss using a
+weighted sum of the sample mean and standard deviation. Their “sample variance penalized”
+objective is convenient to compute and can be used to guarantee fast rates in theory, but a
+lack of convexity makes it hard to minimize in practice. A convex approximation is developed
+by Duchi and Namkoong [11], who show that a sub-class of (empirical) DRO risks can be used
+to approximate the sample mean-SD objective, again yielding fast rates when the (population)
+variance is small enough. The critical limitation to this approach is poor guarantees under
+heavy-tailed losses; while we gain in terms of convexity, the empirical DRO risk of [11] is at least
+as sensitive to outliers as the naive empirical objective (i.e., directly minimizing the sample
+mean and SD), which is already known to result in highly sub-optimal performance guarantees
+under heavy tails [5, 10, 15]. Recent work by Zhai et al. [36] studies a natural strategy for
+robustifying the DRO objective (called DORO), which discards a specified fraction of the
+largest losses. While the impact of outliers can be reduced under the right setting of DORO,
+their approach is limited to non-negative losses, and the impact that such one-sided trimming
+has on the resulting mean-SD sum, our ultimate object of interest, is unknown.
+2
+
+With this context in mind, in this paper we propose a new approach to robustly minimize
+the objective (1) under heavy-tailed losses, without a priori knowledge of anything but the
+fact that variance is finite. Our key technique is based on extending a convex program of Sun
+[34] from one-dimensional mean estimation to our mean-variance objective MSµ(h; λ) under
+general losses. After some motivating background points in §2.1–§2.2, we describe our basic
+approach and summarize our contributions in §2.3–§2.4. Theoretical analysis comes in §3, and
+based upon formal properties of the proposed objective function, we derive a general-purpose
+procedure summarized in Algorithm 1, and tested empirically in §4.
+Our main finding is
+that the simple algorithm we derive works remarkably well on both simulated and real-world
+datasets without any fine-tuning, despite sacrificing the convexity enjoyed by procedures based
+on criteria such as CVaR and DRO. Software and notebooks to reproduce all results in this
+paper are provided in an online repository.1
+2
+Background
+Before we describe our proposed approach to the mean-SD task described in §1, we start with
+a much simpler problem, namely the task of robust mean estimation. This will allow us to
+highlight key technical points from the literature which provide both conceptual and technical
+context for our proposal. Key points from the existing literature are introduced in §2.1–§2.2,
+and building upon this we introduce our method in §2.3–§2.4.
+2.1
+Robust mean estimation
+Let X be a random variable. For the moment, our goal will be to construct an accurate empirical
+estimate of the mean Eµ X, assuming only that the variance Vµ X = Eµ X2 − (Eµ X)2 is both
+defined and finite. We assume access to an independent and identically distributed (IID) sample
+X1, . . . , Xn. Since higher-order moments may be infinite, the tails of X may be “heavy” and
+decidedly non-Gaussian, causing problems for the usual empirical mean. This problem setting
+is now very well-understood; see Lugosi and Mendelson [22] for an authoritative reference.
+One very well-known approach is to use M-estimators [18], namely to design an estimator
+An ≈ Eµ X satisfying
+An ∈ arg min
+a∈R
+b
+n
+n
+�
+i=1
+ρ
+�Xi − a
+b
+�
+(2)
+where ρ: R → R+ is a function that is approximately quadratic near zero, but grows more
+slowly in the limit, i.e., large deviations are penalized in a sub-quadratic manner, where “large”
+is relative to the scaling parameter b > 0, used to control bias. When ρ(·) is convex, differen-
+tiable, and the solution set is non-empty, the condition (2) is equivalent to
+1
+n
+n
+�
+i=1
+ρ′
+�Xi − An
+b
+�
+= 0
+(3)
+and when the derivative ρ′(·) is bounded such that
+− log(1 + −x + γx2) ≤ ρ′(x) ≤ log(1 + x + γx2),
+x ∈ R
+(4)
+for some constant 0 < γ < ∞, then the analytical approach of Catoni [6] tells us that when b2
+scales with Vµ X/n, the deviations |An − Eµ X| enjoy sub-Gaussian tails, namely upper bounds
+1https://github.com/feedbackward/bdd-mv
+3
+
+-2
+0
+2
+2
+0
+2
+-2
+0
+2
+′
+-2
+0
+2
+′′
+Figure 1: From left to right, we plot the graphs of ρ(·), ρ′(·), and ρ′′(·) with ρ as in (6). In the middle plot,
+the dotted curves represent the upper (blue) and lower (dark pink) bounds in (4) with γ = 1.
+of the order O(
+�
+log(1/δ) Vµ X/n) with probability at least 1 − δ. Under these weak assump-
+tions, such guarantees are essentially optimal [10]. While a very important result, for practical
+purposes, the need for knowledge of Vµ X is a significant limitation, since without finite higher-
+order moments, it is not plausible to obtain variance estimates with analogous sub-Gaussian
+guarantees (e.g., impossibility results of [10]). There do exist other robust estimators such
+as median-of-means [22, §2.1] which do not require variance information, and this illustrates
+the fact that knowledge of the variance is sufficient, although not necessary, for sub-Gaussian
+mean estimation under heavy tails.
+2.2
+Good-enough ancillary scaling
+Since sub-Gaussian estimates of the variance Vµ X are not possible under our weak assump-
+tions, it is natural to ask whether there exists a middle-ground, namely whether or not it is
+possible to construct a (data-driven) procedure for setting the scale b > 0 in (2) which is “good
+enough” in the sense that the resulting An is sub-Gaussian, even though the scale itself cannot
+be. An initial (affirmative) answer to this question was given in recent work by Sun [34], whose
+basic idea we briefly review here, with some slight re-formulation for readability and additional
+generality.
+Essentially, the underlying idea in [34] is to utilize the convexity of ρ in (2), and to solve
+for both a ∈ R and b > 0 simultaneously, while penalizing b in such a way as to encourage
+scaling which is “good enough” as mentioned. More precisely, the empirical objective
+�Sn(a, b) ..= βb + b
+n
+n
+�
+i=1
+ρ
+�Xi − a
+b
+�
+(5)
+plays a central role, where 0 < β < 1 is a parameter we can control, and ρ is fixed as
+ρ(x) =
+�
+x2 + 1 − 1,
+x ∈ R
+(6)
+which is differentiable, and satisfies the Catoni condition (4) with γ = 1 (see Figure 1). If we
+fix b > 0, then the solution sets (in a) of both �Sn(a, b) and b × �Sn(a, b) are identical, and it
+should be noted that the re-scaled map x �→ b2ρ(x/b) = b
+√
+x2 + b2 − b2 closely approximates
+x �→ x2/2 as b grows large (Figure 2), and is well-known as the “pseudo Huber” or “smooth
+Huber” function, where b acts as a smoothing parameter.2
+When considering the joint objective �Sn(a, b), from the computational side, one important
+fact is that this function is convex on R × (0, ∞) (see Lemma 8). From the statistical side of
+2Barron [2, §1] gives a summary of this and related functions from the perspective of loss function design.
+This is not the only smoothed variant of the classic Huber function [17], see for example Rey [29, §6.4.4].
+4
+
+1
+0
+1
+0.0
+0.2
+0.4
+x
+b2 (x/b)
+0.1
+2.0
+b value
+Figure 2: Graphs of the smooth Huber function, with ρ as in (6), over a range of smoothing parameters. For
+visual comparison, the graph of x �→ x2/2 is plotted with a thick dashed green curve.
+things, the solutions
+(An, Bn) ∈ arg min
+a∈R,b>0
+�Sn(a, b)
+(7)
+are such that under certain regularity conditions, the deviations |An − Eµ X| are nearly optimal
+(sub-Gaussian, up to poly-logarithmic factors) [34, §3.3].3 The corresponding Bn of course
+cannot give us sub-Gaussian estimates of the variance under such weak assumptions, but it
+does scale in a desirable way [34, §3.2], and when bias is mitigated by setting β sufficiently small
+given the sample size n, the resulting Bn is good enough to provide such guarantees for An,
+which is the ultimate goal anyways. By taking on a slightly more difficult optimization problem,
+it is possible to get away with not having prior knowledge or sub-Gaussian estimates of the
+variance. We use this basic insight as a stepping stone to our approach for learning algorithms
+charged with selecting a decision h such that the loss L(h) has a small mean-variance.
+2.3
+A bridge between two problems
+To develop our proposal, we now return to the more general learning setup, where the test
+data is a random vector Z ∼ µ, test loss is L(h) ..= ℓ(h; Z), and we have n IID training
+points Z1, . . . , Zn yielding losses Li(·) ..= ℓ(·; Zi), i ∈ [n]. If our goal was to simply minimize
+the traditional risk Eµ L(h) over h ∈ H under heavy-tailed losses, then in principle we could
+extend the approach of §2.2 to robustly estimate the test risk using
+(An(h), Bn(h)) ∈ arg min
+a∈R,b>0
+�
+βb + b
+n
+n
+�
+i=1
+ρ
+�Li(h) − a
+b
+��
+(8)
+and design a learning algorithm using (8) as follows:
+Hn ∈ arg min
+h∈H
+An(h).
+(9)
+Under some regularity conditions, the machinery of Brownlees et al. [5] could then be combined
+with pointwise concentration inequalities in [34] to control the tails of Eµ L(Hn) under just finite
+loss variance. Our goal however is not to minimize the expected loss, but rather the mean-SD
+sum (1). Furthermore, the bi-level program inherent in (9) is not computationally congenial
+from the perspective of large-scale machine learning tasks. To ease the computational burden
+3Strictly speaking, the objective used in [34] is �Sn(a, b)/β, but all key results easily translate to our setup.
+5
+
+while at the same time building a bridge between these two problems, we consider a new
+objective function taking the form
+�Cn(h; a, b) ..= αa + βb + λb
+n
+n
+�
+i=1
+ρ
+�Li(h) − a
+b
+�
+(10)
+with parameters α ≥ 0 and β ≥ 0. We call (10) the modified Sun-Huber objective, since ρ is
+fixed as (6), and this form plays a special role in our analysis. Compared with that of (9),
+this objective is a simple function of h, and gradient-based minimizers can be easily applied
+assuming the underlying loss ℓ(·) is sufficiently smooth. On the other hand, it is “biased” in
+the sense that it penalizes not just the loss location (whenever α > 0), but the loss scale as well
+(whenever β > 0). Intuitively, some kind of deviation-driven “bias” is precisely what we need
+from the standpoint of minimizing the mean-SD objective MSµ(h; λ), but it is not immediately
+clear how this objective relates to �Cn(h; a, b), and it is equally unclear if we can just plug this
+new objective into standard machine learning workflows (e.g., using stochastic gradient-based
+optimizers) and achieve the desired effect without a prohibitive amount of manual tuning.
+2.4
+Overview of contributions and limitations
+With our basic idea described and some key questions raised, we summarize the central points
+that characterize the rest of this paper, and also highlight the limitations of this work. Broadly
+speaking, the new proposal here is a class of empirical “risk” minimizers, namely any learning
+algorithm which minimizes the new empirical objective (10). More explicitly, this refers to all
+procedures which returns a triplet satisfying
+(Hn, An, Bn) ∈
+arg min
+h∈H,a∈R,b>0
+�Cn(h; a, b)
+(11)
+where H denotes a set of feasible decisions, and we note that each element of this class is
+characterized by the settings of α, β, and λ used to define �Cn. In analogy with the strat-
+egy employed in §2.2, we do not expect An and Bn to provide sub-Gaussian estimates; we
+simply hope that these estimates are good enough to ensure the mean-SD is smaller and/or
+better-behaved when compared to standard benchmarks such as mean-based empirical risk
+minimization (ERM) and DRO-based algorithms. Theoretically, we are interested in identify-
+ing links between the proposed objective �Cn and loss properties such as Eµ L(h) and Vµ L(h),
+with particular emphasis on how the settings of α, β, and λ influence such links.
+Our main theory-driven contribution is the derivation of a principled approach to determine
+�Cn (i.e., set α and β), before seeing any training data, in such a way that we can balance between
+“biased but robust” ρ-based deviations and “unbiased but outlier-sensitive” squared deviations
+that arise in the loss variance. Details are in §3.1–§3.3, and a concise procedure is summarized
+in Algorithm 1. We do not, however, consider the behavior of MSµ(Hn; λ) for a particular
+implementation of (11) (e.g., SGD) from a theoretical viewpoint; the implementation is left
+abstract. This is where the empirical analysis of §4 comes in. We provide evidence using
+simulated and real data that our procedure can be quite useful, even using a rudimentary
+implementation where we wrap base loss objects and naively pass them to standard stochastic
+gradient-based learning routines, with no manual tweaking of parameters.
+6
+
+3
+Theory
+3.1
+Links to the mean-SD objective
+We would like to make the connection between the proposed objective (10) and the ultimate
+objective (1) a bit more transparent. To do this, we will make use of the population version of
+�Cn, denoted henceforth by Cµ and defined as
+Cµ(h; a, b) ..= αa + βb + λb Eµ ρ
+�L(h) − a
+b
+�
+.
+(12)
+Let us fix the decision h and threshold a, paying close attention to the optimal value of the
+scale b, denoted here by bµ(h, a). More explicitly, consider any positive real number satisfying
+bµ(h, a) ∈ arg min
+b>0
+Cµ(h; a, b).
+(13)
+While it is not explicit in our notation, the optimal scale in (13) depends critically on the
+value of β. Intuitively, a smaller value of β leads to a weaker penalty for taking b large, thus
+encouraging a larger value of bµ(h, a). In fact, one can show that viewing bµ(h, a) as a function
+of the parameter β, in the limit we have (proof in §B)
+lim
+β→0+ bµ(h, a) = ∞.
+(14)
+Combining this with the fact that
+lim
+b→∞ b Eµ ρ
+�L(h) − a
+b
+�
+= 0
+(15)
+also holds (proof in §B), by re-scaling to avoid trivial limits we can obtain a result which
+sharply bounds the proposed learning criterion at the optimal scale using the square root of
+quadratic deviations, thereby establishing a clear link to the desired mean-SD objective (1).
+Proposition 1. Let H be such that Eµ|L(h)|2 < ∞ for each h ∈ H. If we set α = α(β) such
+that α(β)/√β → �α ∈ [0, ∞) as β → 0+, then in this limit, with appropriate re-scaling the
+scale-optimized learning criteria can be bounded above and below as
+�αa + (1/2)
+�
+λ Eµ(L(h) − a)2 ≤ lim
+β→0+ min
+b>0
+Cµ(h; a, b)
+√β
+≤ �αa + 4
+�
+λ Eµ(L(h) − a)2
+for any choice of threshold a ∈ R and weight α ≥ 0.
+In the special case where a = Eµ L(h) and �α > 0, we naturally recover mean-SD sums akin to
+those studied in an ERM framework by Maurer and Pontil [24] and those bounded from above
+using convex surrogates by Duchi and Namkoong [11].
+Of course in practice, we will only ever be working with fixed values of β, and the entire
+point of introducing new criteria (namely �Cn and Cµ) was to give us some control over how
+sensitive our objective is to loss tails. The following result makes the nature of this control
+(through β) more transparent.
+Proposition 2. Let H and L(h) be as stated in Proposition 1. Letting bµ(h, a) be as specified
+in (13), we define a Bernoulli random variable
+I(h; a) ..= I {|L(h) − a| ≤ bµ(h, a)}
+7
+
+for any choice of h ∈ H and a ∈ R. The optimal scale can then be bounded by
+λ
+4β Eµ I(h; a)(L(h) − a)2 ≤ b2
+µ(h, a) ≤ λ
+2β Eµ(L(h) − a)2
+for any choice of 0 < β < λ and a ∈ R.
+While it is difficult to pin down exactly how bµ(h, a) changes as a function of β, Proposition 2
+clearly shows us the appealing property that optimal scale induced by the proposed objective
+function essentially falls between the (tail-sensitive) quadratic deviations and a (tail-insensitive)
+truncated variant, with the truncation threshold loosening as β shrinks.
+3.2
+Guiding the optimal threshold
+Since the preceding Propositions 1–2 both hold for any choice of threshold a ∈ R, they clearly
+hold when both a and b are optimal, i.e., when a and b are set as
+(aµ(h), bµ(h)) ∈ arg min
+a∈R,b>0
+Cµ(h; a, b).
+(16)
+In particular, using first-order conditions, the inclusion (16) is equivalent to the following two
+equalities holding at once:
+Eµ
+�
+�
+L(h) − aµ(h)
+�
+(L(h) − aµ(h))2 + b2µ(h)
+�
+� = α,
+Eµ
+�
+�
+bµ(h)
+�
+(L(h) − aµ(h))2 + b2µ(h)
+�
+� = 1 − β/λ.
+(17)
+Given the context of our analysis in §3.1, let us consider the effect of taking β towards zero.
+For any non-trivial random loss, the second equality asks that bµ(h) grow without bound as
+β → 0+, while |aµ(h)| must be either bounded or grow slower than bµ(h). On the other hand,
+if α is too large (i.e., α > 1) then the first equality will be impossible to satisfy. In addition to
+taking 0 < α < 1, note that if we multiply both sides of the first equality in (17) by bµ(h) and
+apply Proposition 2, then under this optimality condition we must have
+Eµ
+�
+�
+�
+�
+L(h) − aµ(h)
+�
+(L(h)−aµ(h)
+bµ(h)
+)2 + 1
+�
+�
+�
+� ≤ α
+�
+λ
+2β Eµ(L(h) − aµ(h))2.
+(18)
+With this inequality in place, we adopt the following strategy: encourage the optimal location
+to converge as aµ(h) → Eµ L(h) when β → 0+. Since λ > 0 is assumed to be fixed in advance,
+the only way to ensure this using (18) is to set α = α(β) such that
+lim
+β→0+
+α(β)
+√β = 0.
+(19)
+While (19) gives us a rather clear condition for determining α given β, we still do not have a
+principled setting for β. This point will be treated in the following sub-section.
+3.3
+Deriving an algorithm using finite-sample theory
+To complement the preceding analysis and discussion centered around the population objective
+(12), we now return to the empirical objective function �Cn(h; a, b) introduced in (10). We
+maintain the running assumption that the training data Z1, . . . , Zn are an IID sample from µ,
+8
+
+and thus the losses Li(h), i = 1, . . . , n are independent given any fixed h. With h and b > 0
+fixed for the moment, we will now take a closer look at the optimal (empirical) threshold that
+arises from this objective function, namely any random variable An(h, b) satisfying
+An(h, b) ∈ arg min
+a∈R
+�Cn(h; a, b).
+(20)
+Using the property (4) of the smooth Huber-like function ρ, we can demonstrate how data-
+driven thresholds satisfying (20) are concentrated at a point near the expected loss, where α
+and b play a key role in how close this point is to the mean.
+Proposition 3 (Concentration at a shifted location). Taking 0 ≤ α < 1, b > 0, and 0 < δ < 1,
+with large enough n it is always possible to satisfy the condition
+4α
+λ ≤ 4
+�Vµ L(h)
+b2
++ log(2/δ)
+n
+�
+≤ 1 − 4α
+λ ,
+and when this condition is satisfied, the data-driven threshold An(h, b) in (20) satisfies
+����An(h, b) −
+�
+Eµ L(h) − 2α
+λ b
+����� ≤ 2
+�Vµ L(h)
+b
++ b log(2/δ)
+n
+�
+with probability no less than 1 − δ.
+This result can be seen as an extension of [34, Prop. 3.1] for the function (5) used in mean
+estimation to our generalized learning problem, although we use a different proof strategy
+which does not require strong convexity of �Cn (with respect to a).
+With Proposition 3 established, conventional wisdom might incline one to pursue a O(1/√n)
+rate in the upper bound; in this case, setting β ∝ 1/n is a natural strategy since Proposition
+2 tells us that for the population objective, the optimal setting of b scales with
+�
+λ/β. While
+this is natural from the perspective of tight concentration bounds for An(h, b), we argue that a
+different strategy is more appropriate when we actually consider how (Hn, An, Bn) will behave
+in the full joint optimization (11). The most obvious reason for this is that the joint objective
+lacks convexity and smoothness, as the following result summarizes.
+Proposition 4 (Joint objective is non-convex and non-smooth). Even when H is a compact
+convex set and the base loss function ℓ(·; Z) is convex, the mapping (h, a, b) �→ �Cn is not convex
+in general, and is non-smooth in the sense that its gradient is not Lipschitz continuous on
+H × R × (0, ∞).
+In consideration of Proposition 4, standard complexity results for typical optimizers such as
+stochastic gradient descent to achieve a ε-stationary point are on the order of O(ε−4); see
+Davis and Drusvyatskiy [8] for example.4
+With this in mind, setting β ∝ 1/n to achieve
+O(ε−2) sample complexity for error bounds of An(h, b) seems superfluous if in the end the
+dominant complexity for solving the ultimate problem (11) will be of the order O(ε−4). As
+such, in order to match this rate, the more natural strategy is to set β ∝ 1/√n, or more
+precisely to set
+β = β0
+√n
+(21)
+where β0 > 0 is a constant used to ensure 0 < β < λ. This, coupled with α(β) = β to satisfy
+(19) from the previous sub-section, is our proposed setting to determine (α, β) (and thus �Cn)
+using just knowledge of n, and without having observed any data points. This procedure is
+summarized in Algorithm 1, and will be the subject of empirical analysis later in §4.
+4Even if the objective were smooth, the same rates are typical; see for example Ghadimi and Lan [12].
+9
+
+Algorithm 1 Modified Sun-Huber
+Inputs: data Z1, . . . , Zn and parameter λ > 0.
+Set: β = β0/√n, with β0 such that 0 < β < λ.
+{Based on (21).}
+Set: α = β.
+{Satisfies (19).}
+Minimize: �Cn(h; a, b) in (h, a, b) using α and β as above.
+3.4
+Stationary points of mean-variance
+Having established links between the proposed objective and the mean-SD objective, we next
+consider the mean-variance objective
+MVµ(h) ..= Eµ L(h) + Vµ L(h).
+(22)
+This quantity can be expressed as the minimum value of a convex function, namely we have
+MVµ(h) = min
+a∈R
+�
+a + Eµ(L(h) − a)2 + 1
+2
+�
+= aMV(h) + Eµ(L(h) − aMV(h))2 + 1
+2
+(23)
+where on the right-most side we have set aMV(h) ..= Eµ L(h)−1. Assuming the underlying loss
+is differentiable, the gradient with respect to h can be written as
+MV′
+µ(h) = Eµ L′(h) + Eµ L(h) L′(h) − Eµ L(h) Eµ L′(h)
+= Eµ L′(h) + Eµ (L(h) − Eµ L(h)) L′(h)
+= Eµ (L(h) − (Eµ L(h) − 1)) L′(h)
+which implies a stationarity condition of
+MV′
+µ(h) = 0 ⇐⇒ Eµ (L(h) − (Eµ L(h) − 1)) L′(h) = 0.
+(24)
+Similarly, the partial derivative of the learning criterion (12) taken with respect to h is
+∂
+∂h Cµ(h; a, b) = Eµ
+�
+L(h) − a
+�
+(L(h) − a)2 + b2
+�
+L′(h)
+and thus multiplying both sides by b > 0, we obtain a simple stationarity condition of
+∂
+∂h Cµ(h; a, b) = 0 ⇐⇒ Eµ
+�
+�
+L(h) − a
+�
+(L(h)−a
+b
+)2 + 1
+�
+� L′(h) = 0.
+(25)
+With the right threshold setting, obviously the two conditions become very similar as b grows
+large. The following result makes this precise.
+Proposition 5. Let loss function ℓ and data distribution µ be such that the random vector
+L(h) L′(h) is integrable and has a norm with finite mean, i.e., Eµ∥L(h) L′(h)∥ < ∞ for some
+choice of h ∈ H. Then, for any a ∈ R, defining
+f(h; a) ..= lim
+b→∞ b ∂
+∂h Cµ(h; a, b)
+(26)
+the stationary points of the mean-variance objective are related to those of the proposed objective
+(12) through the following equivalence:
+f(h; aMV(h)) = 0 ⇐⇒
+∂
+∂h MVµ(h) = 0
+where MVµ(h) is as defined in (22).
+10
+
+1.0
+0.5
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+Figure 3: Graph of the Legendre transform ρ∗ as given in (28) over (−1, 1).
+3.5
+Comparison with dual form of DRO risk
+Some readers may notice that the proposed (population) objective (12) looks quite similar to
+the dual form of DRO risks:
+DROµ(h; β) ..=
+inf
+a∈R,b>0
+�
+a + βb + b Eµ φ∗
+�L(h) − a
+b
+��
+(27)
+where φ∗ is the Legendre-Fenchel convex conjugate φ∗(x) ..= supu∈R[xu − φ(u)] induced by a
+function φ : R → R, assumed to be convex and lower semi-continuous, with φ(1) = 0 and
+φ(x) = ∞ whenever x < 0 (cf. [33, §3.2]). Given this similarity, one might ask whether or not
+some form of DRO risk can be reverse engineered from our proposed objective. Taking up this
+point briefly, we first note that the conjugate of ρ given by (6) is
+ρ∗(x) ..= sup
+u∈R
+[xu − ρ(u)] = sup
+u∈R
+�
+xu −
+�
+u2 + 1 + 1
+�
+.
+From the non-negative nature of ρ, clearly ρ∗(0) = −ρ(0) = 0. For x ̸= 0, note that taking the
+derivative of concave function u �→ xu − ρ(u) and setting it to zero, we obtain the first-order
+optimality conditions
+u
+√
+u2 + 1 = x ⇐⇒
+sign(x)
+�
+1 + 1/u2 = x ⇐⇒
+1
+x2 = 1 + 1/u2 ⇐⇒ u =
+sign(x)
+�
+1/x2 − 1.
+Plugging this solution in whenever |x| < 1 and doing a bit of algebra readily yields the simple
+closed-form expression
+ρ∗(x) =
+�
+�
+�
+x2
+√
+1−x2 + 1 −
+1
+√
+1−x2 ,
+if 0 ≤ |x| < 1
+∞,
+else.
+(28)
+As can be readily observed from both (28) and Figure 3, this function does not satisfy any
+of the requirements placed on φ except convexity, and thus despite the similar form, the non-
+monotonic nature of ρ is in sharp contrast with monotonicity of typical cases of φ∗ that arise
+in the DRO literature (e.g. [3, §3]), and does not readily imply a “primal” DRO objective that
+can be recovered using ρ∗.
+4
+Empirical analysis
+Our investigation in the previous section led us to Algorithm 1, giving us a principled and
+precise strategy to construct the objective function �Cn, but leaving the actual minimization
+procedure abstract.
+Here we make this concrete by implementing a simple gradient-based
+minimizer of this objective, and comparing this procedure with natural benchmarks from the
+literature.
+11
+
+4.1
+Simulated noisy classification on the plane
+As a simplified and controlled setting to start with, we generate random data points on the
+plane which are mostly linearly separable, save for a single distant outlier (Figure 4). Before
+we consider off-sample generalization, here we focus simply on the training loss distribution
+properties as a function of algorithm iterations.
+Experiment setup
+We generate n = 100 training data points using two Gaussian distri-
+butions on the plane to represent two classes, with each class having the same number of
+points.
+We choose a single point uniformly at random, and perturb it by multiplying the
+scalar -10. We compare our proposed procedure (denoted “Modified Sun-Huber”) with three
+alternatives: traditional mean-based empirical risk minimization (denoted “Vanilla ERM”),
+conditional value-at-risk (CVaR) [7], and the well-studied χ2-DRO risk [11, 13]. In light of
+Algorithm 1, we set λ = log(n)/√n > β = β0/√n, and try a variety of β0 values just for
+reference. For all the aforementioned methods, we set the base loss ℓ(·) to the usual binary
+logistic loss (linear model), and run (batch) gradient descent on the empirical risk objectives
+implied by each of these methods (see §C for details), with a fixed step size of 0.01 over 15,000
+iterations. Alternative settings of step size and iteration number were not tested. All methods
+are initialized at the same point, shown in Figure 4.
+Results and discussion
+In Figure 5, we show the empirical mean-SD trajectories for the
+base loss, over algorithm iterations (log10 scale), for each method of interest. Using our no-
+tation, this is the sample version of MSµ(h; λ) in (1), with λ = 1 fixed. All methods besides
+vanilla ERM have multiple settings that were tested, and the results for each are distinguished
+using curves of different color. Our method tests different values of β0, CVaR tests different
+quantile levels, and DRO tests different robustness levels (details in §C). Since Vanilla ERM
+is designed to optimize the average loss, it is perhaps not surprising that it fails in terms of
+the mean-SD objective. On the other hand, the proposed method (for any choice of β0) is as
+good or better than all the competing methods. As a basic sanity check, in Figure 6 we also
+consider the error rate (average zero-one loss) and model norm trajectories over iterations for
+each method. For each method, we plot just one trajectory, namely the one achieving the best
+final error rate. While our method is not designed to minimize the average loss and typical
+surrogate theory does not apply, we find that the error rate is surprisingly good, albeit with
+slower convergence than the other methods. Note also how the error rate for CVaR matches
+that of Vanilla ERM; this is in fact the CVaR setting with the worst final mean-SD value. On
+the other hand, the proposed method performs well from both perspectives at once.
+4.2
+Classification on real datasets
+We proceed to experiments using real-world datasets, some of which are orders of magnitude
+larger than the simple setup given in §4.1, and which include multi-class classification tasks.
+Experiment setup
+We make use of four well-known datasets, all available from online
+repositories: adult,5 australian,6 cifar10,7 and fashion_mnist.8 For multi-class datasets,
+we extend the binary logistic loss to the usual multi-class logistic regression loss under a linear
+5https://archive.ics.uci.edu/ml/datasets/Adult
+6https://archive.ics.uci.edu/ml/datasets/statlog+(australian+credit+approval)
+7https://www.cs.toronto.edu/~kriz/cifar.html
+8https://github.com/zalandoresearch/fashion-mnist
+12
+
+model, with one linear model for each class. Features for all datasets are normalized to [0, 1],
+with one-hot representations of categorical features. The learning algorithms being compared
+here are the same as described in §4.1, except that now we implement each method using
+mini-batch stochastic gradient descent (batch size 32), and do 30 epochs (i.e., 30 passes over
+the training data). In addition, our proposed “Modified Sun-Huber” method performs almost
+identically for the range of β0 values tested in §4.1, and thus we have simply fixed β0 = 0.9, so
+there is only one trajectory curve this time. On the other hand, we now try a range of step sizes
+for each method, choosing the best step size in terms of average (base) loss value on validation
+data for each method. We run five independent trials, and for each trial we randomly re-shuffle
+the dataset, taking 80% for training, 10% for validation (used to select step sizes), and 10%
+for final testing.
+Results and discussion
+Our main results are shown in Figure 7, where once again we
+plot the trajectory of the mean-SD objective, but this time computed on test data, and given
+as a function of epoch number, rather than individual iterations.
+Since there are multiple
+trials, the curves drawn represent averages taken over trials, and the lightly shaded region
+above/below each curve shows standard deviation over trials. Perhaps surprisingly, the very
+simple implementation of our proposed Algorithm 1 (fixed step size, no regularization) works
+remarkably well on a number of datasets. From the perspective of mean-SD minimization,
+for three our of four datasets, the proposed method is far better than Vanilla ERM, and as
+good or better than even the best settings of CVaR and DRO viewed after the fact. Regarding
+the sub-standard performance observed on fashion_mnist, detailed analysis shows that more
+fine-tuned settings of α and β can readily bring the method up to par; the non-convex and non-
+smooth nature of �Cn naturally means that some tasks will require more careful settings than
+are captured by our Algorithm 1, and indeed will take explicit account of the optimizer to be
+used. We leave both the theoretical grounding and empirical testing of such optimizer-aligned
+mean-SD minimizers for future work.
+13
+
+Figure 4: 2D classification example from §4.1. The red line represents the initial value used by each method.
+102
+1
+2
+Vanilla ERM
+102
+Modified Sun-Huber
+102
+CVaR
+102
+2-DRO risk
+0.1
+0.9
+0.0
+0.8
+0.0
+0.3
+Mean + SD (2D classification with outliers)
+Figure 5: Trajectory of the (empirical) mean-SD objective (1) over iterations. Colors correspond to different
+choices from each class: β0 for Modified Sun-Huber, quantile level for CVaR, and constraint level for DRO.
+101
+103
+0.0
+0.2
+0.4
+Error rate
+101
+103
+0
+2
+4
+Norm
+Trajectory with best final error rate
+Vanilla ERM
+Modified Sun-Huber
+CVaR
+2-DRO risk
+Figure 6:
+From each method class, we show the classification error rate and Euclidean norm trajectories
+corresponding to the setting that achieved the best error rate after the final iteration.
+14
+
+0
+20
+0.7
+0.8
+Vanilla ERM
+0
+20
+Modified Sun-Huber
+0
+20
+CVaR risk
+0
+20
+2-DRO risk
+Mean + SD (dataset: adult)
+0
+20
+0.8
+1.0
+Vanilla ERM
+0
+20
+Modified Sun-Huber
+0
+20
+CVaR risk
+0
+20
+2-DRO risk
+Mean + SD (dataset: australian)
+0
+20
+2.5
+3.0
+3.5
+Vanilla ERM
+0
+20
+Modified Sun-Huber
+0
+20
+CVaR risk
+0
+20
+2-DRO risk
+Mean + SD (dataset: cifar10)
+0
+20
+1.5
+2.0
+2.5
+Vanilla ERM
+0
+20
+Modified Sun-Huber
+0
+20
+CVaR risk
+0
+20
+2-DRO risk
+Mean + SD (dataset: fashion_mnist)
+Figure 7: Mean-SD trajectories on real-world datasets as described in §4.2, given as a function of epochs and
+averaged over multiple independent trials. Coloring for CVaR and DRO is analogous to that of Figure 5.
+15
+
+References
+[1] Ash, R. B. and Doléans-Dade, C. A. (2000). Probability and Measure Theory. Academic
+Press, 2nd edition.
+[2] Barron, J. T. (2019). A general and adaptive robust loss function. In Proceedings of the
+IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4331–4339.
+[3] Ben-Tal, A., Den Hertog, D., De Waegenaere, A., Melenberg, B., and Rennen, G. (2013).
+Robust solutions of optimization problems affected by uncertain probabilities. Management
+Science, 59(2):341–357.
+[4] Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.
+[5] Brownlees, C., Joly, E., and Lugosi, G. (2015). Empirical risk minimization for heavy-tailed
+losses. The Annals of Statistics, 43(6):2507–2536.
+[6] Catoni, O. (2012). Challenging the empirical mean and empirical variance: a deviation
+study. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 48(4):1148–1185.
+[7] Curi, S., Levy, K. Y., Jegelka, S., and Krause, A. (2020). Adaptive sampling for stochastic
+risk-averse learning. In Advances in Neural Information Processing Systems 33 (NeurIPS
+2020), pages 1036–1047.
+[8] Davis, D. and Drusvyatskiy, D. (2019). Stochastic model-based minimization of weakly
+convex functions. SIAM Journal on Optimization, 29(1):207–239.
+[9] Devroye, L., Györfi, L., and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recogni-
+tion. Springer.
+[10] Devroye, L., Lerasle, M., Lugosi, G., and Oliveira, R. I. (2016).
+Sub-Gaussian mean
+estimators. The Annals of Statistics, 44(6):2695–2725.
+[11] Duchi, J. and Namkoong, H. (2019). Variance-based regularization with convex objectives.
+Journal of Machine Learning Research, 20(68):1–55.
+[12] Ghadimi, S. and Lan, G. (2013). Stochastic first- and zeroth-order methods for nonconvex
+stochastic programming. SIAM Journal on Optimization, 23(4):2341–2368.
+[13] Hashimoto, T. B., Srivastava, M., Namkoong, H., and Liang, P. (2018). Fairness with-
+out demographics in repeated loss minimization. In Proceedings of the 35th International
+Conference on Machine Learning (ICML), volume 80 of Proceedings of Machine Learning
+Research, pages 1929–1938.
+[14] Holland, M. J. and Tanabe, K. (2022). Learning criteria going beyond the usual risk.
+arXiv preprint arXiv:2110.04996v2.
+[15] Hsu, D. and Sabato, S. (2016). Loss minimization and parameter estimation with heavy
+tails. Journal of Machine Learning Research, 17(18):1–40.
+[16] Hu, S., Wang, X., and Lyu, S. (2022).
+Rank-based decomposable losses in machine
+learning: A survey. arXiv preprint arXiv:2207.08768v1.
+[17] Huber, P. J. (1964). Robust estimation of a location parameter. The Annals of Mathe-
+matical Statistics, 35(1):73–101.
+16
+
+[18] Huber, P. J. and Ronchetti, E. M. (2009). Robust Statistics. John Wiley & Sons, 2nd
+edition.
+[19] Lee, J., Park, S., and Shin, J. (2020). Learning bounds for risk-sensitive learning. In
+Advances in Neural Information Processing Systems 33 (NeurIPS 2020), pages 13867–13879.
+[20] Li, T., Beirami, A., Sanjabi, M., and Smith, V. (2021). Tilted empirical risk minimization.
+In The 9th International Conference on Learning Representations (ICLR).
+[21] Luenberger, D. G. (1969). Optimization by Vector Space Methods. John Wiley & Sons.
+[22] Lugosi, G. and Mendelson, S. (2019). Mean estimation and regression under heavy-tailed
+distributions: A survey. Foundations of Computational Mathematics, 19(5):1145–1190.
+[23] Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7(1):77–91.
+[24] Maurer, A. and Pontil, M. (2009).
+Empirical Bernstein bounds and sample variance
+penalization. In Proceedings of the 22nd Conference on Learning Theory (COLT).
+[25] Medina, A. M. and Yang, S. (2021). Robust unsupervised learning via L-statistic mini-
+mization. In 38th International Conference on Machine Learning (ICML), volume 139 of
+Proceedings of Machine Learning Research, pages 7524–7533.
+[26] Menon, A. K., Jayasumana, S., Rawat, A. S., Jain, H., Veit, A., and Kumar, S. (2021).
+Long-tail learning via logit adjustment. In The 9th International Conference on Learning
+Representations (ICLR).
+[27] Mohri, M., Rostamizadeh, A., and Talwalkar, A. (2012). Foundations of Machine Learn-
+ing. MIT Press.
+[28] Nesterov, Y. (2004). Introductory Lectures on Convex Optimization: A Basic Course.
+Springer.
+[29] Rey, W. J. J. (1983).
+Introduction to Robust and Quasi-Robust Statistical Methods.
+Springer.
+[30] Rockafellar, R. T. and Uryasev, S. (2000).
+Optimization of conditional value-at-risk.
+Journal of Risk, 2:21–42.
+[31] Rockafellar, R. T. and Uryasev, S. (2013).
+The fundamental risk quadrangle in risk
+management, optimization and statistical estimation. Surveys in Operations Research and
+Management Science, 18(1-2):33–53.
+[32] Royset, J. O. (2022). Risk-adaptive approaches to learning and decision making: A survey.
+arXiv preprint arXiv:2212.00856.
+[33] Shapiro, A. (2017). Distributionally robust stochastic programming. SIAM Journal on
+Optimization, 27(4):2258–2275.
+[34] Sun, Q. (2021). Do we need to estimate the variance in robust mean estimation? arXiv
+preprint arXiv:2107.00118v1.
+[35] Vapnik, V. N. (1999). The Nature of Statistical Learning Theory. Statistics for Engineering
+and Information Science. Springer, 2nd edition.
+17
+
+[36] Zhai, R., Dan, C., Kolter, J. Z., and Ravikumar, P. (2021). DORO: Distributional and
+outlier robust optimization. In 38th International Conference on Machine Learning (ICML),
+volume 139 of Proceedings of Machine Learning Research, pages 12345–12355.
+A
+Technical appendix
+A.1
+Basic facts
+Assuming ρ is defined as in (6), let us consider the function
+f(x, a, b) ..= αa + βb + bρ
+�x − a
+b
+�
+(29)
+= αa + βb +
+�
+(x − a)2 + b2 − b
+(30)
+= αa +
+�
+(x − a)2 + b2 − (1 − β)b.
+(31)
+The partial derivatives are as follows.
+∂xf(x, a, b) =
+x − a
+�
+(x − a)2 + b2
+(32)
+∂af(x, a, b) = α −
+x − a
+�
+(x − a)2 + b2
+(33)
+∂bf(x, a, b) =
+b
+�
+(x − a)2 + b2 − (1 − β)
+(34)
+The corresponding second derivatives are as follows.
+∂2
+xf(x, a, b) =
+1
+�
+(x − a)2 + b2 −
+(x − a)2
+((x − a)2 + b2)3/2 =
+b2
+((x − a)2 + b2)3/2
+(35)
+∂2
+af(x, a, b) =
+1
+�
+(x − a)2 + b2 −
+(x − a)2
+((x − a)2 + b2)3/2 =
+b2
+((x − a)2 + b2)3/2
+(36)
+∂2
+b f(x, a, b) =
+1
+�
+(x − a)2 + b2 −
+b2
+((x − a)2 + b2)3/2 =
+(x − a)2
+((x − a)2 + b2)3/2
+(37)
+The remaining elements of the Hessian of f(x, a, b) follow easily, given as follows.
+∂a∂xf(x, a, b) =
+−1
+�
+(x − a)2 + b2 +
+(x − a)2
+((x − a)2 + b2)3/2 =
+−b2
+((x − a)2 + b2)3/2
+(38)
+∂b∂xf(x, a, b) =
+−b(x − a)
+((x − a)2 + b2)3/2
+(39)
+∂b∂af(x, a, b) =
+b(x − a)
+((x − a)2 + b2)3/2
+(40)
+Lemma 6 (Useful inequalities).
+1
+1 + x ≤ 1 − x
+2,
+0 ≤ x ≤ 1.
+(41)
+(1 + x)c ≥ 1 + cx,
+x ≥ −1, c ∈ R \ (0, 1).
+(42)
+18
+
+A.2
+Convexity and smoothness
+Lemma 7. The map x �→ 1/√1 + x is convex on [0, ∞).
+Lemma 8 (Properties of partial objective). With ρ as in (6) and β ≥ 0, the function
+(x, b) �→ βb + bρ
+�x
+b
+�
+is convex and (1 + max{1 − β, β})-Lipschitz (in ∥·∥1) on R × (0, ∞), but its gradient is not
+(globally) Lipschitz, and thus the function is not smooth.9
+Proof of Lemma 8. For notational convenience, setting 0 < β < 1, let us denote
+g(x, b) ..= βb + bρ(x/b),
+x ∈ R, b > 0
+with ρ as in (6). From the partial derivatives (32) and (34), it is clear that we have
+−1 ≤ ∂xg(x, b) ≤ 1,
+−(1 − β) ≤ ∂bg(x, b) ≤ β
+when evaluated at any choice of x ∈ R and b > 0. It follows that the gradient norm can be
+bounded as
+∥∇g(x, b)∥1 ≤ 1 + max{(1 − β), β}
+and thus g(·) is Lipschitz continuous in ∥·∥1 (and also ∥·∥2).10
+Next, let us denote the Hessian of g(·) evaluated at (x, b) by H. Basic calculus gives us the
+simple form
+H ..=
+1
+(x2 + b2)3/2
+�
+b2
+−xb
+−xb
+x2
+�
+and for any pair of real values u = (u1, u2), we have
+⟨Hu, u⟩ =
+1
+(x2 + b2)3/2 (u1b − u2x)2 ≥ 0.
+(43)
+Since this holds for any choice of x ∈ R and b > 0, the Hessian is thus positive semi-definite,
+implying that g(·) is (jointly) convex [28, Thm. 2.1.4].
+On the other hand, the function g(·) is not smooth. To see this, first note that having
+chosen any u such that ∥u∥ ≤ 1, we have that the (operator) norm is bounded below as
+∥H∥ = sup
+∥u′∥≤1
+�
+sup
+∥u′′∥≤1
+⟨Hu′, u′′⟩
+�
+≥ ⟨Hu, u⟩.
+Then, as a concrete example, consider setting x = b, with u = (u1, u2) such that u1 ̸= u2.
+Recalling the lower bound (43), we have
+∥H∥ ≥
+b2
+(2b2)3/2 (u1 − u2)2 = (u1 − u2)2
+(
+√
+2)3b
+→ ∞
+in the limit as b → 0+.
+As such, the gradient of g(·) cannot be Lipschitz continuous on
+R × (0, ∞), and thus g(·) is not smooth [28, Thm. 2.1.6].
+9We prove that the Hessian’s norm is unbounded, which implies (via Nesterov [28, Thm. 2.1.6]) that the
+convex function of interest cannot be smooth.
+10That bounded gradients imply Lipschitz continuity is a general fact on linear spaces [21, §7.3, Prop. 2].
+19
+
+B
+Additional proofs
+Proof of Proposition 1. To begin, note that the function
+b �→ b Eµ ρ
+�L(h) − a
+b
+�
+= Eµ
+��
+(L(h) − a)2 + b2 − b
+�
+(44)
+is monotonic (non-increasing) on (0, ∞) (follows clearly from (34)). We will use this property
+moving forward. Recalling the upper and lower bounds of Proposition 2, we re-write them as
+clo(β)
+β
+≤ b2
+µ(h, a) ≤ chi
+β
+(45)
+using the shorthand notation
+clo(β) ..= λ
+4 Eµ I(h; a)(L(h) − a)2
+chi ..= λ
+2 Eµ(L(h) − a)2
+and noting that while chi is free of β, clo(β) depends on β through the definition of I(h; a).
+Fixing 0 < β < λ for now and recalling the form of Cµ in (12), the preceding bounds (45) and
+monotonicity of (44) can be used to obtain a lower bound of the form
+min
+b>0 Cµ(h; a, b) ≥ αa +
+�
+βclo(β) + λ√chi Eµ
+�
+�
+�
+(L(h) − a)2
+chi
++ 1
+β −
+�
+1
+β
+�
+� .
+(46)
+Using the fact (14) and applying dominated convergence [1, Thm. 1.6.9], in the limit we have
+lim
+β→0+ clo(β) = λ
+4 Eµ(L(h) − a)2.
+Dividing both sides of (46) by √β, setting α = α(β) as in the proposition statement, and
+taking the limit as β → 0+, we obtain
+lim
+β→0+ min
+b>0
+Cµ(h; a, b)
+√β
+≥ �αa +
+�
+λ
+4 Eµ(L(h) − a)2 + λ Eµ(L(h) − a)2
+2√chi
+= �αa +
+�
+λ
+4 Eµ(L(h) − a)2 +
+�
+λ
+2 Eµ(L(h) − a)2
+= �αa +
+�1
+2 + 1
+√
+2
+� �
+λ Eµ(L(h) − a)2.
+The first inequality uses the fact that for any c > 0, we have
+√
+cx + x2 − x → c/2 as x → ∞,
+and also uses dominated convergence. The remaining equalities just follow from plugging in
+the definition of chi and cleaning up terms. This proves the desired lower bound.
+As for the upper bound of interest, a perfectly analogous argument can be applied. Using
+Proposition 2 again and taking β small enough that
+clo(β) ≥ chi/4
+(47)
+20
+
+holds (always possible), we can obtain upper bounds of the form
+min
+b>0 Cµ(h; a, b) ≤ αa +
+�
+βchi + λ
+�
+clo(β) Eµ
+��
+(L(h) − a)2
+clo(β)
++ 1
+β −
+�
+1
+β
+�
+≤ αa +
+�
+βchi + λ√chi Eµ
+�
+�
+�
+4(L(h) − a)2
+chi
++ 1
+β −
+�
+1
+β
+�
+�
+(48)
+noting that the latter inequality (48) follows from using (47) as well as clo(β) ≤ chi. As with
+the lower bound argument in the preceding paragraph, we set α = α(β), divide both sides by
+√β, and take the limit as β → 0+. This results in
+lim
+β→0+ min
+b>0
+Cµ(h; a, b)
+√β
+≤ �αa +
+�
+λ
+2 Eµ(L(h) − a)2 + 2λ Eµ(L(h) − a)2
+√chi
+= �αa +
+�
+λ
+2 Eµ(L(h) − a)2 + 2
+�
+2λ Eµ(L(h) − a)2
+= �αa +
+�
+2
+√
+2 + 1
+√
+2
+� �
+λ Eµ(L(h) − a)2
+which gives us the desired upper bound. The bounds given in the proposition statement are
+slightly looser, but more readable.
+Proof of Proposition 2. We adapt key elements of the scale control used by Sun [34, §2] to our
+setting. We start by looking at first-order conditions for optimality of b > 0. First, note that
+∂
+∂b Cµ(h; a, b) = β + λ ∂
+∂b
+�
+Eµ
+�
+(L(h) − a)2 + b2 − b
+�
+= β + λ Eµ
+�
+b
+�
+(L(h) − a)2 + b2
+�
+− λ.
+As such, it follows that
+Eµ
+�
+b
+�
+(L(h) − a)2 + b2
+�
+= 1 − β/λ
+(49)
+is equivalent to ∂b Cµ(h; a, b) = 0. Obviously, the left-hand side of (49) is non-negative for all
+b ≥ 0 and bounded above by 1 for all b ≥ 0, a ∈ R, and h ∈ H. Thus (49) can only hold for
+0 ≤ β ≤ λ. Using convexity (Lemma 7) and Jensen’s inequality [1, Thm. 6.3.5], we have
+Eµ
+�
+b
+�
+(L(h) − a)2 + b2
+�
+= Eµ
+�
+�
+1
+�
+(L(h)−a
+b
+)2 + 1
+�
+� ≥
+�
+�
+1
+�
+Eµ(L(h)−a
+b
+)2 + 1
+�
+�
+and thus whenever (49) holds, we know that
+(1 − β/λ)2 ≥
+1
+Eµ(L(h)−a
+b
+)2 + 1
+must also hold. Re-arranging terms, we see that this implies
+b2 ≤ (1 − β/λ)2 Eµ(L(h) − a)2
+1 − (1 − β/λ)2
+.
+21
+
+For readability, set η ..= β/λ, and note that since
+(1 − η)2
+1 − (1 − η)2 = (1 − η)2
+2η − η2 =
+(1 − η)2
+2η(1 − η/2) ≤ (1 − η)2
+2η(1 − η) ≤ 1
+2η
+we can obtain the cleaner (but looser) upper bound
+b2 ≤ Eµ(L(h) − a)2
+2η
+= λ
+2β Eµ(L(h) − a)2
+for any choice of 0 < β ≤ λ and a ∈ R. Since the first-order condition (49) is necessary for
+optimality [28, Thm. 1.2.1], it follows that
+b2
+µ(h, a) ≤ Eµ(L(h) − a)2
+2β
+(50)
+which is the desired upper bound.
+Considering a lower bound next, note first that using the concavity of x �→ √x on R+,
+another application of Jensen’s inequality gives us
+Eµ
+�
+b
+�
+(L(h) − a)2 + b2
+�
+= Eµ
+�
+b2
+(L(h) − a)2 + b2 ≤
+�
+�
+�
+�Eµ
+�
+1
+(L(h)−a
+b
+)2 + 1
+�
+.
+(51)
+Using the inequality 1/(x + 1) ≤ 1 − x/2 for all 0 ≤ x ≤ 1 ((41) in Lemma 6), this suggests
+a natural event to use as a condition.
+More precisely, writing E ..= I {|L(h) − a| ≤ b} for
+readability, note that we have
+1
+(L(h)−a
+b
+)2 + 1
+=
+1 − E
+(L(h)−a
+b
+)2 + 1
++
+E
+(L(h)−a
+b
+)2 + 1
+≤
+1 − E
+(L(h)−a
+b
+)2 + 1
++ E
+�
+1 − 1
+2
+�L(h) − a
+b
+�2�
+=
+�
+1 − E
+(L(h)−a
+b
+)2 + 1
++ E
+�
+�
+��
+�
+≤1
+−E
+2
+�L(h) − a
+b
+�2
+.
+Taking expectation and utilizing (51), whenever (49) holds, we have
+1 − β/λ = Eµ
+�
+b
+�
+(L(h) − a)2 + b2
+�
+≤
+�
+1 − Eµ
+E
+2
+�L(h) − a
+b
+�2
+.
+(52)
+With this established, note that via helper inequality (42), for any β ≤ λ we have
+(1 − β/λ)2 ≥ 1 − 2β/λ
+and thus in light of (52), we may conclude that
+1 − 2β/λ ≤ 1 − Eµ
+E
+2
+�L(h) − a
+b
+�2
+which implies
+λ
+4β Eµ E(h; a, b)(L(h) − a)2 ≤ b2
+22
+
+noting that we have written E(h; a, b) to emphasize the dependence on h, a, and b. Once again
+since the first-order condition (49) is necessary for optimality, we may conclude that
+λ
+4β Eµ E(h; a, bµ(h, a))(L(h) − a)2 ≤ b2
+µ(h, a)
+(53)
+which is the remaining desired inequality.
+Proof of the limit (14). Recall from the proof of Proposition 2 the first-order optimality con-
+dition (49), which is satisfied by any solution bµ(h, a) given by (13), i.e., we have
+Eµ
+�
+�
+bµ(h, a)
+�
+(L(h) − a)2 + (bµ(h, a))2
+�
+� = 1 − β/λ
+(54)
+for any 0 < β ≤ λ. Defining g(β) ..= 1 − β/λ and taking any 0 < β2 < β1 ≤ λ, clearly we have
+g(β1) < g(β2) and thus using the equality (54), we must have that bµ(h, a; β2) ≥ bµ(h, a; β1),
+otherwise it would result in a contradiction of (54). Using this monotonicity, clearly
+E(h; a, bµ(h, a; β1)) ≤ E(h; a, bµ(h, a; β2))
+and thus
+Eµ E(h; a, bµ(h, a; β1))(L(h) − a)2 ≤ E(h; a, bµ(h, a; β2))(L(h) − a)2.
+Applying this to the lower bound in Proposition 2, we have
+lim inf
+β→0+ b2
+µ(h, a) ≥ lim
+β→0+
+λ
+4β Eµ I(h; a)(L(h) − a)2 = ∞
+as desired.
+Proof of the limit (15). Note that we can easily bound the random variable of interest as
+0 ≤ bρ
+�L(h) − a
+b
+�
+=
+�
+(L(h) − a)2 + b2 − b ≤ |L(h) − a|
+(55)
+for any choice of 0 < b < ∞. Some straightforward calculus shows that
+lim
+b→∞ bρ
+�L(h) − a
+b
+�
+= 0
+in a pointwise sense. Since the upper bound in (55) is µ-integrable by assumption, a simple
+application of dominated convergence [1, Thm. 1.6.9] yields
+lim
+b→∞ b Eµ ρ
+�L(h) − a
+b
+�
+= Eµ
+�
+lim
+b→∞ bρ
+�L(h) − a
+b
+��
+= 0
+as desired.
+23
+
+Proof of Proposition 3. From condition (20), since any solution must also be a stationary point
+[28, Thm. 1.2.1], we know that An ..= An(h, b) must satisfy the first-order condition
+λ
+n
+n
+�
+i=1
+ρ′
+�Li(h) − An
+b
+�
+= α
+which is equivalent to
+b
+n
+n
+�
+i=1
+ρ′
+�Li(h) − An
+b
+�
+= α
+λb.
+(56)
+Next we make use of the argument developed by Catoni [6, §2]. First note that fixing any
+a ∈ R and b > 0, we have
+E exp
+� n
+�
+i=1
+ρ′
+�Li(h) − a
+b
+��
+= E
+� n
+�
+i=1
+exp
+�
+ρ′
+�Li(h) − a
+b
+���
+=
+n
+�
+i=1
+Ei exp
+�
+ρ′
+�Li(h) − a
+b
+��
+≤
+n
+�
+i=1
+Ei
+�
+1 + Li(h) − a
+b
++ γ
+b2 (Li(h) − a)2
+�
+=
+�Eµ L(h) − a
+b
++ γ
+b2 Eµ(L(h) − a)2
+�n
+≤ exp
+�n
+b (Eµ L(h) − a) + nγ
+b2 Eµ(L(h) − a)2
+�
+.
+(57)
+The second equality above follows from the independence of the training data, and the first
+inequality uses the upper bound in (4), which is satisfied by ρ given in (6) with γ = 1, though
+we leave γ as is to illustrate how more general results are obtained. The third equality just uses
+the fact that the training data is an IID sample from µ, and the final inequality culminating
+in (57) just uses the bound 1 + x ≤ exp(x). Using Markov’s inequality and taking 0 < δ < 1,
+it is straightforward to show that (57) implies a 1 − δ event (over the draw of Z1, . . . , Zn) in
+which we have
+n
+�
+i=1
+ρ′
+�Li(h) − a
+b
+�
+≤ n
+b (Eµ L(h) − a) + nγ
+b2 Eµ(L(h) − a)2 + log(1/δ).
+Multiplying both sides by b/n, on the same “good” event, we have
+b
+n
+n
+�
+i=1
+ρ′
+�Li(h) − a
+b
+�
+≤ Eµ L(h) − a + γ
+b Eµ(L(h) − a)2 + b log(1/δ)
+n
+= Eµ L(h) − a + γ
+b
+�
+Vµ L(h) + (Eµ L(h) − a)2�
++ b log(1/δ)
+n
+(58)
+where (58) follows from expanding the quadratic term and doing some algebra.
+With the
+equality (56) in mind, subtracting a constant from both sides of (58), note that we equivalently
+have
+b
+n
+n
+�
+i=1
+ρ′
+�Li(h) − a
+b
+�
+− α
+λb ≤ p(a)
+(59)
+24
+
+where we have defined
+p(a) ..= Eµ L(h) − a + γ
+b
+�
+Vµ L(h) + (Eµ L(h) − a)2�
++ b log(1/δ)
+n
+− α
+λb
+(60)
+for readability. Note that p(·) in (60) is a polynomial of degree 2, and can be written as
+p(a) = ua2 + va + w
+(61)
+with coefficients defined as
+u ..= γ
+b
+v ..= (−1)
+�
+1 + 2γ Eµ L(h)
+b
+�
+w ..= Eµ L(h) + γ
+b Eµ|L(h)|2 + b log(1/δ)
+n
+− α
+λb.
+This polynomial has real roots whenever v2 − 4uw ≥ 0, and some algebra shows that this is
+equivalent to
+0 ≤ D ≤ 1, where D ..= 4
+��γ
+b
+�2
+Vµ L(h) + γ log(1/δ)
+n
+− γα
+λ
+�
+.
+(62)
+Assuming this holds, denoting by a+ the smallest root of p(·), i.e., the smallest of satisfying
+p(a+) = 0, the critical fact of interest to us is that An ≤ a+ on the good event of (59). This is
+valid due to two facts: first, the left-hand side of (59) is a decreasing function of a; second, due
+to (56), we know that An is a root of the left-hand side of (59). With this key fact in hand,
+using the quadratic formula we have
+An ≤ a+
+= Eµ L(h) + b
+2γ
+�
+1 −
+√
+1 − D
+�
+= Eµ L(h) + b
+2γ
+�
+1 −
+√
+1 − D
+� �
+1 +
+√
+1 − D
+�
+�
+1 +
+√
+1 − D
+�
+= Eµ L(h) + b
+2γ
+D
+�
+1 +
+√
+1 − D
+�
+≤ Eµ L(h) + b
+2γ D.
+Taking the two ends of this inequality chain together and expanding D, we have
+An ≤ Eµ L(h) − 2(α/λ)b + 2
+�γ
+b Vµ L(h) + b log(1/δ)
+n
+�
+(63)
+with probability no less than 1 − δ, assuming that n, b, and α are such that 0 ≤ D ≤ 1 holds.
+This gives us the desired upper bound.
+To obtain a lower bound, a perfectly analogous argument can be applied. First, using the
+lower bound in (4) and the fact that ρ′(−x) = −ρ′(x), we know that
+ρ′
+�a − Li(h)
+b
+�
+≤ log
+�
+1 + a − Li(h)
+b
++ γ
+b2 (a − Li(h))2
+�
+(64)
+25
+
+for any a ∈ R, b > 0, and i ∈ [n]. Plugging this inequality (64) into an argument analogous to
+the chain of inequalities that led to (58) earlier, it is clear that again on an event of probability
+no less than 1 − δ, we have
+b
+n
+n
+�
+i=1
+ρ′
+�a − Li(h)
+b
+�
+≤ a − Eµ L(h) + γ
+b
+�
+Vµ L(h) + (Eµ L(h) − a)2�
++ b log(1/δ)
+n
+.
+(65)
+Once again the upper bound we can bound this using a polynomial of degree 2, namely
+b
+n
+n
+�
+i=1
+ρ′
+�a − Li(h)
+b
+�
++ α
+λb ≤ q(a)
+(66)
+where we have defined
+q(a) ..= a − Eµ L(h) + γ
+b
+�
+Vµ L(h) + (Eµ L(h) − a)2�
++ b log(1/δ)
+n
++ α
+λb.
+(67)
+Now, since An is a root of the left-hand side of (66) viewed as a function of a, and this function
+is monotonically increasing, it is evident that denoting the largest root of q(·) (when it exists)
+by a−, we have An ≥ a−, a lower bound in contrast to the An ≤ a+ upper bound used earlier.
+For completeness, we write this polynomial as
+q(a) = u′a2 + v′a + w′
+(68)
+with coefficients
+u′ ..= γ
+b
+v′ ..=
+�
+1 − 2γ Eµ L(h)
+b
+�
+w′ ..= (−1) Eµ L(h) + γ
+b Eµ|L(h)|2 + b log(1/δ)
+n
++ α
+λb.
+We have two real roots whenever
+1 ≥ D′ ..= 4
+��γ
+b
+�2
+Vµ L(h) + γ log(1/δ)
+n
++ γα
+λ
+�
+(69)
+holds, and thus we obtain a high probability lower bound on An as follows:
+An ≥ a−
+= Eµ L(h) − b
+2γ
+�
+1 −
+√
+1 − D′
+�
+≥ Eµ L(h) − b
+2γ D′.
+Expanding D′ gives us the lower bound
+An ≥ Eµ L(h) − 2(α/λ)b − 2
+�γ
+b Vµ L(h) + b log(1/δ)
+n
+�
+(70)
+with probability no less than 1 − δ, as desired.
+26
+
+Let us conclude this proof by organizing the technical assumptions. First of all, for the
+two quadratics used in the preceding bounds, we require both (62) and (69) to hold. It is
+straightforward to verify that having these conditions both hold is equivalent to the following:
+4γα
+λ
+≤ 4
+��γ
+b
+�2
+Vµ L(h) + γ log(1/δ)
+n
+�
+≤ 1 − 4γα
+λ .
+(71)
+As such, whenever α, δ, and b are such that (71) holds, using a union bound, it follows that
+with probability no less than 1 − 2δ, we have a bound on
+|An − (Eµ L(h) − 2(α/λ)b)| ≤ 2
+�γ
+b Vµ L(h) + b log(1/δ)
+n
+�
+as desired. The proposition statement takes a cleaner form since we have γ = 1.
+Proof of Proposition 4. The lack of convexity follows from the fact that the composition of
+two convex functions need not be convex when the outermost function is non-monotonic (see
+for example Boyd and Vandenberghe [4, Ch. 3]), and the lack of smoothness follows a fortiori
+from Lemma 8.
+C
+Empirical test appendix
+For reference, we give the population versions of the CVaR and χ2-DRO criteria used in the
+empirical tests of §4. First, it is well known (see Rockafellar and Uryasev [30]) that CVaR at
+quantile level ξ can be represented as
+CVaRµ(h; ξ) = inf
+a∈R
+�
+a +
+1
+1 − ξ Eµ (L(h) − a)+
+�
+(72)
+where (x)+ ..= max{0, x}. Similarly, DRO risk based on the Cressie-Read family of divergence
+functions is formulated (for any c > 1) using
+DROµ(h; η) = inf
+a∈R
+�
+a + (1 + c(c − 1)η)1/c �
+Eµ (L(h) − a)c∗
++
+�1/c∗�
+(73)
+where c∗ ..= c/(c − 1), and χ2-DRO is the special case where c = 2 [11, 13, 36]. The different
+“robustness levels” mentioned in §4 correspond to a re-parameterized quantity �η ∈ (0, 1),
+related to η by the equality η = (1/(1 − �η) − 1)/2. Just as our �Cn(h; a, b) is solved jointly in
+(h, a, b), our empirical tests minimize the empirical versions of (72) and (73) jointly in (h, a).
+27
+

8tFJT4oBgHgl3EQfnyx6/content/tmp_files/2301.11593v1.pdf.txt

@@ -0,0 +1,1890 @@
+arXiv:2301.11593v1  [math.AG]  27 Jan 2023
+THE DONOVAN–WEMYSS CONJECTURE VIA THE
+TRIANGULATED AUSLANDER–IYAMA CORRESPONDENCE
+GUSTAVO JASSO, BERNHARD KELLER, AND FERNANDO MURO
+Abstract. We provide an outline of the proof of the Donovan–Wemyss Con-
+jecture in the context of the Homological Minimal Model Program for three-
+folds. The proof relies on results of August, of Hua and the second-named au-
+thor, Wemyss, and on the Triangulated Auslander–Iyama Correspondence—a
+recent result by the first- and third-named authors.
+Contents
+Introduction
+1
+1.
+Preliminaries
+2
+2.
+The Derived Donovan–Wemyss Conjecture
+7
+3.
+Uniqueness of the 2Z-derived contraction algebra
+8
+4.
+Concluding remarks
+20
+References
+23
+Introduction
+We work over the field C of complex numbers. A compound Du Val (=cDV)
+singularity is a complete local hypersurface
+R ∼= C�x, y, z, t�
+(f + tg) ,
+where C�x, y, z�/(f) is a Kleinian surface singularity and g ∈ C�x, y, z, t� is arbi-
+trary. Introduced by Reid in the early 1980s [Rei83], cDV singularities form an
+important class of three-dimensional singularities in birational geometry and play
+a significant role in the Minimal Model Program (MMP) for threefolds [KM98,
+Sec. 5.3] as well as in the Homological MMP [Wem18].
+We refer the reader
+to [Aug19, Ch. 1] and [Wem21] for introductions to the subject.
+This note is concerned with the following geometric situation: Let R be an iso-
+lated cDV singularity and p: X → Spec(R) a crepant resolution, that is p is a
+proper birational map with smooth source such that the pullback of the dualising
+sheaf of Spec(R) along f is the dualising sheaf of X. It follows that Spec(R) has
+a unique singular point m and the (reduced) exceptional fibre p−1(m) = �n
+i=1 Ci is
+a union of curves, with Ci ∼= P1
+C [VdB04, Lemma 3.4.1]. To these data, Donovan
+and Wemyss [DW16, DW19] associate a (basic, connected) finite-dimensional alge-
+bra Λcon = Λcon(p), the contraction algebra of p, which represents the functor of
+‘simultaneous non-commutative deformations’ of the reduced exceptional fibre. By
+construction, Λcon is a Cn-augmented algebra, and hence in particular determines
+the number n of irreducible components of the exceptional fibre. The contraction
+2020 Mathematics Subject Classification. Primary 14E30; Secondary 13D03.
+Key words and phrases. Minimal model program; compound Du Val singularity; crepant res-
+olution; contraction algebra; Hochschild cohomology; Auslander correspondence.
+1
+
+2
+G. JASSO, B. KELLER, AND F. MURO
+algebra encodes a surprising amount of information stemming from the given geo-
+metric setup. For example, when p contracts a single curve, the contraction algebra
+recovers known invariants such as Reid’s width [Rei83] and the Gopakumar–Vafa
+invariants [Kat08], see [Tod15]. Neither the dimension nor the Gabriel quiver of
+contraction algebras suffice for differentiating cDV singularities [DW16, Table 2].
+In fact, it is well-known that there are continuous families of pairwise non isomor-
+phic cDV singularities (that is ‘cDV singularities have moduli’). Notwithstanding,
+at the risk of stating the obvious, let us point out that the contraction algebra is
+equiped with crucial data in the form of the multiplication law and that this law
+is essential in recovering the above mentioned invariants. Equipped with their al-
+gebra structure, contraction algebras distinguish between non-isomorphic isolated
+cDV singularities that admit a crepant resolution in all known examples. These
+considerations motivate the following remarkable conjecture.
+Conjecture A (Donovan and Wemyss [DW16]). Let R1 and R2 be isolated cDV
+singularities with crepant resolutions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2).
+Then, the contraction algebras Λcon(p1) and Λcon(p2) are derived equivalent if and
+only if there is an isomorphism of algebras R1 ∼= R2.
+The original conjecture was formulated only in the case of single-curve contrac-
+tions; algebraically, this corresponds to the case where the contraction algebras are
+local and thus derived equivalence reduces to mere isomorphism of algebras since
+contraction algebras are basic, see [Zim14, Prop. 6.7.4] for example. In the above
+form, which allows for contracting multiple curves, the conjecture appeared in print
+in [Aug20, Conj. 1.3].
+That the contraction algebras of a given isolated cDV singularity are derived
+equivalent follows by combining results from Wemyss [Wem18] and Dugas [Dug15].
+In this note we provide an outline of the proof of the remaining part of Conjecture A.
+This proof first appeared in the appendix to [JM22] written by the second-named
+author where it is explained how the conjecture follows by combining previous
+results of August [Aug20] and [HK18] with the Triangulated Auslander–Iyama Co-
+rrespondence—the main result in [JM22]. For the sake of concreteness, we restrict
+ourselves to the specific context of the conjecture, with the understanding that
+most concepts and results that are presented here are but special cases of a much
+more general theory that the reader can find in the original sources. We hope that
+this sacrifice in generality makes the proof of the conjecture accessible to a broader
+readership.
+Acknowledgements. The authors thank Michael Wemyss and Zheng Hua for in-
+teresting conversations and email exchanges; in addition, they thank M. W. for sug-
+gesting the terminology ‘2Z-derived contraction algebra’ used in these proceedings.
+The third-named author is algo grateful to Matt Booth for answering questions.
+Financial
+support. F.
+M.
+was
+partially
+supported
+by
+grants
+PID2020-
+117971GB-C21
+funded
+by
+MCIN/AEI/10.13039/501100011033,
+US-1263032
+(US/JUNTA/FEDER, UE), and P20 01109 (JUNTA/FEDER, UE).
+1. Preliminaries
+In this section we collect preliminary definitions and results that are needed in
+our proof of Conjecture A. We use freely the theories of differential graded cate-
+gories [Kel94, Kel06] and A∞-categories [LH03]. We denote the derived category
+of an algebra or, more generally, a DG algebra A by D(A); the perfect derived
+
+THE DONOVAN–WEMYSS CONJECTURE
+3
+category of A, that is the full subcategory of D(A) spanned by its compact objects,
+is denoted by Dc(A). All (DG) modules are right (DG) modules.
+1.1. 2Z-cluster tilting objects. Let T be a triangulated category whose underly-
+ing additive category is Krull–Schmidt and has finite-dimensional morphism spaces.
+Definition 1.1.1 ([IY08, GKO13]). A basic1 object T ∈ T is 2-cluster tilting if the
+following conditions hold:
+(1) The object T is rigid: T(T, T [1]) = 0.
+(2) For each object X ∈ T there exists a triangle T1 → T0 → X → T1[1] with
+T0, T1 ∈ add(T ), where add(T ) is the smallest additive subcategory of T
+containing T that is closed under direct summands.
+We say that T is 2Z-cluster tilting if it is 2-cluster tilting and T ∼= T [2].
+Remark 1.1.2. Clearly, if T ∈ T is a 2- or 2Z-cluster tilting object, then T generates
+T as a triangulated category with split idempotents (which is to say that T is a
+classical generator of T). In particular, if the triangulated category T is algebraic2
+then there exists a DG algebra A and an equivalence of triangulated categories
+T
+∼
+−→ Dc(A),
+T �−→ A.
+Remark 1.1.3. Given a basic 2-cluster tilting object T ∈ T, one can produce a new
+such object by a procedure called mutation that, in a nutshell, replaces a single
+indecomposable direct summand of T by a new one, see [IY08] for a precise defini-
+tion. This process, which can be iterated, is an important reason for introducing
+2-cluster tilting objects into the framework of the Homological MMP, see [Wem18]
+and compare with Section 1.3.
+Remark 1.1.4. In general, 2Z-cluster tilting objects are not invariant under mu-
+tation (however, see [HI11, Sec. 4]). In the context of the Homological MMP this
+problem does not occur since the notions of 2- and 2Z-cluster tilting object coincide,
+see Section 1.2.
+1.2. Maximal Cohen–Macaulay modules and singularity categories. Let
+R be an isolated cDV singularity and
+CM(R) := {M ∈ mod(R)| depth(M) = dim(R)}
+be the category of maximal Cohen–Macaulay R-modules [Yos90, LW12]. The cate-
+gory CM(R) is a Frobenius exact category and, therefore, the stable category CM(R)
+has an induced structure of a triangulated category; moreover, there is a canonical
+equivalence of triangulated categories
+CM(R)
+∼
+−→ Dsg(R) := Db(mod(R))/ Kb(proj(R)),
+where Dsg(R) is the singularity category of R, see [Buc21] for details. We record
+the following facts for later use:
+• [Yos90, Prop. 1.18] Since R is complete local, CM(R) ≃ Dsg(R) is a Krull–
+Schmidt category with finite-dimensional morphism spaces.
+• [Aus78] Since R is 3-dimensional, CM(R) ≃ Dsg(R) is a 2-Calabi–Yau tri-
+angulated category [Kon98, Kel08], that is there is a natural isomorphism
+DHom(X, Y )
+≃
+−→ Hom(Y, X[2]),
+X, Y ∈ CM(R) ≃ Dsg(R),
+where V �→ DV denotes the passage to the C-linear dual.
+1An object in a Krull–Schmidt additive category is basic if all of its indecomposable direct
+summands have multiplicity one.
+2A triangulated category is algebraic if it is equivalent—as a triangulated category—to the
+stable category of a Frobenius exact category [Kel94].
+
+4
+G. JASSO, B. KELLER, AND F. MURO
+• [Eis80] Since R is a hypersurface, CM(R) ≃ Dsg(R) is a 2-periodic triangu-
+lated category, that is there is an isomorphism of exact functors [2] ∼= 1.
+In particular, the notions of 2- and 2Z-cluster tilting object coincide in this
+context.
+• The endomorphism algebra of any basic object X in CM(R) ≃ Dsg(R) is
+a symmetric algebra.
+This is an immediate consequence of the natural
+isomorphisms
+Hom(X, X) ∼= D Hom(X, X[2]) ∼= D Hom(X, X) .
+We also consider the DG category Dsg(R)dg, which is defined as the DG quo-
+tient [Kel99, Dri04] of the canonical DG enhancements of the triangulated cate-
+gories Db(mod(R)) and Kb(proj(R)). By construction,
+H0(Dsg(R)dg) = Dsg(R).
+1.3. Contraction algebras via 2Z-cluster tilting objects. Let R be an isolated
+cDV singularity and p: X → Spec(R) a crepant resolution. As explained in the
+introduction, to this geometric setup Donovan and Wemyss associate a basic finite-
+dimensional algebra Λcon = Λcon(p). We recall an alternative construction of the
+algebra Λcon that is more adapted to the methods we utilise in this note. Given p
+as above, a theorem of Van den Bergh [VdB04] furnishes a tilting bundle
+OX ⊕ N = OX ⊕ N(p) ∈ coh(X)
+and Wemyss proves [Wem18] that there is an isomorphism of algebras
+Λcon ∼= EndR(N)
+between the contraction algebra of p and the stable endomorphism algebra of
+N := H0(N) ∈ CM(R). Remarkably, when viewed as an object of the triangulated
+category CM(R), the R-module N is a 2Z-cluster tilting object. Conversely, given
+a 2Z-cluster tilting object T ∈ CM(R), there exists a crepant resolution of Spec(R)
+whose associated contraction algebra is isomorphic to EndR(T ). We summarise the
+previous discussion in the following theorem.3
+Theorem 1.3.1 ([Wem18]). Let R be an isolated cDV singularity and assume
+that Spec(R) admits a crepant resolution. Then, the contraction algebras of R are
+precisely the endomorphism algebras of 2Z-cluster tilting objects in the triangulated
+category CM(R) ≃ Dsg(R).
+The following theorem of August reduces Conjecture A from a derived equiv-
+alence to an isomorphism problem.
+The proof leverages the characterisation of
+contraction algebras provided by Theorem 1.3.1.
+Theorem 1.3.2 ([Aug20, Thm. 1.4]). Let R be an isolated cDV singularity and
+assume that Spec(R) admits a crepant resolution. The contraction algebras of R
+form a single and complete derived equivalence class of basic algebras.
+Corollary 1.3.3. Let R1 and R2 be isolated cDV singularities with crepant reso-
+lutions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2)
+and corresponding contraction algebras Λ1 = Λcon(p1) and Λ2 = Λcon(p2). If the
+algebras Λ1 and Λ2 are derived equivalent, then there exists a contraction algebra
+Λ of R2 such that Λ ∼= Λ1.
+3Wemyss proves even more: Up to isomorphism on both sides, crepant resolutions of R cor-
+respond bijectively to (basic) 2Z-cluster tilting objects in Dsg(R). In particular, the number of
+isomorphism classes of 2Z-cluster tilting objects in Dsg(R) if finite, for the number of minimal
+models of Spec(R) is finite [KM87].
+
+THE DONOVAN–WEMYSS CONJECTURE
+5
+1.4. Hochschild cohomology. Let A be a graded algebra.
+The bigraded
+Hochschild cochain complex has components
+Cp,q(A, A) := HomC
+�
+A⊗p, A(q)
+�
+,
+p ≥ 0, q ∈ Z,
+where V �→ V (1) is the (vertical) degree shift of graded vector spaces, equipped
+with the Hochschild differential x �→ ∂(x) of bidegree (1, 0), see [Mur20] for the
+precise definition, related structure (described below) and sign conventions. The
+first degree is called horizontal or Hochschild degree and the second is the vertical
+or internal degree; the sum of both is the total degree and we denote it by |x| (we
+also use this notation for the degree of an element in a singly-graded vector space).
+The component of total degree n of the Hochschild complex is
+�
+p+q=n
+Cp,q(A, A) .
+The Hochschild complex is equipped with a brace algebra structure, consisting
+of operations called braces (which we do not describe explicitly here)
+C•,∗(A, A)⊗n+1
+−→ C•,∗(A, A) ,
+x0 ⊗ x1 ⊗ · · · ⊗ xn �−→ x0{x1, . . . , xn},
+that are defined for all n ≥ 1, and satisfy the brace relation
+x{y1, . . . , yp}{z1, . . . , zq}
+=
+�
+0≤i1≤j1≤···≤ip≤jp≤q
+(−1)ǫ{z1, . . . , zi1, y1{zi1+1, . . . , zj1}, zj1+1, . . .
+. . . , zip, yq{zip+1, . . . , zjp}, zjp+1, . . . , zq}.
+Above, ǫ reflects the Koszul sign rule with respect to the total degree shifted by
+−1. Brace operations have horizontal degree −n and vertical degree 0 and the
+n-th brace operation vanishes when x0 has horizontal degree < n. The Hochschild
+complex is a bigraded associative algebra equipped with the cup-product
+x · y = (−1)|x|−1m2{x, y},
+where m2 ∈ C2,0(A, A) is, up to a sign, the multiplication in the graded algebra A.
+The Hochschild complex also has the structure of a (horizontally-shifted) graded
+Lie algebra, with Lie bracket
+[x, y] = x{y} − (−1)(|x|−1)(|y|−1)y{x}
+of horizontal degree −1 and vertical degree 0. The Hochschild differential is given
+by
+∂(x) = [m2, x]
+and satisfies the corresponding Leibniz rules with respect to the previous associative
+product and Lie bracket.
+The relation between brace operations, the Hochschild differential and the cup
+product are encoded in the following straightforward consequences of the brace
+relation.
+
+6
+G. JASSO, B. KELLER, AND F. MURO
+Lemma 1.4.1. The following formula holds for all n ≥ 1:
+∂(x0{x1, . . . , xn}) = ∂(x0){x1, . . . , xn}
++
+n
+�
+i=1
+(−1)
+�i−1
+j=0 |xj|−ix0{x1, . . . , ∂(xi), . . . , xn}
++ (−1)|x0|−1+|x0||x1|x1 · x0{x2, . . . , xn−1}
++
+n−1
+�
+i=1
+(−1)
+�i
+j=0 |xi|−i−1x0{x1, . . . , xi · xi+1, . . . , xn}
++ (−1)
+�n−1
+j=0 |xj|−n−1x0{x1, . . . , xn−1} · xn.
+Lemma 1.4.2. The following formula holds for all n ≥ 1:
+(x · y){z1, . . . , zn} =
+n
+�
+i=0
+(−1)|y| �i
+j=1(|zi|−1)x{z1, . . . , zi} · y{zi+1, . . . , zn}.
+Lemma 1.4.1 for n = 1 proves that the induced associative product in Hochschild
+cohomology
+HH•,∗(A, A) = H•,∗(C•,∗(A, A))
+(the cohomology of the Hochschild complex) is graded commutative with respect to
+the total degree. This products satisfies the following compatibility relation with
+the (horizontally shifted) Lie algebra structure,
+[x, y · z] = [x, y] · z + (−1)(|x|−1)|y|y · [x, z],
+and hence Hochschild cohomology is a Gernstenhaber algebra. For this we use both
+Lemmas 1.4.1 and 1.4.2, for n = 2 and n = 1 respectively. The Hochschild complex
+C•,∗(A, M) and Hochschild cohomology HH•,∗(A, M) are defined, more generally,
+for M an A-bimodule, but it does not have any multiplicative structure in this
+general case. For the existence of a graded associative algebra structure it suffices
+that M be an associative algebra in A-bimodules, see also [Mur22, Sec. 1].
+1.5. Minimal A∞-algebras. We now describe minimal A∞-algebras and their
+morphisms in terms of the Hochschild complex. A minimal A∞-algebra structure
+on a graded algebra A is a Hochschild cochain
+m = (0, 0, 0, m3, . . . , mn, . . . ) ∈ C•,∗(A, A)
+of total degree 2 such that the Maurer–Cartan equation
+(1.5.1)
+∂(m) + m{m} = ∂(m) + 1
+2[m, m] = 0
+is satisfied. The pair (A, m) is also denoted
+(A, m3, . . . , mn, . . . ).
+If g : A′ → A is a graded algebra isomorphism, then
+m ∗ g = (0, 0, 0, g−1m3g⊗3, . . . , g−1mng⊗n, . . . )
+is a minimal A∞-algebra structure on A′. If
+m′ = (0, 0, 0, m′
+3, . . . , m′
+n, . . . ) ∈ C•,∗(A, A)
+is another minimal A∞-algebra structure on A, an A∞-isomorphism with identity
+linear part
+f : (A, m) −→ (A, m′)
+is a Hochschild cochain
+f = (0, 0, f2, f3, . . . , fn, . . . ) ∈ C•,∗(A, A)
+
+THE DONOVAN–WEMYSS CONJECTURE
+7
+of total degree 1 such that the following Hochschild cochain vanishes
+(1.5.2)
+∂(f) + f · f +
+�
+r≥0
+m′{f, r. . ., f} − m − f{m}.
+More generally, an A∞-isomorphism between minimal A∞-algebras
+f : (A, m) −→ (A′, m′)
+consists of an isomorphism of graded algebras
+f1 : A −→ A′
+and a Hochschild cochain
+(0, 0, f2, f3, . . . , fn, . . . ) ∈ C•,∗(A, A′)
+of total degree 1 such that
+(0, 0, f −1
+1 f2, f −1
+1 f3, . . . , f −1
+1 fn, . . . ): (A, m) −→ (A, m′ ∗ f1)
+is an A∞-isomorphism with identity linear part.
+2. The Derived Donovan–Wemyss Conjecture
+In this section we discuss one of the main results in [HK18]—crucial to our proof
+of the Dononvan–Wemyss conjecture—and explain how it implies a derived version
+of the conjecture (Corollary 2.2.4).
+2.1. 2Z-derived contraction algebras. By means of the equivalence of triangu-
+lated categories CM(R) ≃ Dsg(R), the contraction algebra associated to a crepant
+resolution p of an isolated cDV singularity can be promoted to the DG algebra
+Λcon = Λcon(p) := REnd(N)
+given by the derived endomorphism algebra of the corresponding 2Z-cluster tilt-
+ing object N = N(p) ∈ Dsg(R)dg. By construction H0(Λcon) ∼= Λcon and, as a
+consequence of the 2-periodicity of the singularity category of R,
+H•(Λcon) ∼= Λcon[ı±1] = Λcon ⊗C C[ı±1],
+|ı| = −2.
+We refer to the DG algebra Λcon as the 2Z-derived contraction algebra of p. The
+(soft) non-positive truncation Λ≤0
+con = τ ≤0Λcon of Λcon is quasi-isomorphic to the
+derived contraction algebra of p considered for example in [Boo19, Boo21, HK18],
+and there is an isomorphism of graded algebras
+H•(Λ≤0
+con) ∼= Λcon[ı] = Λcon ⊗C C[ı],
+|ı| = −2.
+The 2Z-derived contraction algebra Λcon is a localisation of Λ≤0
+con (see [Boo21,
+Thms. 6.4.6 and 7.2.3] and [HK18, Thm. 4.17]) and Λcon can also be interpreted
+as a non-connective variant of Λ≤0
+con.
+Notice also that, since 2Z-cluster tilting objects are in particular classical gen-
+erators, there is a canonical quasi-equivalence of DG categories
+Dc(Λcon)dg
+∼
+−→ Dsg(R)dg
+that induces an equivalence of triangulated categories
+Dc(Λcon)
+∼
+−→ Dsg(R).
+Although we do not need this fact in the sequel, we mention that there is an
+equivalence of triangulated categories4 [HK18, Lemma 5.12]
+C(Λ≤0
+con) := Dc(Λ≤0
+con)/ Dfd(Λ≤0
+con) ≃ Dsg(R),
+4For a DG algebra A, we denote by Dfd(A) the full subcategory of D(A) spanned by the DG
+A-modules with finite-dimensional total cohomology.
+
+8
+G. JASSO, B. KELLER, AND F. MURO
+that is compatible with the canonical DG enhancements on either side. The cate-
+gory C(Λ≤0
+con) is known as the Amiot cluster category of Λ≤0
+con [Ami07] and, indeed,
+establishing a link between birational geometry and the theory of cluster categories
+was one of the objectives in [HK18].
+2.2. The Derived Donovan–Wemyss Conjecture. The following theorem of
+Hua and the second-named author settles a derived version of Conjecture A.
+Theorem 2.2.1 ([HK18, Thm. 5.9]). Let R = C�x, y, z, t�/(f) be an isolated cDV
+singularity. Then, there is an isomorphism of algebras
+HH0(Dsg(R)dg) ∼=
+C�x, y, z, t�
+(f, ∂xf, ∂yf, ∂zf, ∂tf)
+between the 0-th Hochschild cohomology of the DG category Dsg(R)dg and the Tyu-
+rina algebra of R. In particular, if R′ is a further isolated cDV singularity such
+that the DG categories Dsg(R)dg and Dsg(R′)dg are quasi-equivalent, then there is
+an isomorphism of algebras R ∼= R′.
+Remark 2.2.2. The proof of Theorem 2.2.1 relies on a comparison result [Kel18,
+Kel19] between the singular Hochschild cohomology (=Hochschild–Tate cohomol-
+ogy) of R and the Hochschild cohomology of the DG category Dsg(R)dg.
+The
+appearance of the Tyurina algebra stems from an earlier result of the Buenos Aires
+Cyclic Homology Group [GGRV92, Thm. 3.2.7]. That the Tyurina algebra, together
+with the dimension of R, determines the isomorphism type of the singularity is a
+theorem of Mather and Yau [MY82].
+Remark 2.2.3. In [Dyc11], Dyckerhoff shows that the 0-th Hochschild cohomology
+of Dsg(R)dg—viewed as a Z/2-graded DG category—is isomorphic to the Milnor
+algebra
+C�x, y, z, t�
+(∂xf, ∂yf, ∂zf, ∂tf)
+of the singularity (which does not determine the isomorphism type of the singularity,
+even if one knows the dimension). Thus, in Theorem 2.2.1 it is crucial to consider
+Dsg(R)dg as a Z-graded DG category.
+Corollary 2.2.4 (Derived Donovan–Wemyss Conjecture). Let R1 and R2 be iso-
+lated cDV singularities with crepant resolutions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2).
+If the 2Z-derived contraction algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic,
+then there is an isomorphism of algebras R1 ∼= R2.
+Proof. Indeed, if the DG algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic, then
+the DG categories
+Dc(Λcon(p1))dg ≃ Dsg(R1)dg
+and
+Dc(Λcon(p2))dg ≃ Dsg(R2)dg
+are quasi-equivalent. Theorem 2.2.1 then implies the existence of an isomorphism
+of algebras R1 ∼= R2.
+□
+3. Uniqueness of the 2Z-derived contraction algebra
+In this section we prove that 2Z-derived contraction algebras are determined
+up to quasi-isomorphism by their zeroth cohomology plus a minimal amount of
+additional algebraic data (see Corollary 3.4.6 for the precise statement). Before
+that, we formulate a closely related result (Theorem 3.1.1) that states that two 2Z-
+derived contraction algebras whose zeroth cohomologies are isomorphic as algebras
+must be quasi-isomorphic, and use this result to prove Conjecture A.
+
+THE DONOVAN–WEMYSS CONJECTURE
+9
+3.1. Proof of the Donovan–Wemyss Conjecture. In view of Corollaries 1.3.3
+and 2.2.4, Conjecture A is an immediate consequence of the following theorem, the
+proof of which is given in Section 3.4.
+Theorem 3.1.1. Let R1 and R2 be isolated cDV singularities with crepant resolu-
+tions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2).
+If the contraction algebras Λ(p1) and Λ(p2) are isomorphic, then the 2Z-derived
+contraction algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic.
+Remark
+3.1.2.
+Theorem 3.1.1
+is
+a
+special
+case
+of
+[JM22,
+Thm.
+5.1.10],
+see Section 4.2.
+Proof of Conjecture A using Theorem 3.1.1. Let R1 and R2 be isolated cDV sin-
+gularities with crepant resolutions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2)
+whose corresponding contraction algebras Λcon(p1) and Λcon(p2) are derived equiva-
+lent. In view of Corollaries 1.3.3 and 2.2.4, we may and we will assume that Λcon(p1)
+and Λcon(p2) are isomorphic and hence, by Theorem 3.1.1, the 2Z-derived contrac-
+tion algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic. Finally, Corollary 2.2.4
+yields the desired algebra isomorphism R1 ∼= R2.
+□
+3.2. The restricted universal Massey product. The proof of Theorem 3.1.1
+makes use of an invariant of the 2Z-derived contraction algebra, a certain Hochschild
+cohomology class of bidegree (4, −2) that we call the restricted universal Massey
+product. As we explain below, this invariant is induced by the first possibly non-
+trivial higher operation on a minimal A∞-algebra model of the 2Z-derived contrac-
+tion algebra.
+Setting 3.2.1. Fix an isolated cDV singularity R that admits a crepant resolution
+p: X → Spec(R), and let Λ = Λcon(p) be the corresponding 2Z-derived contraction
+algebra so that H0(Λ) ∼= Λ = Λcon(p) is the contraction algebra defined by Donovan
+and Wemyss. For simplicity, we treat the isomorphism of graded algebras
+H•(Λ) ∼= Λ[ı±1] = Λ ⊗C C[ı±1],
+|ı| = −2,
+as an identification.
+Kadeishvili’s Homotopy Transfer Theorem [Kad82] provides us with a minimal
+A∞-algebra structure, unique up to A∞-isomorphism with identity linear part,
+B = (Λ[ı±1], m3, m4, m5, · · · )
+on the cohomology algebra Λ[ı±1]. Since Λ[ı±1] is concentrated in even degrees and,
+by definition,
+mn : Λ[ı±1]⊗n −→ Λ[ı±1]
+is a morphism of degree 2 − n, we conclude that mn = 0 whenever n is odd. We
+write
+B = (Λ[ı±1], m4, m6, m8, · · · )
+as a way to record this observation. We refer to B as a minimal (A∞-algebra)
+model of the DG algebra Λ and fix it for the rest of the section.
+Remark 3.2.2. The passage from DG to A∞-algebras is a matter of technical
+convenience: The homotopy theories of non-unital DG and of A∞-algebras are
+equivalent, [LV12, Thm. 11.4.8]. In particular, two non-unital DG algebras are
+quasi-isomorphic if and only if their minimal models are A∞-isomorphic [LV12,
+Thms. 11.4.9 and 10.3.10]. Here, we are exclusively interested in unital DG algebras,
+
+10
+G. JASSO, B. KELLER, AND F. MURO
+but this is not a problem since by [Mur14, Prop. 6.2] two unital DG algebras are
+quasi-isomorphic as non-unital DG algebras if and only if they are quasi-isomorphic
+as unital DG algebras.
+Consider now the bigraded Hochschild cochain complex
+Cp,q�
+Λ[ı±1], Λ[ı±1]
+� := HomC
+�
+Λ[ı±1]⊗p, Λ[ı±1](q)
+�
+,
+p ≥ 0, q ∈ Z,
+recalled in Section 1.4. Since m3 = 0, the A∞-equations imply that ∂(m4) = 0
+([LH03, Lemme B.4.1]); hence we obtain a class
+(3.2.3)
+{m4} =
+�
+mΛ
+4
+�
+∈ HH4,−2�
+Λ[ı±1], Λ[ı±1]
+�
+that we call the universal Massey product (of length 4). It follows from the definition
+of A∞-morphism ([LH03, Lemme B.4.2]) that the class {m4} does not depend on
+the choice of minimal model for Λ and hence the universal Massey product can and
+will be regarded as an invariant of the latter DG algebra.
+Consider now the graded-algebra morphism j : Λ ֒→ Λ[ı±1] given by the inclusion
+of the degree 0 part. The morphism j induces a restriction morphism on Hochschild
+cohomology5
+j∗: HH•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+−→ HH•,∗�
+Λ, Λ[ı±1]
+�
+,
+where the space on the right is the Hochschild cohomology of Λ, viewed as a graded
+algebra concentrated in degree 0, with coefficients in the graded Λ-bimodule Λ[ı±1].
+In particular, since the degree −2 component Λ · ı of Λ[ı±1] is isomorphic to the
+diagonal Λ-bimodule, we obtain a class
+(3.2.4)
+j∗{m4} = j∗�
+mΛ
+4
+�
+∈ HH4,−2�
+Λ, Λ[ı±1]
+�
+= HH4(Λ, Λ · ı) = Ext4
+Λe(Λ, Λ) ,
+where Λe = Λ ⊗C Λop is the eveloping algebra of Λ; we call the class j∗{m4}
+the restricted universal Massey product (of length 4) and, as with the unrestricted
+version, we regard it as an invariant of the 2Z-derived contraction algebra Λ. Notice
+also that the previous discussion applies verbatim to any DG algebra A whose
+cohomology is isomorphic to the graded algebra Λ[ı±1], so that we may associate
+to A its restricted universal Massey product j∗�
+mA
+4
+�
+.
+The following theorem is the first main step towards the proof of Theorem 3.1.1.
+Theorem 3.2.5. The restricted universal Massey product j∗{m4}, when viewed as
+an element of the space Ext4
+Λe(Λ, Λ) of Yoneda extensions of Λ-bimodules, can be
+represented by an exact sequence
+0 → Λ → P3 → P2 → P1 → P0 → Λ → 0
+with projective middle terms. In particular, Ω4
+Λe(Λ) ∼= Λ in the stable category of
+Λ-bimodules.
+Proof. The first claim is a special case of [JM22, Cor. 4.5.17]. Indeed, by definition,
+the 2Z-derived contraction algebra Λ is the derived endomorphism algebra of a 2Z-
+cluster tilting object in Dc(Λ) ≃ Dsg(R), which is one of the equivalent conditions
+in loc. cit. The second claim follows immediately from the first.
+□
+Remark 3.2.6. The proof of [JM22, Cor. 4.5.17], and hence that of Theorem 3.2.5, is
+non-trivial. In the special case of the contraction algebra, it is possible that detailed
+knowledge of the first non-trivial higher operation m4 of some minimal model of
+the 2Z-derived contraction algebra allows for establishig the desired property of the
+restricted universal Massey product j∗{m4} directly. The approach taken in [JM22],
+which deals with an abstract and more general situation, rather leverages the fact
+that Λ is the endomorphism algebra of a 2Z-cluster tilting object T ∈ Dsg(R).
+5In fact, the morphism j∗ is surjective, see [JM22, Prop. 4.6.9] and take σ = 1 and d = 2
+(which is even and hence no signs occur in the formulas therein).
+
+THE DONOVAN–WEMYSS CONJECTURE
+11
+The upshot is that the additive closure add(T ) of T has an induced structure of a
+so-called 4-angulated category, that is add(T ) is equipped with a natural class of
+diagrams □GKO, called 4-angles, of the form6
+T1 → T2 → T3 → T4 → T1[2]
+that satisfies axioms analogous to those of triangulated categories [GKO13]. On
+the other hand, an extension of Λ-bimodules
+0 → Λ → P3 → P2 → P1 → P0 → Λ → 0
+with P0, P1, P2 projective-injective (but perhaps not P3) that represents the class
+j∗{m4} ∈ Ext4
+Λe(Λ, Λ) yields a class of 4-angles □j∗{m4} defined in terms of certain
+exactness properties [Ami07, Lin19]; the class □j∗{m4} is a priori not known to
+form a 4-angulation of add(T ). The crux of the argument is then to prove that
+□GKO = □j∗{m4}
+so that the class □j∗{m4} is indeed a 4-angulation of add(T ); this agreement relies
+on a delicate analysis of the relationship between Toda brackets, Massey products
+and the classes □GKO and □j∗{m4}. Finally, in view of the exactness properties
+defining the class □j∗{m4} (now known to be 4-angulation), a theorem of Auslander
+and Reiten [AR91] for detecting projective bimodules implies that P3 must be a
+projective Λ-bimodule, which is what Theorem 3.2.5 claims. The reader is referred
+to [JM22] for details.
+Recall that the contraction algebra is Frobenius (in fact, symmetric). Conse-
+quently, its enveloping algebra is also a Frobenius algebra and we may consider the
+Hochschild–Tate cohomology
+HH•,∗�
+Λ, Λ[ı±1]
+� := Ext•,∗
+Λe
+�
+Λ, Λ[ı±1]
+�
+defined in terms of the extension spaces in the stable category of graded Λ-
+bimodules; thus,
+HH>0,∗�
+Λ, Λ[ı±1]
+�
+= HH>0,∗�
+Λ, Λ[ı±1]
+�
+and there is a surjection
+HH0,∗�
+Λ, Λ[ı±1]
+�
+։ HH0,∗�
+Λ, Λ[ı±1]
+�
+.
+The multiplication on Λ[ı±1] endows HH•,∗�
+Λ, Λ[ı±1]
+�
+with the structure of a bi-
+graded algebra, see [Mur22, Sec. 5] for details.
+Corollary 3.2.7. The restricted universal Massey product j∗{m4}, when viewed
+as an element of the Hochschild–Tate cohomology HH•,∗�
+Λ, Λ[ı±1]
+�
+, is a unit.
+Proof. Immediate from Theorem 3.2.5 and [Mur22, Prop. 5.7 and Rmk. 5.8], which
+characterises the units in HH•,∗�
+Λ, Λ[ı±1]
+�
+of positive Hochschild (=horizontal) de-
+gree.
+□
+Remark 3.2.8. In Corollary 3.2.7 it is essential to pass from Hochschild to
+Hochschild–Tate cohomology in order to have units of positive Hochschild degree.
+6Recall that [2] ∼
+= 1 in Dsg(R).
+
+12
+G. JASSO, B. KELLER, AND F. MURO
+3.3. Hochschild cohomology of the graded contraction algebra. In this
+section we compute the Hochschild cohomology of the graded algebra
+Λcon[ı±1] = Λ[ı±1] = Λ ⊗C C[ı±1],
+|ı| = −2,
+that we call graded contraction algebra, in terms of the Hochschild cohomology of
+the Dononvan–Wemyss contraction algebra Λcon = Λ.
+First, notice that ı lies in the (graded) centre of Λ[ı±1], which is
+Z(Λ[ı±1]) = HH0,∗�
+Λ[ı±1], Λ[ı±1]
+�
+;
+hence
+ı ∈ HH0,−2�
+Λ[ı±1], Λ[ı±1]
+�
+.
+We introduce the fractional Euler derivation
+¯δ ∈ C1,0�
+Λ[ı±1], Λ[ı±1]
+�
+,
+which acts by the formula
+¯δ : a �−→ |a|
+2 a,
+where we observe that |a|
+2 is an integer since Λ[ı±1] is concentrated in even degrees.
+It is a cocycle with cohomology class
+δ ∈ HH1,0�
+Λ[ı±1], Λ[ı±1]
+�
+.
+Proposition 3.3.1. The following statements hold:
+(1) There is an isomorphism of graded commutative algebras
+HH•,∗�
+Λ[ı±1], Λ[ı±1]
+� ∼= HH•(Λ, Λ) [ı±1, δ].
+The graded Lie algebra structure on the right hand side is induced by the
+(usual) Lie algebra structure on HH•,∗(Λ, Λ) by setting
+[ı, HH•(Λ, Λ)] = 0,
+[ı, ı] = 0,
+[δ, HH•(Λ, Λ)] = 0,
+[δ, ı] = −ı.
+(2) There is an isomorphism of graded algebras
+HH•,∗�
+Λ, Λ[ı±1]
+� ∼= HH•(Λ, Λ) [ı±1].
+Moreover, the morphism
+j∗ : HH•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+−→ HH•,∗�
+Λ, Λ[ı±1]
+�
+induced by the inclusion j : Λ ֒→ Λ[ı±1] of the degree 0 part is the apparent
+natural projection with kernel the graded ideal generated by δ.
+(3) There is an isomorphism of graded algebras
+HH•,∗�
+Λ, Λ[ı±1]
+� ∼= HH•(Λ, Λ) [ı±1].
+Furthermore, the comparison map
+HH•,∗�
+Λ, Λ[ı±1]
+�
+−→ HH•,∗�
+Λ, Λ[ı±1]
+�
+is the apparent extension of the comparison map HH•(Λ, Λ) → HH•(Λ, Λ).
+Proof. All of the forthcoming claims follow from the proof of [JM22, Prop. 4.6.9]
+for σ = 1Λ and d = 2.
+(1) The Hochschild complex C•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+contains the subcomplex
+C•,∗
+C[ı±1]
+�
+Λ[ı±1], Λ[ı±1]
+�
+of C[ı±1]-linear cochains; this subcomplex is also an associative subalgebra and a
+Lie subalgebra of the C-linear Hochschild complex.
+The composite
+C•,∗
+C[ı±1]
+�
+Λ[ı±1], Λ[ı±1]
+�
+i֒→ C•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+j∗
+−→ C•,∗�
+Λ, Λ[ı±1]
+�
+,
+
+THE DONOVAN–WEMYSS CONJECTURE
+13
+of the inclusion of the C[ı±1]-linear Hochschild cochains into the C-linear ones with
+the restriction of scalars along the inclusion j : Λ ֒→ Λ[ı±1] of the degree 0 part is
+an isomorphism of DG algebras. The target, unlike the source, does not a priori
+carry any Lie algebra structure. Nevertheless, there is an obvious isomorphism of
+DG algebras
+C•,∗�
+Λ, Λ[ı±1]
+� ∼= C•(Λ, Λ) [ı±1]
+that we regard as an identification, and the composite isomorphism
+C•,∗
+C[ı±1]
+�
+Λ[ı±1], Λ[ı±1]
+� ∼= C•(Λ, Λ) [ı±1]
+is also a Lie algebra map when we regard the target as a Lie algebra extension of
+C•(Λ, Λ) with ı a central element (in the Lie-algebra sense).
+The morphism
+C•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+−→ C���(Λ, Λ) [ı±1, ¯δ],
+x �−→ j∗(x) − ı−1 · j∗([x, ı]) · ¯δ,
+is a quasi-isomorphism of DG algebras with quasi-inverse
+C•(Λ, Λ) [ı±1] ⊕ C•(Λ, Λ) [ı±1] · ¯δ −→ C•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+,
+x + y · ¯δ �−→ i(x) + i(y) · ¯δ.
+(The latter is just a morphism of complexes since ¯δ2 ̸= 0 in the target, it only
+vanishes in cohomology.)
+In fact, the relevant composite equals the identity of
+C•(Λ, Λ) [ı±1, ¯δ].
+The Lie bracket formulas in the statement of the proposition
+follow from the definition of the fractional Euler class, C[ı±1]-linear cochains and
+degree considerations.
+(2) The statement follows easily from the previous computations.
+(3) The statement is consequece of the fact that HH•,∗(Λ, Λ) is obtained from
+HH•,∗(Λ, Λ) by inverting any element of
+HH4(Λ, Λ) = Ext4
+Λe(Λ, Λ)
+representing the 4-periodicty of Λ, and similarly when the coefficients lie in Λ[ı±1].
+□
+Below, we use the isomorphisms in Proposition 3.3.1 as identifications.
+Corollary 3.3.2. Let u ∈ HH4(Λ, Λ) be a unit. There exists a unique Hochschild
+class
+m ∈ HH4,−2�
+Λ[ı±1], Λ[ı±1]
+�
+such that
+j∗(m) = u · ı,
+1
+2[m, m] = 0.
+Proof. The first equation in the statement is equivalent to m being of the form
+(3.3.3)
+m = (u + x · δ) · ı
+for some x ∈ HH3(Λ, Λ). Using the relations in a Gerstenhaber algebra, the second
+equation is equivalent to
+0 = ([u, u] − 2u · x) · ı2 − 2[x, u] · ı2 · δ.
+This means that both summands must vanish. For the first one, this is equivalent
+to
+x = 1
+2u−1[u, u].
+This is takes place in the piece of HH•(Λ, Λ) that agrees with HH•(Λ, Λ), and is
+compatible with the second summand since
+0 = 1
+2[[u, u], u] = [ux, u] = u[x, u] − [u, u]x = u[x, u] − u−1[u, u]2 = u[x, u],
+
+14
+G. JASSO, B. KELLER, AND F. MURO
+so [x, u] = 0. The first step follows from the graded Jacobi identity and we also
+use that [u, u]2 = 0 since [u, u] has odd total degree and Hochschild cohomology is
+graded commutative.
+□
+The following result should be compared with equation (3.3.3); its proof is similar
+to that of Corollary 3.3.2.
+Corollary 3.3.4. Let u ∈ HH4(Λ, Λ) be a unit such that [u, u] = 0. Given
+(x + y · δ) · ıq ∈ HHp,−2q�
+Λ[ı±1], Λ[ı±1]
+�
+with p ≥ 2, x ∈ HHp(Λ, Λ) and y ∈ HHp−1(Λ, Λ), if [u · ı, (x + y · δ) · ıq] = 0 then
+(x + y · δ) · ıq = [u · ı, u−1 · δ · x · ıq−1].
+We obtain the following more precise information on a minimal A∞-model of the
+2Z-derived contraction algebra.
+Proposition 3.3.5. The 2Z-derived contraction algebra has a minimal A∞-model
+(Λ[ı±1], m4, m6, · · · )
+such that mn is C[ı±1]-linear for all n ≥ 4. In particular, {m4} = u · ı for some
+unit u ∈ HH4(Λ, Λ) satisfying [u, u] = 0.
+Proof. The first part follows from [HK18].
+The rest is a direct consecuence of
+Proposition 3.3.1 and the fact that [{m4}, {m4}] = 0, which follows from (1.5.1).
+□
+3.4. Proof of Theorem 3.1.1. The introduction of the restricted universal
+Massey product of Λ is justified by the following result and the subsequent corol-
+lary. Theorem 3.4.1 is an immediate consequence of [JM22, Thm. B], and the latter
+theorem is obtained an application of the obstruction theory for the existence of
+A∞-structures developed by the third-named author in [Mur20]. In this note we
+give a direct proof of Theorem 3.4.1 that leverages our detailed knowledge of the
+relationship between the Hochschild cohomology of the contraction algebra and
+that if its graded variant (see Section 3.3), although part of the techniques used to
+prove [JM22, Thm. B] are utilised in some guise.
+Theorem 3.4.1. Let A be a DG algebra such that H•(A) = Λ[ı±1] as graded
+algebras. If
+j∗�
+mA
+4
+�
+= j∗�
+mΛ
+4
+�
+∈ HH•,∗�
+Λ, Λ[ı±1]
+�
+,
+then A is quasi-isomorphic to the 2Z-derived contraction algebra Λ via a quasi-iso-
+morphism that induces the identity in cohomology.
+Proof. Let
+(Λ[ı±1], m4, m6, . . . )
+be a minimal model for the 2Z-derived contraction algebra as in Proposition 3.3.5,
+and
+(Λ[ı±1], m′
+4, m′
+6, . . . )
+a minimal model for A. Inductively, we will construct an A∞-isomorphism with
+identity linear part
+f = (0, 0, 0, f3, 0, f5, . . . ): (Λ[ı±1], m4, m6, . . . ) −→ (Λ[ı±1], m′
+4, m′
+6, . . . ),
+and this clearly suffices to prove the claim. Notice that, necessarily, f2n = 0 for all
+n ≥ 0 since Λ[ı±1] is concentrated in even degrees.
+We proceed as follows. For all n ≥ 0 we define a Hochschild cochain of total
+degree 1
+f (n) = (0, 0, 0, f (n)
+3
+, 0, . . . , f (n)
+2n+1, 0, . . . )
+
+THE DONOVAN–WEMYSS CONJECTURE
+15
+such that f (n) coincides with f (n−1) up to Hochschild degree 2n − 2 and
+(3.4.2)n
+∂(f (n)) + f (n) · f (n) +
+�
+r≥0
+m′{f (n), r. . ., f (n)} − m − f (n){m}
+vanishes up to Hochschild degree 2n + 2. If we achieve this goal, then we can take
+f = (0, 0, 0, f (3)
+3 , 0, . . . , f (n)
+2n���3, 0, . . . ).
+Indeed, f coincides with f (n) up to Hochschild degree 2n − 2, so (1.5.2) coincides
+with (3.4.2)n up to Hochschild degree 2n − 1. In particular (1.5.2) vanishes up to
+Hochschild degree 2n − 1 for all n ≥ 0. Therefore (1.5.2) fully vanishes, so f is
+indeed an A∞-isomorphism with identity linear part.
+We start with f (0) = 0.
+With this choice, (3.4.2)0 clearly vanishes up to
+Hochschild degree 2.
+Below, when defining f (n) we will only specify f (n)
+2n−1 and f (n)
+2n+1 since in smaller
+Hochschild degrees they are determined by f (n−1) and in higher Hochschild degrees
+they are irrelevant. Moreover, we will also use that (3.4.2)n−1 and (3.4.2)n agree
+(and hence both vanish) up to Hochschild degree 2n − 1.
+Since j∗{m4} = j∗{m′
+4}, then {m4} = {m′
+4} by Corollary 3.3.2, so there exists
+f (1)
+3
+such that
+(3.4.3)
+∂(f (1)
+3 ) + m′
+4 − m4 = 0.
+This proves that (3.4.2)1 vanishes up to Hochschild degree 4.
+Assume we have constructed up to f (n−1) for some n ≥ 2. Let us see how to
+construct f (n). We know by [LH03, Lemme B.4.2] that the Hochschild degree 2n+2
+part of (3.4.2)n−1, that we simply denote by a, is an obstruction cocycle (∂(a) = 0)
+which vanishes in cohomology if and only if there exists f (n)
+2n+1 such that, taking
+f (n)
+2n−1 = f (n−1)
+2n−1 , (3.4.2)n vanishes up to Hochschild degree 2n + 2. Indeed, any
+f (n)
+2n+1 such that a + ∂(f (n)
+2n+1) = 0 would do. We claim that
+(3.4.4)
+[m4, a] + ∂(b − f n−1
+3
+{a})
+vanishes, where b is the Hochschild degree 2n + 4 part of (3.4.2)n−1. We prove this
+claim below. Now, we deduce from Proposition 3.3.5 and Corollary 3.3.4 that there
+exist Hochschild cochains c2n−1 and c2n+1 such that
+a + ∂(c2n+1) + [m4, c2n−1] = 0,
+∂(c2n−1) = 0.
+If we set
+f (n)
+2n−1 = f (n−1)
+2n−1 + c2n−1,
+f (n)
+2n+1 =
+�
+c5 + f (1)
+3 {c3} − 1
+2c3{c3},
+n = 2;
+c2n+1 + f3{c2n−1},
+n > 2;
+we complete the induction step since the Hochschild degree 2n part of (3.4.2)n is,
+∂(c2n−1) = 0,
+and its Hochschild degree 2n + 2 part is, for n = 2,
+a + ∂
+�
+c5 + f (1)
+3 {c3} − 1
+2c3{c3}
+�
++ c3 · c3 + m′
+4{c3} − c3{m4}
+= a + ∂(c5) + ∂
+�
+f (1)
+3
+�
+{c3} + f (1)
+3 {∂(c3)} + m′
+4{c3} − c3{m4}
+= a + ∂(c5) + (m4 − m′
+4){c3} + m′
+4{c3} − c3{m4}
+= a + ∂(c5) + [m4, c3] = 0,
+
+16
+G. JASSO, B. KELLER, AND F. MURO
+where we use that ∂(c3{c3}) = 2c3 · c3 by Lemma 1.4.1, and for n > 2,
+a + ∂
+�
+c2n+1 + f3{c2n−1}
+�
++ m′
+4{c2n−1} − c2n−1{m4}
+= a + ∂(c2n+1) + ∂
+�
+f (n−1)
+3
+�
+{c2n−1} + f (n−1)
+3
+{∂(c2n−1)} + m′
+4{c2n−1} − c2n−1{m4}
+= a + ∂(c2n+1) + (m4 − m′
+4){c2n−1} + m′
+4{c2n−1} − c2n−1{m4}
+= a + ∂(c2n+1) + [m4, c2n−1] = 0.
+We finish the proof with the vanishing of (3.4.4). In what follows, let us write
+Ξ = (3.4.2)n−1 and f = f (n−1), so as not to overload notation. Note that (3.4.4) is
+the Hochschild degree 2n + 5 part of
+(3.4.5)
+[m, Ξ] + ∂(Ξ − f{Ξ}).
+This cochain vanishes in Hochschild degrees < 2n + 5.
+We now start a series of computations. We number most terms for bookkeeping
+purposes. In the first equation we use Lemma 1.4.1,
+∂(Ξ) =
+1
+∂(f) · f −
+2
+f · ∂f +
+3
+�
+r≥0
+∂(m′){f, r. . ., f}
+−
+4
+�
+r≥1
+r
+�
+i=1
+m′{f, i−1
+. . ., ∂(f), r−i
+. . ., f}
+−
+5
+�
+r≥1
+f · m′{f, r−1
+. . ., f} −
+6
+�
+r≥2
+r−1
+�
+i=1
+m′{f, i−1
+. . ., f 2, r−i−1
+. . . , f}
++
+7
+�
+r≥1
+m′{f, r−1
+. . ., f} · f −
+8
+∂(m) −
+9
+∂(f){m} −
+10
+f{∂(m)} +
+11
+f · m −
+12
+m · f
+Since m and m′ are A∞-algebra structures,
+8 = −m{m},
+10 = −f{m{m}},
+3 = −
+�
+r≥0
+m′{m′}{f, r. . ., f}
+= −
+�
+r≥0
+�
+0≤i≤j≤r
+m′{f,
+i. . ., m′{f, j−i
+. . ., f}, r−j
+. . ., f},
+= −
+13
+�
+r≥0
+m′{m′{f, r. . ., f}} −
+14
+�
+r≥1
+�
+0≤i≤j≤r
+j−i<r
+m′{f,
+i. . ., m′{f, j−i
+. . ., f}, r−j
+. . ., f}
+
+THE DONOVAN–WEMYSS CONJECTURE
+17
+Here we also use the brace relation. We also split the following summations in two
+parts,
+4 =
+15
+m′{∂(f)} +
+16
+�
+r≥2
+r
+�
+i=1
+m′{f, i−1
+. . ., ∂(f), r−i
+. . ., f},
+6 =
+17
+m′{f 2} +
+18
+�
+r≥3
+r−1
+�
+i=1
+m′{f, i−1
+. . ., f 2, r−i−1
+. . . , f} .
+Consider the following cochain, that we decompose using the brace relation,
+19
+�
+r≥0
+m′{f, r. . ., f}{m} =
+�
+r≥0
+r
+�
+i=0
+m′{f,
+i. . ., m, r−i
+. . ., f}
++
+�
+r≥1
+r
+�
+i=0
+m′{f, i−1
+. . ., f{m}, r−i
+. . ., f}
+=
+20
+m′{m} +
+21
+�
+r≥1
+r
+�
+i=0
+m′{f,
+i. . ., m, r−i
+. . ., f} +
+22
+m′{f{m}}
++
+23
+�
+r≥2
+r
+�
+i=0
+m′{f, i−1
+. . ., f{m}, r−i
+. . ., f} .
+Consider also the following cochain, which is computed by using Lemma 1.4.2,
+24
+f 2{m} =
+25
+f · f{m} −
+26
+f{m} · f .
+Since Ξ vanishes up to Hochschild degree 2n + 1 and its Hochschild degree 2n + 2
+part is a cocycle, the following cochains vanish up to Hochschild degree 2n + 5,
+�
+r≥2
+r
+�
+i=1
+m′{f, i−1
+. . ., Ξ, r−i
+. . ., f} = 16 + 18 + 14 − 21 − 23 ,
+∂(f){Ξ} + m′{Ξ} − m{Ξ},
+f{∂(Ξ)}.
+Notice that
+Ξ · f = 1 + f 3 + 7 − 12 − 26 ,
+f · Ξ = 2 + f 3 + 5 − 11 − 25 ,
+m′{Ξ} = 15 + 17 + 13 − 20 − 22 ,
+Ξ{m} = 9 + 24 + 19 + 8 + 10 .
+
+18
+G. JASSO, B. KELLER, AND F. MURO
+Using all the above, we obtain the following relations, where ≡ stands for con-
+gruence modulo cochains vanishing in Hochschild degrees < 2n + 5,
+(3.4.5) = m{Ξ} + Ξ{m} + ∂(Ξ) − ∂(f){Ξ} − f{∂(Ξ)} + f · Ξ − Ξ · f
+≡ m{Ξ} −
+�
+9 + 24 + 19 + 8 + 10
+�
++
+�
+1 − 2 + 3 − 4 − 5 − 6 + 7 − 8 − 9 − 10 + 11 − 12
+�
++
+�
+15 + 17 + 13 − 20 − 22
+�
+− m{Ξ} − 0
++
+�
+2 + f 3 + 5 − 11 − 25
+�
+−
+�
+1 + f 3 + 7 − 12 − 26
+�
++
+�
+16 + 18 + 14 − 21 − 23
+�
+= 0.
+This finally concludes the proof.
+□
+Corollary 3.4.6. Let A be a DG algebra such that H•(A) = Λ[ı±1] as graded
+algebras. If the restricted universal Massey product
+j∗�
+mA
+4
+�
+∈ HH•,∗�
+Λ, Λ[ı±1]
+�
+is a unit, then A is quasi-isomorphic to the 2Z-derived contraction algebra Λ.
+Proof. Firstly, we observe that the group of graded-algebra automorphisms of Λ[ı±1]
+acts on the right of HH•,∗�
+Λ, Λ[ı±1]
+�
+by conjugation.
+In particular, the group
+Z(Λ)× of units of the centre of Λ acts on HH•,∗�
+Λ, Λ[ı±1]
+�
+via the graded-algebra
+automorphisms
+gu : x �−→ xui,
+|x| = 2i,
+where u ∈ Z(Λ)×. The induced action on
+HH4,−2�
+Λ, Λ[ı±1]
+� ∼= Ext4
+Λe(Λ, Λ)
+has the following explicit description: Given a unit u ∈ Z(Λ)× and an exact se-
+quence of Λ-bimodules
+η: 0 → Λ
+f−→ X3 → X2 → X1 → X0 → Λ → 0,
+we let [η] · u be the class of the exact sequence
+0 → Λ
+f ′
+−→ X3 → X2 → X1 → X0 → Λ → 0,
+where f ′ := u−1f.
+The above action clearly restricts to the set of units in
+HH•,∗�
+Λ, Λ[ı±1]
+�
+of bidegree (4, −2), since these are precisely the classes that can be
+represented by an exact sequence with projective(-injective) middle terms [Mur22,
+Rmk. 5.8], and is in fact transitive on this set. To prove the latter claim on the
+transitivity of the action we appeal to [Che21, Cor. 2.3], which allows us to lift
+stable bimodule isomorphisms Λ ≃ Λ to honest bimodule isomorphisms Λ ∼= Λ, see
+the proof of [JM22, Prop. 2.2.16].
+Let u ∈ Z(Λ)× be a unit such that
+j∗�
+mA
+4
+�
+· u = j∗�
+mΛ
+4
+�
+∈ HH4,−2�
+Λ, Λ[ı±1]
+�
+.
+Given a minimal model
+(Λ[ı±1], mA
+4 , mA
+6 , mA
+8 , · · · )
+of the DG algebra A we define a new minimal A∞-algebra structure
+(Λ[ı±1], mA
+4 , mA
+6 , mA
+8 , · · · ) = (Λ[ı±1], mA
+4 , mA
+6 , mA
+8 , · · · ) ∗ gu
+with n-ary opearations mA
+n := g−1
+u mA
+4 g⊗n
+u
+(notice that mA
+4 ̸= mA
+4 as soon as u ̸= 1).
+It is easy to see that
+j∗�
+mA
+4
+�
+= j∗�
+mA
+4
+�
+· u = j∗�
+mΛ
+4
+�
+
+THE DONOVAN–WEMYSS CONJECTURE
+19
+and that there is an isomorphism of A∞-algebras
+(Λ[ı±1], mA
+4 , mA
+6 , mA
+8 , · · · ) ⇝ (Λ[ı±1], mA
+4 , mA
+6 , mA
+8 , · · · ).
+Thus, A is quasi-isomorphic to any DG algebra model B of the minimal A∞-
+algebra on the right-hand side, see Remark 3.2.2, and by Theorem 3.4.1 the DG
+algebras B and Λ are quasi-isomorphic. Consequently, the DG algebras A and Λ
+are isomorphic, which is what we needed to prove.
+□
+We are ready to prove Theorem 3.1.1.
+Proof of Theorem 3.1.1. Let R1 and R2 be isolated cDV singularities with crepant
+resolutions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2)
+whose corresponding contraction algebras Λ1 = Λ(p1) and Λ2 = Λ(p2) are isomor-
+phic. We need to prove that the 2Z-derived contraction algebras Λ1 = Λcon(p1)
+and Λ2 = Λcon(p2) are quasi-isomorphic. Recall that
+H•(Λ1) ∼= Λ1[ı±1]
+and
+H•(Λ2) ∼= Λ2[ı±1],
+|ı| = −2.
+In view of the assumption that Λ1 ∼= Λ2, we obtain a chain of isomorphisms of
+graded algebras
+H•(Λ1) ∼= Λ1[ı±1] ∼= Λ2[ı±1] ∼= H•(Λ2).
+In particular, we may and we will identify minimal models of Λ1 and Λ2 via the
+above isomorphism. For simplicity, let Λ = Λ1 ∼= Λ2. Corollary 3.2.7 shows that
+the restricted universal Massey products
+j∗�
+mΛ1
+4
+�
+∈ HH4,−2�
+Λ, Λ[ı±1]
+�
+and
+j∗�
+mΛ2
+4
+�
+∈ HH4,−2�
+Λ, Λ[ı±1]
+�
+are units in the Hochschild–Tate cohomology HH•,∗�
+Λ, Λ[ı±1]
+�
+. Corollary 3.4.6 im-
+plies that the DG algebras Λ1 and Λ2 are quasi-isomorphic, which is what we
+needed to prove.
+□
+We also have the following important corollary.
+Corollary 3.4.7. Let R be an isolated cDV singularity that admits a crepant res-
+olution. Then, the singularity category Dsg(R) admits a unique DG enhancement
+in the sense of [BK90].
+Proof. Let A be a DG enhancement of Dsg(R). By definition, this means that A
+is a pre-triangulated DG category and there exists an equivalence of triangulated
+categories
+H0(A) ≃ Dsg(R).
+Let T ∈ Dsg(R) be a 2Z-cluster tilting object and A the derived endomorphism alge-
+bra of T computed by means of the DG enhancement A. In particular Dc(A)dg ≃ A
+since T is a classical generator. By definition, H•(A) ∼= H•(Λ) where Λ is the de-
+rived endomorphism algebra of T computed by means of the canonical DG enhance-
+ment of Dsg(R). The proofs of Theorem 3.2.5 and Corollary 3.2.7 only rely on the
+fact that T ∈ Dsg(R) is a 2Z-cluster tilting object, see Remark 3.2.6. Consequently,
+the restricted universal Massey product j∗�
+mA
+4
+�
+is a unit in HH•,∗�
+Λ, Λ[ı±1]
+�
+and,
+by Corollary 3.4.6, the DG algebras A and Λ are quasi-isomorphic. Therefore, the
+DG categories
+A ≃ Dc(A)dg
+and
+Dsg(R)dg
+are quasi-equivalent. This shows that every DG enhancement of Dsg(R) is equiva-
+lent to the canonical DG enhancement and the claim follows.
+□
+
+20
+G. JASSO, B. KELLER, AND F. MURO
+Remark 3.4.8. Corollary 3.4.7 is stronger that Conjecture A, as it shows that the
+DG category Dsg(R)dg is determined by Λcon up to isomorphism in Hmo, the Morita
+category of small DG categories [Tab05].
+4. Concluding remarks
+In this section we collect some observations, some of which follow easily from
+the results in the previous sections.
+4.1. Formality of contraction algebras. Recall that a DG algebra A is formal
+if there is a quasi-isomorphism A ≃ H•(A), where the graded algebra H•(A) is
+viewed as a DG algebra with vanishing differential. We observe that 2Z-derived
+contraction algebras are almost never formal.7
+Theorem 4.1.1. Let R be an isolated cDV singularity with a crepant resolution
+and Λcon a 2Z-derived contraction algebra for R.
+The following statements are
+equivalent:
+(1) The 2Z-derived contraction algebra Λcon is formal.
+(2) There is an isomorphism of algebras Λcon ∼= C.
+(3) There is an isomorphism of algebras
+R ∼= C�x, y, z, t�/(xy − zt),
+so that Spec(R) is the base of the Atiyah flop [Ati58].
+If the above equivalent conditions hold, then there is a quasi-isomorphism
+Λcon ≃ C[ı±1],
+|ı| = −2,
+where the graded algebra C[ı±1] is equipped with the trivial differential.
+Proof. (1)⇒(2) If the 2Z-derived contraction algebra Λcon is formal, then its re-
+stricted universal Massey product j∗{m4} is represented by the trivial sequence in
+HH4,−2�
+Λcon, Λcon[ı±1]
+�
+= Ext4
+Λecon(Λcon, Λcon). In view of Theorem 3.2.5, this can
+only happen if Λcon is projective as a Λcon-bimodule or, equivalently, if the algebra
+Λcon is semisimple. Since contraction algebras are basic and connected, we must
+have an isomorphism of algebras Λcon ∼= C, which is what we needed to prove.
+(2)⇒(1) If Λcon ∼= C, then there is an isomorphism of graded algebras
+H•(Λcon) ∼= Λcon[ı±1] ∼= C[ı±1],
+|ı| = −2.
+It is well-known (and easy to prove using Kadeishvili’s Theorem [Kad88], for exam-
+ple) that the latter graded algebra is intrinsically formal, that is every DG algebra
+with cohomology algebra C[ı±1] is formal. In particular, Λcon is formal.8
+(2)⇔(3) In view of the validity of Conjecture A, it is enough to observe that C is
+indeed isomorphic to the contraction algebra of the Atiyah flop [DW16, Table 2].
+□
+Remark 4.1.2. The (non-)formality of the derived contraction algebra Λ≤0
+con is inves-
+tigated in [Boo19, Ch. 9] where minimal models of this DG algebra are computed
+in various examples, see also [Boo18, Boo21, Boo22].
+7This fact will come as no surprise to the experts, but we did not find a proof of it in the
+literature.
+8Alternatively, one can prove this fact using the results in this note as follows: Since the
+enveloping algebra
+Λe
+con ∼
+= C ⊗C C ∼
+= C
+is semisimple, the Hochschild–Tate cohomology HH•,∗�
+Λcon, Λcon[ı±1]
+�
+vanishes in positive
+Hochschild degrees. Then, by Theorem 3.4.1, the 2Z-derived contraction algebra Λcon is quasi-
+isomorphic to its cohomology algebra H•(Λcon) for the condition on the agreement of the corre-
+sponding universal Massey products is trivially satisfied. Of course, this is essentially the same
+proof as the one using Kadeishvili’s Theorem, which is indeed a special case of [JM22, Thm. B].
+
+THE DONOVAN–WEMYSS CONJECTURE
+21
+Remark 4.1.3. More generally, Kadeishvili’s Theorem [Kad88] can be used to prove
+that the Laurent polynomial algebra K[ı±1] = K ⊗C C[ı±1] is intrinsically formal
+if K is a finite-dimensional algebra of projective dimension at most 2 as a K-
+bimodule [Sai20, Cor. 4.2].
+This K, however, cannot be a contraction algebra
+unless K = C since contraction algebras are basic and connected, and a symmetric
+C-algebra has finite projective dimension as a bimodule over itself if and only if it
+is separable (semisimple).
+4.2. Relationship to the Triangulated Auslander–Iyama Correspondence.
+Let T be a triangulated category whose underlying additive category is Krull–
+Schmidt and has finite-dimensional morphism spaces.
+A basic object T ∈ T is
+d-cluster tilting, d ≥ 1, if the following conditions are satisfied [IY08, Bel15]:
+• The object T is d-rigid: T(T, T [i]) = 0 for all 0 < i < d.
+• For each object X ∈ T there exists a diagram
+Td−2
+· · ·
+T1
+T0
+Td−1
+Xd−2
+· · ·
+X2
+X1
+X
++1
++1
++1
+in which Ti ∈ add(T ), 0 ≤ i < d, the oriented triangles denote exact triangles in
+T and the unoriented triangles commute. The object T is dZ-cluster tilting if it
+is d-cluster tilting and T ∼= T [d]. Theorem 3.1.1 is a special case of the theorem
+below. For a finite-dimensional algebra Λ, we let proj(Λ) be the category of finite-
+dimensional projective Λ-modules. For example, if Λ = T(T, T ), then the Yoneda
+functor
+T ⊇ add(T )
+∼
+−→ proj(Λ),
+X �−→ T(T, X),
+is an equivalence (of additive categories).
+Theorem
+4.2.1
+(Triangulated
+Auslander–Iyama
+Correspondence
+[JM22,
+Thm. 5.1.10]). Let d ≥ 1.
+There is a bijective correspondence between the
+following:
+(1) Quasi-isomorphism classes of DG algebras A that satisfy the following:
+• The algebra H0(A) is a basic finite-dimensional algebra.
+• The free DG A-module A ∈ Dc(A) is a dZ-cluster tilting object.
+(2) Equivalence classes of pairs (Λ, I) consisting of
+• a basic finite-dimensional self-injective algebra Λ and
+• an invertible Λ-bimodule I such that Ωd+2
+Λe (Λ) ∼= I in the stable category
+of Λ-bimodules.
+The correspondence is given by the formula A �→ (H0(A), H−d(A)).
+Remark 4.2.2. The case d = 1 of Theorem 4.2.1 is one way to formulate the main
+result in [Mur22]. Indeed, an object T ∈ T is 1-cluster tilting if and only if it is
+1Z-cluster tilting if and only if add(T ) = T; the latter condition means that T is a
+triangulated category of finite type in the terminology of [Mur22].
+Remark 4.2.3. Let (Λ, I) be a pair as in Theorem 4.2.1(2). Since the algebra Λ is
+assumed to be basic, the map
+Aut(Λ) −→ Pic(Λ),
+σ �−→ [1Λσ]
+from the group of algebra automorphisms of Λ to the Picard group of invertible
+Λ-bimodules induces an isomorphism of groups [Bol84, Prop. 3.8]
+Out(Λ)
+∼
+−→ Pic(Λ),
+[σ] �−→ [1Λσ],
+where Out(Λ) = Aut(Λ)/ Inn(Λ) is the group of outer automorphisms of Λ. In par-
+ticular, there exists σ ∈ Aut(Λ) such that I ∼= 1Λσ as Λ-bimodules. The condition
+
+22
+G. JASSO, B. KELLER, AND F. MURO
+Ωd+2
+Λe (Λ) ≃ 1Λσ expresses the fact that the algebra Λ is twisted (d + 2)-periodic
+with respect to σ. When σ = 1 or, equivalently, I ∼= Λ, the algebra Λ is said to be
+(d+ 2)-periodic. For example, contraction algebras are known to be 4-periodic. We
+refer the reader to [ES08] and the references therein for information on (twisted)
+periodic algebras.
+Theorem 3.1.1
+follows
+from
+the
+injectivity
+of
+the
+correspondence
+in Theorem 4.2.1 with d = 2; the proof of Theorem 3.1.1 outlined in this
+note effectively goes through the proof of the latter in the special case of
+2Z-derived contraction algebras.
+We record the resulting characterisation of
+contraction algebras for the sake of completeness.
+Theorem 4.2.4. Let R be an isolated cDV singularity with a crepant resolution
+p: X → Spec(R).
+Up to quasi-isomorphism, the 2Z-derived contraction algebra
+Λ = Λcon(p) is the unique DG algebra with the following properties:
+(1) Λ ∈ Dc(Λ) is a 2Z-cluster tilting object.
+(2) There is an isomorphism of algebras H0(Λ) ∼= Λ = Λcon(p).
+(3) There is an isomorphism of Λ-bimodules H−2(Λ) ∼= Λ.
+In other words, Λ is determined up to quasi-isomorphism by its image (Λ, Λ) under
+the Triangulated Auslander–Iyama Correspondence.
+Proof. That the 2Z-derived contraction algebra satisfies the first two properties
+follows from Theorem 1.3.1 since, by definition, Λ is the derived endomorphism
+algebra of a 2Z-cluster tilting object in Dsg(R) ≃ Dc(Λ). Therefore Λ belongs to
+the class of DG algebras in Theorem 4.2.1(1) with d = 2. Given that Λ is 4-periodic,
+the third property follows. Moreover, Λ is determined up to quasi-isomorphism by
+its image (Λ, Λ) under the Triangulated Auslander–Iyama Correspondence, which
+is what we needed to prove.
+□
+4.3. Isolated cDV singularities with non-smooth minimal models. Con-
+traction algebras are defined for an arbitrary cDV singularity R that is neither
+isolated nor admits a crepant resolution. However, contraction algebras are finite-
+dimensional if and only if R defines an isolated singularity [DW19, Summary 5.6],
+and this finite-dimensionality is crucial to our approach. On the other hand, R
+admits a crepant resolution if and only if the singularity category Dsg(R) admits a
+2Z-cluster tilting object, see [BIKR08, Thm. 5.4] and the references therein. If, on
+the other hand, the minimal models9 of R are singular, then the contraction algebras
+of R are the endomorphism algebras of maximal rigid objects in Dsg(R) [Wem18],
+that is objects T ∈ Dsg(R) such that
+add(T ) = {X ∈ Dsg(R)| Hom(T ⊕ X, (T ⊕ X)[1]) = 0}.
+It is easy to verify that 2-cluster tilting objects are maximal rigid, but the converse is
+false in general [BIKR08, BMV10]; moreover, if there exists a 2-cluster tilting object
+then every maximal rigid object is also 2-cluster tilting [BIRS09, Thm. II.1.8].10 In
+any case, one may still define the 2Z-derived contraction algebras of R as the derived
+endomorphism algebras of maximal rigid objects in Dsg(R), computed in terms of
+the canonical DG enhancement of the latter triangulated category. Note, however,
+that Theorem 4.2.1 does not cover the case of maximal rigid objects that are not
+9In the context of the MMP in dimension three and higher, minimal models play the role of
+minimal resolutions of surfaces. We do not recall the technical definition in this note, but only
+mention that minimal models are permitted to have ‘mild’ singularities as long as they remain
+‘closer’ to the original space than a smooth resolution (which always exists by a famous theorem
+of Hironaka [Hir64].)
+10The reader should compare this statement with the following geometric fact: If one minimal
+model of Spec(R) is smooth, then all of its minimal models are also smooth [Kol89, Cor. 4.11].
+
+THE DONOVAN–WEMYSS CONJECTURE
+23
+2Z-cluster tilting and hence it cannot be applied to prove a version of Theorem 4.2.4
+for isolated cDV singularities that do not admit a crepant resolution. Finally, we
+mention that the apparent variant of Conjecture A does not hold for isolated cDV
+singularities whose minimal models are not smooth, see [Boo21, Ex 8.4.2] for an
+explicit counterexample.
+References
+[Ami07]
+Claire Amiot. On the structure of triangulated categories with finitely many indecom-
+posables. Bull. Soc. Math. France, 135(3):435–474, 2007.
+[AR91]
+Maurice Auslander and Idun Reiten. On a theorem of E. Green on the dual of the trans-
+pose. In Representations of finite-dimensional algebras (Tsukuba, 1990), volume 11 of
+CMS Conf. Proc., pages 53–65. Amer. Math. Soc., Providence, RI, 1991.
+[Ati58]
+M. F. Atiyah. On analytic surfaces with double points. Proc. Roy. Soc. London Ser.
+A, 247:237–244, 1958.
+[Aug19]
+Jenny August. The tilting theory of contraction algebras. Ph. D. thesis, University of
+Edinburgh, 2019.
+[Aug20]
+Jenny August. On the finiteness of the derived equivalence classes of some stable
+endomorphism rings. Math. Z., 296(3-4):1157–1183, 2020.
+[Aus78]
+Maurice Auslander. Functors and morphisms determined by objects. In Representation
+theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), pages 1–244.
+Lecture Notes in Pure Appl. Math., Vol. 37. Dekker, New York, 1978.
+[Bel15]
+Apostolos Beligiannis. Relative homology, higher cluster-tilting theory and categorified
+Auslander-Iyama correspondence. J. Algebra, 444:367–503, 2015.
+[BIKR08]
+Igor Burban, Osamu Iyama, Bernhard Keller, and Idun Reiten. Cluster tilting for
+one-dimensional hypersurface singularities. Adv. Math., 217(6):2443–2484, 2008.
+[BIRS09]
+A. B. Buan, O. Iyama, I. Reiten, and J. Scott. Cluster structures for 2-Calabi-Yau
+categories and unipotent groups. Compos. Math., 145(4):1035–1079, 2009.
+[BK90]
+A. I. Bondal and M. M. Kapranov. Framed triangulated categories. Mat. Sb.,
+181(5):669–683, 1990.
+[BMV10]
+Aslak Bakke Buan, Robert J. Marsh, and Dagfinn F. Vatne. Cluster structures from
+2-Calabi-Yau categories with loops. Math. Z., 265(4):951–970, 2010.
+[Bol84]
+Michael L. Bolla. Isomorphisms between endomorphism rings of progenerators. J. Al-
+gebra, 87(1):261–281, 1984.
+[Boo18]
+Matt Booth. Noncommutative deformation theory, the derived quotient, and DG sin-
+gularity categories, November 2018, 1810.10060 [math.AG].
+[Boo19]
+Matt Booth. The derived contraction algebra, 2019, 1911.09626 [math.AG].
+[Boo21]
+Matt Booth. Singularity categories via the derived quotient. Adv. Math., 381:Paper
+No. 107631, 56, 2021.
+[Boo22]
+Matt Booth. The derived deformation theory of a point. Mathematische Zeitschrift,
+300(3):3023–3082, March 2022.
+[Buc21]
+Ragnar-Olaf Buchweitz. Maximal Cohen-Macaulay modules and Tate cohomology, vol-
+ume 262 of Mathematical Surveys and Monographs. American Mathematical Society,
+Providence, RI, [2021] ©2021. With appendices and an introduction by Luchezar L.
+Avramov, Benjamin Briggs, Srikanth B. Iyengar and Janina C. Letz.
+[Che21]
+Justin Chen. Surjections of unit groups and semi-inverses. J. Commut. Algebra,
+13(3):323–331, 2021.
+[Dri04]
+Vladimir Drinfeld. DG quotients of DG categories. J. Algebra, 272(2):643–691, 2004.
+[Dug15]
+Alex Dugas. A construction of derived equivalent pairs of symmetric algebras. Proc.
+Amer. Math. Soc., 143(6):2281–2300, 2015.
+[DW16]
+Will Donovan and Michael Wemyss. Noncommutative deformations and flops. Duke
+Math. J., 165(8):1397–1474, 2016.
+[DW19]
+Will Donovan and Michael Wemyss. Contractions and deformations. Amer. J. Math.,
+141(3):563–592, 2019.
+[Dyc11]
+Tobias Dyckerhoff. Compact generators in categories of matrix factorizations. Duke
+Math. J., 159(2):223–274, 2011.
+[Eis80]
+David Eisenbud. Homological algebra on a complete intersection, with an application
+to group representations. Trans. Amer. Math. Soc., 260(1):35–64, 1980.
+[ES08]
+Karin Erdmann and Andrzej Skowro´nski. Periodic algebras. In Trends in representa-
+tion theory of algebras and related topics, EMS Ser. Congr. Rep., pages 201–251. Eur.
+Math. Soc., Z¨urich, 2008.
+
+24
+G. JASSO, B. KELLER, AND F. MURO
+[GGRV92] Jorge Alberto Guccione, Juan Jose Guccione, Maria Julia Redondo, and Orlando Eu-
+genio Villamayor. Hochschild and cyclic homology of hypersurfaces. Advances in Math-
+ematics, 95(1):18–60, September 1992.
+[GKO13]
+Christof Geiss, Bernhard Keller, and Steffen Oppermann. n-angulated categories. J.
+Reine Angew. Math., 675:101–120, 2013.
+[HI11]
+Martin Herschend and Osamu Iyama. Selfinjective quivers with potential and 2-
+representation-finite algebras. Compos. Math., 147(6):1885–1920, 2011.
+[Hir64]
+Heisuke Hironaka. Resolution of singularities of an algebraic variety over a field of
+characteristic zero. I, II. Ann. of Math. (2) 79 (1964), 109–203; ibid. (2), 79:205–326,
+1964.
+[HK18]
+Zheng Hua and Bernhard Keller. Cluster categories and rational curves, 2018,
+1810.00749 [math.AG].
+[IY08]
+Osamu Iyama and Yuji Yoshino. Mutation in triangulated categories and rigid Cohen-
+Macaulay modules. Invent. Math., 172(1):117–168, 2008.
+[JM22]
+Gustavo Jasso and Fernando Muro. The Triangulated Auslander–Iyama Correspon-
+dence, with an appendix by B. Keller, 2022, 2208.14413 [math.RT].
+[Kad82]
+T. V. Kadeishvili. The algebraic structure in the homology of an A(∞)-algebra. Soob-
+shch. Akad. Nauk Gruzin. SSR, 108(2):249–252 (1983), 1982.
+[Kad88]
+T. V. Kadeishvili. The structure of the A(∞)-algebra, and the Hochschild and Harrison
+cohomologies. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 91:19–27,
+1988.
+[Kat08]
+Sheldon Katz. Genus zero Gopakumar-Vafa invariants of contractible curves. J. Dif-
+ferential Geom., 79(2):185–195, 2008.
+[Kel94]
+Bernhard Keller. Deriving DG categories. Ann. Sci. ´Ecole Norm. Sup. (4), 27(1):63–
+102, 1994.
+[Kel99]
+Bernhard Keller. On the cyclic homology of exact categories. J. Pure Appl. Algebra,
+136(1):1–56, 1999.
+[Kel06]
+Bernhard Keller. On differential graded categories. In International Congress of Math-
+ematicians. Vol. II, pages 151–190. Eur. Math. Soc., Z¨urich, 2006.
+[Kel08]
+Bernhard Keller. Calabi-Yau triangulated categories, 2008.
+[Kel18]
+Bernhard Keller. Singular Hochschild cohomology via the singularity category. C. R.
+Math. Acad. Sci. Paris, 356(11-12):1106–1111, 2018.
+[Kel19]
+Bernhard Keller. Corrections to “Singular Hochschild cohomology via the singularity
+category” [C. R. Acad. Sci. Paris, Ser. I 356 (2018) 1106–1111]. C. R. Math. Acad.
+Sci. Paris, 357(6):533–536, 2019.
+[KM87]
+Yujiro Kawamata and Kenji Matsuki. The number of the minimal models for a 3-fold
+of general type is finite. Math. Ann., 276(4):595–598, 1987.
+[KM98]
+J´anos Koll´ar and Shigefumi Mori. Birational geometry of algebraic varieties, volume
+134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge,
+1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998
+Japanese original.
+[Kol89]
+J´anos Koll´ar. Flops. Nagoya Math. J., 113:15–36, 1989.
+[Kon98]
+Maxim Kontsevich. Triangulated categories and geometry. Course at the ´Ecole Nor-
+male Sup´erieure, Paris. Notes taken by J. Bella¨ıche, J.-F. Dat, I. Marin, G. Racinet
+and H. Randriambololona, 1998.
+[LH03]
+Kenji Lef`evre-Hasegawa. Sur les A-infini cat´egories, 2003, math/0310337.
+[Lin19]
+Zengqiang Lin. A general construction of n-angulated categories using periodic injec-
+tive resolutions. J. Pure Appl. Algebra, 223(7):3129–3149, 2019.
+[LV12]
+Jean-Louis Loday and Bruno Vallette. Algebraic operads, volume 346 of Grundlehren
+der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sci-
+ences]. Springer, Heidelberg, 2012.
+[LW12]
+Graham J. Leuschke and Roger Wiegand. Cohen-Macaulay representations, volume
+181 of Mathematical Surveys and Monographs. American Mathematical Society, Prov-
+idence, RI, 2012.
+[Mur14]
+Fernando Muro. Moduli spaces of algebras over nonsymmetric operads. Algebr. Geom.
+Topol., 14(3):1489–1539, 2014.
+[Mur20]
+Fernando Muro. Enhanced A∞-obstruction theory. J. Homotopy Relat. Struct.,
+15(1):61–112, 2020.
+[Mur22]
+Fernando Muro. Enhanced Finite Triangulated Categories. J. Inst. Math. Jussieu,
+21(3):741–783, 2022.
+[MY82]
+John N. Mather and Stephen S. T. Yau. Classification of isolated hypersurface singu-
+larities by their moduli algebras. Invent. Math., 69(2):243–251, 1982.
+
+THE DONOVAN–WEMYSS CONJECTURE
+25
+[Rei83]
+Miles Reid. Minimal models of canonical 3-folds. In Algebraic varieties and analytic
+varieties (Tokyo, 1981), volume 1 of Adv. Stud. Pure Math., pages 131–180. North-
+Holland, Amsterdam, 1983.
+[Sai20]
+Shunya
+Saito.
+Tilting
+objects
+in
+periodic
+triangulated
+categories,
+2020,
+arXiv:2011.14096 [math.RT].
+[Tab05]
+Gon¸calo Tabuada. Invariants additifs de DG-cat´egories. Int. Math. Res. Not.,
+(53):3309–3339, 2005.
+[Tod15]
+Yukinobu
+Toda.
+Non-commutative
+width
+and
+Gopakumar-Vafa
+invariants.
+Manuscripta Math., 148(3-4):521–533, 2015.
+[VdB04]
+Michel Van den Bergh. Three-dimensional flops and noncommutative rings. Duke
+Math. J., 122(3):423–455, 2004.
+[Wem18]
+Michael Wemyss. Flops and clusters in the homological minimal model programme.
+Invent. Math., 211(2):435–521, 2018.
+[Wem21]
+Michael Wemyss. A lockdown survey on cDV singularities, 2021, arXiv:2103.16990
+[math.AG].
+[Yos90]
+Yuji Yoshino. Cohen-Macaulay modules over Cohen-Macaulay rings, volume 146 of
+London Mathematical Society Lecture Note Series. Cambridge University Press, Cam-
+bridge, 1990.
+[Zim14]
+Alexander Zimmermann. Representation theory, volume 19 of Algebra and Applica-
+tions. Springer, Cham, 2014. A homological algebra point of view.
+(G. Jasso) Lund University, Centre for Mathematical Sciences, S¨olvegatan 18A, 22100
+Lund, Sweden
+Email address: gustavo.jasso@math.lu.se
+URL: https://www.maths.lu.se/staff/gustavo-jasso/
+(B. Keller) Universit´e Paris Cit´e, UFR de Math´ematiques, Case 7012, Bˆatiment Sophie
+Germain, 8 place Aur´elie Nemours, 75013 Paris Cedex 13, France
+Email address: bernhard.keller@imj-prg.fr
+URL: https://webusers.imj-prg.fr/~bernhard.keller/
+(F. Muro) Universidad de Sevilla,
+Facultad de Matem´aticas,
+Departamento de
+´Algebra, Calle Tarfia s/n, 41012 Sevilla, Spain
+Email address: fmuro@us.es
+URL: https://personal.us.es/fmuro/
+

99E4T4oBgHgl3EQfDgse/content/tmp_files/2301.04869v1.pdf.txt

@@ -0,0 +1,1726 @@
+Parallel Interior-Point Solver for Block-Structured Nonlinear
+Programs on SIMD/GPU Architectures
+Fran¸cois Pacauda, Michel Schanenb, Sungho Shinb, Daniel Adrian Maldonadob,
+Mihai Anitescub
+a Centre Automatique et Syst`emes, Mines Paris - PSL, Paris, France; b Mathematics and
+Computer Science Department, Argonne National Laboratory, Lemont, USA
+ARTICLE HISTORY
+Compiled January 13, 2023
+ABSTRACT
+We investigate how to port the standard interior-point method to new exascale
+architectures for block-structured nonlinear programs with state equations. Compu-
+tationally, we decompose the interior-point algorithm into two successive operations:
+the evaluation of the derivatives and the solution of the associated Karush-Kuhn-
+Tucker (KKT) linear system. Our method accelerates both operations using two
+levels of parallelism. First, we distribute the computations on multiple processes
+using coarse parallelism. Second, each process uses a SIMD/GPU accelerator lo-
+cally to accelerate the operations using fine-grained parallelism. The KKT system
+is reduced by eliminating the inequalities and the state variables from the corre-
+sponding equations, to a dense matrix encoding the sensitivities of the problem’s
+degrees of freedom, drastically minimizing the memory exchange. We demonstrate
+the method’s capability on the supercomputer Polaris, a testbed for the future exas-
+cale Aurora system. Each node is equipped with four GPUs, a setup amenable to our
+two-level approach. Our experiments on the stochastic optimal power flow problem
+show that the method can achieve a 50x speed-up compared to the state-of-the-art
+method.
+1. Introduction
+Solving complex engineering problems often resorts to the solution of large-scale block-
+structured nonlinear programs. As such, there has been a long interest in designing
+efficient nonlinear optimization algorithms, particularly by using parallel computing.
+Parallelism can happen at two levels. At first, coarse parallelism splits the program
+into large computational chunks, usually dispatched to multiple processors using a
+message-passing interface in distributed memory. In this paradigm, the parallel al-
+gorithm is designed to minimize the communication between the different processes.
+In a complementary direction, fine-grained parallelism breaks down the program into
+small tasks, fast to compute in shared memory. This method requires a large num-
+ber of processors to be efficient, and it is usually better on SIMD architectures with
+low communication overhead, as provided by Graphical Processing Units (GPUs). In
+the mathematical optimization community, coarse parallelism has traditionally been
+used to solve large-scale block-structured optimization problems, as encountered in
+dynamic or stochastic nonlinear programs. On the contrary, fine-grained parallelism
+has gained attraction only recently, with the renewed interests for machine learning
+arXiv:2301.04869v1  [math.OC]  12 Jan 2023
+
+applications and stochastic gradient algorithms. In this work, we combine coarse and
+fine-grained parallelism to solve block-structured nonlinear problems on new exascale
+architectures, where the solution algorithm is streamlined on different GPUs using
+CUDA-aware MPI.
+1.1. Literature review
+In his pioneering work [39, 40], Robert Schnabel identified three practical approaches
+to run optimization algorithms in parallel: (i) parallelize the function evaluations; (ii)
+parallelize the linear algebra; and (iii) parallelize the optimization algorithm itself.
+The first attempt to parallelize the evaluations has been to streamline the com-
+putation of the derivatives using finite-differences [29]. Soon, it has been noted that
+parallelizing the forward pass in automatic differentiation (AD) is also straightforward,
+provided that we can propagate the tangents (encoding the first-order sensitivity) in
+parallel [20]. Unfortunately, doing the same in the reverse pass is not trivial, as ad-
+joining a mutable code leads to race conditions (e.g., every read becomes a write
+operation). This has led to extensive research on adapting automatic differentiation
+to parallel environments [4, 19, 27]. Now, most state-of-the-art differentiable tools em-
+ploy a Domain Specific Language (DSL) constraining the user to specific differentiable
+operations. In particular, this approach has been adopted mainly in machine learning,
+leading to the development of fast AD libraries efficiently generating the derivatives
+efficiently on hardware accelerators such as GPUs or TPUs [3, 32].
+The parallelization of linear algebra is usually more involved, as most large-scale op-
+timization methods fall back on the solution of sparse indefinite Karush-Kuhn-Tucker
+(KKT) systems [30]. In the 1980s, preliminary results were obtained by running itera-
+tive methods in parallel, using block-Krylov [36] or block-truncated Newton methods
+[28]. However, block iterative algorithms are quickly limited by the lack of generic
+preconditioners for KKT systems. The 1990s witnessed the emergence of the interior-
+point methods (IPM), together with the development of large-scale sparse direct linear
+solvers [12, 38]. In IPM, a significant portion of the time is spent solving a sequence
+of (indefinite) KKT systems, hence the method directly benefits from efficient sparse
+linear solvers able to run in parallel [1, 13]. In the 2000s, it was shown that, for block-
+structured optimization problems as we consider here, the layout of the optimization
+problem can be exploited further in a Schur complement approach to solve the Newton
+step in parallel [2, 9, 17, 22, 33, 44, 45]. These developments led to the development
+of mature decomposition-based parallel nonlinear solvers for scenario-based problems
+in the 2010s [8, 16, 34, 41, 46].
+Eventually, running an optimization algorithm fully in parallel generally requires a
+subtle combination of (i) and (ii), often devolving to a software engineering problem.
+The challenge is to evaluate the derivatives and solve the resulting KKT system each in
+parallel; all this while minimizing the communication between the different processes.
+This has led to the development of different prototypes for MPI-parallel modelers [10,
+21, 34, 43], most of them extending a specific AD backend [5, 14, 15]. Such approaches
+have been successfully applied to solve large-scale block-structured nonlinear problems,
+as encountered in stochastic programming and dynamic optimization.
+2
+
+1.2. Contributions
+In this article, we introduce a new parallel algorithm to solve block-structured non-
+linear programs involving state equations on exascale supercomputers. Our algorithm
+uses the parallel interior-point solver MadNLP [41], using two layers of parallelism to
+streamline both the evaluation of the derivatives and the solution of the KKT system.
+This framework targets new exascale supercomputers, where each node is assigned
+to multiple GPUs connected with a unified memory (designed to have fast memory
+exchange between the different GPUs).
+We demonstrate the capability of the algorithm on scenario-based power flow prob-
+lems (block-OPF), here formulated as two-stage stochastic nonlinear programs. The
+scenarios can be stochastic or represent contingencies (which can be interpreted as
+stochastic outcomes with uniform distribution), as is the case of the very widely used
+security-constrained AC optimal power flow (SC-ACOPF) problem [7]. SC-ACOPF is
+one of the core analyses undertaken in the planning, operational planning, and real-
+time operation of transmission systems [7]. SC-ACOPF is run several times a day by
+many operators in the US and the world. For brevity, we will refer to such problems
+as stochastic.
+The block structure of such problems is given by the different scenarios associ-
+ated with the stochastic problem, leading to potential parallelism in both the evalu-
+ation of the derivatives and the solution of the resulting block-angular KKT system.
+The parallel solution of the block-OPF problem with a Schur complement approach
+has been studied extensively both with PIPS-NLP [8, 37] (multiprocessing) and with
+Beltistos [23, 25] (multiprocessing + factorization of the dense Schur complement on
+the GPU). Compared to the state-of-the-art solver Beltistos, our approach carries out
+almost all computation on the GPUs including a global CUDA-aware MPI reduction,
+from the evaluation of the derivatives to the assembling of the Schur complement. We
+test our implementation on the pre-exascale supercomputer Polaris, where each node
+is equipped with 4 A100 GPUs, and we solve block-OPF problems with up to 9,251
+nodes.
+2. Problem statement
+In systems engineering, it is common to encounter optimization problems with rela-
+tively few degrees of freedom – ”controls”. Then, the goal is to appropriately fix the
+values for the degrees of freedom, e.g., by minimizing a given operational cost while
+satisfying the physical equations of the problem. In that context, the internal state of
+the system is described by a state variable x ∈ Rnx, whose values depend on the cur-
+rent controls u ∈ Rnu associated with the problem’s degrees of freedom. If the problem
+is well-posed, this translates to the state equation g(x, u) = 0, where the function g
+exhibits the physical structure of the problem (e.g., a differential equation encoding a
+dynamics, or a nonlinear network flow associated with static balance equations). When
+the system faces uncertainties, it is often appropriate to choose a control u feasible
+under a finite set of conditions (or scenarios). That is, the control u must satisfy N
+different state equations
+gi(xi, u) = 0
+for all
+i = 1, · · · , N,
+(1)
+3
+
+where the state xi now depend on the current scenario i. The variables xi can be assim-
+ilated into a recourse variable. The N functions g1, · · · , gN define the block structure
+of the problem.
+2.1. Block-structured nonlinear programs
+In addition to satisfying the N state equations (1), we aim at minimizing the average
+operating costs on the N different scenarios. The corresponding problem formulates
+as a two-stage nonlinear program, which, in our case, is a nonlinear program with
+partially separable structure [11]:
+min
+x1,··· ,xN,
+u
+N
+�
+i=1
+fi(xi, u)
+s.t.
+�
+�
+�
+�
+�
+xi ≥ 0 ,
+u ≥ 0
+gi(xi, u) = 0 ,
+hi(xi, u) ≤ 0 ,
+∀i = 1, · · · , N ,
+(2)
+with fi : Rnx × Rnu → R, gi : Rnx × Rnu → Rnx, hi : Rnx × Rnu → Rm smooth
+functions encoding the objective, the state equations, and the operational constraints,
+respectively. We note that the number of variables (N × nx + nu) and constraints
+(N × (m + nx)) are linearly proportional to the number of blocks N.
+In addition, if we introduce local control variables u1, · · · , uN with the additional
+coupling constraint u1 = · · · = uN = u, we get a problem with a separable structure,
+solvable using the primal decomposition method; at the expense of increasing the
+search space [11, 35].
+By introducing slack variables s1, · · · , sN, we rewrite (2) in standard form:
+min
+x1,··· ,xN,
+s1,··· ,sN,
+u
+N
+�
+i=1
+fi(xi, u)
+s.t.
+�
+�
+�
+�
+�
+u ≥ 0 ,
+xi ≥ 0 ,
+si ≥ 0
+gi(xi, u) = 0 ,
+hi(xi, u) + si = 0 ,
+∀i = 1, · · · , N .
+(3)
+We define yi ∈ Rnx the multipliers (or adjoints) associated to the equality con-
+straints gi(xi, u) = 0, zi ∈ Rm the multipliers associated to the operational constraints
+hi(xi, u) + si = 0, as well as λ, κi, νi the three multipliers associated to the respective
+bound constraints u ≥ 0, xi ≥ 0, si ≥ 0. The Lagrangian associated to (3) is:
+L(x, u, s; y, z, λ, µ, ν) :=
+N
+�
+i=1
+�
+fi(xi, u) + y⊤
+i gi(xi, u) + z⊤
+i
+�
+hi(xi, u) + si
+�
+− κixi − νisi
+�
+− λu ,
+(4)
+with x := (x1, · · · , xN), s := (s1, · · · , sN), y := (y1, · · · , yN), z := (z1, · · · , zN). To
+simplify the notations, we define the extended objective function and the extended
+constraints:
+f(x, u) :=
+�
+i=1
+fi(xi, u) ,
+g(x, u) :=
+�
+��
+g1(x1, u)
+...
+gN(xN, u)
+�
+�� ,
+h(x, u) :=
+�
+��
+h1(x1, u)
+...
+hN(xN, u)
+�
+�� .
+4
+
+We assume the functions f, g, h are twice differentiable. We denote
+H = ∂(x,u)h(x, u) ∈ RNm×(Nnx+nu)
+Jacobian of the inequality cons.
+G = ∂(x,u)g(x, u) ∈ RNnx×(Nnx+nu)
+Jacobian of the equality cons.
+W = ∇2
+(x,u)L(x, u, s; ·) ∈ R(Nnx+nu)×(Nnx+nu)
+Hessian of Lagrangian.
+2.2. Interior-point method
+The interior-point method (IPM) [30, Chapter 19] is a classical approach to solve (3).
+2.2.1. KKT system
+The Karush-Kuhn-Tucker (KKT) equations associated to (3) can be expressed as
+∇xfi + (Gi
+x)⊤yi + (Hi
+x)⊤zi − κi = 0,
+∀i = 1, · · · , N
+(5a)
+N
+�
+i=1
+�
+∇ufi + (Gi
+u)⊤yi + (Hi
+u)⊤zi
+�
+− λ = 0,
+(coupling)
+(5b)
+zi − νi = 0,
+∀i = 1, · · · , N
+(5c)
+gi(xi, u) = 0,
+∀i = 1, · · · , N
+(5d)
+hi(xi, u) + si = 0,
+∀i = 1, · · · , N
+(5e)
+Xiκi = 0, (xi, κi) ≥ 0,
+∀i = 1, · · · , N
+(5f)
+Siνi = 0, (si, νi) ≥ 0,
+∀i = 1, · · · , N
+(5g)
+Uλ = 0, (u, λ) ≥ 0,
+(5h)
+where U = diag(u), Xi = diag(xi), Si = diag(si).
+The interior-point method uses a homotopy parameter µ > 0 to replace the
+complementarity constraints (5f)-(5g)-(5h) by the smooth approximations: Xiκi =
+µenx, Siνi = µem, Uλ = µenu (en being the vector of all ones of dimension n). The
+resulting (smooth) system of nonlinear equations can be solved iteratively using New-
+ton method, where at each iteration, the descent direction is updated by solving the
+following augmented linear system:
+�
+���
+W + Σp
+0
+G⊤
+H⊤
+0
+Σs
+0
+I
+G
+0
+0
+0
+H
+I
+0
+0
+�
+���
+�
+���
+pd
+ps
+py
+pz
+�
+��� = −
+�
+���
+r1
+r2
+r3
+r4
+�
+���
+(6)
+with r1 =
+�
+∇xf + G⊤
+x y + H⊤
+x z − µX−1enx
+∇uf + G⊤
+u y + H⊤
+u z − µU−1enu
+�
+, r2 = z − µS−1em, r3 = g(x, u), r4 =
+h(x, u) + s. The primal descent direction pd decomposes as pd = (px1, · · · , pxN, pu).
+2.2.2. Block angular structure
+The linear system (6) is sparse and symmetric indefinite, and can be factorized using
+the Bunch-Kaufman algorithm. However, it is often beneficial to exploit its block-
+angular structure. Indeed, both the Hessian of the Lagrangian and the Jacobians have
+5
+
+a block-angular structure, given as
+W =
+�
+����
+Wx1x1
+Wx1u
+...
+...
+WxNxN
+WxNu
+Wux1
+. . .
+WuxN
+Wuu
+�
+���� ,
+G =
+�
+��
+G1
+x1
+G1
+u
+...
+...
+GN
+xN
+GN
+u
+�
+�� .
+By reordering the linear system (6), we can expose the block-angular structure of the
+KKT system as:
+�
+����
+A1
+B⊤
+1
+...
+...
+AN
+B⊤
+N
+B1
+. . .
+BN
+A0
+�
+����
+(7)
+with
+A0 = Wuu,
+Ai =
+�
+���
+Wxixi + Σxi
+0
+G⊤
+xi
+H⊤
+xi
+0
+Σsi
+0
+I
+Gxi
+0
+0
+0
+Hxi
+I
+0
+0
+�
+��� ,
+Bi =
+�
+�
+Wxiu
+(Gi
+u)⊤
+(Hi
+u)⊤
+�
+�
+⊤
+.
+The block-angular structure (7) can be exploited to solve the KKT linear system in
+parallel using a Schur complement approach. In that case, the submatrices Ai can be
+factorized independently to assemble the Schur complement in parallel [8].
+2.3. Condensation and reduction
+Instead of reordering the augmented KKT system (6) as a block angular matrix (7),
+we propose an alternative approach based on successive condensation and reduction
+of the KKT system, following the method introduced in [31]. If the structure is well-
+defined, we show that we can condense the KKT system (6) to a dense matrix with
+size nu × nu in two steps: first, by removing the inequality constraints in (6), then by
+exploiting the structure of the equality constraints to reduce the condensed system to
+a dense matrix. The condensation and reduction steps are illustrated in Figure 1.
+2.3.1. Condensation step
+The condensation step allows reducing the size of the KKT system drastically if the
+number of inequality constraints is large1.
+Proposition 2.1 (Condensed KKT system). The linear system (6) is equivalent to
+�
+K + Σp
+G⊤
+G
+0
+� �
+pd
+py
+�
+= −
+�
+r1 + H⊤(Σsr4 − r2)
+r3
+�
+,
+(8)
+where K ∈ R(Nnx+nu)×(Nnx+nu) is the condensed matrix K := W + H⊤ΣsH. The
+1It is equivalent to the normal equations in linear programming [30, Chapter 16, p.412]
+6
+
+Figure 1.. Successive reductions for a block-structured nonlinear problem with N = 3: Aug-
+mented system (6), Condensed system (8), Reduced system (11).
+descent directions ps and pz are recovered as
+�
+pz = Σs
+�
+Hpd + r4
+�
+− r2 ,
+ps = −Σ−1
+s
+�
+r2 + pz
+�
+.
+(9)
+Proof. See [31, Theorem 2.2].
+The condensed matrix K inherits the block-angular structure of the Hessian of the
+Lagrangian W.
+Proposition 2.2. The condensed matrix K = W +H⊤ΣsH has a block-angular struc-
+ture, given as
+K =
+�
+����
+Kx1x1
+Kx1u
+...
+...
+KxNxN
+KxNu
+Kux1
+. . .
+KuxN
+Kuu
+�
+����
+(10)
+where we have defined the condensed blocks Kxixi := Wxixi + (Hi
+xi)⊤ΣsiHi
+xi, Kuxi :=
+Wuxi + (Hi
+u)⊤ΣsiHi
+xi and Kuu := Wuu + �N
+i=1(Hi
+u)⊤ΣsiHi
+u.
+Proof. This is proved by induction.
+2.3.2. Reduction step
+In addition, we can exploit the structure of the equality constraints g1, · · · , gN to
+further reduce the size of the linear system (8) down to a dense matrix with size
+nu × nu. Equation (10) exhibits the structure w.r.t. the state x and the control u, we
+7
+
+3755 x 3755
+1193 x 1193
+107 x 107rewrite as such the condensed KKT system (8) as
+�
+�����������
+Kx1x1
+Kx1u
+(G1
+x1)⊤
+...
+...
+...
+KxNxN
+KxNu
+(GN
+xN)⊤
+Kux1
+. . .
+KuxN
+Kuu
+(G1
+u)⊤
+. . .
+(G1
+u)⊤
+G1
+x1
+G1
+u
+...
+...
+GN
+xN
+GN
+u
+�
+�����������
+�
+�����������
+px1...
+pxN
+pu
+p1
+y...
+pN
+y
+�
+�����������
+= −
+�
+����������
+ˆr1
+1...
+ˆrN
+1
+ˆr2
+ˆr1
+3...
+ˆrN
+3
+�
+����������
+,
+where we have renamed the right-hand-side in (8) as ˆr.
+Proposition 2.3 (Reduction). Assume that for all i = 1, · · · , N the Jacobian matri-
+ces Gi
+x ∈ Rnx×nx are invertible. Then the linear system (8) is equivalent to
+ˆKuu pu = −ˆr2 +
+N
+�
+i=1
+�
+(Gi
+u)⊤(Gi
+x)−⊤ˆri
+1 +
+�
+Kuxi − (Gi
+u)⊤(Gi
+x)−⊤Kxixi
+�
+(Gi
+x)−1ˆri
+3
+�
+(11)
+with ˆKuu := Z⊤KZ and Z ∈ R(nu+Nnx)×nu is the reduction operator defined as
+Z =
+�
+����
+−(G1
+x)−1G1
+u
+...
+−(GN
+x )−1GN
+u
+I
+�
+���� .
+(12)
+The descent directions px and py are recovered as
+�
+pi
+x = −(Gi
+x)−1�
+ˆri
+3 + Gi
+upu
+�
+pi
+y = −(Gi
+x)−⊤�
+ˆri
+1 + Kxixipi
+x + Kxiupu
+�
+.
+(13)
+Proof. See [31, Theorem 2.1].
+The reduction (11) is equivalent to a Schur complement approach applied to the
+condensed KKT system (8). In Proposition (2.1), we have shown that the condensed
+matrix K has a block-angular structure. The associated condensed KKT system (8)
+is also inheriting a block-angular structure in the form of (7), where the blocks are
+given by
+A0 = Kuu ,
+Ai =
+�
+Kxixi
+(Gi
+x)⊤
+Gi
+x
+0
+�
+,
+Bi =
+�
+Kxiu
+Gi
+u
+�⊤
+.
+(14)
+Proposition 2.4. Assume that for each i = 1, · · · , N the Jacobian Gi
+x is invertible.
+Let Suu = A0 − �N
+i=1 BiA−1
+i B⊤
+i
+be the Schur complement associated to the block-
+angular system (7) with the matrices (Ai, Bi) defined in (14). Then, the Schur com-
+plement Suu is equal to the reduced matrix �Kuu defined in (11): Suu = Z⊤KZ.
+8
+
+Proof. First, note that if the Jacobian Gi
+x is invertible, then the block matrix Ai
+defined in (14) is also invertible, with
+A−1
+i
+=
+�
+0
+(Gi
+x)−1
+(Gi
+x)−⊤
+−(Gi
+x)−⊤Kxixi(Gi
+x)−1
+�
+.
+(15)
+Using (14)-(15), we expand the expression of the terms in the sum constituting the
+Schur complement Suu:
+BiA−1
+i B⊤
+i =
+�
+Kuxi
+(Gi
+u)⊤� �
+0
+(Gi
+x)−1
+(Gi
+x)−⊤
+−(Gi
+x)−⊤Kxixi(Gi
+x)−1
+� �
+Kxiu
+Gi
+u
+�
+,
+= (Gi
+u)⊤(Gi
+x)−⊤Kxiu + Kuxi(Gi
+x)−1(Gi
+u) − (Gi
+u)⊤(Gi
+x)−⊤Kxixi(Gi
+x)−1Gi
+u .
+Hence, the Schur complement Suu = A0 − �N
+i=1 BiA−1
+i B⊤
+i expands as
+Suu = Kuu −
+N
+�
+i=1
+�
+(Gi
+u)⊤(Gi
+x)−⊤Kxiu + Kuxi(Gi
+x)−1(Gi
+u) − (Gi
+u)⊤(Gi
+x)−⊤Kxixi(Gi
+x)−1Gi
+u
+�
+= Z⊤KZ .
+We recover the expression of the reduced matrix �Kuu in Proposition 2.3.
+2.4. Discussion
+Hence, we can interpret the reduction step as a Schur complement approach. Forming
+the Schur complement has always been the bottleneck when solving distributed block
+angular problems in parallel [8, 26]. Its reduction operation involves large memory
+transfers between the processes, with the number of transfers being on the order
+of O(log(p)), where p is the number of processes. Due to the quasi-shared memory
+architecture on GPUs, the reduction can be implemented efficiently [31]. In the next
+section, we propose to extend [31] to assemble the reduced matrix �Kuu using two levels
+of parallelism, using both MPI and CUDA, thus reducing the reliance on distributed
+memory.
+3. Parallel implementation
+In the previous section, we have detailed the structure of block-angular nonlinear
+programs and presented the condensation and reduction steps for the KKT system.
+The loose coupling between the blocks is favorable for parallelizing the evaluation of
+the derivatives and the solution of the block-angular KKT system. Globally, we can
+distribute the computation on different processes using MPI (coarse parallelism). Lo-
+cally, we can further streamline the computation using GPU accelerators (fine-grained
+parallelism). This paradigm, with its two levels of parallelism, is directly in line with
+what is currently offered by the new exascale architectures, where each node has 4 to
+8 GPUs, all sharing a unified memory for fast communication. We present in §3.1 how
+we streamline the evaluation of the model using automatic differentiation, and in §3.2
+how we parallelize the solution of the KKT system.
+9
+
+3.1. Parallel automatic-differentiation
+First, we present how to evaluate the model in parallel using automatic differentiation
+[18]. We illustrate the procedure in Figure 2. The goal of the algorithm is to streamline
+the evaluation of the N scenarios on N/M GPUs, M being the number of scenarios
+evaluated locally on each GPU (we suppose here that N is a multiple of M).
+root
+g1, · · · , g4
+g13, · · · , g16
+g1
+g4
+· · ·
+g13
+g16
+· · ·
+· · ·
+· · ·
+Figure 2.. Parallel evaluation of the derivatives for g1, · · · gN on 4 GPUs: we have a total of
+N = 16 scenarios, each GPU evaluating M = 16/4 = 4 scenarios locally.
+3.1.1. Local parallelism
+The first level of parallelism streamlines the evaluation of the model on SIMD/GPU
+devices. We have designed our implementation to run entirely on the GPU device, to
+avoid any data transfer between the host and the device.
+3.1.1.1. Block evaluation. We suppose that the nonlinear functions (fi, gi, hi)
+share the same structure, its expressions yielding the same Abstract Syntax Tree (AST)
+for all i = 1, · · · , M. We illustrate the block evaluation on a simple abstract tree, but
+the reasoning extends to more complicated structures. We suppose that for all i, the
+functions fi, gi, hi depend linearly on a nonlinear basis matrix ψ : Rnx × Rnu → Rnb:
+that is, there exists three sparse matrices Lf, Lg, Lh such that
+fi(xi, u) = Lfψ(xi, u) ,
+gi(xi, u) = Lgψ(xi, u) ,
+hi(xi, u) = Lhψ(xi, u) .
+(16)
+Suppose we aim to evaluate the M functions g1, · · · , gM in batch for the states
+x1, · · · , xM. The structure (16) is directly amenable for SIMD evaluation. We de-
+note by XM = (x1, · · · , xM) ∈ Rnx×M the dense matrix obtained by concatenat-
+ing the M states together. By using a proper GPU kernel or a parallel modeler,
+we can evaluate the basis in a SIMD fashion and build the matrix Ψ(XM, u) :=
+�
+ψ(x1, u), · · · , ψ(xM, u)
+�
+∈ Rnb×M. Then, evaluating the functions g1, · · · , gM simul-
+taneously translates to the evaluation of one SpMM product:
+�
+g1(x1, u), · · · , gM(xM, u)
+�
+= LgΨ(XM, u) ∈ Rnx×M .
+(17)
+The total memory required in the two successive operations is O((nx + nb) × M), and
+depends linearly on the number of blocks M. We note the SpMM operations are generally
+implemented efficiently in the vendor library (cusparse for CUDA, rocSPARSE for
+AMDGPU).
+10
+
+3.1.1.2. First-order derivatives. Suppose that for a given i we have a differen-
+tiable implementation gbi : Rnd → Rnx associated to the function gi. We aim to
+evaluate the Jacobian-matrix products (∇gi)D for p tangents encoded in a matrix
+D ∈ Rnd×p using forward-mode AD and operator overloading. This operation trans-
+lates to propagating forward a vector of dual numbers. Denoting by d ∈ Dnd
+p
+the dual
+number encoding the p tangents stored in D, evaluating (∇gi)D simply amounts to
+call gbi(d) and extract the results in the dual numbers returned as a result. As Gi
+is sparse, we can apply the technique of Jacobian coloring [18] to compress the inde-
+pendent columns of the sparse matrix Gi and reduces the number of required seeding
+tangents p needed to evaluate the full Jacobian.
+Suppose now we want to evaluate the sparse Jacobians G1, · · · , GM in batch. As
+the functions gi are based on the same AST, their respective Jacobians G1, · · · , GM
+are sharing the same sparsity pattern. By seeding a matrix of dual numbers DM =
+(d1, · · · , dM) ∈ Dnd×M
+p
+, we can use the same operation as (17) to streamline the
+evaluation of the M Jacobian-vector products using the SIMD kernel Ψ(·) and SpMM
+operations:
+�
+gb1(d1), · · · , gbM(dM)
+�
+:= LgΨ(DM) ∈ Dnx×M
+p
+.
+(18)
+Once the results are evaluated, it remains to uncompress the dual outputs to build
+the M sparse Jacobians G1, · · · , GM. Hence, we can streamline the evaluation of the
+Jacobian along with the number of tangents p and the number of blocks M. This comes
+at the expense of increasing memory usage to O((nx + nb + nd) × M × p) (to store the
+dual matrices associated to the input, the intermediate basis Ψ and the output).
+3.1.1.3. Second-order derivatives. The evaluation of the second-order deriva-
+tives follows the same procedure, using forward-over-reverse AD. For each i, we sup-
+pose available an adjoint function adj gbi : Rnd × Rnx → Rnd which for any pri-
+mal x ∈ Rnd and adjoint y ∈ Rnx evaluates the Jacobian-transpose vector product
+(Gi(x))⊤y (reverse-mode). Using forward-mode AD on top of adj gbi, we can compute
+the second-order derivatives y⊤∇2gi(x)V for p directions V by calling adj gbi(x, y).
+Using Hessian coloring, we can compress the independent columns of the sparse matrix
+y⊤∇2g(x) and reduce the number of seeding tangents p required to evaluate the full
+Hessian. We note that in general obtaining an adjoint adj gbi running in parallel is
+nontrivial due to potential race conditions incurred by the control flow reversal of the
+original code.
+Computing the Hessian y⊤∇2gi(x) in parallel for i = 1, · · · M amounts to defining
+two matrices of dual numbers XM = (X1, · · · , XM) ∈ Dnd×M
+p
+, Y M = (y1, · · · , yM) ∈
+Dnx×M
+p
+and evaluate ∇Ψ(XM)⊤L⊤
+g Y M. The dual outputs are uncompressed to build
+the M sparse Hessians (as the sparsity pattern of the Hessians is different than those
+of the Jacobians, the matrix XM employed here is different than the one used in (18)).
+The total memory required to store the duals is O((2nx +nd +nb)×M ×p). For more
+details, we refer to the vector forward mode as described in [18].
+3.1.2. Global parallelism
+Now, if we have several GPUs at our disposal, we can push the parallelism further
+by distributing the evaluations using multiprocessing and a Message Passing Interface
+(MPI) library. Coming back at our original problem (2), we illustrate in Figure 2 how to
+dispatch the evaluation of the N nonlinear constraints g1, · · · , gN (the same reasoning
+11
+
+applies to the objectives f1, · · · , fN and the inequality constraints h1, · · · , hN). We use
+the streamlined implementation described in the previous subsection to evaluate the
+constraints in a batch of size M: the first GPU evaluates the constraints g1, · · · , gM,
+the second GPU evaluates gM+1, · · · , g2M, and so on. In total, the evaluation of the
+N constraints requires N/M GPUs (if M = 1, each GPU evaluate one constraint; if
+M = N, we use only one GPU evaluating all the constraints).
+The implementation has been designed to minimize the communication between
+the different processes: each batch g1, · · · , gM stores the data it needs locally, the only
+data exchange with the other processes being the vector of input and the vector of
+output. In addition, we will see in the next section we do not have to transfer the first-
+and second-order information if a parallel linear solver is being used.
+3.2. Parallel KKT solver
+By exploiting the block-angular structure of the KKT system, we can solve the New-
+ton step in parallel using a Schur complement approach. The challenge lies in the
+computation of the Schur complement matrix S = A0 − �N
+i=1 BiA−1
+i B⊤
+i . Each prod-
+uct BiA−1
+i B⊤
+i requires the factorization of the matrix Ai and the solution of a linear
+system with multiple (sparse) right-hand-side A−1
+i Bi. State-of-the-art methods are
+evaluating the Schur complement using an incomplete augmented factorization ap-
+plied on the auxiliary matrix
+�
+Ai
+B⊤
+i
+Bi
+0
+�
+, as currently implemented in the Pardiso
+linear solver [33]. Here, we use an alternative approach building on the reduced KKT
+system §2.3.2 (equivalent to the Schur complement approach). As the reduction can
+be streamlined on GPU accelerators [31], this approach can assemble the Schur com-
+plement in parallel using CUDA-aware MPI. We illustrate the parallel computation of
+the Schur complement in Figure 3.
+root
+(Assembling)
+K, G
+K, G
+K, G
+K, G
+AD
+AD
+AD
+AD
+ˆK1:4
+uu
+ˆK5:8
+uu
+ˆK9:12
+uu
+ˆK13:16
+uu
+(Reduction)
+Z⊤KZ
+Z⊤KZ
+Z⊤KZ
+Z⊤KZ
+MPI AllReduce
++
+(Schur compl.)
+ˆKuu
+Figure 3.. Parallel computation of the Schur complement.
+Assembling the sparse matrices. Using the procedure introduced in §3.1.1, we
+evaluate locally the Jacobians G1, · · · , GM the Jacobians H1, · · · , HM and the Hes-
+sians W 1, · · · , W M. Using dedicated kernels, we uncompress the results in the block-
+12
+
+angular sparse Jacobians
+G1:M
+x
+=
+�
+��
+G1
+x1
+...
+GM
+xM
+�
+�� , G1:M
+u
+=
+�
+��
+G1
+u...
+GM
+u
+�
+�� , H1:M =
+�
+��
+H1
+x1
+H1
+u
+...
+...
+HM
+xM
+HM
+u
+�
+�� ,
+and sparse Hessian
+W 1:M =
+�
+����
+Wx1x1
+Wx1u
+...
+...
+WxMxM
+WxMu
+Wux1
+. . .
+WuxM
+Wuu
+�
+���� .
+Once the sparse matrices are obtained, we recover the condensed matrix K1:M =
+W 1:M + (H1:M)⊤Σ(H1:M) (Proposition 2.1) using one SpGEMM operation and we fac-
+torize the matrix G1:M
+x
+using a sparse LU factorization (potentially running in batch
+as the matrices G1
+x1, · · · , GM
+xM are sharing the same sparsity pattern). Once the ma-
+trix G1:M
+x
+is factorized as PG1:M
+x
+Q = LU (P, Q being two permutation matrices),
+computing (G1:M
+x
+)−1b translates to two backsolves (SpSV) and two matrix-vector mul-
+tiplications (SpMV), as (G1:M
+x
+)−1b = QU−1L−1Pb.
+Local reduction. Once the sparse matrices are built, we evaluate locally the reduced
+matrix �K1:M
+uu
+on the GPU, using div(nu, nbatch) + 1 matrix-matrix product �K1:M
+uu V
+(with V ∈ Rnu×nbatch a dense matrix encoding nbatch vectors of the Cartesian basis of
+Rnu). The evaluation of one batched matrix-matrix product �K1:M
+uu V = (Z⊤K1:MZ)V
+proceeds in three steps
+(1) Solve Tx = −(G1:M
+x
+)−1(G1:M
+u
+V ).
+(2) Evaluate
+�
+Lx
+Lu
+�
+:=
+�
+K1:M
+xx
+K1:M
+xu
+K1:M
+ux
+K1:M
+uu
+� �
+Tx
+V
+�
+.
+(3) Set �K1:M
+uu V = Lu − G1:M
+u
+(G1:M
+x
+)−⊤Lx.
+In total, we need 2 SpSM and 3 SpMM operations in the first step, 1 SpMM in the second
+step, and 2 SpSM and 3 SpMM operations in the third step, giving a total of 4 SpSM
+and 7 SpMM operations. More than the computation, the reduction is limited by the
+memory, as we have to store the three buffers Lx, Tx, Tu with a total size of (2M ×
+nx + nu) × nbatch. If nx is too large, it is in our interest to reduce M (by using more
+GPUs) or to reduce nbatch (at the expense of computing more matrix-matrix product
+�K1:M
+uu V ).
+Global reduction. Once we obtain the locally reduced matrices �KnM+1:(n+1)M
+uu
+for n = 0, · · · , N/M − 1, we can assemble the global reduced matrix
+�Kuu =
+�N/M−1
+n=0
+�KnM+1:(n+1)M
+uu
+using one all reduce (MPI Allreduce) operation. The size of
+the reduced matrix �Kuu is nu × nu, hence limiting the memory transfer required in
+the algorithm.
+13
+
+3.3. Discussion
+We have presented a practical way to assemble the Schur complement on multi-GPU
+architectures. The parallelism occurs both at the local level (SIMD evaluations on the
+GPUs) and at the global level (distributed computation with MPI). The algorithm
+has the advantage of assembling the sparse Jacobians and Hessians only locally, as
+the reduction occurs before proceeding to the memory transfer with MPI Allreduce.
+The reduced matrix has a dimension nu × nu, which compresses the memory transfer
+significantly if the number of degrees of freedom nu is small. However, this comes at the
+expense of storing a vector of dual numbers (whose memory is linearly proportional
+to the number of blocks M evaluated locally and the number of tangents p being
+employed to evaluate the sparse derivatives) and additional buffers in the reduction
+algorithm. In the next section, we will test an implementation of the algorithm on
+CUDA GPUs, and show that the algorithm is practical.
+4. Numerical results
+We demonstrate the capabilities of the algorithm we introduced in Section §3 on the
+supercomputer Polaris, using CUDA-aware MPI to dispatch the solution on multiple
+GPUs. We present in §4.1 the stochastic optimal power flow problem, and give in §4.2
+detailed assessments of the algorithms we have introduced earlier in §3. Eventually,
+we present in §4.3 a benchmark comparing our parallel solution algorithm with a
+state-of-the-art solution method running on the CPU.
+4.1. Settings
+4.1.1. Case study: the block-structured optimal power flow
+The stochastic optimal power flow problem aims at finding an optimal dispatch for the
+generators u. The solution u should minimize the operational costs while satisfying the
+physical constraints (power flow equations g(x, u) = 0, here playing the role of the state
+equations) and operational constraints (line flow constraints h(x, u) ≤ 0) on a given
+set of scenarios. Each scenario is assigned given load parameters (energy demands) and
+potential contingencies (line tripping). The values of the state x depend on the local
+scenario we are in, the state x being the recourse variable in our case. As such, the
+problem has a partially separable structure as introduced in Problem (2), the control
+u being shared across all scenarios. We refer to [7] for the original presentation of the
+stochastic optimal power flow problem and to [8, 23, 24, 26] for practical algorithms
+solving the stochastic optimal power flow problem (some also focus on the multistage
+setting, which is not covered in this article). For our benchmark, we look at reference
+instances provided by MATPOWER [47], whose characteristics are detailed in Table 1.
+We recall that in our case, the size of the Schur complement matrix ˆKuu is given by
+the number of controls nu.
+4.1.2. Implementation
+The algorithm has been implemented entirely in Julia 1.8. The Schur complement
+approach has been developed as an extension of the nonlinear optimization solver
+MadNLP [41], using CUDA-aware MPI as provided in [6]. We have used the package
+14
+
+Name
+#bus
+#lines
+#gen
+nx
+nu
+case118
+118
+186
+54
+181
+107
+case1354pegase
+1,354
+1,991
+260
+2,447
+519
+case2869pegase
+2,869
+4,582
+510
+5,227
+1,019
+case9241pegase
+9,241
+16,049
+1,445
+17,036
+2,889
+Table 1.. MATPOWER instances used in the benchmark.
+ExaPF as a nonlinear modeler for the optimal power flow problem. All the results
+presented here have been generated on the supercomputer Polaris equipped with a
+total of 560 nodes, each node having with 1 CPU and 4 A100 GPUs.
+4.2. Assessment of the parallel implementation
+4.2.1. Assessing the performance of the parallel automatic differentation
+We first assess the performance of the parallel automatic differentiation we introduced
+in §3.1 in a multi-GPU setting. We compare the performance we obtain with a CPU
+implementation. We use case1354pegase as a representative instance, and display the
+time spent in the automatic differentiation as we increase the total number of scenarios
+N. The results are displayed in Figure 4.
+We observe that the computation time depends linearly on the number of scenarios,
+as expected. For N = 8, it is not worthwhile dispatching the evaluation on multiple
+GPUs as the problem is small enough to be evaluated on a single GPU. For N = 512,
+the evaluation time is 12.3s on the CPU, compared to 0.50, 0.41, 0.31, and 0.28s using
+1, 2, 4 and 8 GPUs, respectively. Hence, we get a 40x speed-up when evaluating the
+derivatives in a multi-GPU setting, and it is not worthwhile to use more than 4 GPUs
+(one node).
+8
+16
+32
+64
+128
+256
+512
+N scenarios
+10
+1
+100
+101
+Evaluation time [s]
+case1354pegase
+CPU
+1 GPUs
+2 GPUs
+4 GPUs
+8 GPUs
+Figure 4.. Time spent to evaluate the model and its derivatives with automatic differentiation.
+15
+
+4.2.2. Assessing the performance of the parallel KKT solver
+We proceed to the same performance analysis to assess the performance of the parallel
+KKT solver detailed in §3.2. We compare the time required to evaluate the full solution
+of the KKT system afresh (including reduction time, factorization time and backsolve
+time) on case1354pegase as we increase the number of scenarios N. As a reference,
+we give the time taken by the sparse linear solvers HSL MA27 (single-threaded) and
+HSL MA57 (multi-threaded). The results are displayed in Figure 5.
+On the left, we display the evolution of the time spent in the linear solver as we
+increase the number of scenarios. For N = 512, we observe that we get a linear speed-
+up as we increase the number of GPUs: using 8 GPUs, the parallel KKT solver is
+40x faster than using HSL MA27 on the CPU. Interestingly, we observe that HSL
+MA57 is not faster than HSL MA27, despite being multithreaded. This is consistent
+with the observation made in [42], and illustrates the difficulty of parallelizing ef-
+fectively the sparse LDL factorization (Bunch-Kaufman). On the right, we display a
+performance profile detailing the time spent in MA27 and the parallel KKT solver on
+case1354pegase with N = 512 scenarios. We observe that most of the time in HSL
+MA27 is spent on factorizing the sparse augmented KKT system (6). On the other
+side, the factorization of the dense reduced matrix ˆKuu is trivial using LAPACK on
+the GPU; the bottleneck in the parallel KKT solver is the reduction algorithm itself.
+Fortunately, the reduction algorithm can run in parallel: we get a linear speed-up as
+we increase the number of GPUs used in the reduction algorithm.
+8
+16
+32
+64
+128
+256
+512
+N scenarios
+10
+1
+100
+101
+Time [s]
+Linear solver time against N
+ma27
+ma57
+1 GPUs
+2 GPUs
+4 GPUs
+8 GPUs
+ma27
+1 GPU
+2 GPUs
+4 GPUs
+8 GPUs
+0
+2
+4
+6
+8
+Time [s]
+Performance profile for N = 512
+Factorization
+Backsolve
+Reduction
+Figure 5.. Time spent to solve the KKT system for case1354pegase.
+4.2.3. Assessing the memory consumption
+We have observed in §3.1 that the total memory required to store the duals is O((2nx+
+nd + nb) × M × p), with M being the number of scenarios stored locally (M = N on
+1 GPU, M = N/2 on 2 GPUs) and p the number of tangents. We display in Table 2
+the memory taken by the automatic differentiation backend and by the parallel KKT
+solver for case1354pegase as we increase the number of scenarios N. We note that
+storing the duals is expensive in terms of memory, with up to 10.9GB for N = 512 on
+one GPU (as a reference, each NVIDIA A100 GPU on Polaris has 40GB of memory
+available). By evaluating the model on different processes with MPI, we can split the
+memory consumption on the different GPUs we are using, leading to better use of the
+resource at our disposal.
+16
+
+1 GPU
+2 GPUs
+N
+AD
+KKT solver
+AD
+KKT solver
+8
+171.1
+92.3
+85.5
+48.1
+16
+342.2
+181.5
+171.1
+93.1
+32
+684.3
+360.0
+342.2
+183.2
+64
+1,368.7
+716.8
+684.3
+363.2
+128
+2,737.3
+1,430.5
+1,368.7
+723.4
+256
+5,474.7
+2,858.0
+2,737.3
+1,443.6
+512
+10,949.3
+5,712.8
+5,474.7
+2,884.1
+Table 2.. Memory consumption in MB
+4.3. Parallel solution of the block-structured OPF problem
+We analyze the parallel performance of our implementation on block-structured OPF
+problems.
+4.3.1. Assessing the parallel performance w.r.t. the number of scenarios
+First, we are interested in the scaling of the parallel algorithm in relation to the total
+number of scenarios N. We consider the case118 instance, and increase the number
+of scenarios N from 8 up to 2,048. For each N, we solve the block-structured OPF
+problem with MadNLP using our parallel KKT solver, and we compare with the
+performance we obtained with HSL MA27. The results are displayed in Figure 6. We
+observe that the solver HSL MA27 is initially faster than our parallel KKT solver, as
+the problem is too small to benefit from parallelism. However, as soon as N ≥ 16 the
+parallel KKT solver becomes competitive with HSL MA27. The relative performance is
+improving as we increase the number of scenarios N: for N = 512, we get a 68x speed-
+up when using 8 GPUs, compared to the reference HSL MA27 (10.4s versus 712s).
+Interestingly, using 2 nodes (=8 GPUs) does not lead to any speed-up compared to
+a single node (=4 GPUs) if N ≤ 256; this setting is attractive only when the size of
+the problem becomes sufficiently large (N ≥ 1024) to compensate for the additional
+memory exchange.
+4.3.2. Assessing the parallel performance w.r.t. the size of the problem
+Second, we increase the size of the problems. We set a fixed number of scenarios
+N = 8, and look at the time to solution for case1354pegase, case2869pegase and
+case9241pegase. We detail the respective dimension of each problem in Table 3. We
+display the results in Figure 7, and give the detailed benchmark in Table 4. On the
+left (a), we display the total time required to find the solution of the three instances as
+a function of the number of GPUs; on the right (b), we show the performance profile
+associated to case9241pegase. In (a), we observe that overall the parallel algorithm is
+faster than the CPU implementation. The parallel algorithm scales well as we increase
+the number of GPUs we are using, the parallel algorithm being 35x faster than the
+reference when using 8 GPUs to solve case9241pegase. In (b), we detail the time
+17
+
+8
+16
+32
+64
+128
+256
+512
+1,024 2,048
+N scenarios
+100
+101
+102
+103
+Time to solution [s]
+case118
+ma27 (CPU)
+polaris (1 GPUs)
+polaris (2 GPUs)
+polaris (4 GPUs)
+polaris (8 GPUs)
+Figure 6.. Time to solve the block-structured OPF problem case118 as a function of the
+number of scenarios N.
+spent in the different operations for case9241pegase: the time spent to factorize the
+Schur complement with Lapack (using cusolve) is constant as the size of the Schur
+complement remains the same as we increase the number of GPUs. We observe that
+the time spent in the AD decreases linearly with the number of GPUs exploited, but
+the relative time spent in AD is negligible (less than 5% of the total time). Most of
+the time is spent in the parallel reduction, as discussed earlier in §4.2.2.
+N
+nvar
+ncon
+ˆKuu (mb)
+1354pegase
+8
+20,095
+53,520
+2.1
+2869pegase
+8
+42,835
+119,216
+7.9
+9241pegase
+8
+139,177
+404,640
+63.7
+1354pegase
+512
+1,253,383
+4,425,280
+2.1
+Table 3.. Dimension of the instances we have used in our benchmark.
+4.3.3. Assessing the parallel performance on a very large-scale instance
+We finish our numerical experiments by solving a very large-scale instance:
+case1354pegase with N = 512 scenarios. The dimension of the resulting optimization
+problem is displayed in Table 3: the problem has more than 1 million variables, and 4
+millions constraints. We solve this instance on resp. 1 node, 2, 4 and 8 nodes (resp. 4,
+8, 16 and 32 GPUs). The results are displayed in Figure 8. We observe that the scaling
+is almost perfect when we use 2 nodes (8 GPUs) instead of a single node (4 GPUs)
+but we do not observe the same behavior when we increase the number of nodes to 4
+and 8. On that instance, the gain we get when using 8 nodes (32 GPUs) is marginal
+18
+
+CPU
+1 GPU
+2 GPU
+4 GPU
+8 GPU
+100
+101
+102
+103
+Time to solution [s]
+Benchmark
+1354pegase
+2849pegase
+9241pegase
+1 GPU
+2 GPUs
+4 GPUs
+8 GPUs
+10
+1
+100
+101
+102
+Time [s]
+Performance profile, 9241pegase
+Total time
+linear scaling
+AD
+Factorization
+Figure 7.. For a fixed number of scenarios N = 8, (a) total time spent solving the block-
+OPF case1354pegase, case2869pegase and case9241pegase with MadNLP (b) performance
+profile for case9241pegase with varying number of GPUs.
+1354pegase
+2869pegase
+9241pegase
+#it
+AD
+KKT
+Tot.
+#it
+AD
+KKT
+Tot.
+#it
+AD
+KKT
+Tot.
+CPU
+44
+2.6
+4.2
+7.0
+77
+11.9
+27.4
+40.3
+136
+205.6
+771.8
+984.1
+1 GPU
+44
+0.3
+1.8
+2.1
+93
+1.1
+11.7
+12.8
+98
+5.5
+112.3
+117.8
+2 GPUs
+44
+0.3
+1.1
+1.4
+93
+0.8
+7.4
+8.2
+98
+3.4
+56.8
+60.2
+4 GPUs
+44
+0.3
+1.0
+1.3
+93
+0.8
+5.7
+6.5
+98
+2.3
+35.8
+38.1
+8 GPUs
+44
+0.2
+1.0
+1.2
+93
+0.6
+5.1
+5.7
+98
+1.4
+26.4
+27.7
+Table 4.. Detailed results
+compared to when using 4 nodes (16 GPUs): the solving time only decreases from
+67s to 58s. This corroborate our observations: it is better to pack all the computa-
+tion on a single node to use four A100 GPUs connected together via unified memory
+(NVLINK has a transfer rate of 600GB/s). When we have to use more than 2 nodes,
+the memory transfers are more involved as they have to pass through the network of
+the supercomputer.
+5. Conclusion
+We show promising results for leveraging massively parallel SIMD architectures like
+GPUs for block-structured nonlinear programs. The parallelism is applied to both the
+derivative evaluation and the solution of the KKT linear system. The main operation
+in the KKT algorithm is the assembling of the Schur complement, the factorization of
+the dense Schur complement being fast to carry on the GPU.
+At all levels, the method benefits significantly from the massive parallelism, achiev-
+ing a speedup of around 40 for the derivatives compared to a sequential CPU im-
+plementation. The speedup is very application dependent, not least on the Hessian
+coloring and the problem’s structure. The assembling of the Schur complement is bot-
+tlenecked by a distributed reduction operation bound by the interconnect’s latency
+and throughput between GPUs. Current, so-called super nodes with multiple GPUs
+19
+
+4
+8
+16
+32
+#GPUs
+102
+3 × 101
+4 × 101
+6 × 101
+2 × 102
+Time to solution [s]
+Benchmark 1354pegase (N=512)
+Measured
+linear scaling
+Figure 8.. Solving case1354pegase with N = 512
+connected via fast networks like NVLINK greatly accelerate this operation. Lastly, our
+method is limited by the memory capacity of the GPU accelerators as it grows linearly
+with the number of problem blocks. In the context of ACOPF we are confident that
+upcoming GPUs will provide enough memory to solve a large number of scenarios in
+parallel, even for the largest grid instances (e.g., Eastern Interconnection with 70,000
+nodes).
+With the upcoming release of the Aurora supercomputer, these SIMD architectures
+will allow new science in regimes that were impossible with previous CPU architec-
+tures.
+Acknowledgment
+This material was based upon work supported by the U.S. Department of Energy, Of-
+fice of Science, Office of Advanced Scientific Computing Research (ASCR) under Con-
+tract DE-AC02-06CH11347 and by NSF through award CNS-1545046. The authors
+gratefully acknowledge the funding support from the Applied Mathematics Program
+within the U.S. Department of Energy’s (DOE) Office of Advanced Scientific Com-
+puting Research (ASCR) as part of the project ExaSGD. This research used resources
+of the Argonne Leadership Computing Facility, which is a DOE Office of Science User
+Facility supported under Contract DE-AC02-06CH11357.
+References
+[1] Amestoy, P. R., Duff, I. S., and L’excellent, J.-Y. (2000). Multifrontal parallel distributed
+symmetric and unsymmetric solvers. Computer methods in applied mechanics and engineer-
+ing, 184(2-4):501–520.
+[2] Birge, J. R. and Qi, L. (1988).
+Computing block-angular Karmarkar projections with
+applications to stochastic programming. Management science, 34(12):1472–1479.
+[3] Bradbury, J., Frostig, R., Hawkins, P., Johnson, M. J., Leary, C., Maclaurin, D., Necula, G.,
+20
+
+Paszke, A., VanderPlas, J., Wanderman-Milne, S., and Zhang, Q. (2018). JAX: composable
+transformations of Python+NumPy programs.
+[4] B¨ucker, H. M., Lang, B., an Mey, D., and Bischof, C. H. (2001). Bringing together auto-
+matic differentiation and OpenMP. In Proceedings of the 15th international conference on
+Supercomputing, pages 246–251.
+[5] Bussieck, M. R. and Meeraus, A. (2004). General algebraic modeling system (GAMS). In
+Modeling languages in mathematical optimization, pages 137–157. Springer.
+[6] Byrne, S., Wilcox, L. C., and Churavy, V. (2021). MPI. jl: Julia bindings for the Message
+Passing Interface. In Proceedings of the JuliaCon Conferences, volume 1, page 68.
+[7] Capitanescu, F., Ramos, J. M., Panciatici, P., Kirschen, D., Marcolini, A. M., Platbrood,
+L., and Wehenkel, L. (2011).
+State-of-the-art, challenges, and future trends in security
+constrained optimal power flow. Electric power systems research, 81(8):1731–1741.
+[8] Chiang, N., Petra, C. G., and Zavala, V. M. (2014). Structured nonconvex optimization
+of large-scale energy systems using PIPS-NLP. In 2014 Power Systems Computation Con-
+ference, pages 1–7. IEEE.
+[9] Choi, I. C. and Goldfarb, D. (1993). Exploiting special structure in a primal—dual path-
+following algorithm. Mathematical Programming, 58(1):33–52.
+[10] Colombo, M., Grothey, A., Hogg, J., Woodsend, K., and Gondzio, J. (2009). A structure-
+conveying modelling language for mathematical and stochastic programming. Mathematical
+Programming Computation, 1(4):223–247.
+[11] DeMiguel, V. and Nogales, F. J. (2008). On decomposition methods for a class of partially
+separable nonlinear programs. Mathematics of Operations Research, 33(1):119–139.
+[12] Duff, I. S. (2004). MA57—a code for the solution of sparse symmetric definite and indef-
+inite systems. ACM Transactions on Mathematical Software (TOMS), 30(2):118–144.
+[13] Duff, I. S. and Van Der Vorst, H. A. (1999). Developments and trends in the parallel
+solution of linear systems. Parallel Computing, 25(13-14):1931–1970.
+[14] Dunning, I., Huchette, J., and Lubin, M. (2017). JuMP: A modeling language for math-
+ematical optimization. SIAM review, 59(2):295–320.
+[15] Fourer, R., Gay, D. M., and Kernighan, B. W. (1990). A modeling language for mathe-
+matical programming. Management Science, 36(5):519–554.
+[16] Gondzio, J. and Grothey, A. (2009). Exploiting structure in parallel implementation of
+interior point methods for optimization. Computational Management Science, 6(2):135–160.
+[17] Gondzio, J. and Sarkissian, R. (2003). Parallel interior-point solver for structured linear
+programs. Mathematical Programming, 96(3):561–584.
+[18] Griewank, A. and Walther, A. (2008). Evaluating derivatives: principles and techniques
+of algorithmic differentiation. SIAM.
+[19] Hovland, P. and Bischof, C. (1998). Automatic differentiation for message-passing parallel
+programs. In Proceedings of the First Merged International Parallel Processing Symposium
+and Symposium on Parallel and Distributed Processing, pages 98–104. IEEE.
+[20] Hovland, P. D. (1997). Automatic differentiation of parallel programs. University of Illinois
+at Urbana-Champaign.
+[21] Huchette, J., Lubin, M., and Petra, C. (2014). Parallel algebraic modeling for stochas-
+tic optimization. In 2014 First Workshop for High Performance Technical Computing in
+Dynamic Languages, pages 29–35. IEEE.
+[22] Jessup, E. R., Yang, D., and Zenios, S. A. (1994). Parallel factorization of structured
+matrices arising in stochastic programming. SIAM journal on Optimization, 4(4):833–846.
+[23] Kardoˇs, J., Kourounis, D., and Schenk, O. (2019). Two-level parallel augmented Schur
+complement interior-point algorithms for the solution of security constrained optimal power
+flow problems. IEEE Transactions on power systems, 35(2):1340–1350.
+[24] Kardoˇs, J., Kourounis, D., and Schenk, O. (2020). Structure-exploiting interior point
+methods. In Parallel Algorithms in Computational Science and Engineering, pages 63–93.
+Springer.
+[25] Kardoˇs, J., Kourounis, D., Schenk, O., and Zimmerman, R. (2022). Beltistos: A robust
+interior point method for large-scale optimal power flow problems. Electric Power Systems
+21
+
+Research, 212:108613.
+[26] Lubin, M., Petra, C. G., Anitescu, M., and Zavala, V. (2011). Scalable stochastic opti-
+mization of complex energy systems. In SC’11: Proceedings of 2011 International Conference
+for High Performance Computing, Networking, Storage and Analysis, pages 1–10. IEEE.
+[27] Moses, W. S., Churavy, V., Paehler, L., H¨uckelheim, J., Narayanan, S. H. K., Schanen, M.,
+and Doerfert, J. (2021). Reverse-mode automatic differentiation and optimization of GPU
+kernels via Enzyme. In Proceedings of the International Conference for High Performance
+Computing, Networking, Storage and Analysis, pages 1–16.
+[28] Nash, S. G. and Sofer, A. (1989). Block truncated-Newton methods for parallel optimiza-
+tion. Mathematical Programming, 45(1):529–546.
+[29] Nash, S. G. and Sofer, A. (1991). A general-purpose parallel algorithm for unconstrained
+optimization. SIAM Journal on Optimization, 1(4):530–547.
+[30] Nocedal, J. and Wright, S. J. (2006). Numerical optimization. Springer series in operations
+research. Springer, New York, 2nd edition.
+[31] Pacaud, F., Shin, S., Schanen, M., Maldonado, D. A., and Anitescu, M. (2022).
+Ac-
+celerating condensed interior-point methods on SIMD/GPU architectures. arXiv preprint
+arXiv:2203.11875.
+[32] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z.,
+Gimelshein, N., Antiga, L., et al. (2019). Pytorch: An imperative style, high-performance
+deep learning library. Advances in neural information processing systems, 32.
+[33] Petra, C. G., Schenk, O., Lubin, M., and G¨artner, K. (2014). An augmented incomplete
+factorization approach for computing the Schur complement in stochastic optimization.
+SIAM Journal on Scientific Computing, 36(2):C139–C162.
+[34] Rodriguez, J. S., Parker, R., Laird, C. D., Nicholson, B., Siirola, J. D., and Bynum, M.
+(2021). Scalable parallel nonlinear optimization with PyNumero and Parapint. Optimization
+Online.
+[35] Ruszczynski, A. (1993).
+Interior point methods in stochastic programming.
+Working
+paper.
+[36] Saad, Y. (1980).
+On the rates of convergence of the Lanczos and the block-Lanczos
+methods. SIAM Journal on Numerical Analysis, 17(5):687–706.
+[37] Schanen, M., Gilbert, F., Petra, C. G., and Anitescu, M. (2018). Toward multiperiod
+ac-based contingency constrained optimal power flow at large scale. In 2018 Power Systems
+Computation Conference (PSCC), pages 1–7. IEEE.
+[38] Schenk, O. and G¨artner, K. (2004). Solving unsymmetric sparse systems of linear equa-
+tions with PARDISO. Future Generation Computer Systems, 20(3):475–487.
+[39] Schnabel, R. B. (1985). Parallel computing in optimization. In Computational Mathe-
+matical Programming, pages 357–381. Springer.
+[40] Schnabel, R. B. (1995). A view of the limitations, opportunities, and challenges in parallel
+nonlinear optimization. Parallel computing, 21(6):875–905.
+[41] Shin, S., Coffrin, C., Sundar, K., and Zavala, V. M. (2021). Graph-based modeling and
+decomposition of energy infrastructures. IFAC-PapersOnLine, 54(3):693–698.
+[42] Tasseff, B., Coffrin, C., W¨achter, A., and Laird, C. (2019). Exploring benefits of lin-
+ear solver parallelism on modern nonlinear optimization applications.
+arXiv preprint
+arXiv:1909.08104.
+[43] Watson, J.-P., Woodruff, D. L., and Hart, W. E. (2012). PySP: modeling and solving
+stochastic programs in python. Mathematical Programming Computation, 4(2):109–149.
+[44] Word, D. P., Kang, J., Akesson, J., and Laird, C. D. (2014). Efficient parallel solution
+of large-scale nonlinear dynamic optimization problems. Computational Optimization and
+Applications, 59(3):667–688.
+[45] Zavala, V. M., Laird, C. D., and Biegler, L. T. (2008).
+Interior-point decomposition
+approaches for parallel solution of large-scale nonlinear parameter estimation problems.
+Chemical Engineering Science, 63(19):4834–4845.
+[46] Zhu, Y., Word, D., Siirola, J., and Laird, C. D. (2009). Exploiting modern computing
+architectures for efficient large-scale nonlinear programming. In Computer Aided Chemical
+22
+
+Engineering, volume 27, pages 783–788. Elsevier.
+[47] Zimmerman, R. D., Murillo-S´anchez, C. E., and Thomas, R. J. (2010). MATPOWER:
+Steady-state operations, planning, and analysis tools for power systems research and edu-
+cation. IEEE Transactions on Power Systems, 26(1):12–19.
+Government License: The submitted manuscript
+has been created by UChicago Argonne, LLC,
+Operator of Argonne National Laboratory (“Ar-
+gonne”). Argonne, a U.S. Department of Energy
+Office of Science laboratory, is operated under
+Contract No. DE-AC02-06CH11357. The U.S.
+Government retains for itself, and others acting
+on its behalf, a paid-up nonexclusive, irrevoca-
+ble worldwide license in said article to repro-
+duce, prepare derivative works, distribute copies
+to the public, and perform publicly and display
+publicly, by or on behalf of the Government.
+The Department of Energy will provide pub-
+lic access to these results of federally sponsored
+research in accordance with the DOE Public
+Access Plan. http://energy.gov/downloads/doe-
+public-access-plan.
+23
+

9tFLT4oBgHgl3EQfCC72/content/tmp_files/2301.11974v1.pdf.txt

@@ -0,0 +1,2684 @@
+arXiv:2301.11974v1  [math.OC]  27 Jan 2023
+Augmenting Bi-objective Branch and Bound
+by Scalarization-Based Information
+Julius Bauß∗ and Michael Stiglmayr
+University of Wuppertal, School of Mathematics and Natural Sciences,
+IMACM, Gaußstr. 20, 42119 Wuppertal, Germany
+{bauss,stiglmayr}@math.uni-wuppertal.de
+While Branch and Bound based algorithms are a standard approach to solve single-
+objective (mixed-)integer optimization problems, multi-objective Branch and Bound meth-
+ods are only rarely applied compared to the predominant objective space methods. In this
+paper we propose modifications to increase the performance of multi-objective Branch
+and Bound algorithms by utilizing scalarization-based information. We use the hypervol-
+ume indicator as a measure for the gap between lower and upper bound set to implement
+a multi-objective best-first strategy. By adaptively solving scalarizations in the root node
+to integer optimality we improve both, upper and lower bound set. The obtained lower
+bound can then be integrated into the lower bounds of all active nodes, while the deter-
+mined solution is added to the upper bound set. Numerical experiments show that the
+number of investigated nodes can be significantly reduced by up to 83% and the total
+computation time can be reduced by up to 80%.
+Keywords: multi-objective optimization, mult-iobjective branch and bound, integer pro-
+gramming, hypervolume indicator
+1 Introduction
+Many optimization problems occurring in real-word applications include a conflict of interests and
+goals, or secondary objectives, in a word, they are multi-objective. Thus, there is (in general) not
+one solution that optimizes all objectives at once. Following the a posteriori paradigm of decision
+making, we aim at determining the set of so-called efficient solutions or the images, the so-called
+non-dominated points, which cannot be improved in one objective without deterioration in at least
+one other objective. Thus, efficient solutions are reasonable choices for decision makers.
+As we are considering specifically bi-objective integer linear programs and their solution with
+multi-objective Branch and Bound methods, the following literature survey will also focus on this
+and closely related topics. A comprehensive introduction to multi-objective optimization in general
+is given, e.g., in Steuer (1986); Ehrgott (2005).
+Solution approaches for multi-objective optimization problems are often categorized in: objective
+space and decision space methods. Objective space methods scalarize the underlying problem, i. e.,
+it is replaced by a series of single-objective problems to determine successively the set of efficient
+∗Corresponding author
+1
+
+solutions.
+In the case of multi-objective integer programming, these scalarized problems can be
+solved with commercial integer programming solvers like CPLEX or Gurobi. The utilization of these
+optimized, single-criteria solvers are a major advantage and one of the reasons why those methods
+are predominant in multi-objective optimization.
+There are numerous objective space methods and a popular one is the ε-constraint method that
+was introduced for two objectives by Haimes et al. (1971). In every iteration the first objective is
+optimized with an updated constraint to ensure an improvement regarding the second objective. In
+Laumanns et al. (2006) an extension to three and more objectives is presented. Many approaches
+based on the ε-constraint method have been published in the last decades, for example Boland et al.
+(2017) and Kirlik and Sayın (2014) combine the method with reduction of dimension in the tri-
+respectively multi-dimensional case.
+The weighted sum scalarization is an objective space method based on the optimization of a
+weighted sum of the objective functions using non-negative weights.
+Note that not all efficient
+solutions can be determined as optimal solutions of the weighted sum scalarization using suit-
+able weights (see, e. g. Aneja and Nair, 1979).
+Efficient solutions which can be obtained by us-
+ing weighted sum scalarization are denoted as supported efficicent and their corresponding non-
+dominated points are located on the boundary of the convex hull of feasible image points. Extensions
+of the weighted sum method to the multi-objective case are proposed in Przybylski et al. (2010a),
+Özpeynirci and Köksalan (2010), Bö¸kler and Mutzel (2015), and Przybylski et al. (2019).
+Ulungu and Teghem (1995) introduced the so-called two-phase method for bi-objective problems.
+In the first phase the extreme supported non-dominated points are generated with an algorithm sim-
+ilar to the initial weighted sum approach. In the second phase the remaining non-dominated points
+are generated by searching in triangles defined by two consecutive extreme supported non-dominated
+points. In Przybylski et al. (2008) and Tuyttens et al. (2000) problem specific algorithms are sug-
+gested for the second phase, while in Przybylski et al. (2010b) a two-phase method for problems
+with more than two objectives is proposed.
+The augmented weighted Tchebycheff method, first presented in Steuer and Choo (1983), mini-
+mizes the augmented weighted Tchebycheff distance between a predefined reference point and the
+set of feasible image points. Dächert et al. (2012) suggested an adaptive choice of the augmentation
+term for the bi-objective case.
+In Boland et al. (2015a), Boland et al. (2015b) (for the bi-objective case), Dächert and Klamroth
+(2014), and Klamroth et al. (2015) (for the tri- respectively multi-objective case) search region split-
+ting methods are proposed. In this class of objective space methods, the search region (based on the
+already determined non-dominated points) is splitted into so-called search zones on which scalariza-
+tions are solved indpendently.
+Besides their advantages, objective space methods share a shortcoming: In each iteration a scalar-
+ized integer program is solved from scratch. Even though in some objective space methods starting
+solutions can be transfered from previous iterations, a large number of very similar problems has
+to be solved. In order to avoid this effort, decision space methods, mainly the Branch and Bound
+method, have been increasingly investigated in the recent years.
+Klein and Hannan (1982) developed one of the first Branch and Bound algorithms for multi-
+objective integeger programs with a typical one tree structure. In Kiziltan and Yucaoğlu (1983) a
+general Branch and Bound framework for multi-objective integer programs with binary variables
+is presented. Ulungu and Teghem (1997) and Visée et al. (1998) proposed problem specific Branch
+and Bound approaches for bi-objective Knapsack problems, where the latter approach is integrated
+in a two-phase method.
+Mavrotas and Diakoulaki (1998) extend the Branch and Bound approach to multi-objective mixed
+integer programs. Parts of the algorithm are refined in Mavrotas and Diakoulaki (2005). In Vincent et al.
+(2013) this algorithm is improved and it is shown that the original algorithm is not correct because
+the final dominance test is incomplete.
+In Belotti et al. (2012) a Branch and Bound method is
+presented that can handle bi-objective mixed integer programs with continious variables in both
+objective functions.
+2
+
+The Branch and Bound method proposed in Sourd and Spanjaard (2008) uses a set of points as
+lower bound instead of just using a single point. Furthermore hyperplanes are used to fathom nodes
+by dominance. In Stidsen et al. (2014) this idea is continued. They use hyperplanes as a lower
+bound set that are generated by solving weighted sum scalarizations. Additionally they present
+the so-called Pareto branching and the slicing technique. With Pareto branching it is possible to
+divide the objective space to possibly ignore parts of it in specific nodes.
+Slicing partitions the
+objective space in equally large parts and a respective slice can be fathomed if it is dominated by
+an already found integer point.
+In Stidsen and Andersen (2018) this algorithm is improved and
+an approach to parallelize the algorithm is presented. Based on this, the idea Pareto branching is
+further investigated in Parragh and Tricoire (2019) and Gadegaard et al. (2019) for the bi-objective
+case and Forget et al. (2022) for the tri-objective case. A self-contained survey of multi-objective
+Branch and Bound approaches is given in Przybylski and Gandibleux (2017).
+In this paper we present a bi-objective Branch and Bound algorithm that is augmented by
+scalarization-based information. We make use of optimized single-objective solvers for scalar in-
+teger programs and integrate the resulting information into the bi-objective Branch and Bound by
+improving lower and upper bounds. Furthermore, we propose a new adaptive node selection strategy,
+which relies on objective space information. In our numerical analysis we show the effectiveness of
+these improvements by comparing them with a generic multi-objective Branch and Bound algorithm,
+which we use as our baseline algorithm.
+The remainder of the article is organized as follows: In Section 2, we introduce notations and def-
+initions for multi-objective optimization. In Section 3, we present a general multi-objective Branch
+and Bound framework and its key components. Furthermore, we describe a specific (however stan-
+dard) multi-objective Branch and Bound algorithm, which will be used as baseline implementation
+in our numerical tests. In Section 4, we present augmentations of the multi-objective Branch and
+Bound, that utilize objective space information to improve the node selection as well as the compu-
+tation of upper and lower bounds. We provide numerical results in Section 5 and in Section 6, we
+outline conclusions and outlooks for further research.
+2 Preliminaries
+We introduce a general multi-objective integer linear program which can be written in the form:
+min
+�
+z1(x), . . . , zp(x)
+�⊤
+s.t.
+A x ≤ b
+x ≥ 0
+x ∈ Zn.
+(MOILP)
+Thereby, z(x) := (z1(x), . . . , zp(x))⊤ = C ·x ∈ Rp (with p ≥ 2) denotes the objective function vector,
+with C ∈ Rp×n the matrix of objective coefficients. The set of feasible solutions X := {x ∈ Zn : A ≤
+b, x ≥ 0} is a subset of the decision space Rn, while its image Y := {C x: x ∈ X} is a subset of the
+objective space Rp.
+We use the Pareto concept of optimality which relies on the componentwise order. Let y1, y2 ∈ Rp,
+then we define the corresponding dominance relations as follows:
+• y1 ≦ y2, i.e., y1 weakly dominates y2 if y1
+k ≤ y2
+k for k = 1, ..., p,
+• y1 < y2, i.e., y1 strictly dominates y2 if y1
+k < y2
+k for k = 1, ..., p,
+• y1 ≤ y2, i.e., y1 dominates y2 if y1 ≤ y2 and y1 ̸= y2.
+A feasible solution x ∈ X is called efficient if there is no other solution ˆx ∈ X dominating it, i.e.,
+z(ˆx) ≤ z(x). A feasible solution x ∈ X is called weakly efficient if there is no ˆx ∈ X such that
+z(ˆx) < z(x). The set of efficient solutions is denoted by XE. By YN = {z(x) ∈ Y : x ∈ XE} we
+3
+
+denote the set of the non-dominated points in the objective space. Moreover, for any set Q ⊆ Rp
+we denote by QN the set of its non-dominated points (i.e., q ∈ QN
+⇐⇒ ∄q′ ∈ Q: q′ ≤ q). For a
+comprehensive introduction to multi-objective optimization see, e. g., Ehrgott (2005).
+In this article we consider a minimal complete set as solution of a multi-objective optimization
+problem. A minimal complete set denotes the set of all non-dominated points YN and one efficient
+solution for each non-dominated point. See Serafini (1987) for a comparison of solution concepts in
+multi-objective optimization.
+A standard solution approach in multi-objective optimization is the weighted sum scalarization
+given in (WSλ).
+min WSλ(x) := λ⊤z(x) =
+p
+�
+i=1
+λi zi(x)
+s.t. x ∈ X
+(WSλ)
+Obviously, every optimal solution of the weighted sum scalarization for λ ∈ Rp
+> := {λ ∈ Rp : λ > 0}
+is efficient for (MOILP). However, in general not all efficient solutions are optimal solutions of a
+corresponding weighted sum problem. An efficient solution x′ ∈ XE is called supported if there is a
+weighting vector λ′ ∈ Rp
+> such that x′ is optimal for (WSλ) for λ = λ′, otherwise x′ is unsupported.
+Note that the non-dominated points corresponding to supported efficient solutions are located on
+the boundary of the convex hull of Y , while the unsupported non-dominated points are located in
+its (relative) interior.
+As already mentioned in the introduction the computation of upper and lower bounds on the
+non-dominated set is a crucial component of any multi-objective Branch and Bound algorithm. The
+tightest componentwise upper and lower bounds of YN are the ideal point yI and the Nadir point
+yN given by:
+yI
+k = min
+y∈Y yk
+and
+yN
+k = max
+y∈YN yk
+for k = 1, . . . p.
+Obviously, yI ≦ y ≦ yN holds for every y ∈ YN, i.e.; YN is contained in the hyperbox spanned by
+the corner points yI and yN. However, these single point bounds are in general very weak except for
+the degenerate case of yI = yN. This motivates to consider bound sets instead of bounds consisting
+of a single point. We will rely on the definition of bound sets proposed in Ehrgott and Gandibleux
+(2007). Let Rp
+≧ := {y ∈ Rp : y ≧ 0}, then
+• A lower bound set L ⊂ Rp for YN is a
+– Rp
+≧-closed (i.e., the set L + Rp
+≧ is closed),
+– Rp
+≧-bounded (i.e., there exists a y ∈ Rp such that L ⊂ y + Rp
+≧)
+– stable set (i.e., L ⊂ (L + Rp
+≧)N),
+such that YN ⊂ (L + Rp
+≧).
+• An upper bound set U ⊂ Rp for YN is a
+– Rp
+≧-closed,
+– Rp
+≧-bounded,
+– stable sets,
+such that YN ⊂ cl
+�
+(U + Rp
+≧)∁�
+.
+The upper bound and lower bound that we will define for our branch and bound framework in
+Section 3 will suit these definitions. We say a lower bound L is weakly dominated by an upper
+bound U if for all l ∈ L there exists an u ∈ U such that u ≦ l.
+4
+
+In the following we restrict ourselves to bi-objective binary linear optimization problems, i. e.,
+problems with two linear objective functions and variables x ∈ {0, 1}n:
+min
+z(x) =
+�
+z1(x), z2(x)
+�⊤
+s.t.
+A x ≤ b
+x ∈ {0, 1}n.
+(BO01LP)
+3 A Generic Multi-objective Branch and Bound Framework
+In this section we present a generic multi-objective Branch and Bound framework, which we specify
+and augment by using scalarization based information in the then following sections.
+Branch and Bound methods follow a “divide and conquer” paradigm. A problem that is too hard
+to be solved directly, is divided into smaller and thus easier subproblems. Thereby, subproblems are
+associated with nodes in a tree data structure according to their descent, i.e., node i is a descendant
+node of node j iff the feasible set of the subprobem associated with node i is a subset of the feasible
+set of the subproblem associated with node j. The corresponding subproblems of the child nodes are
+created by subdividing the feasible set of the corresponding (sub)problem of the parent node. Starting
+with the root node, to which the original optimization problem is associated, the algorithm selects in
+each iteration one active node and updates its lower bound and upper bound. Then the active node
+can be fathomed if the corresponding subproblem is either solved or irrelevant for the determination
+of a minimal complete set. If we cannot prune we subdivide the corresponding problem into new
+subproblems and create corresponding child nodes (branching). For a more detailed introduction
+and survey of multi-objective Branch and Bound algorithms see Przybylski and Gandibleux (2017).
+A recent survey of single-objective Branch and Bound frameworks is given e.g. in Morrison et al.
+(2016). In the following we specify the lower bound, upper bound, branching rule and node selection
+we use in our framework.
+Lower bound:
+Lower bound sets are often determined by solving relaxations of the respective
+subproblem. Like in the single-objective case, the most frequently used relaxations are linear and
+convex relaxations. In order to solve the linear relaxation we are using in our framework, we apply
+Benson’s outer approximation algorithm (Benson, 1998). The algorithm is initiated with a lower
+bound, which is improved in every iteration by generating cuts. Due to the outer approximation
+structure the algorithm can be aborted at any time returning a valid lower bound.
+Alternatively, linear (or convex) relaxations can be obtained using a dichotomic scheme (see, for
+example, Aneja and Nair, 1979; Özpeynirci and Köksalan, 2010; Przybylski et al., 2010a).
+Upper bound:
+The upper bound set, in the following denoted by U, is stored in the form of a so-
+called incumbent list. Throughout the run of the algorithm, it contains all integer feasible solutions
+and their corresponding outcome vectors that are not dominated by another feasible solution found
+so far. In every iteration the extreme supported solutions of the computed lower bound sets are
+checked for integer feasibility. An integer feasible solution ¯x ∈ X is then appended to the incumbent
+list, if there is no x ∈ U dominating ¯x, i. e., C(x) ≤ C(¯x). If a new solution ¯x is added to the
+incumbent list U all solutions x ∈ U which are dominated by ¯x (C(¯x) ≤ C(x)) are removed from
+it. Note that an update of the incubent list requires a subsequent update of the list of local upper
+bounds. A detailed description of local upper bounds, their computation and update in an arbitrary
+number of criteria is given in Klamroth et al. (2015). In this framework we start with an empty
+upper bound set. However, it is also possible to initialize the incumbent list by heuristic methods, or
+by solving scalarizations like, e.g., in the two-phase method (Ulungu and Teghem, 1995; Visée et al.,
+1998).
+U ⊎ {¯x} :=
+�
+U
+if ∃x ∈ U : C(x) ≤ C(¯x)
+{¯x} ∪ {x ∈ U : C(¯x) ≰ C(x)}
+otherwise
+5
+
+Node selection:
+In every iteration of the algorithm an unexplored node is selected from the tree of
+subproblems. This node is called active node. The order in which the nodes of the tree are considered
+has a significant impact on the number of created nodes that have to be explored and thus on the
+computation time.
+Two types of strategies need to be distinguished: static strategies and dynamic strategies. The two
+most common examples of static strategies are the depth-first strategy and the breadth-first strategy.
+Most multi-objective Branch and Bound algorithms in literature follow a depth-first strategy. Thus,
+we use this strategy for our baseline implementation as well.
+In contrast to the single-objective case, dynamic node selection strategies are rarely applied in the
+multi-objective case. The usage of dynamic strategies for the choice of the active node can be seen
+in (Stidsen et al., 2014), (Belotti et al., 2012) and (Jesus et al., 2021), for example.
+Fathoming:
+In order to avoid the total enumeration of all feasible solutions, nodes are fathomed
+if the respective subproblem is either solved to optimality or does not contain solutions which are
+necessary to determine a minimal complete set. In particular, there are three different situations in
+which a node can be fathomed:
+i) Fathoming by infeasibility: If the LP-relaxation of a subproblem is infeasible then the corre-
+sponding subproblem is infeasible as well, since the feasible set of the subproblem is a subset
+of the feasible set of its relaxation.
+ii) Fathoming by optimality: Similar to the single-objective case we can fathom a node by opti-
+mality if the lower bound L is equal to the upper bound U. This implies the subproblem is
+solved to optimality and the associated node must not be subdiveded further. However, this
+can happen in the multi-objective case only if the lower and upper bound consist of the same
+single point, namely the ideal point.
+iii) Fathoming by dominance: A node can be fathomed by dominance if all feasible solutions of this
+subproblem are dominated by points in the incumbent list. In order to check dominance for all
+feasible outcome vectors of a subproblem we compare the lower bound L of the corresponding
+node to the current upper bound U. If for all l ∈ L there is a point in the incumbent list u ∈ U
+with u ≦ l then all feasible points in the subtree are dominated by the current incumbent list.
+In other words, if there is no local upper bound defined by U above the computed lower bound
+the node can be fathomed by dominance.
+Branching:
+As mentioned in the beginning of this section, one of the key aspects of Branch and
+Bound is iterative subdivision into smaller subproblems. Thereby subproblems are associated with
+nodes in a tree, such that the subproblem associated to a child node is obtained by one branching
+step. Since we consider binary optimization problems (BO01LP), we can divide a (sub)problem into
+two new subproblems by fixing a specific variable to 0 and respectively to 1 in the other subproblem.
+This results in a binary Branch and Bound tree.
+The branching rule determines which variable is selected as branching variable in each iteration.
+Thereby, one distinguishes between static and dynamic strategies. Static strategies determine an
+order of the variables in advance. In each iteration of the algorithm the next variable in this list is
+used as branching variable. With dynamic strategies the branching variable is selected by consid-
+ering information obtained from previous iterations, i.e., from the solution of (linear) relaxations of
+(sub)problems.
+The basic idea of static strategies for single-objective problems is to sort the variables, beginning
+with the most promising according to the objective function values (see, e. g., (Kellerer et al., 2004)).
+However, this cannot be easily extended to the multi-objective case due to conflicting objective
+functions. Nevertheless there are some approaches to extend static strategies to the multi-objective
+case (see for example (Ulungu and Teghem, 1997) and (Bazgan et al., 2009)).
+6
+
+In contrast to most of the published papers which apply static strategies we use a dynamic strategy
+as proposed in Belotti et al. (2012). By solving the linear relaxation of a (sub)problem we obtain
+the lower bound set L. For all extreme points of L we check how often a variable is fractional in the
+corresponding solutions. As branching variable we choose the one which is most often fractional.
+4 Using Objective Space Information in Multi-objective Branch
+and Bound
+In this section, we propose modifications which improve the computational efficiency of bi-objective
+Branch and Bound algorithms in two critical aspects.
+One of the weaknesses of multi-objective
+Branch and Bound as compared to its single-objective counterpart is the bounding procedure. While
+any feasible solution ¯x ∈ X dominates w.r.t. one (linear) objective a half-space in decision space (i.e.,
+{x ∈ Rn : c⊤x ≥ c⊤¯x}), the set of feasible solutions which are dominated by a solution ¯x in p ≥ 2
+objective functions (C ∈ Rp×n) forms a cone {x ∈ Rn : C x ≧ C ¯x}. The cone of dominated solutions
+is smaller the more the objective functions are in conflict, leading also to a larger number of efficient
+solutions. This implies that a significant part of the Branch and Bound tree has to be enumerated
+and only a small number of branches can be pruned by dominance. Despite of this general problem
+in multi-objective optimization, this asks for good bounding procedures to avoid the unnecessary
+evaluation of dominated branches. This however, requires good solutions in the incumbent list as
+well as tight lower bounds.
+In order to achieve this, we suggest a new branching strategy and the hybridization of Branch and
+Bound with objective space methods. We determine scalarized subproblems adapted to the state of
+the Branch and Bound and solve these to integer optimality.
+4.1 Branching Strategy
+The branching strategy comprises two subsequent decisions: the choice of the active node and its
+branching into subproblems, i. e. the decision on which variable the subproblem is branched. This
+second step is denoted as branching rule. We discuss these two steps together since the order in
+which the nodes are considered has a significant impact on the branched variable. Instead of the
+static depth- or breadth-first we use a dynamic node selection strategy, while we rely on the most
+fractional rule as branching rule.
+The basic idea of our strategy is quite simple and a natural extension of choosing the largest gap
+in the single-objective case (see, for example, Dechter and Pearl, 1985). For every created node we
+compute the approximate hypervolume gap between lower and upper bound. We use the definition of
+hypervolume proposed in Zitzler and Thiele (1999). In every iteration we choose the node with the
+largest hypervolume gap as active node (cf. Jesus et al. (2021)). Note that when a node is created
+during the branching process, the approximated hypervolume gap of the parent node is assigned to
+it. We distinguish two variants of the hypervolume gap: the total hypervolume gap and the local
+hypervolume gap. While the total hypervolume gap measures the volume of the search region, i. e.
+the volume between lower and upper bound set, the local hypervolume gap approach considers only
+the volume of the largest search zone, i. e. the gap between a local upper bound and the lower bound
+set. For a more detailed definition of search regions and search zones we refer to Klamroth et al.
+(2015).
+Figure 1 illustrates the two different approaches. Here, z1, . . . , z4 ∈ K ⊂ U are points of the
+incumbent list and lu1, . . . , lu3 are their corresponding local upper bounds, where K is a subset
+of the incumbent list containing just the points above the lower bound of node ¯n. The green line
+represents the lower bound. Figure 1a shows how to measure the total hypervolume gap of a node ¯n,
+in the following denoted by thg(¯n). For this approach we consider the approximated search region
+of the corresponding node. Since there is a natural order in the bi-objective case, it is possible to
+consider the approximated search zone of the first local upper bound, i.e. the local upper bound
+7
+
+z1
+z2
+lu1
+lu2
+lu3
+z1
+z2
+z3
+z4
+(a)
+z1
+z2
+A
+lu1
+lu2
+lu3
+z1
+z2
+z3
+z4
+(b)
+z1
+z2
+B
+lu1
+lu2
+lu3
+z1
+z2
+z3
+z4
+(c)
+z1
+z2
+C
+lu1
+lu2
+lu3
+z1
+z2
+z3
+z4
+(d)
+Figure 1: Example of computation of the two different approximated hypervolume gap approaches.
+8
+
+with the smallest z1-value. Therefore we define the two spanning points, which, together with the
+corresponding local upper bound, define a triangle. The spanning points of a local upper bound lu
+are defined by spi(lu) := {l ∈ L: l3−i = lu3−i}, i = 1, 2. So, the approximate hypervolume gap of lu
+is given by
+hg(lu) := 1
+2
+��sp1(lu)1 − lu1
+�� ·
+��sp2(lu)2 − lu2
+��.
+For the remaining local upper bounds we compute the hypervolume of slices as shown in the illus-
+tration. The hypervolume of the slice of lui, i = 1, . . . , |K| − 1 is defined as
+sl(lui) :=
+��zi
+2 − sp2(lui−1)2
+�� +
+��lui
+2 − sp2(lui)2
+��
+2
+·
+��zi
+1 − lui
+1
+��.
+So, the total (approximated) hypervolume gap, which is assigned to node ¯n, is given by
+thg(¯n) := hg(lu1) + sl(lu2) + . . . + sl(lu|K|−1).
+The Figures 1b, 1c and 1d show the computation of the local hypervolume gap. The local hyper-
+volume gap of a node ¯n is considered as the largest approximated hypervolume gap of a local upper
+bound corresponding to points in K. Therefore, the the local hypervolume gap, which is assigned
+to node ¯n, is defined by
+lhg(¯n) :=
+max
+i=1,...,|K|−1hg(lui).
+In the given example, B is the largest approximated hypervolume and therefore is assigned to node
+¯n.
+Note that in our presented algorithms in Section 4.2 the local upper bound is initialized with the
+point (∞, ∞)⊤. Therefore, it is possible to apply the new branching strategies immediately at the
+beginning of the algorithm. Obviously, this approximation may neglect significantly large parts of
+the search regions and search zones. However, the idea of the approximated hypervolume gap eases
+computation and saves time. The efficiency of these new dynamic branching strategies is shown in
+Section 5.
+4.2 Augmenting Branch and Bound with IP Scalarizations
+In this subsection, we introduce a method to incorporate scalarizations into Branch and Bound. We
+build a hybrid Branch and Bound algorithm combining the partial enumeration of decision space
+with objective space information by solving scalarizations to integer optimality.
+An integer optimal solution ¯x of a scalarization can be used to update upper and lower bound.
+Obviously, the corresponding image point z(¯x) can be added to the incumbent list. Moreover, a
+scalarizing function and its optimal solution ¯x define a level set, which can be included in the lower
+bound set for all descendant nodes. In order to utilize these improved lower bounds in all nodes we
+solve the IP scalarizations in the root node.
+4.2.1 Using Weighted Sum Scalarization
+During the run of the Branch and Bound algorithm, a strategy triggers the IP solution of weighted
+sum scalarizations in the root node. Thus, we solve problem (WSλ) for for adaptively chosen values
+of λ ∈ R2
+>. Although we solve the IP scalarization in the root node the parameter λ is gained from
+the currently active node. Thereby, λ is determined by the largest approximated local hypervolume
+gap in the active node. This gap is spanned by two points in the incumbent list together with their
+local upper bound. Note that these points spanning the largest gap are already determined if the
+local hypervolume gap branching strategy is applied. The corresponding value of λ is determined by
+computing the normal to the hyperplane that is defined by those two points. Once λ is obtained,
+we can solve problem (WSλ) with a single-objective integer linear programming solver. Let ¯xλ be
+9
+
+the optimal solution of the weighted sum scalarization with weighting vector λ, then z(¯xλ) is a
+supported non-dominated point of (BO01LP). Thus, we can add this point to the incumbent list (if
+it was not found in previous iterations) and filter the resulting list for non-dominance. Moreover,
+the solution of integer scalarizations can also be used to tighten the lower bound set, since the level
+set {z ∈ R2 : λ⊤z = WSλ(¯xλ)} provides the valid inequality λ⊤z(x) ≥ WSλ(¯xλ) for all x ∈ X.
+z1
+z2
+lu1
+lu2
+lu3
+z1
+z2
+z3
+z4
+z(¯xλ)
+(a)
+z1
+z2
+lu4
+lu5
+z1
+z5
+z4
+(b)
+z1
+z2
+lu4
+lu5
+z1
+z5
+z4
+(c)
+z1
+z2
+lu4
+lu5
+z1
+z5
+z4
+(d)
+Figure 2: Example of updating the lower and upper bound with the usage of the weighted sum
+scalarization.
+Figure 2 illustrates the update of the lower and upper bound set. In Figure 2a, z1, . . . , z4 indicate
+points that are currently in the incumbent list U¸ and lu1, . . . , lu3 are the corresponding local upper
+bounds.
+The point z(¯xλ) is obtained by solving a weighted sum scalarization (WSλ) to integer
+optimality. Since the new point is not contained in the incumbent list so far, we can update the
+upper bound as it is shown in Figure 2b. The new incumbent list then reads as U := {z(¯xλ)} ∪ {z ∈
+U : z(¯xλ) � z}. Moreover, the lower bound set L can be updated by integrating the blue hyperplane
+into the lower bound set, i. e. L := {z ∈ L + R2
+≧ : λ⊤z ≥ WS(¯xλ)}N as it is shown in Figure 2c and
+2d. In this situation, both —the lower and upper bound— are updated, which is not the case in
+10
+
+general.
+The example illustrates the benefits of hybridizing multi-objective Branch and Bound with IP
+scalarizations. Due to weak bounding, nodes may not be fathomed by dominance even if they do
+not contain additional non-dominated points. The tighter upper bound increases the probability of
+fathoming a node by dominance in later iterations of the algorithm. Also, the lower bound might be
+improved. Since we are solving an IP scalarization in the root node, the obtained optimal level set is
+a valid inequality for all subproblems. We combine our new branching strategy and the augmentation
+with IP scalarizations to our first hybrid Branch and Bound approach.
+Hybrid Branch and Bound Algorithm using Weighted Sum Scalarization
+• Lower bound: linear relaxation
+• Upper bound: incumbent list
+• Node selection: node with the largest total/local hypervolume gap
+• Branching rule: most fractional
+• Adaptively solve weighted sum scalarizations in the root node to integer optimality to improve
+lower and upper bounds by objective space information
+Instead of using a static depth-first strategy (as in the general Branch and Bound framework in
+Section 3) we apply the dynamic strategy based on the hypervolume gap (c.f. Section 4.1). Even
+though the extreme points of the lower bound sets might be updated by the weighted sum scalar-
+ization, the branching variable is selected based on the original lower bounds. This is due to the
+fact that the preimages of such intersection points of IP scalarizations and the lower bound set
+are in general not available. Note that the weighted sum IP scalarizations are included adaptively
+into the Branch and Bound. The description of their algorithmic control, however, is postponed to
+Section 4.3.
+In order to conclude the description of the proposed hybrid Branch and Bound algorithm using
+weighted sum scalarizations, we want to briefly discuss its advantages and shortcomings. Firstly, it
+is easy to determine the scalarization parameter λ and to integrate the hyperplane into the lower
+bound set. Its advantage, however, is that the lower bound remains convex. Therefore, the check
+for fathoming by dominance remains intuitive. Unfortunately, the weighted sum scalarization can
+only find supported efficient solutions and the lower bound cannot be improved beyond the convex
+hull of YN. This motivates us to consider in the following the augmented weighted Tchebycheff
+scalarization, a scalarization approach which can determine also unsupported efficient solutions.
+4.2.2 Using Augmented Weighted Tchebycheff Scalarization
+We start by defining the weighted Tchebycheff norm: Let wi > 0, i = 1, . . . , p be positive weights
+with �p
+i=1 wi = 1. Then the weighted Tchebycheff norm of a vector z ∈ Rp is defined by
+∥z∥w
+∞ := max
+i=1,...,p
+�
+wi |zi|
+�
+.
+(1)
+So, the weighted Tchebycheff scalarization of a multi-objective optimization problem (MOILP) with
+respect to a given reference point s ∈ Rp can be written as:
+min
+�
+∥z(x) − s∥w
+∞ : x ∈ X
+�
+.
+(2)
+If the reference point is chosen such that s < z(x) for all x ∈ X, every efficient solution can be deter-
+mined as optimal solution of the weighted Tchebycheff scalarization (2) by variation of w ∈ Rp
++ (see,
+e.g., Miettinen, 1998). Nevertheless, optimal solutions of the weighted Tchebycheff scalarization cor-
+respond in general only to weakly efficient solutions of the multi-objective problem (Steuer and Choo,
+1983; Miettinen, 1998). This shortcoming is compensated by an additive augmentation term in the
+augmented weighted Tchebycheff norm
+∥z∥w
+τ := ���z∥w
+∞ + τ ∥z∥1 ,
+(3)
+11
+
+where ∥z∥1 = |z1| + . . . + |zp| denotes the L1-norm, wi ≥ 0, i = 1, . . . , p, �p
+i=1 wi = 1 and τ >
+0.
+Steuer and Choo (1983) proposed the augmented weighted Tchebycheff scalarization given in
+(AWT w
+τ ).
+min AWT w
+τ (x) := ∥z(x) − s∥w
+τ
+s.t. x ∈ X
+(AWT w
+τ )
+Thereby, the augmentation term makes the augmented weighted Tchebycheff norm a strongly mono-
+tone norm and thus the objective function of (AWT w
+τ ) a strongly increasing achievement scalar-
+izing function (Miettinen, 1998). Consequently, every optimal solution of (AWT w
+τ ) is efficient for
+(MOILP).
+Note that an appropriate choice of the parameter τ is difficult in general. On the one hand, too
+small values of τ may lead to numerical difficulties. On the other hand, non-supported efficient
+solutions might be suboptimal for (AWT w
+τ ) if the value of τ is too large. However, for bi-objective
+integer programming Dächert et al. (2012) propose an adaptive method to determine an optimal
+value of τ. We use this proposed parameters w1, w2 and τ for our method.
+As a reference point s we use the local ideal point of two adjacent non-dominated points. Since the
+augmented weighted Tchebycheff scalarization can only determine non-dominated points (and the
+corresponding efficient solutions) which are (strictly) dominated by the reference point, we obtain a
+non-dominated point in this box.
+The goal to improve the lower bound set beyond the convex hull of non-dominated points is the
+motivation to solve augmented weighted Tchebycheff scalarizations to integer optimality. Figure
+3 shows an example how such an update of the bounds could look like. Here, z1 and z2 are two
+known non-dominated points (obtained with the weighted sum IP scalarization).
+Point z3 is a
+non-supported non-dominated point that has not been found yet in Figure 3a. By using the local
+ideal point of z1 and z2 as the reference point s, Figure 3b illustrates how the non-dominated point
+z3 is found by applying the augmented weighted Tchebycheff scalarization. In Figure 3c and 3d
+the resulting improvements of the lower and upper bound are shown. Obviously the lower bound
+is improved beyond the convex hull of YN. We now define our second hybrid Branch and Bound
+approach:
+Hybrid Branch and Bound Algorithm using augmented weighted Tchebycheff Scalarization
+• Lower bound: linear relaxation
+• Upper bound: incumbent list
+• Node selection: node with the biggest total/local hypervolume gap
+• Branching rule: most fractional
+• Adaptively solve weighted sum and augmented weighted Tchebycheff scalarizations in the root
+node to integer optimality to improve lower and upper bounds by objective space information
+Additionally to the weighted sum scalarization, we use the augmented weighted Tchebycheff scalar-
+ization. Since two adjacent non-dominated points are required as input of the augmented weighted
+Tchebycheff scalarization, we cannot rely on points in the incumbent list, which are only non-
+dominated so far. In fact, we apply augmented weighted Tchebycheff IP scalarizations only to boxes
+spanned by points obtained as optimal solutions of the weighted sum scalarization. Thus, we do not
+rely on parameters from the currently active node, but solve the augmented weighted Tchebycheff
+scalarization in the largest area defined by two adjacent known non-dominated points.
+When using augmented weighted Tchebycheff IP scalarizations, the lower bound can become
+tighter than the convex hull of the set of non-dominated points, which reduces the area where
+new non-dominated points can be found. Additionally, we can find non-supported non-dominated
+points in early stages of the algorithm. This improves the upper bound in the beginning resulting
+in a higher chance of fathoming a node by dominance. However, this also implies that the lower
+12
+
+z1
+z2
+z1
+z2
+z3
+lu1
+(a)
+z1
+z2
+z1
+z2
+z3
+lu1
+s
+(b)
+z1
+z2
+z1
+z2
+z3
+lu2
+lu3
+s
+(c)
+z1
+z2
+z1
+z2
+z3
+lu2
+lu3
+(d)
+Figure 3: Example of updating the lower and upper bound with the usage of the augmented weighted
+Tchebycheff scalarization.
+bound gets non-convex in general, which makes the fathoming tests significantly harder, and the
+lower bound improves only locally.
+4.3 Algorithmic Control of IP Scalarizations
+In the previous subsections we did not specify when to solve IP scalarizations, which implies a sig-
+nificant computational cost itself. However, this might be the most crucial part within the presented
+methods. Obviously, we aim at gaining as much information as possible by solving IP scalarizations.
+More objective space information will lead to tighter bounds that reduce the number of created
+nodes, due to a higher probability of fathoming by dominance and smaller search zones. Moreover, a
+reduced number of created nodes will reduce the total computation time. At the same time, solving
+overly many IP scalarizations will have a negative impact on the computation time. Furthermore,
+at a certain point the lower and upper bound will not improve anymore when solving additional IP
+13
+
+scalarizations.
+So, there exists a trade-off between the reduction of the number of created subproblems and the
+decrease of the computation time. The difficulty is to find an appropriate condition to trigger an IP
+scalarization. Obviously, solving IP scalarizations more frequently in the beginning of the Branch
+and Bound algorithm is very promising.
+The earlier the lower and upper bounds are improved
+the more nodes might be fathomed. Moreover, solving the IP scalarization when the active node
+has weak bounds will lead to stronger improvements than in later stages of the algorithm. This
+is complemented by our adaptive branching strategy, which tends to select subproblems with weak
+lower bounds first.
+The hybrid Branch and Bound algorithm using augmented weighted Tchebycheff scalarization
+entails also another problem. The augmented weighted Tchebycheff scalarization improves the lower
+bound just locally. If we use this scalarization at the beginning of the algorithm instead of the
+weighted sum scalarization, this could lead to an increase of created nodes. Once again, the intuitive
+idea is to start with the weighted sum IP scalarization more frequently in the beginning of the
+algorithm. This ensures that the lower bound improves globally at early stages of the Branch and
+Bound. The augmented weighted Tchebycheff scalarization should be used in later stages of the
+algorithm to find non-supported non-dominated points and to improve the lower bound locally. The
+efficiency of this idea and other approaches will be shown in the next section where we present
+numerical test results.
+5 Numerical Results
+All algorithms were implemented in Julia 1.7.1 and the linear relaxations were solved with Bensolve
+2.1. The numerical tests were executed on a single core of a 3.20 GHz Intel® Core™ i7-8700 CPU
+processor in a computer with 32 GB RAM, running under openSUSE linux Leap 15.3.
+We present numerical results of our new approaches and compare them to the general Branch
+and Bound framework presented in Section 3 which we use as baseline implementation. We consider
+three different types of problems: knapsack problems, assignment problems and discrete facility
+location problems. Multiple combinations of parameter settings are used to solve these test problems.
+Thereby, we compare the average number of explored nodes, the average number of solved IPs and
+the average computation time for 20 instances per problem size. The different evaluated approaches
+are
+• the generic bi-objective Branch and Bouch (BB),
+• bi-objective Branch and Bound using the local (BS1) respectively global (BS2) hypervolume
+gap as node selection criterion,
+• hybrid Branch and Bound including weighted sum IP scalarizations (WS), and
+• different combinations of the hybrid Branch and Bound algorithm using weighted sum IP scalar-
+ization (M1.α.β) and hybrid Branch and Bound algorithm using weighted sum and augmented
+weighted Tchebycheff IP scalarization (M2.α.β.γ).
+The parameter α ∈ {1, 2, 3} controls how often IP scalarizations are applied. Since the algorithmic
+control of IP scalarization is selected for each value of α problem dependent, it is described in
+detail in the following subsections. However, the larger the parameter α is chosen, the fewer IP
+scalarizations are applied. With β we distinguish between the local (β = 1) and the global (β = 2)
+hypervolume gap strategy. In the hybrid Branch and Bound algorithm using augmented weighted
+Tchebycheff scalarization we also distinguish between integrating the objective space information
+of the augmented weighted Tchebycheff into the lower bound (γ = 1) or not (γ = 2). Note that
+the tested parameter values are not optimized but have shown to provide good results on our test
+instances.
+14
+
+5.1 Bi-objective Multidimensional Knapsack Problems
+We consider bi-objective, multidimensional knapsack problems with one, two and three linear re-
+strictions (i. e. m = 1, 2, 3). For every problem size we randomly generate 20 instances of the form
+max
+n
+�
+i=1
+ck
+i xi
+k = 1, 2
+s.t.
+n
+�
+i=1
+wi xi ≤ b
+n
+�
+i=1
+vij xi ≤ dj
+j = 1, ..., m − 1
+x ∈ {0, 1}n
+with ck
+i ∈ [50, 100], wi ∈ [5, 15], b = 5 n, vij ∈ [5, 15] and dj =
+� r n
+2
+�
+with r ∈ [5, 15]. Depending on
+the parameter α we specify when and how often IP scalarizations are solved. In M1.1.β and WS
+we apply the weighted sum scalarization every 10-th iterations. In M1.2.β we apply it every 10-th
+iteration but only within the first n2 iterations. In M1.3.β we apply the weighted sum scalarization
+every 10-th iteration within the first n2/3 iterations, every n-th iteration within the next n2/3
+iterations and every 2n-th iteration within the third n2/3 iterations. In M2.1.β.γ we apply the
+weighted sum scalarization every 10-th iteration and every 50-th iteration the augmented weighted
+Tchebycheff scalarization is used instead. In M2.2.β.γ we operate like in M1.2.β but after the first
+n2 iterations we apply the augmented weighted Tchebycheff scalarization every 50-th iteration. In
+M2.3.β.γ we operate like in M1.3.β but after the first n2 iterations we apply the augmented weighted
+Tchebycheff scalarization every 50-th iteration. If a scalarization cannot be applied or the same IP
+scalarization has already been solved before, no IP scalarization is solved in that iteration.
+First of all, we notice that our branching strategies have a huge impact on the number of explored
+nodes and the computation time in knapsack problems. We observe that in general the local hyper-
+volume gap strategy works better than the global hypervolume gap strategy. With the local strategy
+we can reduce the number of explored nodes by up to 76% (Table 1c and 2b) and the computation
+time by up to 73% (Table 1c). Although the local strategy works better the global hypervolume
+gap strategy has also a significant impact. The number of explored nodes can be reduced by up to
+58% (Table 2c) and the computation time by up to 52% (Table 2c). The number of nodes and the
+computation time is reduced in all our approaches and we can notice that combinations with the
+local hypervolume strategy work better.
+By limiting the number of solved weighted sum IPs (i. e. in M1.2.β, M1.3.β, M2.2.β.γ and
+M2.3.β.γ) we notice two consequences. The number of nodes increases while the number of solved
+IPs decreases. Although the number of nodes (and thus the number of considered subproblems) is
+increasing, the total computation time decreases. This implies that the reduced computation time
+to solve IP scalarizations compensates the increase of nodes, which results in a trade-off between the
+number of explored nodes and the computation time. Another interesting aspect can be observed in
+M2.α.β.1 and M2.α.β.2. The computation time can be reduced if we do not integrate the augmented
+weighted Tchebycheff objective level set into the lower bound. This can be explained by the fact that
+the lower bound improvements of augmented weighted Tchebycheff are only local and do not com-
+pensate the computation time needed to integrate the information. The intuitive assumption that
+the number of explored nodes will then rise significantly is false. So, both our branching strategies
+work better, if we do not consider the local updates of the lower bound.
+We can reach a reduction of the explored nodes by up to 83% (Table 2b) and a reduction of the
+computation time by up to 80% (Table 2b) in the best case. The strategies M2.1.1.1 and M2.1.1.2
+seem to work best for knapsack problems. In most cases these two strategies have the largest impact
+on the number of explored nodes. Nevertheless, M2.1.1.2 achieves for all instance sizes the best
+computation times, since computation time is saved by not integrating the augmented weighted
+15
+
+knapsack problem, m = 1, n = 50
+version
+nodes
+time (s)
+solved
+IPs
+BB
+27916.3
+18.153
+0.0
+BS1
+11788.1
+8.339
+0.0
+WS
+14270.7
+10.507
+33.75
+M1.1.1
+10789.7
+8.452
+26.4
+M1.2.1
+10793.5
+8.188
+21.2
+M1.3.1
+10795.7
+8.116
+17.95
+M2.1.1.1
+9888.5
+10.873
+48.7
+M2.2.1.1
+10140.3
+9.437
+32.65
+M2.3.1.1
+10521.0
+8.774
+25.65
+M2.1.1.2
+9840.1
+8.396
+45.55
+M2.2.1.2
+10130.6
+8.422
+32.35
+M2.3.1.2
+10401.8
+8.288
+26.25
+BS2
+16739.8
+11.397
+0.0
+M1.1.2
+11026.3
+8.861
+26.1
+M1.2.2
+11024.5
+8.860
+19.85
+M1.3.2
+11047.4
+8.645
+16.05
+M2.1.2.1
+10071.8
+10.907
+45.85
+M2.2.2.1
+10421.4
+9.587
+31.8
+M2.3.2.1
+10583.2
+9.448
+24.45
+M2.1.2.2
+9994.1
+8.940
+46.65
+M2.2.2.2
+10413.4
+8.820
+32.55
+M2.3.2.2
+10568.9
+8.727
+25.15
+(a) Knapsack problem with m = 1 constraint
+and n = 50 variables
+knapsack problem, m = 1, n = 80
+version
+nodes
+time (s)
+solved
+IPs
+BB
+153938.9
+186.330
+0.0
+BS1
+36392.0
+50.952
+0.0
+WS
+58825.7
+79.545
+54.0
+M1.1.1
+34337.7
+50.431
+41.65
+M1.2.1
+34333.9
+50.312
+33.1
+M1.3.1
+34307.1
+50.505
+26.35
+M2.1.1.1
+31643.7
+81.625
+100.2
+M2.2.1.1
+32708.9
+68.939
+76.2
+M2.3.1.1
+32986.3
+69.848
+63.6
+M2.1.1.2
+31274.5
+46.721
+102.85
+M2.2.1.2
+32795.7
+48.576
+76.3
+M2.3.1.2
+33025.8
+48.358
+63.4
+BS2
+90976.0
+116.847
+0.0
+M1.1.2
+39745.1
+59.321
+45.25
+M1.2.2
+40083.2
+59.350
+31.2
+M1.3.2
+39918.1
+58.999
+24.5
+M2.1.2.1
+31905.8
+80.505
+99.7
+M2.2.2.1
+34496.9
+79.444
+84.0
+M2.3.2.1
+34571.7
+72.955
+65.15
+M2.1.2.2
+32074.9
+48.510
+104.85
+M2.2.2.2
+34169.8
+51.464
+87.15
+M2.3.2.2
+34943.3
+51.887
+63.1
+(b) Knapsack problem with m = 1 constraint
+and n = 80 variables
+knapsack problem, m = 1, n = 100
+version
+nodes
+time (s)
+solved
+IPs
+BB
+297345.3
+484.676
+0.0
+BS1
+68920.5
+128.967
+0.0
+WS
+128080.8
+224.587
+66.95
+M1.1.1
+67369.1
+128.665
+54.1
+M1.2.1
+67370.1
+128.924
+39.95
+M1.3.1
+67353.3
+128.993
+32.9
+M2.1.1.1
+58214.2
+198.683
+156.85
+M2.2.1.1
+61533.1
+179.516
+123.0
+M2.3.1.1
+62127.3
+177.621
+104.55
+M2.1.1.2
+58151.3
+112.575
+158.1
+M2.2.1.2
+61490.6
+118.600
+120.1
+M2.3.1.2
+61762.6
+118.306
+108.65
+BS2
+187306.9
+318.524
+0.0
+M1.1.2
+73766.2
+144.684
+54.75
+M1.2.2
+74065.4
+144.677
+37.9
+M1.3.2
+73865.0
+144.306
+31.0
+M2.1.2.1
+59512.7
+200.754
+158.05
+M2.2.2.1
+64489.2
+192.211
+127.0
+M2.3.2.1
+64330.5
+187.803
+114.65
+M2.1.2.2
+60470.8
+118.479
+157.75
+M2.2.2.2
+64943.8
+127.428
+123.5
+M2.3.2.2
+64711.1
+126.525
+113.5
+(c) Knapsack problem with m = 1 constraint
+and n = 100 variables
+knapsack problem, m = 2, n = 50
+version
+nodes
+time (s)
+solved
+IPs
+BB
+32655.6
+25.8684
+0.0
+BS1
+10982.3
+9.6578
+0.0
+WS
+14180.9
+13.1749
+33.25
+M1.1.1
+9784.7
+9.5159
+26.6
+M1.2.1
+9782.5
+9.2684
+19.65
+M1.3.1
+9791.1
+9.1580
+14.85
+M2.1.1.1
+8900.7
+12.6639
+47.75
+M2.2.1.1
+9407.5
+11.5112
+34.5
+M2.3.1.1
+9507.5
+11.0256
+24.8
+M2.1.1.2
+8892.9
+9.2702
+47.0
+M2.2.1.2
+9370.2
+9.2053
+33.05
+M2.3.1.2
+9484.3
+9.1161
+24.75
+BS2
+15639.2
+13.1246
+0.0
+M1.1.2
+10665.3
+10.8423
+28.5
+M1.2.2
+10671.3
+10.5141
+19.4
+M1.3.2
+10854.2
+10.5765
+15.7
+M2.1.2.1
+9045.3
+12.8916
+48.9
+M2.2.2.1
+9629.7
+12.1913
+34.9
+M2.3.2.1
+9814.7
+11.6068
+28.1
+M2.1.2.2
+9030.0
+9.5854
+48.2
+M2.2.2.2
+9608.5
+9.7621
+36.05
+M2.3.2.2
+9787.6
+9.6722
+28.35
+(d) Knapsack problem with m = 2 constraints
+and n = 50 variables
+Table 1: Numerical results of the bi-objective, multidimensional knapsack problems
+16
+
+knapsack problem, m = 2, n = 80
+version
+nodes
+time (s)
+solved
+IPs
+BB
+159911.4
+287.925
+0.0
+BS1
+41092.0
+88.121
+0.0
+WS
+63215.0
+130.338
+55.0
+M1.1.1
+37799.1
+82.654
+43.9
+M1.2.1
+37835.8
+82.544
+30.55
+M1.3.1
+37811.3
+82.369
+24.75
+M2.1.1.1
+31615.0
+115.164
+102.55
+M2.2.1.1
+34772.5
+102.965
+72.5
+M2.3.1.1
+35127.1
+100.706
+60.6
+M2.1.1.2
+31590.2
+69.290
+104.7
+M2.2.1.2
+34977.7
+77.063
+72.45
+M2.3.1.2
+35170.3
+77.279
+61.3
+BS2
+115223.3
+224.926
+0.0
+M1.1.2
+43581.8
+97.039
+47.8
+M1.2.2
+43744.8
+97.874
+29.1
+M1.3.2
+45481.1
+102.689
+23.45
+M2.1.2.1
+32388.8
+116.173
+106.25
+M2.2.2.1
+36453.2
+120.161
+78.3
+M2.3.2.1
+35942.7
+116.972
+69.05
+M2.1.2.2
+33264.8
+74.207
+104.75
+M2.2.2.2
+36578.1
+81.915
+77.2
+M2.3.2.2
+35505.1
+77.971
+69.55
+(a) Knapsack problem with m = 2 constraints
+and n = 80 variables
+knapsack problem, m = 2, n = 100
+version
+nodes
+time (s)
+solved
+IPs
+BB
+428526.3
+1074.21
+0.0
+BS1
+100962.6
+326.98
+0.0
+WS
+166108.1
+464.71
+67.25
+M1.1.1
+98831.5
+323.54
+54.8
+M1.2.1
+99313.5
+325.32
+38.95
+M1.3.1
+98770.6
+322.65
+32.6
+M2.1.1.1
+69951.9
+402.48
+149.35
+M2.2.1.1
+73433.8
+379.33
+119.6
+M2.3.1.1
+73424.7
+371.63
+102.95
+M2.1.1.2
+70172.3
+212.40
+153.15
+M2.2.1.2
+72824.8
+219.73
+121.55
+M2.3.1.2
+73651.5
+221.96
+107.5
+BS2
+271110.8
+720.46
+0.0
+M1.1.2
+113605.9
+381.46
+57.45
+M1.2.2
+117188.3
+394.57
+36.45
+M1.3.2
+113665.3
+378.63
+29.85
+M2.1.2.1
+70603.0
+404.28
+150.8
+M2.2.2.1
+77836.0
+400.18
+121.4
+M2.3.2.1
+76818.2
+399.89
+110.8
+M2.1.2.2
+72316.9
+219.91
+148.4
+M2.2.2.2
+78135.2
+240.57
+121.3
+M2.3.2.2
+77073.0
+235.49
+112.45
+(b) Knapsack problem with m = 2 constraints
+and n = 100 variables
+knapsack problem, m = 3, n = 50
+version
+nodes
+time (s)
+solved
+IPs
+BB
+54430.4
+51.5026
+0.0
+BS1
+15260.1
+17.9276
+0.0
+WS
+18112.9
+20.5208
+36.2
+M1.1.1
+13522.4
+16.8431
+28.8
+M1.2.1
+13530.7
+16.4735
+19.0
+M1.3.1
+13576.5
+16.4442
+14.95
+M2.1.1.1
+12345.3
+22.1289
+51.15
+M2.2.1.1
+12973.5
+19.7592
+34.5
+M2.3.1.1
+13014.0
+19.6348
+28.8
+M2.1.1.2
+12241.1
+16.1190
+53.55
+M2.2.1.2
+12934.3
+16.2933
+35.15
+M2.3.1.2
+12908.3
+16.1009
+30.35
+BS2
+22597.3
+24.5736
+0.0
+M1.1.2
+14645.7
+18.6425
+29.55
+M1.2.2
+14573.2
+18.0521
+16.75
+M1.3.2
+14597.0
+17.9793
+14.5
+M2.1.2.1
+12617.9
+22.3518
+54.65
+M2.2.2.1
+13324.9
+20.7655
+33.4
+M2.3.2.1
+13252.3
+20.4366
+30.55
+M2.1.2.2
+12682.4
+16.8670
+56.65
+M2.2.2.2
+13180.2
+16.7601
+33.6
+M2.3.2.2
+13274.4
+16.8497
+32.15
+(c) Knapsack problem with m = 3 constraints
+and n = 50 variables
+knapsack problem, m = 3, n = 80
+version
+nodes
+time (s)
+solved
+IPs
+BB
+263971.6
+724.999
+0.0
+BS1
+81609.9
+287.899
+0.0
+WS
+121360.8
+376.247
+56.4
+M1.1.1
+80406.5
+282.897
+47.35
+M1.2.1
+79971.5
+279.885
+32.2
+M1.3.1
+80089.6
+279.686
+26.55
+M2.1.1.1
+54187.1
+340.389
+115.7
+M2.2.1.1
+55915.9
+328.411
+85.45
+M2.3.1.1
+58486.8
+330.347
+66.9
+M2.1.1.2
+53396.0
+164.755
+116.0
+M2.2.1.2
+56572.6
+174.131
+86.25
+M2.3.1.2
+57452.9
+176.657
+67.85
+BS2
+140681.4
+390.578
+0.0
+M1.1.2
+92175.8
+334.414
+50.45
+M1.2.2
+96339.3
+350.148
+29.05
+M1.3.2
+96099.5
+348.915
+24.0
+M2.1.2.1
+54119.5
+349.261
+112.5
+M2.2.2.1
+59379.8
+344.899
+89.1
+M2.3.2.1
+60326.6
+349.874
+74.15
+M2.1.2.2
+54595.6
+176.090
+119.65
+M2.2.2.2
+62851.9
+211.373
+88.9
+M2.3.2.2
+61200.8
+205.413
+76.6
+(d) Knapsack problem with m = 3 constraints
+and n = 80 variables
+Table 2: Numerical results of the bi-objective, multidimensional knapsack problems
+17
+
+Tchebycheff objective space information into the lower bound. Note that with rising numbers of
+variables and constraints the hybridization techniques have larger impact on the performance of the
+Branch and Bound algorithm.
+5.2 Bi-objective Assignment Problems
+We consider bi-objective assignment problems having n = ℓ2 variables,
+max
+ℓ
+�
+i=1
+ℓ
+�
+j=1
+ck
+ij xij
+k = 1, 2
+s.t.
+ℓ
+�
+i=1
+xij = 1
+j = 1, ..., ℓ
+ℓ
+�
+j=1
+xij = 1
+i = 1, ..., ℓ
+x ∈ {0, 1}ℓ×ℓ
+where the cost coefficients ck
+ij ∈ [50, 100]. The algorithmic strategy for the solution of IP scalariza-
+tions depending on the value of the parameter α is chosen similarly to the previous case of knapsack
+problems. However, we adapt the boundaries due to the different number of nodes to explore in as-
+signment problems. In M1.1.β weighted sum scalarizations are solved every 10-th iteration to integer
+optimality. In M1.2.β we apply the weighted sum every 10-th iteration within the first n·ℓ iterations.
+In M1.3.β we apply the weighted sum every 10-th iteration within the first n · ℓ/3 iterations, every
+ℓ-th iteration in the next n · ℓ/3 iterations and every n-th iteration in the third n · ℓ/3 iterations.
+For M.2.α.β.γ we use the same algorithmic strategy as in hybrid Branch and Bound for knapsack
+problems. If a scalarization cannot be applied or an IP with identical objective function has been
+solved prior, no IP is solved in that iteration.
+Due to the total unimodularity of the assignment problem, the weighted sum scalarizations do in
+general not improve the lower bound sets of subproblems. However, in situations where the weighted
+sum IP scalarization generates a supported efficient solution, whose corresponding non-dominated
+point is not an extreme point of the lower bound set, the local upper bounds move closer to the lower
+bound set. This reduces the gap between upper and lower bound and may lead to a decrease of the
+explored subproblems. Note that this update of the upper bound set may also have the contrary
+effect (the number of considered subproblems increases), since it can change the order in which the
+subproblems are considered. Nevertheless, the weighted sum IP scalarizations are necessary to find
+non-dominated points on which the augmented weighted Tchebycheff scalarization can be applied.
+Our branching strategies have a significant impact on the number of explored nodes and the
+computation time. Again, the local hypervolume gap performs better than the global hypervolume
+gap strategy. With the local strategy we can reduce the number of explored nodes by up to 39%
+(Table 3c) and the computation time by up to 33% (Table 3c). Using the global hypervolume gap
+strategy we can reduce the number of explored nodes by up to 12% (Table 3b) and the computation
+time by up to 12% (Table 3b). We reach a reduction of the explored nodes by up to 46% (Table 3d)
+and a reduction of the computation time by up to 42% (Table 3d), in the best case. Again, the
+strategies M2.1.1.1 and M2.1.1.2 seem to work the best for assignment problems in terms of explored
+nodes. Nevertheless, M2.1.1.2 leads to a better computation time which can be explained by the
+same argument as before. Furthermore, strategy BS1 works very well and is able to compete with
+the previously mentioned strategies with respect to number of nodes and computation time without
+solving a single IP scalarization.
+18
+
+assignment problem, n = 100
+version
+nodes
+time (s)
+solved
+IPs
+BB
+3117.0
+5.1507
+0.0
+BS1
+2422.2
+4.1182
+0.0
+WS
+3117.0
+5.2631
+19.2
+M1.1.1
+2425.1
+4.1937
+13.95
+M1.2.1
+2425.1
+4.1564
+11.95
+M1.3.1
+2425.1
+4.1996
+8.4
+M2.1.1.1
+2418.2
+4.4063
+19.8
+M2.2.1.1
+2420.5
+4.2726
+14.4
+M2.3.1.1
+2419.5
+4.2242
+9.65
+M2.1.1.2
+2420.5
+4.3474
+19.9
+M2.2.1.2
+2420.5
+4.2014
+14.55
+M2.3.1.2
+2423.0
+4.2055
+9.8
+BS2
+2948.9
+4.9644
+0.0
+M1.1.2
+2519.8
+4.4349
+14.1
+M1.2.2
+2518.7
+4.4211
+11.05
+M1.3.2
+2516.2
+4.4203
+7.65
+M2.1.2.1
+2499.7
+4.6173
+20.4
+M2.2.2.1
+2518.7
+4.4661
+12.2
+M2.3.2.1
+2516.2
+4.4214
+8.35
+M2.1.2.2
+2498.1
+4.5400
+19.55
+M2.2.2.2
+2518.7
+4.4451
+12.25
+M2.3.2.2
+2516.2
+4.3972
+8.4
+(a) Assignment problem with n = 100 variables
+assignment problem, n = 144
+version
+nodes
+time (s)
+solved
+IPs
+BB
+9661.9
+28.2667
+0.0
+BS1
+6255.4
+18.9362
+0.0
+WS
+9610.8
+28.1363
+33.35
+M1.1.1
+6274.5
+18.9323
+22.3
+M1.2.1
+6274.5
+18.9869
+14.3
+M1.3.1
+6274.5
+18.9318
+9.85
+M2.1.1.1
+6210.9
+20.0216
+46.0
+M2.2.1.1
+6279.9
+19.4670
+23.55
+M2.3.1.1
+6271.2
+19.1542
+12.65
+M2.1.1.2
+6171.0
+19.4994
+43.6
+M2.2.1.2
+6261.5
+19.2851
+24.65
+M2.3.1.2
+6270.5
+18.8676
+12.45
+BS2
+8448.0
+24.7742
+0.0
+M1.1.2
+6781.8
+20.6825
+24.05
+M1.2.2
+6779.4
+20.5940
+13.25
+M1.3.2
+6734.2
+20.4270
+9.3
+M2.1.2.1
+6472.8
+20.8732
+44.1
+M2.2.2.1
+6724.2
+20.6838
+16.8
+M2.3.2.1
+6682.3
+20.4212
+12.6
+M2.1.2.2
+6500.3
+20.2536
+42.55
+M2.2.2.2
+6737.7
+20.5214
+16.75
+M2.3.2.2
+6689.0
+20.3987
+12.95
+(b) Assignment problem with n = 144 variables
+assignment problem, n = 225
+version
+nodes
+time (s)
+solved
+IPs
+BB
+26810.5
+160.712
+0.0
+BS1
+16142.2
+107.909
+0.0
+WS
+26810.5
+163.666
+46.85
+M1.1.1
+16150.0
+107.842
+32.8
+M1.2.1
+16151.2
+109.687
+19.7
+M1.3.1
+16164.7
+109.073
+12.05
+M2.1.1.1
+15424.1
+111.181
+87.4
+M2.2.1.1
+15964.3
+110.004
+43.05
+M2.3.1.1
+16112.2
+110.372
+19.5
+M2.1.1.2
+15554.1
+106.748
+89.5
+M2.2.1.2
+16008.4
+107.641
+41.35
+M2.3.1.2
+16106.5
+109.735
+19.25
+BS2
+24566.2
+152.341
+0.0
+M1.1.2
+17499.1
+117.481
+34.15
+M1.2.2
+17591.4
+118.804
+16.75
+M1.3.2
+17287.8
+116.681
+11.3
+M2.1.2.1
+16095.7
+115.012
+85.8
+M2.2.2.1
+17048.8
+116.275
+31.55
+M2.3.2.1
+17093.3
+117.976
+17.75
+M2.1.2.2
+16183.0
+110.920
+79.55
+M2.2.2.2
+17104.7
+117.885
+32.2
+M2.3.2.2
+17070.0
+117.557
+16.65
+(c) Assignment problem with n = 225 variables
+assignment problem, n = 324
+version
+nodes
+time (s)
+solved
+IPs
+BB
+76643.0
+798.644
+0.0
+BS1
+47311.6
+527.471
+0.0
+WS
+75179.2
+786.036
+61.75
+M1.1.1
+47327.0
+530.055
+47.75
+M1.2.1
+47332.8
+523.562
+21.9
+M1.3.1
+47375.6
+524.610
+15.25
+M2.1.1.1
+40978.0
+477.927
+157.5
+M2.2.1.1
+43760.5
+497.812
+82.0
+M2.3.1.1
+44343.9
+499.678
+52.35
+M2.1.1.2
+41294.4
+457.120
+159.05
+M2.2.1.2
+43864.6
+487.051
+81.25
+M2.3.1.2
+44499.7
+489.744
+52.05
+BS2
+69042.4
+723.853
+0.0
+M1.1.2
+49367.5
+565.719
+48.45
+M1.2.2
+49053.0
+553.130
+22.95
+M1.3.2
+49221.4
+554.594
+15.2
+M2.1.2.1
+41840.5
+489.647
+150.2
+M2.2.2.1
+45621.0
+520.469
+67.15
+M2.3.2.1
+46915.9
+533.692
+39.35
+M2.1.2.2
+42726.6
+474.220
+156.15
+M2.2.2.2
+45901.6
+513.111
+67.05
+M2.3.2.2
+46902.8
+526.440
+43.45
+(d) Assignment problem with n = 324 variables
+Table 3: Numerical results of the bi-objective assignment problems
+19
+
+5.3 Bi-objective Discrete Facility Location Problems
+We consider discrete facility location problems of the form
+max
+ℓ
+�
+i=1
+q
+�
+j=1
+ck
+ij xij +
+q
+�
+j=1
+f k
+j yj
+k = 1, 2
+s.t.
+q
+�
+j=1
+xij ≤ 1
+i = 1, ..., ℓ
+xij ≤ yj
+i = 1, ..., ℓ, j = 1, ..., q
+x ∈ {0, 1}ℓ×q
+y ∈ {0, 1}q
+where ℓ is the number of customers and q the number of facilities. We randomly generate coordinates
+of ℓ customers and q facilities in a square with length 200. The costs of the first objective function
+correspond to the l1-distances between the customers and facilities, while the costs of the second
+objective function are randomly generated (i. e. c2
+ij ∈ [1, 200]) and f k
+j ∈ [200, 400]. The number
+of variables is n = (ℓ + 1) q.
+We restrict the numerical tests to problems where the number of
+facilities is 20% of the number of customers. Again, we need to specify when and how often integer
+scalarizations are applied: We use the same methods as before but adapt the boundaries due to the
+different number of nodes to explore in discrete facility location problems. In M1.1.β we apply the
+weighted sum IP scalarization every 10-th iteration. In M1.2.β we apply the weighted sum every
+10-th iteration within the first n2/4 iterations. In M1.3.β we apply the weighted sum scalarization
+every 10-th iteration in the first n2/4 iterations, every n/2-th iteration in the next n2/4 iterations
+and every n-th iteration in the third n2/4 iterations. In M.2.α.β.γ we operate analogous to the
+methods used for knapsack and assignment problems. If a scalarization cannot be applied or an IP
+with identical objective function has been solved prior, no IP is solved in that iteration.
+Again, both new branching strategies have an impact on the number of explored nodes and the
+computation time. The local strategy, once more, leads to better results, namely reduction of the
+explored nodes by up to 52% (Table 4d) and reduction of the computation time by up to 45%
+(Table 4d). With the global hypervolume gap strategy we can reach a reduction of the explored
+nodes by up to 24% (Table 4d) and a reduction of the computation time by up to 21% (Table 4d).
+In the best case we can reach a reduction of the explored nodes by up to 57% (Table 4d) and of
+the computation time by up to 50% (Table 4d). Once again, M2.1.1.2 seems to be the best choice
+with respect to the number of explored nodes and with a rising number of variables it is also the
+best choice regarding the computation time. With a smaller number of variables, BS1 leads to good
+results with respect to both aspects without solving a single IP.
+5.4 Summary
+In all of the three tested problem classes (knapsack, assignment, discrete facility location) a signif-
+icant reduction of the number of explored nodes and the computation time can be realized with
+all presented combinations of the hybrid Branch and Bound approach. With increasing problem
+size (number of variables) the impact of the presented augmentations increases. Furthermore, the
+approaches perform better on problems where the gap between YN and the solution of the linear
+relaxation is larger compared to totally unimodular problems.
+The local hypervolume gap strategy for the node selection outperforms the global hypervolume gap
+strategy in our numerical tests. A reason for this is that in the global hypervolume gap strategy many
+small search zones can add up to a large gap although the lower bound might be quite close to the non-
+dominated points. The local hypervolume gap strategy chooses the node with the largest search zone,
+which has the biggest potential to reduce this gap. Moreover, the local hypervolume gap strategy
+aims at an uniform distribution of points in the incumbent list. In our numerical test, M2.1.1.2 turn
+20
+
+facility location problem, n = 48
+version
+nodes
+time (s)
+solved
+IPs
+BB
+1431.1
+1.0851
+0.0
+BS1
+999.8
+0.7640
+0.0
+WS
+1431.1
+1.4550
+15.35
+M1.1.1
+1000.2
+0.8354
+10.95
+M1.2.1
+1000.2
+0.8219
+8.8
+M1.3.1
+1000.2
+0.8243
+6.95
+M2.1.1.1
+999.7
+0.9361
+13.2
+M2.2.1.1
+1000.2
+0.8361
+9.85
+M2.3.1.1
+1000.2
+0.8079
+7.85
+M2.1.1.2
+999.3
+0.8536
+12.8
+M2.2.1.2
+1000.2
+0.8182
+9.85
+M2.3.1.2
+1000.2
+0.8037
+7.85
+BS2
+1268.6
+0.9714
+0.0
+M1.1.2
+1041.8
+0.8654
+12.3
+M1.2.2
+1041.8
+0.8527
+9.35
+M1.3.2
+1041.8
+0.8592
+7.35
+M2.1.2.1
+1036.5
+0.9722
+15.65
+M2.2.2.1
+1041.8
+0.8761
+10.25
+M2.3.2.1
+1041.8
+0.8735
+7.6
+M2.1.2.2
+1034.6
+0.9678
+15.1
+M2.2.2.2
+1041.8
+0.8843
+10.4
+M2.3.2.2
+1041.8
+0.8654
+7.6
+(a) Facility location problem with 15 customers
+and 3 facilities
+facility location problem, n = 84
+version
+nodes
+time (s)
+solved
+IPs
+BB
+7949.1
+12.6393
+0.0
+BS1
+4626.7
+7.9303
+0.0
+WS
+7490.2
+12.2058
+35.0
+M1.1.1
+4627.2
+8.1249
+26.45
+M1.2.1
+4627.2
+8.1149
+18.2
+M1.3.1
+4626.4
+8.0610
+13.85
+M2.1.1.1
+4527.8
+9.2394
+49.0
+M2.2.1.1
+4601.6
+8.4311
+26.45
+M2.3.1.1
+4610.4
+8.2569
+18.9
+M2.1.1.2
+4526.1
+8.4415
+45.3
+M2.2.1.2
+4591.1
+8.1289
+24.25
+M2.3.1.2
+4605.2
+8.1312
+19.8
+BS2
+7084.4
+11.5822
+0.0
+M1.1.2
+4873.8
+8.7719
+26.35
+M1.2.2
+4874.8
+8.7534
+17.45
+M1.3.2
+4894.1
+8.7031
+12.6
+M2.1.2.1
+4673.6
+9.5755
+49.65
+M2.2.2.1
+4832.0
+8.9275
+23.6
+M2.3.2.1
+4812.8
+8.8050
+17.4
+M2.1.2.2
+4628.9
+8.7019
+46.15
+M2.2.2.2
+4825.9
+8.7491
+24.4
+M2.3.2.2
+4807.5
+8.5984
+17.5
+(b) Facility location problem with 20 customers
+and 4 facilities
+facility location problem, n = 130
+version
+nodes
+time (s)
+solved
+IPs
+BB
+17461.9
+51.6307
+0.0
+BS1
+10684.0
+34.5157
+0.0
+WS
+16795.9
+50.9317
+51.35
+M1.1.1
+10753.9
+35.4356
+37.1
+M1.2.1
+10753.9
+35.2764
+25.5
+M1.3.1
+10753.9
+35.1007
+19.15
+M2.1.1.1
+10104.4
+38.3909
+89.65
+M2.2.1.1
+10678.1
+35.7434
+39.35
+M2.3.1.1
+10722.3
+35.6262
+27.7
+M2.1.1.2
+10103.0
+34.5003
+85.95
+M2.2.1.2
+10691.2
+35.3730
+39.05
+M2.3.1.2
+10718.6
+35.1690
+27.45
+BS2
+15474.7
+46.5130
+0.0
+M1.1.2
+11548.8
+38.4891
+39.3
+M1.2.2
+11601.4
+38.5848
+24.6
+M1.3.2
+11535.1
+38.2917
+18.0
+M2.1.2.1
+10684.3
+39.8639
+81.5
+M2.2.2.1
+11381.8
+39.1988
+37.15
+M2.3.2.1
+11336.0
+38.7754
+28.45
+M2.1.2.2
+10695.5
+36.9098
+78.25
+M2.2.2.2
+11341.6
+38.2646
+39.55
+M2.3.2.2
+11352.1
+38.0063
+25.9
+(c) Facility location problem with 25 customers
+and 5 facilites
+facility location problem, n = 186
+version
+nodes
+time (s)
+solved
+IPs
+BB
+67369.3
+373.238
+0.0
+BS1
+31844.1
+203.145
+0.0
+WS
+62192.8
+349.741
+69.95
+M1.1.1
+32106.6
+206.244
+53.05
+M1.2.1
+32097.9
+206.851
+33.4
+M1.3.1
+32148.5
+207.199
+23.75
+M2.1.1.1
+28384.2
+224.486
+186.8
+M2.2.1.1
+31074.5
+218.314
+101.15
+M2.3.1.1
+32004.8
+209.608
+50.1
+M2.1.1.2
+28558.8
+186.010
+172.7
+M2.2.1.2
+30946.4
+202.329
+97.75
+M2.3.1.2
+32011.8
+207.715
+47.5
+BS2
+50759.0
+292.344
+0.0
+M1.1.2
+35150.1
+230.687
+55.5
+M1.2.2
+35789.8
+233.967
+30.8
+M1.3.2
+35255.0
+233.163
+21.95
+M2.1.2.1
+29704.3
+230.016
+172.65
+M2.2.2.1
+33959.7
+236.351
+81.75
+M2.3.2.1
+34228.0
+233.553
+50.2
+M2.1.2.2
+30412.6
+202.719
+167.4
+M2.2.2.2
+34511.8
+229.970
+69.95
+M2.3.2.2
+34405.0
+229.021
+47.1
+(d) Facility location problem with 30 customers
+and 6 facilities
+Table 4: Numerical results of the bi-objective integer facility location problem
+21
+
+out to be the best choice in most cases with respect to the number of explored nodes and computation
+time. In this version, we use the local hypervolume gap strategy for the choice of the active node,
+every 10-th iteration the weighted sum IP scalarization is applied and every 50-th iteration we
+apply the augmented weighted Tchebycheff scalarization instead. Futhermore, the objective space
+information gained by the augmented weighted Tchebycheff scalarization is not used to update the
+lower bound set, since its local improvements do not compensate the increased computation time.
+Although we need to solve more IPs than in most other approaches, the computation time is the
+lowest compared to the others. So, using the augmented weighted Tchebycheff scalarization in the
+beginning of the Branch and Bound works best. Due to the likelihood of finding non-supported non-
+dominated points in the early stages of the algorithm, the upper bound can be further improved.
+This results to a higher probability of fathoming a node by dominance. Nevertheless, with version
+BS1 we also achive a remarkable reduction in terms of the number of explored nodes and computation
+time by using the local hypervolume gap strategy for node selection.
+6 Conclusion and Outlook
+In this paper, we propose two approaches to incorporate objective space information in bi-objective
+Branch and Bound. By using the local or global (approximated) hypervolume gap as a node selection
+criterion, we adapt the run of the Branch and Bound algorithm to the problem instance. Additionally,
+we adaptively solve scalarizations to integer optimality to improve the lower and the upper bound
+set by the obtained objective space information. Our numerical results show the effectiveness of
+both approaches and in particular of their combination. The dynamic branching rule based on the
+local (approximated) hypervolume gap has large impact on the number of explored subproblems,
+is compuationally efficient and can be easily integrated in other multi-objective Branch and Bound
+algorithms.
+While we tested in this paper the individual contributions of our augmentations on a generic
+bi-objective Branch and Bound, we will continue to extend our ideas to multiple dimensions and
+integrate them into a competetive multi-objective Branch and Bound framework. Particularly in
+higher dimensions, it may be promising to combine our approaches with objective space branching.
+Acknowledgment
+The authors thankfully acknowledge financial support by Deutsche Forschungsgemeinschaft, project
+number KL 1076/11-1.
+References
+Y. P. Aneja and K. P. K. Nair. Bicriteria transportation problem. Management Science, 25(1):
+73–78, 1979. doi: 10.1287/mnsc.25.1.73.
+C. Bazgan, H. Hugot, and D. Vanderpooten. Solving efficiently the 0–1 multi-objective knapsack
+problem. Computers & Operations Research, 36(1):260–279, 2009. doi: 10.1016/j.cor.2007.09.009.
+P. Belotti, B. Soylu, and M. M. Wiecek. A branch-and-bound algorithm for biobjective mixed-integer
+programs. Technical report, https://optimization-online.org/?p=12266, 2012.
+H. P. Benson. An outer approximation algorithm for generating all efficient extreme points in the
+outcome set of a multiple objective linear programming problem. Journal of Global Optimization,
+13(1):1–24, 1998. doi: 10.1023/A:1008215702611.
+22
+
+N. Boland, H. Charkhgard, and M. Savelsbergh. A criterion space search algorithm for biobjective
+integer programming: The balanced box method. INFORMS Journal on Computing, 27(4):735–
+754, 2015a. doi: 10.1287/ijoc.2015.0657.
+N. Boland, H. Charkhgard, and M. Savelsbergh. A criterion space search algorithm for biobjective
+mixed integer programming: The triangle splitting method. INFORMS Journal on Computing,
+27(4):597–618, 2015b. doi: 10.1287/ijoc.2015.0646.
+N. Boland, H. Charkhgard, and M. Savelsbergh. The quadrant shrinking method: A simple and
+efficient algorithm for solving tri-objective integer programs. European Journal of Operational
+Research, 260(3):873–885, 2017. doi: 10.1016/j.ejor.2016.03.035.
+F. Bö¸kler and P. Mutzel. Output-sensitive algorithms for enumerating the extreme nondominated
+points of multiobjective combinatorial optimization problems. In Algorithms - ESA 2015, pages
+288–299. Springer Berlin Heidelberg, Berlin Heidelberg, 2015. doi: 10.1007/978-3-662-48350-3_25.
+K. Dächert and K. Klamroth.
+A linear bound on the number of scalarizations needed to solve
+discrete tricriteria optimization problems. Journal of Global Optimization, 61(4):643–676, 2014.
+doi: 10.1007/s10898-014-0205-z.
+K. Dächert, J. Gorski, and K. Klamroth. An augmented weighted Tchebycheff method with adap-
+tively chosen parameters for discrete bicriteria optimization problems. Computers & Operations
+Research, 39(12):2929–2943, 2012. doi: 10.1016/j.cor.2012.02.021.
+R. Dechter and J. Pearl. Generalized best-first search strategies and the optimality of A∗. Journal
+of the ACM, 32(3):505–536, 1985. doi: 10.1145/3828.3830.
+M. Ehrgott. Multicriteria Optimization. Springer, Berlin Heidelberg, 2005. ISBN 978-3-540-21398-7.
+doi: 10.1007/3-540-27659-9.
+M. Ehrgott and X. Gandibleux. Bound sets for biobjective combinatorial optimization problems.
+Computers & Operations Research, 34(9):2674–2694, 2007. doi: 10.1016/j.cor.2005.10.003.
+N. Forget, S. L. Gadegaard, K. Klamroth, L. R. Nielsen, and A. Przybylski. Branch-and-bound and
+objective branching with three or more objectives. Computers & Operations Research, 148:106012,
+dec 2022. doi: 10.1016/j.cor.2022.106012.
+S. L. Gadegaard, L. R. Nielsen, and M. Ehrgott.
+Bi-objective branch-and-cut algorithms based
+on LP relaxation and bound sets. INFORMS Journal on Computing, 31(4):790–804, 2019. doi:
+10.1287/ijoc.2018.0846.
+Y. Haimes, L. Lasdon, and D. Wismer. On a bicriterion formation of the problems of integrated
+system identification and system optimization. IEEE Transactions on Systems, Man, and Cyber-
+netics, pages 296–297, 1971. doi: 10.1109/TSMC.1971.4308298.
+A. D. Jesus, L. Paquete, B. Derbel, and A. Liefooghe. On the design and anytime performance
+of indicator-based branch and bound for multi-objective combinatorial optimization.
+In Pro-
+ceedings of the Genetic and Evolutionary Computation Conference, Lille, 2021. ACM.
+doi:
+10.1145/3449639.3459360.
+H. Kellerer, U. Pferschy, and D. Pisinger. Knapsack Problems. Springer, Berlin Heidelberg, 2004.
+doi: 10.1007/978-3-540-24777-7.
+G. Kirlik and S. Sayın. A new algorithm for generating all nondominated solutions of multiobjective
+discrete optimization problems. European Journal of Operational Research, 232(3):479–488, 2014.
+doi: 10.1016/j.ejor.2013.08.001.
+23
+
+G. Kiziltan and E. Yucaoğlu. An algorithm for multiobjective zero-one linear programming. Man-
+agement Science, 29(12):1444–1453, 1983. doi: 10.1287/mnsc.29.12.1444.
+K. Klamroth, R. Lacour, and D. Vanderpooten. On the representation of the search region in multi-
+objective optimization. European Journal of Operational Research, 245(3):767–778, 2015. doi:
+10.1016/j.ejor.2015.03.031.
+D. Klein and E. Hannan. An algorithm for the multiple objective integer linear programming prob-
+lem. European Journal of Operational Research, 9(4):378–385, 1982. doi: 10.1016/0377-2217(82)
+90182-5.
+M. Laumanns, L. Thiele, and E. Zitzler.
+An efficient, adaptive parameter variation scheme for
+metaheuristics based on the epsilon-constraint method. European Journal of Operational Research,
+169(3):932–942, 2006. doi: 10.1016/j.ejor.2004.08.029.
+G. Mavrotas and D. Diakoulaki. A branch and bound algorithm for mixed zero-one multiple objective
+linear programming. European Journal of Operational Research, 107(3):530–541, 1998. doi: 10.
+1016/s0377-2217(97)00077-5.
+G. Mavrotas and D. Diakoulaki. Multi-criteria branch and bound: A vector maximization algorithm
+for mixed 0-1 multiple objective linear programming. Applied Mathematics and Computation, 171
+(1):53–71, 2005. doi: 10.1016/j.amc.2005.01.038.
+K. Miettinen. Nonlinear Multiobjective Optimization. Springer, New York, 1998. ISBN 0792382781.
+doi: 10.1007/978-1-4615-5563-6.
+D. R. Morrison, S. H. Jacobson, J. J. Sauppe, and E. C. Sewell. Branch-and-bound algorithms: A
+survey of recent advances in searching, branching, and pruning. Discrete Optimization, 19:79–102,
+2016. doi: 10.1016/j.disopt.2016.01.005.
+Ö. Özpeynirci and M. Köksalan. An exact algorithm for finding extreme supported nondominated
+points of multiobjective mixed integer programs. Management Science, 56(12):2302–2315, 2010.
+doi: 10.1287/mnsc.1100.1248.
+S. N. Parragh and F. Tricoire. Branch-and-bound for bi-objective integer programming. INFORMS
+Journal on Computing, 31(4):805–822, 2019. doi: 10.1287/ijoc.2018.0856.
+A. Przybylski and X. Gandibleux. Multi-objective branch and bound. European Journal of Opera-
+tional Research, 260(3):856–872, 2017. doi: 10.1016/j.ejor.2017.01.032.
+A. Przybylski, X. Gandibleux, and M. Ehrgott. Two phase algorithms for the bi-objective assignment
+problem. European Journal of Operational Research, 185(2):509–533, 2008. doi: 10.1016/j.ejor.
+2006.12.054.
+A. Przybylski, X. Gandibleux, and M. Ehrgott. A recursive algorithm for finding all nondominated
+extreme points in the outcome set of a multiobjective integer programme. INFORMS Journal on
+Computing, 22(3):371–386, 2010a. doi: 10.1287/ijoc.1090.0342.
+A. Przybylski, X. Gandibleux, and M. Ehrgott. A two phase method for multi-objective integer
+programming and its application to the assignment problem with three objectives. Discrete Opti-
+mization, 7(3):149–165, 2010b. doi: 10.1016/j.disopt.2010.03.005.
+A. Przybylski, K. Klamroth, and R. Lacour. A simple and efficient dichotomic search algorithm for
+multi-objective mixed integer linear programs, 2019.
+P. Serafini. Some considerations about computational complexity for multi objective combinatorial
+problems. In Recent Advances and Historical Development of Vector Optimization, pages 222–232.
+Springer Berlin Heidelberg, Berlin Heidelberg, 1987. doi: 10.1007/978-3-642-46618-2_15.
+24
+
+F. Sourd and O. Spanjaard. A multiobjective branch-and-bound framework: Application to the
+biobjective spanning tree problem. INFORMS Journal on Computing, 20(3):472–484, 2008. doi:
+10.1287/ijoc.1070.0260.
+R. E. Steuer. Multiple Criteria Optimization: Theory, Computation and Application. John Wiley,
+New York, 1986. ISBN 978-0894643934. doi: 10.1002/oca.4660100109.
+R. E. Steuer and E.-U. Choo. An interactive weighted Tchebycheff procedure for multiple objective
+programming. Mathematical Programming, 26(3):326–344, 1983. doi: 10.1007/BF02591870.
+T. Stidsen and K. A. Andersen. A hybrid approach for biobjective optimization. Discrete Optimiza-
+tion, 28:89–114, 2018. doi: 10.1016/j.disopt.2018.02.001.
+T. Stidsen, K. A. Andersen, and B. Dammann.
+A branch and bound algorithm for a class of
+biobjective mixed integer programs. Management Science, 60(4):1009–1032, 2014. doi: 10.1287/
+mnsc.2013.1802.
+D. Tuyttens, J. Teghem, P. Fortemps, and K. V. Nieuwenhuyze. Performance of the mosa method
+for the bicriteria assignment problem. Journal of Heuristics, 6(3):295–310, 2000. doi: 10.1023/a:
+1009670112978.
+E. Ulungu and J. Teghem. The two phases method: An efficient procedure to solve bi-objective
+combinatorial optimization problems. Foundations of Computing and Decision Sciences, 20:149–
+156, 1995.
+E. Ulungu and J. Teghem.
+Solving multi-objective knapsack problem by a branch-and-bound
+procedure.
+In Multicriteria Analysis, pages 269–278. Springer, Berlin Heidelberg, 1997.
+doi:
+10.1007/978-3-642-60667-0_26.
+T. Vincent, F. Seipp, S. Ruzika, A. Przybylski, and X. Gandibleux. Multiple objective branch and
+bound for mixed 0-1 linear programming: Corrections and improvements for the biobjective case.
+Computers & Operations Research, 40(1):498–509, 2013. doi: 10.1016/j.cor.2012.08.003.
+M. Visée, J. Teghem, M. Pirlot, and E. Ulungu. Two-phases method and branch and bound proce-
+dures to solve the bi–objective knapsack problem. Journal of Global Optimization, 12(2):139–155,
+1998. doi: 10.1023/A:1008258310679.
+E. Zitzler and L. Thiele. Multiobjective evolutionary algorithms: a comparative case study and the
+strength pareto approach. IEEE Transactions on Evolutionary Computation, 3(4):257–271, 1999.
+doi: 10.1109/4235.797969.
+25
+

BNE3T4oBgHgl3EQfTwqq/content/tmp_files/2301.04445v1.pdf.txt

@@ -0,0 +1,1153 @@
+MNRAS 000, 1–10 (2023)
+Preprint 12 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Hi intensity mapping with MeerKAT: forecast for delay power spectrum
+measurement using interferometer mode
+Ming Zhang
+ID1, Yichao Li
+ID1★, Jing-Fei Zhang
+ID1, Xin Zhang
+ID1,2,3†
+1Key Laboratory of Cosmology and Astrophysics (Liaoning Province) & Department of Physics, College of Sciences, Northeastern University, Shenyang 110819, China
+2Key Laboratory of Data Analytics and Optimization for Smart Industry (Ministry of Education), Northeastern University, Shenyang 110819, China
+3National Frontiers Science Center for Industrial Intelligence and Systems Optimization, Northeastern University, Shenyang 110819, China
+12 January 2023
+ABSTRACT
+Neutral hydrogen (Hi) intensity mapping (IM) survey is generally regarded as a promising tool to explore the expansion history
+of the universe. In this work, we investigate the capability of MeerKAT Hi IM observation with interferometric mode to estimate
+the power spectrum and constrain cosmological parameters in typical dark energy models. Besides, a novel approach of ‘delay
+spectrum’ is employed, which can achieve separating the weak Hi signal from the foreground in the frequency space. We find
+that the different survey fields have a great influence on the fractional errors on power spectrum Δ𝑃/𝑃 in a limited observational
+time 10 h. With the integration time increasing from 10 h to 10000 h, Δ𝑃/𝑃 becomes distinctly smaller until the cosmic
+variance begins to dominate. In the total 10000 h observation, the lower Δ𝑃/𝑃 in low 𝑘 can be achieved when tracking 100
+points for MeerKAT L-band and 10 points for MeerKAT UHF-band. Through simulating 10000 h Hi IM survey, we obtain
+𝜎(Ωm) = 0.044 and 𝜎(𝐻0) = 2.8 km s−1 Mpc−1 with MeerKAT L-band, which are worse than the results of 𝜎(Ωm) = 0.028
+and 𝜎(𝐻0) = 2.0 km s−1 Mpc−1 with MeerKAT UHF-band in the ΛCDM model. However, in the 𝑤CDM and CPL models,
+MeerKAT shows a limited capability of constraining dark-energy equation of state, even though combined with Planck data. Our
+analysis is shown to be a useful guide for the near future MeerKAT observations in Hi IM survey.
+Key words: techniques: interferometric – cosmology: large scale structure of Universe – cosmology: cosmological parameters
+– radio lines: general
+1 INTRODUCTION
+In the last two decades, the accurate measurement of the cosmic
+microwave background (CMB) brings us to the era of precision cos-
+mology. Another promising method for reaching precision cosmol-
+ogy is the cosmic large scale structure (LSS) survey. At present, the
+cosmic LSS survey has made significant progress with galaxy red-
+shift survey in constraining cosmological parameters, e.g. the 2dF
+Galaxy Redshift Survey (Colless et al. 2001; Cole et al. 2005), the
+6dF Galaxy Survey (Jones et al. 2009; Beutler et al. 2011), the Wig-
+gleZ Dark Energy Survey (Blake et al. 2011; Drinkwater et al. 2010),
+the Baryon Oscillation Spectroscopic Survey (Alam et al. 2017) and
+the Dark Energy Survey (DES) (Abbott et al. 2018). In addition, the
+next generation galaxy survey targeting at even larger and deeper
+universe, such as Dark Energy Spectroscopic Instrument (Dey et al.
+2019), Large Synoptic Survey Telescope (Ivezić et al. 2019; Chisari
+et al. 2019) and Euclid (Amendola et al. 2018), will significantly
+improve the measurement precision in the near future.
+Besides, neutral hydrogen (Hi) is widely regarded as a promising
+tracer of the underlying dark matter distribution of the late Universe.
+Hi can be detected with radio telescope via its 21 cm line which
+★ E-mail: liyichao@mail.neu.edu.cn
+† E-mail: zhangxin@mail.neu.edu.cn
+arises from the spin-flip transition of ground state hydrogen atom.
+Nevertheless, it is known that detecting Hi signal in individual galaxy
+at higher redshfit requires good angular resolution and sensitivity,
+which relies on large radio interferometer in the near future, such as
+Square Kilometre Array (SKA). However, Hi survey of the cosmic
+LSS can be quickly carried out using existing radio telescopes via
+the intensity mapping (IM) methodology, which observes the total
+Hi intensity of the galaxies in a voxel (Battye et al. 2004; McQuinn
+et al. 2006; Loeb & Wyithe 2008; Chang et al. 2008; Wyithe et al.
+2008; Bagla et al. 2010; Seo et al. 2010; Lidz et al. 2011; Ansari
+et al. 2012). A variety of related studies show that Hi IM survey has a
+great performance in cosmology studies, e.g. constraints on the basic
+and dark-energy cosmological parameters (Pritchard & Loeb 2012;
+Bull et al. 2015; Pourtsidou et al. 2017; Olivari et al. 2018; Obuljen
+et al. 2018; Sprenger et al. 2019; Xu et al. 2018; Cheng et al. 2020;
+Xu & Zhang 2020; Jin et al. 2020; Xiao et al. 2021; Zhang et al.
+2021; Jin et al. 2021; Wu & Zhang 2022; Scelfo et al. 2022; Berti
+et al. 2022; Wu et al. 2022a,b), primordial non-Gaussianity (Camera
+et al. 2013; Xu et al. 2015; Li & Ma 2017; Ballardini et al. 2019;
+Karagiannis et al. 2020; Cunnington et al. 2020; Karagiannis et al.
+2021; Viljoen et al. 2021) and neutrino mass (Villaescusa-Navarro
+et al. 2015; Zhang et al. 2020).
+Hi IM LSS detection was first reported by measuring the cross-
+correlation function between the Hi map observed with Green Bank
+© 2023 The Authors
+arXiv:2301.04445v1  [astro-ph.CO]  11 Jan 2023
+
+2
+M. Zhang et al.
+Telescope (GBT) and the DEEP2 optical redshift survey (Chang
+et al. 2010). The cross-correlation power spectrum between Hi IM
+survey and optical galaxy survey was also detected with the GBT
+and the WiggleZ Dark Energy Survey (Masui et al. 2013). Anderson
+et al. (2018) reported the results of Hi IM maps cross-correlated with
+2dF galaxy survey. Tramonte & Ma (2020) presented the feasibility
+of measuring Hi maps with Parkes radio telescope and 2dF galaxy
+survey. A cross-correlation signal detected with Parkes Hi IM and
+WiggleZ redshift data was discussed in Li et al. (2021a). Wolz et al.
+(2022) gave a joint analysis of GBT IM and eBOSS survey. To date,
+the auto-correlation power spectrum is still not detected because of
+systematics and foreground contamination.
+Several large radio telescopes or interferometers, such as Five
+hundred-meter Aperture Spherical radio Telescope (FAST; Nan
+et al. 2011), Baryon acoustic oscillations In Neutral Gas Obser-
+vations (BINGO; Dickinson 2014; Wuensche et al. 2020) and SKA
+(Maartens et al. 2015; Bacon et al. 2020), are built or planned for
+Hi survey, which are expected to detect the auto-correlation power
+spectrum. In addition, interferometers have inherent advantages of
+being less sensitive to systematics, which can be regarded as a com-
+plementary approach to single-dish IM experiments. The smallest
+𝑘-modes accessible to an interferometer is determined by the short-
+est baselines. Therefore, a compacted radio interferometer array is
+required for probing the cosmic LSS, especially for the scales of
+baryon acoustic oscillation (BAO). To date, several interferometers,
+such as Canadian Hydrogen Intensity Mapping Experiment (CHIME;
+Bandura et al. 2014), Hydrogen Intensity and Real-time Analysis eX-
+periment (HIRAX; Newburgh et al. 2016) and Tianlai (Chen 2012;
+Wu et al. 2021), are designed with short baselines to probe the BAO.
+Besides, the MeerKAT radio telescope array with 64 dish anten-
+nas of 13.5 m diameter is a precursor of SKA, which is located in
+the Northern Cape Province of South Africa. MeerKAT has already
+been operating and producing preliminary results (Pourtsidou 2018;
+Wang et al. 2021; Knowles et al. 2021; Terni de Gregory et al. 2021;
+de Villiers & Cotton 2022). The MeerKAT Large Area Synoptic
+Survey (MeerKLASS; Santos et al. 2017; Irfan et al. 2021) proposed
+Hi IM with single-dish mode (Li et al. 2021b; Wang et al. 2021)
+and reported the cross-correlation power spectrum detection with
+the optical galaxy survey (Cunnington et al. 2022). In addition, the
+small field deep surveys using MeerKAT interferometric mode, such
+as the MeerKAT International GHz Tiered Extragalactic Exploration
+(MIGHTEE; Jarvis et al. 2018; Paul et al. 2021; Maddox, N. et al.
+2021), also provided the Hi cube. In this paper, we investigate the
+performance of Hi IM survey in measuring power spectrum using
+MeerKAT interferometric mode with different survey strategies and
+forecast the constraints on cosmological parameters in typical dark
+energy models including ΛCDM, 𝑤CDM and CPL models (Cheval-
+lier & Polarski 2001; Linder 2003).
+In this work, it is worth mentioning that we employ a novel ap-
+proach, i.e. the ‘delay spectrum’ analysis, which is first applied in the
+PAPER observation (Parsons & Backer 2009). It is known that the
+cosmic Hi signal is contaminated by the bright foreground emissions
+in the same frequency ranges, such as the synchrotron and free-free
+emission from the Galaxy and extra-galactic point sources. Gen-
+erally, the foreground contamination components have the smooth
+frequency spectra, which can be separated from the cosmic Hi fluc-
+tuation in the ‘delay spectrum’ space (Parsons et al. 2012; Liu et al.
+2014a,b; Liu & Shaw 2020).
+This paper is organized as follows. In Section 2, we provide the de-
+tailed description of estimating Hi signal power spectrum, MeerKAT
+survey strategy and its system noise, foregrounds and shot noise. In
+Section 3, we present the constraint results of the power spectrum
+and cosmological parameters. Finally, the conclusion is given in Sec-
+tion 4. In our simulation, we assume a flat ΛCDM model and keep
+all cosmological parameters fixed to Planck 2018 results (Aghanim
+et al. 2020).
+2 METHODOLOGY
+2.1 Hi delay spectrum
+Hi IM observation directly measures Hi brightness temperature. The
+sky brightness temperature can be defined as
+𝑇(𝜽, 𝜈) = ¯𝑇(𝜈)[1 + Δ𝑇(𝜽, 𝜈)] ,
+(1)
+where 𝜽 is the position vector on the sky, 𝜈 is the observation fre-
+quency, ¯𝑇(𝜈) and Δ𝑇(𝜽, 𝜈) denote the isotropic and fluctuating com-
+ponents of the Hi brightness temperature distribution, respectively.
+Radio interferometers detect Hi signals by measuring their visi-
+bilities, which are the cross-correlation signals between each pair of
+antennas. Assuming the flat-sky approximation, the visibility for a
+pair of antennae is given by
+𝑉(𝒖, 𝜈) =
+∫
+𝐴(𝜽, 𝜈)Δ𝑇(𝜽, 𝜈)𝑒−𝑖2𝜋𝒖·𝜽𝑑Ω ,
+(2)
+where 𝒖 = 𝜈𝒃/𝑐 is the baseline vector in units of wavelength, cor-
+responding to each antenna pair, where 𝒃 is the baseline vector in
+physical units and 𝑐 is the speed of light. 𝐴(𝜽, 𝜈) denotes the primary
+beam response of the telescope in the direction of 𝜽 and dΩ repre-
+sents the solid angle element. The visibility function can be Fourier
+transferred to the ‘delay spectrum’ space,
+˜𝑉(𝒖, 𝜏) =
+∫
+𝑉(𝒖, 𝜈)𝑒−𝑖2𝜋𝜈𝜏d𝜈,
+(3)
+where 𝜏 = 1/𝛿𝜈 is the corresponding delay of frequency interval 𝛿𝜈.
+Following McQuinn et al. (2006), Parsons et al. (2012) and Liu &
+Shaw (2020), Hi power spectrum can be obtained from measured
+visibilities in the form of ‘delay spectrum’
+𝑃D(𝑘⊥, 𝑘 ∥) ≡ 𝐴𝑒
+𝜆2𝐵
+𝑟2𝑟𝜈
+𝐵
+�� ˜𝑉(𝒖, 𝜏)
+��2
+� 𝜆2
+2𝑘B
+�2
+,
+(4)
+where 𝐴𝑒 and 𝐵 are the effective antenna area and bandwidth, re-
+spectively, 𝜆 denotes the wavelength at the center of the band, 𝑟 is
+the comoving distance to the redshift 𝑧 corresponding to 𝜆, 𝑟𝜈 is the
+comoving width along the line-of-sight (LoS) corresponding to the
+redshift range determined by 𝐵, and 𝑘B is the Boltzmann constant.
+Here, 𝑘⊥ and 𝑘 ∥ are the Fourier wave vectors perpendicular and par-
+allel to the LoS, respectively. They are related to the interferometric
+variables via
+𝑘⊥ = 2𝜋|𝒖|
+𝑟
+;
+𝑘 ∥ = 2𝜋𝜏𝜈21𝐻(𝑧)
+𝑐(1 + 𝑧)2
+.
+(5)
+where 𝜈21 = 1420 MHz is the rest-frame frequency of the 21 cm line.
+𝐻(𝑧) denotes the Hubble parameter as a function of redshift 𝑧.
+There are various advantages of this ‘delay spectrum’ method
+(Parsons et al. 2012; Vedantham et al. 2012; Paul et al. 2016). The
+different spectral behaviors between Hi signal and foreground make
+it possible to isolate the latter in the Fourier space. In addition, the
+Fourier conjugate variable is associated with the LoS cosmological
+distance, therefore the ‘delay spectrum’ constructed in this method
+can recover the cosmological 3D Hi power spectrum (Parsons et al.
+2012; Liu et al. 2014a,b).
+MNRAS 000, 1–10 (2023)
+
+Hi delay spectrum with MeerKAT interferometer mode
+3
+2.2 Hi signal power spectrum
+The mean sky brightness temperature of Hi 21 cm emission can be
+given by (Santos et al. 2015, 2017)
+¯𝑇𝑏(𝑧) ≈ 566ℎ
+� 𝐻0
+𝐻(𝑧)
+� � ΩHi(𝑧)
+0.003
+�
+(1 + 𝑧)2 𝜇K ,
+(6)
+where 𝐻0 = 100ℎ km s−1 Mpc−1 is the Hubble constant, ΩHi(𝑧) is
+the fractional density of Hi, which can be written as
+ΩHi(𝑧) =
+𝜌Hi(𝑧)
+𝜌𝑐,0(1 + 𝑧)3 ,
+(7)
+where 𝜌𝑐,0 is the critical density today and the proper Hi density is
+calculated by
+𝜌Hi(𝑧) =
+∫ 𝑀max
+𝑀min
+d𝑀 d𝑛
+d𝑀 (𝑀, 𝑧)𝑀Hi(𝑀, 𝑧) ,
+(8)
+where 𝑀 denotes the dark matter halo mass, d𝑛/d𝑀 is the proper
+halo mass function and 𝑀Hi(𝑀, 𝑧) denotes the Hi mass in a halo of
+mass 𝑀 at redshift 𝑧. Throughout this paper, we assume a simple
+power-law model of the halo mass following Santos et al. (2015),
+i.e., 𝑀Hi(𝑀) = 𝐴𝑀 𝛼 with 𝐴 ∼ 220 and 𝛼 = 0.6 that can fit both
+low- and high-redshift observations within reasonable accuracy.
+Considering the redshift space distortion (RSD) effect (Kaiser
+1987), Hi signal power spectrum can be written as
+𝑃Hi(𝑘, 𝜇, 𝑧) = ¯𝑇2
+𝑏 (𝑧)𝐹RSD(𝑘, 𝜇)𝑃(𝑘, 𝑧) ,
+(9)
+where 𝜇 ≡
+𝑘 ∥/𝑘 and the matter power spectrum 𝑃(𝑘, 𝑧)
+=
+𝐷2(𝑧)𝑃(𝑘, 𝑧 = 0) with 𝐷(𝑧) being the growth factor and 𝑃(𝑘, 𝑧 = 0)
+being the matter power spectrum at z = 0 which can be obtained by
+CAMB (Lewis et al. 2000). 𝐹RSD(𝑘, 𝜇) represents the RSD effect, and
+its form can be expressed as
+𝐹RSD(𝑘, 𝜇) =
+�
+𝑏2
+Hi(𝑧) + 𝑓 𝜇2�2
+exp
+�
+−𝑘2𝜇2𝜎2
+NL
+�
+.
+(10)
+Here 𝑏Hi(𝑧) is the Hi bias, written as
+𝑏Hi(𝑧) = 𝜌−1
+Hi (𝑧)
+∫ 𝑀max
+𝑀min
+d𝑀 d𝑛
+d𝑀 𝑀Hi(𝑀, 𝑧)𝑏(𝑀, 𝑧),
+(11)
+where 𝑏(𝑀, 𝑧) is the halo bias. 𝑓 ≡ dln𝐷/dln𝑎 is the linear growth
+rate with 𝑎 being the scale factor. 𝜎NL is the nonlinear dispersion
+scale with a middling value of 𝜎NL = 7Mpc. In this paper, for con-
+venience, ΩHi(𝑧) and 𝑏Hi(𝑧) are employed with the fitting functions
+following Santos et al. (2017).
+2.3 MeerKAT noise power spectrum
+The total thermal noise power spectrum can be written as (Bull et al.
+2015)
+𝑃N(𝑘, 𝜇, 𝑧) = 𝑟2(𝑧)𝑟𝜈(𝑧)
+𝑇2sys𝜆4
+𝑛pol𝜈21𝑡int𝐴2𝑒𝑛(𝒖)
+,
+(12)
+where 𝑛pol = 2 denotes the number of polarization. 𝑡int is the inte-
+gration time. The ratio of MeerKAT effective aperture and system
+temperature, 𝐴𝑒/𝑇sys, is frequency dependent. Currently, there are
+two frequency bands available for observation, i.e., the L-band (900–
+1700 MHz) and the UHF-band (580–1000 MHz). Because of the
+serious RFI contamination in the L-band frequency range, only the
+frequency range of 900–1200 MHz (0.18 < 𝑧 < 0.58) is used in our
+analysis. The full UHF-band is used in this analysis, corresponding
+500
+750
+1000
+1250
+1500
+ν [MHz]
+0
+2
+4
+6
+8
+10
+Ae/T sys [m2/K]
+MeerKAT L-band
+MeerKAT UHF-band
+Figure 1. The sensitivity designs for MeerKAT receivers, shown as 𝐴𝑒/𝑇sys
+for L-band and UHF-band.
+to 0.42 < 𝑧 < 1.45. In Fig. 1, we show 𝐴𝑒/𝑇sys for L-band and
+UHF-band.1
+Here 𝑛(𝒖) is the baseline density referring to the detailed 𝑢𝑣
+coverage of a particular observation in the 𝑢𝑣 plane. We employ the
+actual MeerKAT antenna coordinates and track the COSMOS field
+(RA=10h01m, Dec=+02d12m) following Paul et al. (2021). In a strict
+sense, 𝑛(𝒖) is also a function of frequency. We break the frequency
+range into a couple of Δ𝜈 = 60 MHz sub-bands. The 𝑢𝑣 coverage is
+assumed to be uniform within each sub-band and simulated according
+to the center frequency of each sub-band. The simulated 𝑢𝑣 coverage
+corresponding to the sub-bands in L-band centering at 𝑧 = 0.3 and
+in UHF-band centering at 𝑧 = 1.2 are shown in left and right panels
+of Fig. 2, respectively. For both cases, we assume 10 h tracking
+observation of the COSMOS field spanning over two days (the start
+time is 14:15 and 13:33 at UTC, respectively). The 𝑢𝑣 plane is
+segmented on to a discrete grid with cell size Δ𝑢 = Δ𝑣 = 60𝜆. The
+color represents the number of 𝑢𝑣 points within the grid. It is clear
+that there are more samples in the short 𝑢𝑣 distance region at low
+frequency band than high frequency band.
+Because of the uniform 𝑢𝑣 coverage assumption across the sub-
+band, the baseline density 𝑛(𝒖) and the corresponding total thermal
+noise power spectrum 𝑃N are only the functions of 𝑘⊥. According to
+Eq. (5), 𝑘⊥ is proportional to the 𝑢𝑣 distance, i.e. |𝒖| =
+√
+𝑢2 + 𝑣2. The
+circular averaged 𝑢𝑣 coverage within a |𝒖| shell of width Δ|𝒖| = 100𝜆
+are shown in Fig. 3, where the left panel shows the distribution
+corresponding to the sub-bands centering at 𝑧 = 0.3 and the right
+panel shows the one centering at 𝑧 = 1.2. Since |𝑘⊥| is proportional
+to the 𝑢𝑣 distance, the more densely populated 𝑢𝑣 points at smaller
+distance mean the higher sensitivity at the smaller |𝑘⊥| modes.
+It is known that the 𝑢𝑣 coverage also depends on the pointing direc-
+tion. In order to investigate the influence of the different sky zones,
+we also show in Fig. 3 the numbers of 𝑢𝑣 points with 10 h track-
+ing at different declinations: Dec = +30◦, +02◦, −30◦, −60◦, −90◦
+respectively. The case of Dec = +02◦ is the same as tracking the
+COSMOS field and the case of Dec = −30◦ corresponds to tracking
+a field that the transit line passes near Zenith the MeerKAT site. It
+is obvious that when the field is targeted farther from the zenith, the
+number of short baselines rises substantially for both the L-band and
+1 http://public.ska.ac.za/meerkat/meerkat-schedule
+MNRAS 000, 1–10 (2023)
+
+4
+M. Zhang et al.
+Figure 2. The distribution of baselines on a two-dimensional (2D) 𝑢𝑣 plane for 10 h tracking of the COSMOS field with sub-bands in MeerKAT L-band centering
+at 𝑧 = 0.3 (left panel) and in UHF-band centering at 𝑧 = 1.2 (right panel). The 𝑢𝑣 plane is segmented on to a discrete grid with cell-size Δ𝑢 = Δ𝑣 = 60𝜆. The
+color signifies the number of 𝑢𝑣 points on the grid.
+Figure 3. The average number of baselines as a function of 𝑢𝑣 distance, |𝒖| =
+√
+𝑢2 + 𝑣2, with bin size of Δ|𝒖| = 100𝜆, for 10 h tracking with sub-bands in
+MeerKAT L-band centering at 𝑧 = 0.3 (left panel) and in UHF-band centering at 𝑧 = 1.2 (right panel).
+UHF-band, which potentially increases the sensitivity at the smaller
+|𝑘⊥| modes.
+2.4 The foreground wedge and shot noise
+The foreground contamination, which is several orders of magnitude
+stronger than Hi signal, is the major challenge in recovering the Hi
+LSS. Since foreground spectrum is smooth across frequency chan-
+nels, it only contaminates the power spectrum close to the smallest
+𝑘 ∥. However, the property of the interferometer response function
+will cause foreground leakage into the high-𝑘⊥ modes. Therefore,
+we exclude the 𝑘⊥–𝑘 ∥ space modes within the foreground wedge
+(Datta et al. 2010; Morales et al. 2012; Liu et al. 2014a,b; Pober
+2015; Seo & Hirata 2016) that can be expressed as
+𝑘 ∥ < 𝑟(𝑧)𝐻(𝑧) sin(𝜃)
+𝑐(1 + 𝑧)
+𝑘⊥ ,
+(13)
+where 𝜃 denotes the field of view of the interferometer.
+In addition, shot noise needs to be taken into account in Hi IM
+survey. Because of Poisson fluctuations in halo number, the shot
+noise power spectrum is written as (Bull et al. 2015)
+𝑃shot
+Hi (𝑧) =
+� ¯𝑇𝑏(𝑧)
+𝜌Hi(𝑧)
+�2 ∫ 𝑀max
+𝑀min
+d𝑀 d𝑛
+d𝑀 𝑀2
+Hi(𝑀) .
+(14)
+Here Hi mass model is consistent with the description in the Hi
+signal power spectrum. Since shot noise is very low according to our
+calculation, it makes a very small contribution to the total noise.
+3 RESULTS
+In this section, we present the results of Hi IM survey analysis. In
+Section 3.1, we give a detailed analysis of the power spectrum in the
+different survey strategies. The relative errors on the BAO features
+and the constraints on cosmological parameters in different dark
+energy models are showed in Section 3.2.
+MNRAS 000, 1–10 (2023)
+
+uv coverage at z = 0.3 for RA=10:00:28.60 Dec=2:12:21.0
+104
+10
+103
+5
+(Y)
+0
+102
+101
+-10
+-15
+100
+-10
+0
+10
+u (k入)uv coverage at z = 1.2 for RA=10:00:28.60 Dec=2:12:21.0
+104
+10
+103
+5
+0
+102
+5
+101
+-10
+-15
+100
+-10
+0
+10
+u (k入)5000
+Dec=十30°
+Dec=-90°
+4000
+Number of u points
+Dec=+02°
+Dec=-60°
+3000
+Dec=-30°
+L-band
+2000
+1000
+102
+103
+10
+u distance in ^ （z = 0.312000
+Dec=±30°
+10000
+Dec=-90°
+Dec=+02°
+S
+Dec=-60°
+8000
+Dec=-30°
+UHF-band
+TO
+6000
+4000
+2000
+0
+102
+103
+10
+uv distance in ^ (z = 1.2Hi delay spectrum with MeerKAT interferometer mode
+5
+(a) Hi power spectrum
+(b) Total power spectrum
+Figure 4. 2D power spectrum at 𝑧 = 0.3. Left panel: Hi signal power spectrum 𝑃Hi. Right panel: Total power spectrum 𝑃tot with MeerKAT L-band 10 h
+observation.
+(a) Hi power spectrum
+(b) Total power spectrum
+Figure 5. 2D power spectrum at 𝑧 = 1.2. Left panel: Hi signal power spectrum 𝑃Hi. Right panel: Total power spectrum 𝑃tot with MeerKAT UHF-band 10 h
+observation.
+3.1 Power spectrum estimation
+The Hi LSS carries a significant quantity of cosmic information.
+However, it is extremely weak comparing to the brilliant foreground
+contamination. The 2D Hi power spectrum 𝑃Hi at 𝑧 = 0.3 (for
+MeerKAT L-band) is shown in the left panel of Fig. 4. The scales
+available for Hi IM in interferometric mode observation are limited
+by the detailed configuration. In summary, the scales for Hi IM survey
+are:
+𝑘min
+∥
+= 2𝜋/(𝑟𝜈Δ𝜈/𝜈21),
+𝑘max
+∥
+= 1/𝜎NL,
+𝑘min
+⊥
+= 2𝜋|𝒖|min/𝑟,
+𝑘max
+⊥
+= 2𝜋|𝒖|max/𝑟.
+(15)
+In principle, only the scales between minimum and maximum can be
+probed by a certain instrument, and the sensitivity to scales depends
+on the 𝑢𝑣 coverage. Adopting only 10 h observation, the total power
+spectrum 𝑃tot at the same redshift, which consists of the contributions
+of Hi signal, MeerKAT thermal noise, foregrounds and shot noise,
+is shown in the right panel of Fig. 4.
+The power spectrum at 𝑧 = 1.2 (for MeerKAT UHF-band) are
+shown in Fig. 5, where the Hi power spectrum 𝑃Hi is in the left panel
+and the total power spectrum 𝑃tot is in the right panel, respectively.
+Note that, in the right panel of Fig. 5, the brown part denotes the
+foreground wedge. We find that Hi signal is completely covered by
+the thermal noise and foregrounds for both MeerKAT L-band and
+UHF-band, which makes it difficult to obtain Hi signal directly.
+The Hi detection is quantified with the relative error of the power
+spectrum
+� Δ𝑃
+𝑃
+�2
+=
+�
+1
+8𝜋2𝑉bin
+∫
+𝑘2d𝑘d𝜇
+� 𝑃Hi(𝑘, 𝜇)
+𝑃tot(𝑘, 𝜇)
+�2�−1
+,
+(16)
+where 𝑉bin = 𝑆area𝑟2𝑟𝜈 Δ𝜈
+𝜈21 is the survey volume of each redshift bin
+with the survey area 𝑆area = 𝜋
+�
+1
+2
+𝜆
+13.5𝑚
+�2 �
+180
+𝜋
+�2
+.
+Firstly, we investigate the influence on the 𝑃(𝑘) error when track-
+ing the source at the different declinations. As is shown in Fig. 3,
+when tracking the source at Dec = +30◦, +02◦, −30◦, −60◦, −90◦,
+completely different numbers of 𝑢𝑣 points are obtained, e.g. there are
+more 𝑢𝑣 points in the shorter 𝑢𝑣 distance for the case of Dec = +30◦.
+In the top panel of Fig. 6, the relative errors of power spectrum with
+different tracking declinations are shown in different colors. For all
+the cases, we assume 10 h observation time. The results with L-band
+and UHF-band are shown in solid and dashed lines, respectively.
+Here, we divide the whole range of 𝑘 into 10 logarithmic bins. It is
+clear that the power spectrum uncertainty is reduced by more than
+MNRAS 000, 1–10 (2023)
+
+104
+10-1
+102
+ku [Mpc-1]
+6 × 10-2
+100
+4 × 10-2
+10-2
+3 × 10-2
+100
+10'
+102
+k [Mpc-1]107
+10-1
+105
+ku [Mpc-1]
+6 × 10-2
+103
+4 × 10-2
+3 × 102
+101
+10°
+10
+102
+k [Mpc-1]104
+10-1
+102
+ku [Mpc-1]
+100
+10-2
+10-
+100
+101
+k[Mpc-1]107
+10-1
+105
+ku [Mpc-1]
+103
+100
+101
+10
+10
+k}[Mpc-1]6
+M. Zhang et al.
+10−1
+100
+k [Mpc−1]
+100
+101
+102
+∆P/P
+10 hours
+L-band at Dec=+30◦
+L-band at Dec=+02◦
+L-band at Dec=−30◦
+L-band at Dec=−60◦
+L-band at Dec=−90◦
+UHF-band at Dec=+30◦
+UHF-band at Dec=+02◦
+UHF-band at Dec=−30◦
+UHF-band at Dec=−60◦
+UHF-band at Dec=−90◦
+10−1
+100
+101
+102
+k [Mpc−1]
+10−2
+100
+102
+104
+106
+∆P/P
+1 point
+L-band 10 hours
+L-band 100 hours
+L-band 1000 hours
+L-band 10000 hours
+UHF-band 10 hours
+UHF-band 100 hours
+UHF-band 1000 hours
+UHF-band 10000 hours
+10−1
+100
+101
+102
+k [Mpc−1]
+10−2
+10−1
+100
+101
+102
+103
+104
+∆P/P
+10000 hours
+L-band 1 point
+L-band 10 points
+L-band 100 points
+L-band 1000 points
+UHF-band 1 point
+UHF-band 10 points
+UHF-band 100 points
+UHF-band 1000 points
+Figure 6. Fractional errors on 𝑃(𝑘) obtained with MeerKAT L-band and UHF-band. Top panel: 10 h observations at the different declinations. Middle panel:
+Different observation times of tracking the COSMOS field. Bottom panel: Tracking different numbers of points in a 10000 h observation.
+MNRAS 000, 1–10 (2023)
+
+Hi delay spectrum with MeerKAT interferometer mode
+7
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+z
+10−2
+10−1
+100
+σ(DA)/DA
+L-band 10 points
+UHF-band 100 points
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+z
+10−1
+100
+101
+σ(H)/H
+L-band 10 points
+UHF-band 100 points
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+z
+10−1
+100
+101
+σ(fσ8)/(fσ8)
+L-band 10 points
+UHF-band 100 points
+Figure 7. Fractional errors on 𝐷𝐴(𝑧), 𝐻 (𝑧) and 𝑓 𝜎8(𝑧) obtained with MeerKAT L-band and UHF-band.
+a factor of two when tracking Dec = +30◦ compared to tracking
+Dec = −30◦ or Dec = −60◦. The results show that, with limited
+observation time, the tracking declination has a obvious influence on
+the results of the constraints on the power spectrum.
+Next, in order to assess the influence of integration time, we in-
+crease the integration time by assuming observations on the same
+field at the same local sidereal time as the existing data on different
+days, which means that we obtain the same 𝑢𝑣 points from multiple
+days coherently to increase the sensitivity of the same 𝑘 modes. In
+addition to the current 10 h observation, we further consider 100,
+1000 and 10000 h observations. In the middle panel of Fig. 6, we
+show the fractional errors on the power spectrum 𝑃(𝑘) for 10, 100,
+1000 and 10000 hours observation of tracking the COSMOS field
+with MeerKAT L-band (in blue) and UHF-band (in red). The results
+with different integration times are shown with different color satu-
+rations. It can be seen that, compared to L-band, using the UHF-band
+could measure smaller 𝑘 modes down to ∼ 0.1, which makes it pos-
+sible to detect cosmological LSS on the larger scales. In addition, it
+is expected that the lower Δ𝑃/𝑃 can be obtained with the observa-
+tion time increasing as is shown in the middle panel of Fig. 6. With
+MeerKAT UHF-band 10 h observation, the value of Δ𝑃/𝑃 could
+reach 1 roughly. The values of Δ𝑃/𝑃 are distinctly reduced when
+tracking 1000 h, approximately reaching 0.1. However, we find that
+when the integration time increases from 1000 h to 10000 h, the
+reduction of Δ𝑃/𝑃 is not significant at low 𝑘. It is mainly because
+the cosmic variance, which is limited by the survey volume, plays
+the dominating role.
+Therefore, we consider tracking multiple points equally in the total
+10000 h observation. In this case, compared to tracking one point
+with 10000 h, the survey volume 𝑉bin and the thermal noise power
+spectrum 𝑃N are increased by a factor of the number of points 𝑁.
+In our analysis, we calculate the additional fractional error on 𝑃(𝑘)
+for 𝑁 = 10, 100 and 1000 in the 10000 h observation, as shown
+in the bottom panel of Fig. 6. In order to constrain cosmological
+parameters, we expect to obtain lower Δ𝑃/𝑃 in low 𝑘. We find that
+the lower values of Δ𝑃/𝑃 in low 𝑘 are obtained when tracking 100
+points for MeerKAT L-band in the total 10000 h observation, while
+tracking 10 points for MeerKAT UHF-band. Therefore, we employ
+these two survey strategies for MeerKAT L-band and UHF-band,
+respectively, in the next subsection.
+3.2 Cosmological parameters
+In this subsection, we explore the capability of MeerKAT Hi IM
+survey with interferometer mode of constraining cosmological pa-
+rameters using the Fisher matrix method. Given the power spectrum
+measurement at a given redshift, the Fisher matrix for a set of ob-
+servables {𝑝} can be written as
+𝐹𝑖 𝑗 =
+1
+8𝜋2𝑉bin
+∫ 1
+−1
+d𝜇
+∫ 𝑘max
+𝑘min
+𝑘2d𝑘 𝜕𝑃tot
+𝜕𝑝𝑖
+𝜕𝑃tot
+𝜕𝑝 𝑗
+.
+(17)
+Here, we take the set of observables as {𝐷 𝐴(𝑧𝑖), 𝐻(𝑧𝑖), 𝑓 𝜎8(𝑧𝑖),
+𝑏𝜎8(𝑧𝑖), 𝜎NL} in each redshift bin 𝑧𝑖. The nuisance parameters
+𝑏𝜎8(𝑧𝑖) and 𝜎NL can be marginalized by selecting the submatrix of
+𝐹−1
+𝑖 𝑗 with only the appropriate columns and rows. Therefore, we can
+derive the measurement errors on 𝐷 𝐴(𝑧), 𝐻(𝑧) and 𝑓 𝜎8(𝑧).
+For MeerKAT L-band (900-1200 MHz) and UHF-band (580-1000
+MHz), we divide these frequency bands into some bins with equal
+bandwidth Δ𝜈 = 60 MHz and then obtain the estimates for the
+measurement errors on observables in the corresponding redshift
+bins. We plot the fractional measurement errors on 𝐷 𝐴(𝑧), 𝐻(𝑧)
+and 𝑓 𝜎8(𝑧) with MeerKAT 10000 h observation in Fig. 7. We find
+that the survey with interferometer mode has a better measurement
+on 𝐷 𝐴(𝑧), of which the fractional errors can reach roughly 10%
+for MeerKAT L-band and UHF-band. Comparatively speaking, the
+fractional measurement errors on 𝐻(𝑧) and 𝑓 𝜎8(𝑧) seem slightly
+larger, though MeerKAT UHF-band preforms slightly better than
+MeerKAT L-band.
+Next, from the cosmological measurements on 𝐷 𝐴(𝑧), 𝐻(𝑧) and
+𝑓 𝜎8(𝑧), we can constrain the various dark enenrgy models, including
+the ΛCDM, 𝑤CDM and CPL models, by performing a Markov Chain
+Monte Carlo (MCMC) analysis. The 1𝜎 errors of the cosmological
+parameters are summarized in Table 1. In addition, The 1𝜎 and
+2𝜎 posterior distribution contours for cosmological parameters are
+shown in Figs. 8–10.
+In the flat ΛCDM model, we obtain 𝜎(Ωm) = 0.044 and 𝜎(𝐻0) =
+2.8 km s−1 Mpc−1 with MeerKAT L-band and 𝜎(Ωm) = 0.028
+and 𝜎(𝐻0) = 2.0 km s−1 Mpc−1 with MeerKAT UHF-band. We
+find that UHF-band performs better than L-band in constraining
+Ωm and 𝐻0. Recently, Cunnington (2022) gave a result of 𝐻0 =
+69.1+8.4
+−5.7 km s−1 Mpc−1 with MeerKAT UHF-band 4000 h survey
+with single-dish mode. It is found that we give a better constraint
+on 𝐻0 with MeerKAT UHF-band interferometric mode although we
+use a longer observational time of 10000 h. In comparison with other
+radio telescopes, MeerKAT L-band and BINGO perform similarly,
+while MeerKAT UHF-band performs nearly as well as FAST in
+constraining Ωm and 𝐻0 in the flat ΛCDM model (Wu & Zhang
+2022). Compared to the Stage-III dark energy experiments, such as
+DES, we find that MeerKAT UHF-band gives a smaller error on Ωm
+than DES with Ωm = 0.339+0.032
+−0.031 in the ΛCDM model (Abbott et al.
+2020).
+In the 𝑤CDM model, in order to help break the parameter degen-
+MNRAS 000, 1–10 (2023)
+
+8
+M. Zhang et al.
+Table 1. The 1𝜎 errors of the cosmological parameters in the ΛCDM, 𝑤CDM, and CPL models using MeerKAT L-band and UHF-band or in combination
+with Planck data. Note that here 𝐻0 is in units of km s−1 Mpc−1.
+Error
+ΛCDM
+𝑤CDM
+CPL
+L-band
+UHF-band
+Planck+L-band
+Planck+UHF-band
+Planck+L-band
+Planck+UHF-band
+𝜎(Ωm)
+0.044
+0.028
+0.030
+0.024
+0.092
+0.046
+𝜎(𝐻0)
+2.8
+2.0
+3.5
+2.6
+6.1
+4.1
+𝜎(𝑤)
+−
+−
+0.12
+0.08
+−
+−
+𝜎(𝑤0)
+−
+−
+−
+−
+1.1
+0.6
+𝜎(𝑤𝑎)
+−
+−
+−
+−
+4.3
+2.0
+0.2
+0.3
+0.4
+0.5
+Ωm
+60
+65
+70
+75
+H0 [km s−1 Mpc−1]
+L-band
+UHF-band
+Figure 8. Constraints on Ωm and 𝐻0 with MeerKAT L-band and UHF-band
+in the ΛCDM model.
+eracy, we combine the BAO data from MeerKAT IM with Planck
+TT,TE,EE+lowE power spectrum (Aghanim et al. 2020) in the
+MCMC analysis. The 1𝜎 and 2𝜎 measurement error contours for
+Ωm, 𝐻0 and dark-energy equation of state parameter 𝑤 are shown in
+Fig. 9. We obtain 𝜎(Ωm) = 0.030, 𝜎(𝐻0) = 3.5 km s−1 Mpc−1
+and 𝜎(𝑤) = 0.12 with Planck+L-band and 𝜎(Ωm) = 0.024,
+𝜎(𝐻0) = 2.6 km s−1 Mpc−1 and 𝜎(𝑤) = 0.08 with Planck+UHF-
+band. It can be seen that MeerKAT UHF-band combined with Planck
+data gives tighter constraints on cosmological parameters in the
+𝑤CDM model, with the conclusion the same as in the ΛCDM model.
+For dark energy, we find that MeerKAT has a very limited capability
+of constraining 𝑤, and the error on 𝑤 is still larger even though in
+combination with Planck data.
+Finally, we forecast the constraints on cosmological parameters
+in the CPL model. The 1𝜎 and 2𝜎 measurement error contours are
+shown in Fig. 10. We focus on the DE equation of state parameters 𝑤0
+and 𝑤𝑎. We obtain 𝜎(𝑤0) = 1.1 and 𝜎(𝑤𝑎) = 4.3 with Planck+L-
+band, and 𝜎(𝑤0) = 0.6 and 𝜎(𝑤𝑎) = 2.0 with Planck+MeerKAT
+UHF-band. Note that dark energy dominates the evolution of the
+universe in the redshift range of 𝑧 ≲ 0.4. MeerKAT L-band has
+a very limited constraining power for dark energy at the range of
+0.18 < 𝑧 < 0.58, and MeerKAT UHF-band only surveys at the range
+of 0.42 < 𝑧 < 1.45. Therefore, MeerKAT in interferometer mode
+−1.5
+−1.0
+w
+0.2
+0.3
+0.4
+Ωm
+60
+70
+80
+H0 [km s−1 Mpc−1]
+60
+70
+80
+H0 [km s−1 Mpc−1]
+0.3
+0.4
+Ωm
+Planck+L-band
+Planck+UHF-band
+Figure 9. Constraints on Ωm, 𝐻0 and 𝑤 with MeerKAT L-band and UHF-
+band in combination with Planck data in the 𝑤CDM model.
+cannot give stringent constraints on dark energy. But we still keep
+optimistic since the precise measurements on dark energy would be
+achieved by the future larger radio telescopes, such as HIRAX and
+SKA (Wu & Zhang 2022; Wu et al. 2022a,b).
+4 CONCLUSIONS
+In this work, we give a detailed analysis on measuring the Hi IM
+delay power spectrum using the MeerKAT interferometer mode. We
+also discuss the capability of MeerKAT interferometer mode of con-
+straining cosmological parameters.
+We use the Fisher matrix method to estimate the Hi power spec-
+trum with MeerKAT IM observation. We find that the different survey
+fields have the distinct impacts on determining the power spectrum
+errors in the limited observational time of 10 hours. As the obser-
+vational time increases from 10 h to 10000 h, the power spectrum
+errors are reduced evidently until the cosmic variance begins to dom-
+inate. We also discuss the different survey strategies and find that the
+lower fractional errors on power spectrum at low 𝑘 are obtained when
+tracking 100 points for L-band and tracking 10 points for UHF-band
+in a total 10000 h observation.
+We obtain the measurement errors on 𝐷 𝐴(𝑧), 𝐻(𝑧) and 𝑓 𝜎8(𝑧)
+MNRAS 000, 1–10 (2023)
+
+Hi delay spectrum with MeerKAT interferometer mode
+9
+−2
+0
+2
+w0
+0.2
+0.4
+0.6
+Ωm
+50
+60
+70
+80
+H0 [km s−1 Mpc−1]
+−10
+−5
+0
+wa
+−10
+−5
+0
+wa
+50
+60
+70
+80
+H0 [km s−1 Mpc−1]
+0.4
+0.6
+Ωm
+Planck+L-band
+Planck+UHF-band
+Figure 10. Constraints on Ωm, 𝐻0, 𝑤0 and 𝑤𝑎 with MeerKAT L-band and
+UHF-band in combination with Planck data in the CPL model.
+by using the Fisher matrix, and then use these measurements to
+constrain cosmological parameters in typical dark energy models, in-
+cluding ΛCDM, 𝑤CDM and CPL models, by performing the MCMC
+analysis. We obtain 𝜎(Ωm) = 0.028 and 𝜎(𝐻0) = 2.0 km s−1
+Mpc−1 with MeerKAT UHF-band which are better than the results
+of 𝜎(Ωm) = 0.044 and 𝜎(𝐻0) = 2.8 km s−1 Mpc−1 with MeerKAT
+L-band in the ΛCDM model. However, MeerKAT has a very limited
+constraining power for dark-energy equation of state, such as 𝑤 in
+the 𝑤CDM model and 𝑤0 and 𝑤𝑎 in the CPL model, even though in
+combination with Planck data.
+Though MeerKAT L-band and UHF-band Hi IM surveys in in-
+terferometer mode have a very limited constraining power for dark
+energy, our analysis still provide a useful guide for the near future
+MeerKAT survey. It is expected that the future larger radio telescope
+arrays, such as SKA, will have a much better and powerful perfor-
+mance on cosmological research. In addition, MeerKAT baselines
+are not short enough for detecting large cosmological scales, but the
+measurements with MeerKAT interferometer mode on these scales
+are still very useful in detecting Hi content of galaxies, obtaining the
+cross-correlation between Hi content and star formation rates (Wolz
+et al. 2016), constraining warm dark matter (Carucci et al. 2015) and
+breaking the degeneracy between ΩHi and 𝑏Hi (Chen et al. 2021).
+These aspects deserve further detailed investigations in the future.
+ACKNOWLEDGEMENTS
+We thank Peng-Ju Wu and Li-Yang Gao for helpful discussions.
+This work was supported by the National SKA Program of China
+(Grants Nos. 2022SKA0110200 and 2022SKA0110203) and the Na-
+tional Natural Science Foundation of China (Grants Nos. 11975072,
+11875102, and 11835009).
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the corresponding author.
+REFERENCES
+Abbott T. M. C., et al., 2018, Phys. Rev. D, 98, 043526
+Abbott T. M. C., et al., 2020, Phys. Rev. D, 102, 023509
+Aghanim N., et al., 2020, Astron. Astrophys., 641, A6
+Alam S., et al., 2017, Mon. Not. Roy. Astron. Soc., 470, 2617
+Amendola L., et al., 2018, Living Rev. Rel., 21, 2
+Anderson C. J., et al., 2018, Mon. Not. Roy. Astron. Soc., 476, 3382
+Ansari R., et al., 2012, Astron. Astrophys., 540, A129
+Bacon D. J., et al., 2020, Publ. Astron. Soc. Austral., 37, e007
+Bagla J. S., Khandai N., Datta K. K., 2010, Mon. Not. Roy. Astron. Soc., 407,
+567
+Ballardini M., Matthewson W. L., Maartens R., 2019, Mon. Not. Roy. Astron.
+Soc., 489, 1950
+Bandura K., et al., 2014, Proc. SPIE Int. Soc. Opt. Eng., 9145, 22
+Battye R. A., Davies R. D., Weller J., 2004, Mon. Not. Roy. Astron. Soc.,
+355, 1339
+Berti M., Spinelli M., Haridasu B. S., Viel M., Silvestri A., 2022, JCAP, 01,
+018
+Beutler F., et al., 2011, Mon. Not. Roy. Astron. Soc., 416, 3017
+Blake C., et al., 2011, Mon. Not. Roy. Astron. Soc., 418, 1707
+Bull P., Ferreira P. G., Patel P., Santos M. G., 2015, Astrophys. J., 803, 21
+Camera S., Santos M. G., Ferreira P. G., Ferramacho L., 2013, Phys. Rev.
+Lett., 111, 171302
+Carucci I. P., Villaescusa-Navarro F., Viel M., Lapi A., 2015, JCAP, 07, 047
+Chang T.-C., Pen U.-L., Peterson J. B., McDonald P., 2008, Phys. Rev. Lett.,
+100, 091303
+Chang T.-C., Pen U.-L., Bandura K., Peterson J. B., 2010, Nature, 466, 463
+Chen X., 2012, Int. J. Mod. Phys. Conf. Ser., 12, 256
+Chen Z., Wolz L., Spinelli M., Murray S. G., 2021, Mon. Not. Roy. Astron.
+Soc., 502, 5259
+Cheng G., Ma Y.-Z., Wu F., Zhang J., Chen X., 2020, Phys. Rev. D, 102,
+043517
+Chevallier M., Polarski D., 2001, Int. J. Mod. Phys. D, 10, 213
+Chisari N. E., et al., 2019, Astrophys. J. Suppl., 242, 2
+Cole S., et al., 2005, Mon. Not. Roy. Astron. Soc., 362, 505
+Colless M., et al., 2001, Mon. Not. Roy. Astron. Soc., 328, 1039
+Cunnington S., 2022, Mon. Not. Roy. Astron. Soc., 512, 2408
+Cunnington S., Camera S., Pourtsidou A., 2020, Mon. Not. Roy. Astron. Soc.,
+499, 4054
+Cunnington S., et al., 2022, arXiv e-prints, p. arXiv:2206.01579
+Datta A., Bowman J. D., Carilli C. L., 2010, Astrophys. J., 724, 526
+Dey A., et al., 2019, Astron. J., 157, 168
+Dickinson C., 2014, in 49th Rencontres de Moriond on Cosmology. pp 139–
+142 (arXiv:1405.7936)
+Drinkwater M. J., et al., 2010, Mon. Not. Roy. Astron. Soc., 401, 1429
+Irfan M. O., et al., 2021, Mon. Not. Roy. Astron. Soc., 509, 4923
+Ivezić v., et al., 2019, Astrophys. J., 873, 111
+Jarvis M. J., et al., 2018, PoS, MeerKAT2016, 006
+Jin S.-J., He D.-Z., Xu Y., Zhang J.-F., Zhang X., 2020, JCAP, 03, 051
+Jin S.-J., Wang L.-F., Wu P.-J., Zhang J.-F., Zhang X., 2021, Phys. Rev. D,
+104, 103507
+Jones D. H., et al., 2009, Mon. Not. Roy. Astron. Soc., 399, 683
+Kaiser N., 1987, Mon. Not. Roy. Astron. Soc., 227, 1
+Karagiannis D., Slosar A., Liguori M., 2020, JCAP, 11, 052
+Karagiannis D., Fonseca J., Maartens R., Camera S., 2021, Phys. Dark Univ.,
+32, 100821
+Knowles K., et al., 2021, Mon. Not. Roy. Astron. Soc., 504, 1749
+Lewis A., Challinor A., Lasenby A., 2000, Astrophys. J., 538, 473
+Li Y.-C., Ma Y.-Z., 2017, Phys. Rev. D, 96, 063525
+Li L., Staveley-Smith L., Rhee J., 2021a, Res. Astron. Astrophys., 21, 030
+Li Y., Santos M. G., Grainge K., Harper S., Wang J., 2021b, MNRAS, 501,
+4344
+Lidz A., Furlanetto S. R., Oh S. P., Aguirre J., Chang T.-C., Dore O., Pritchard
+J. R., 2011, Astrophys. J., 741, 70
+Linder E. V., 2003, Phys. Rev. Lett., 90, 091301
+Liu A., Shaw J. R., 2020, Publ. Astron. Soc. Pac., 132, 062001
+Liu A., Parsons A. R., Trott C. M., 2014a, Phys. Rev. D, 90, 023018
+MNRAS 000, 1–10 (2023)
+
+10
+M. Zhang et al.
+Liu A., Parsons A. R., Trott C. M., 2014b, Phys. Rev. D, 90, 023019
+Loeb A., Wyithe S., 2008, Phys. Rev. Lett., 100, 161301
+Maartens R., Abdalla F. B., Jarvis M., Santos M. G., 2015, PoS, AASKA14,
+016
+Maddox, N. et al., 2021, A&A, 646, A35
+Masui K. W., et al., 2013, Astrophys. J. Lett., 763, L20
+McQuinn M., Zahn O., Zaldarriaga M., Hernquist L., Furlanetto S. R., 2006,
+Astrophys. J., 653, 815
+Morales M. F., Hazelton B., Sullivan I., Beardsley A., 2012, Astrophys. J.,
+752, 137
+Nan R., et al., 2011, Int. J. Mod. Phys. D, 20, 989
+Newburgh L. B., et al., 2016, Proc. SPIE Int. Soc. Opt. Eng., 9906, 99065X
+Obuljen A., Castorina E., Villaescusa-Navarro F., Viel M., 2018, JCAP, 05,
+004
+Olivari L. C., Dickinson C., Battye R. A., Ma Y.-Z., Costa A. A., Remazeilles
+M., Harper S., 2018, Mon. Not. Roy. Astron. Soc., 473, 4242
+Parsons A. R., Backer D. C., 2009, Astron. J., 138, 219
+Parsons A. R., Pober J. C., Aguirre J. E., Carilli C. L., Jacobs D. C., Moore
+D. F., 2012, Astrophys. J., 756, 165
+Paul S., et al., 2016, Astrophys. J., 833, 213
+Paul S., Santos M. G., Townsend J., Jarvis M. J., Maddox N., Collier J. D.,
+Frank B. S., Taylor R., 2021, Mon. Not. Roy. Astron. Soc., 505, 2039
+Pober J. C., 2015, Mon. Not. Roy. Astron. Soc., 447, 1705
+Pourtsidou A., 2018, PoS, MeerKAT2016, 037
+Pourtsidou A., Bacon D., Crittenden R., 2017, Mon. Not. Roy. Astron. Soc.,
+470, 4251
+Pritchard J. R., Loeb A., 2012, Rept. Prog. Phys., 75, 086901
+Santos M. G., et al., 2015, PoS, AASKA14, 019
+Santos M. G., et al., 2017, in MeerKAT Science: On the Pathway to the SKA.
+(arXiv:1709.06099)
+Scelfo G., Spinelli M., Raccanelli A., Boco L., Lapi A., Viel M., 2022, JCAP,
+01, 004
+Seo H.-J., Hirata C. M., 2016, Mon. Not. Roy. Astron. Soc., 456, 3142
+Seo H.-J., Dodelson S., Marriner J., Mcginnis D., Stebbins A., Stoughton C.,
+Vallinotto A., 2010, Astrophys. J., 721, 164
+Sprenger T., Archidiacono M., Brinckmann T., Clesse S., Lesgourgues J.,
+2019, JCAP, 02, 047
+Terni de Gregory B., et al., 2021, Mon. Not. Roy. Astron. Soc., 504, 2924
+Tramonte D., Ma Y.-Z., 2020, Mon. Not. Roy. Astron. Soc., 498, 5916
+Vedantham H., Shankar N. U., Subrahmanyan R., 2012, Astrophys. J., 745,
+176
+Viljoen J.-A., Fonseca J., Maartens R., 2021, JCAP, 11, 010
+Villaescusa-Navarro F., Bull P., Viel M., 2015, Astrophys. J., 814, 146
+Wang J., et al., 2021, Mon. Not. Roy. Astron. Soc., 505, 3698
+Wolz L., Tonini C., Blake C., Wyithe J. S. B., 2016, Mon. Not. Roy. Astron.
+Soc., 458, 3399
+Wolz L., et al., 2022, Mon. Not. Roy. Astron. Soc., 510, 3495
+Wu P.-J., Zhang X., 2022, JCAP, 01, 060
+Wu F., et al., 2021, Mon. Not. Roy. Astron. Soc., 506, 3455
+Wu P.-J., Shao Y., Jin S.-J., Zhang X., 2022a, arXiv e-prints, p.
+arXiv:2202.09726
+Wu P.-J., Li Y., Zhang J.-F., Zhang X., 2022b, arXiv e-prints, p.
+arXiv:2212.07681
+Wuensche C. A., et al., 2020, Exper. Astron., 50, 125
+Wyithe S., Loeb A., Geil P., 2008, Mon. Not. Roy. Astron. Soc., 383, 1195
+Xiao L., Costa A. A., Wang B., 2021, Mon. Not. Roy. Astron. Soc., 510, 1495
+Xu Y., Zhang X., 2020, Sci. China Phys. Mech. Astron., 63, 270431
+Xu Y., Wang X., Chen X., 2015, Astrophys. J., 798, 40
+Xu X., Ma Y.-Z., Weltman A., 2018, Phys. Rev. D, 97, 083504
+Zhang J.-F., Wang B., Zhang X., 2020, Sci. China Phys. Mech. Astron., 63,
+280411
+Zhang M., Wang B., Wu P.-J., Qi J.-Z., Xu Y., Zhang J.-F., Zhang X., 2021,
+Astrophys. J., 918, 56
+de Villiers M. S., Cotton W. D., 2022, Astron. J., 163, 135
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–10 (2023)
+

ENFRT4oBgHgl3EQfAze2/content/tmp_files/2301.13463v1.pdf.txt

@@ -0,0 +1,974 @@
+arXiv:2301.13463v1  [hep-ph]  31 Jan 2023
+Higgs production at next generation e+e− colliders
+Deniz YILMAZ1∗, Mehmet SAHIN2, Dogukan Hazar YAVUZ1
+1Physics Engineering Department, Ankara University, Ankara,Turkey
+2Department of Computer Engineering, Usak University, Usak,Turkey
+February 1, 2023
+Abstract
+In this study, Higgs production processes, Higgsstrahlung and vector boson (W and Z) fusion
+processes, were investigated for four different future lepton colliders (CEPC, ILC, CLIC, and FCC-
+ee). The cross sections for each production process and corresponding backgrounds were calculated
+considering the ISR and beamstrahlung effects. Various cuts and the b-tagging method were used
+to reduce the background.
+Finally, the number of events for each collider was determined, and
+significance calculations were performed. In our calculations, high event numbers were obtained for
+all four colliders for the Higgsstrahlung, W, and Z fusion process. This shows that electron-positron
+colliders will play an important role in future Higgs physics research.
+1
+Introduction
+The discovery of the Higgs boson at the Large Hadron Collider (LHC) [1, 2] confirmed the electroweak
+symmetry breaking mechanism of the Standard Model (SM) [3, 4, 5, 6]. However, there is still some
+unknown about the observed Higgs boson: is it the fundamental scalar of the SM, or a more complex
+object, or part of an extended Higgs sector? Studying the properties of the Higgs boson at the LHC
+and in future colliders is crucial to understanding its true nature. Up to now, some properties of the
+Higgs boson have been measured at the LHC with an accuracy of about 10% [7, 8, 9, 10]. Although the
+LHC Run 2 to be developed will examine it with higher data, because of the complexity of the internal
+structure of the proton, the LHC will not be sensitive enough to examine the properties of Higgs.
+Electron-positron colliders, which will be installed to precisely measure the properties of the Higgs
+particle, have unique capabilities for the measurement of Higgs boson parameters, including the Higgs
+total cross section, decay width, branching ratios, Higgs width, and determination of Higgs couplings.
+Therefore, today, four e+e− colliders are being designed to study the properties of the Higgs boson
+and other standard model (SM) particles with high precision: the International Linear Collider (ILC)
+[11], with a center of mass energy of 250 – 500 GeV, Compact Linear Collider (CLIC) [12] with center
+of mass energies of 380 – 1500 – 3000 GeV, Circular Electron Positron Collider (CEPC) with center of
+mass energies between 90 and 250 GeV [13] and the Future e+e− Circular Collider (FCC-ee) [14], which
+will be located in a new tunnel at CERN at 240 GeV center of mass energy. The main beam parameters
+of these colliders [11, 12, 13, 14] are given in Table 1. The integrated luminosities given here are annual
+values.
+Table 1: The main collider parameters
+Parameters
+CEPC
+FCC-ee
+ILC
+CLIC
+Center of mass energy (GeV)
+240
+240
+250
+500
+380
+1500
+3000
+Number of particles per bunch (1010)
+15
+18
+2
+2
+0.52
+0.37
+0.37
+Horizontal beam size at IP (σx) (µm)
+20.9
+13.7
+0.516
+0.474
+0.149
+0.06
+0.04
+Vertical beam size at IP (σy) (nm)
+60
+36
+7.66
+5.86
+2.9
+1.5
+1
+Bunch length (mm)
+4.4
+5.3
+0.3
+0.3
+0.07
+0.044
+0.044
+Luminosity (105pb−1 )
+6
+17
+1.35
+1.8
+1.5
+3.7
+5.9
+In the electron-positron collider, Higgs bosons are produced by the Higgsstrahlung and vector boson
+(W and Z) fusion processes [15, 16, 17, 18, 19, 20, 21]. In this study, these three processes were examined
+∗dyilmaz@eng.ankara.edu.tr
+1
+
+Figure 1: The Feynmann diagrams of the Higgs production processes
+and calculations were performed using CalcHEP [22, 23]. In the electron-positron collider, it is important
+to consider the effects of ISR and Beamstrahlung [24, 25]. The parameters listed in Table 1 were used
+to calculate the ISR and the beamstraghlung effects. In section 2 cross sections are given for these three
+processes. Section 3 provides signal and background analyses, the number of events for each collider, and
+the significance calculations. Finally, conclusion is provided in the section 4.
+2
+Higgs Production at the electron - positron colliders
+The main production processes of Higgs at the e+e− colliders are the Higgsstrahlung and W/Z fusion
+mechanisms given below, as shown in Figure 1.
+Higgs − strahlung
+e+e− → ZH
+Wfusion
+e+e− → νeνeH
+Zfusion
+e+e− → e+e−H
+The cross section for the Higgsstrahlung process can be written as
+σ(e+e− → ZH) = G2
+F M 4
+Z
+96πs (η2
+e + a2
+e)κ1/2 κ + 12M 2
+Z/s
+(1 − M 2
+Z/s)2
+(1)
+where ae = −1, ηe = −1 + 4sin2θW are the Z charges of the electron and κ = (1 − (MH + MZ)2/s)(1 −
+(MH − MZ)2/s) is the usual two particle phase space function. The total cross section for the vector
+boson fusion mechanism is
+σ(e+e− → V V → llH) = G2
+F m4
+V
+64
+√
+2π3
+� 1
+xH
+dx
+� 1
+x
+dyT (x, y)
+[1 + (y − x)/xV ]2 .
+(2)
+T (x, y) = (2x
+y3 − 3x + 1
+y2
++ x + 2
+y
+− 1)[
+z
+z + 1 − log(z + 1)] + xz2(1 − y)
+y3(z + 1) ,
+where V denotes the vector bosons W or Z and xH = m2
+H/s, xV = m2
+V /s and z = y(x − xH)/xxV (√s
+is the center-of-mass energy) [26].
+200
+250
+300
+350
+400
+450
+500
+√s [GeV]
+0
+0,05
+0,1
+0,15
+0,2
+0,25
+σ [pb]
+Higgsstrahlung
+W Fusion
+Z Fusion
+Figure 2: The cross sections of the Higgs production mechanisms as a function of center-of-mass energy.
+2
+
+200
+250
+300
+350
+400
+450
+500
+√s [GeV]
+0,05
+0,1
+0,15
+0,2
+0,25
+σ [pb]
+CEPC
+ILC
+FCC-ee
+CLIC
+Higgsstrahlung
+200
+250
+300
+350
+400
+450
+500
+√s [GeV]
+0,01
+0,02
+0,03
+0,04
+0,05
+0,06
+0,07
+σ [pb]
+CEPC
+ILC
+FCC-ee
+CLIC
+W Fusion
+200
+250
+300
+350
+400
+450
+500
+√s [GeV]
+0,001
+0,002
+0,003
+0,004
+0,005
+0,006
+0,007
+σ [pb]
+CEPC
+ILC
+FCC-ee
+CLIC
+Z Fusion
+Figure 3: The cross section comparison for Higgsstrahlung, W Fusion and Z Fusion processes for four
+e+e−colliders.
+The behavior of the production cross-sections of the Higgs boson calculated by the Higgsstrahlung
+and the W/Z fusion mechanisms using the CalcHEP simulation program, depending on the center of
+mass energy, are shown in Figure 2 and Figure 3. The relevant production cross sections as a function of
+the center of mass energy are shown in Figure 2. As shown in Figure 2, the Higgsstrahlung suppresses the
+vector boson production processes for moderate values of the energy due to the additional electroweak
+coupling. With the increase in energy, the cross sections of the vector boson procecesses increase log-
+arithmically and become dominant. At a center of mass energy of about 250 GeV, Higgs bosons are
+predominantly produced from the ZH process as seen in the same figure. In the Figure 3, the cross
+sections are shown as a function of the center of mass energy for each production mechanisms for four
+electron-positron colliders with the ISR and the beamstrahlung effects of each colliders.
+3
+Signal and Background Analyses
+Because the Higgs boson’s decay rate to bb is greater than the decay rate to other quarks and leptons
+[27, 28, 29, 30, 31], the bb decay mode of Higgs (H −→ bb) is considered in all production processes in
+this study. Since the cross sections of the background processes corresponding to the leptonic decays of
+the Z boson are less than the background cross sections corresponding to the other decays, the leptonic
+decays of the Z boson in the Higgsstrahlung process are taken into account. The signal processes are
+given below.
+Signal 1:
+Higgsstrahlung
+e+e− → ZH → llbb
+Zfusion
+e+e− → e+e−bb
+Signal 2:
+Wfusion
+e+e− → νeνebb
+3
+
+Here, l and l are e−, µ− and e+, µ+, respectively. The corresponding background processes analysed here
+are as follows:
+For signal 1:
+i)
+e+e− → ZZ → llJJ,
+ii)
+e+e− → e+e−Z → e+e−JJ,
+iii)
+e+e− → tt → W +JW −J → llJJνlνl,
+For signal 2:
+e+e− → JJ,
+here, J represents the quark and antiquark: J = d, d, u, u, s, s, c, c, b, b. The transverse momentum (PT ),
+pseudo rapidity (η) and invariant mass (Minv) distributions of the final state particles were investigated
+by using CalcHEP program in order to find the cut values to distinguish the signal from the background
+in the FCC-ee collider with a center of mass energy of 240 GeV. The background iii process corresponding
+to Signal 1 is not included in the calculations for 240 GeV, as it starts to contribute at 350 GeV and
+greater center of mass energies. Because the transverse momentum, pseudo rapidity, and invariant mass
+distributions of the final state particles in the signal and background processes will exhibit similar behavior
+for other colliders, the cut values obtained can be used for CEPC, ILC, and CLIC. Transverse momentum
+distribution plots for the final state particles of signal 1 and the corresponding background processes i
+and ii are shown in Figure 4, while the graphs of signal 2 are shown in Figure 5. As can be seen from
+Figure 4 and 5, when a transverse momentum cut of 35 GeV is applied to the e−, e+, µ−, µ+, and two
+jets (J) in the final state particles of signal 1 and signal 2 and the corresponding background processes,
+the signal will almost not change, but the background will be significantly reduced.
+Pseudorapidity plots for signal 1, signal 2, and the corresponding backgrounds are shown in Figure
+6 and 7. As can be seen from the figures, cut regions of −2.5 < ηJ,J < 2.5, −2.5 < ηe−,e+ < 2.5,
+−2.5 < ηµ−,µ+ < 2.5 will be appropriate for e−, e+, µ−, µ+ and two jets (J) in the final state particles
+of signal 1 and signal 2 and the corresponding background processes.
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+√s=240 GeV
+(1/σ)dσ/dpT (1/GeV)
+pT (GeV)
+signal 1
+background i
+background ii
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+√s=240 GeV
+(1/σ)dσ/dpT (1/GeV)
+pT (GeV)
+signal 1
+background i
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+√s=240 GeV
+(1/σ)dσ/dpT (1/GeV)
+pT (GeV)
+signal 1
+background i
+background ii
+Figure 4: Transverse momentum distribution plots for the e−/e+ (upper left ), µ−/µ+ (upper right)
+and J/J (bottom) final state particles of signal 1 and the corresponding bacground processes in FCC-ee
+collider with 240 GeV center of mass energy.
+4
+
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+√s=240 GeV
+(1/σ)dσ/dpT (1/GeV)
+pT (GeV)
+signal 2
+background
+Figure 5: Transverse momentum distribution plots for the b/b and J/J final state particles of signal 2
+and the corresponding bacground processes in FCC-ee collider with 240 GeV center of mass energy.
+An Emiss
+T
+cut value of >15 GeV was also used for neutrinos in our calculations.
+Invariant mass distribution plots for signal 1, signal 2 and their corresponding background processes
+are shown in Figure 8. As can be seen from the figures, in the calculations, it would be appropriate to
+exclude the 80 GeV < Minv(e−, e+) <100 GeV and 80 GeV < Minv(µ−, µ+) < 100 GeV regions for the
+ll final states in signal and background processes. In addition, only the 115 GeV < Minv(J, J) <135
+GeV region was included in the calculations for two final jet states in the signal and background pro-
+cesses. These included and excluded invariant mass regions allow the signal to be distinguished from the
+background.
+In addition to these cut values, the separation cuts of ∆R(l, J) >0.5 and ∆R(l, J) >0.5 distinguish
+the final state leptons and antileptons from the jets, while the ∆R(J, J) >0.5 separation cut was used to
+distinguish the final state jets from each other.
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+-6
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+√s=240 GeV
+(1/σ)dσ/dη
+ηe
+signal 1
+background i
+background ii
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+-6
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+√s=240 GeV
+(1/σ)dσ/dη
+ηµ
+signal 1
+background i
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+-6
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+√s=240 GeV
+(1/σ)dσ/dη
+ηJ
+signal 1
+background i
+background ii
+Figure 6: Pseudorapidity distribution plots for the e−/e+ (upper left ), µ−/µ+ (upper right) and J/J
+(bottom) final state particles of signal 1 and the corresponding bacground processes in FCC-ee collider
+with 240 GeV center of mass energy.
+5
+
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+-6
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+√s=240 GeV
+(1/σ)dσ/dη
+ηJ
+signal 2
+background 
+Figure 7: Pseudorapidity distribution plots for the b/b and J/J final state particles of signal 2 and the
+corresponding bacground processes in FCC-ee collider with 240 GeV center of mass energy.
+All the cut values obtained are listed in Table 2, and these cut values were used in the calculations for
+the four colliders. In addition to the cut values in Table 2, because the Higgs boson decays to bb in our
+signal processes, it is possible to further reduce the background cross section value using the b-tagging
+method [27]: 68% is used for the b-tagging identification rate, and a 1% ratio is used for misidentification
+rate with light quarks as b quarks.
+The following equation is used to calculate the significance of the
+10-8
+10-6
+10-4
+10-2
+100
+ 0
+ 25
+ 50
+ 75
+ 100
+ 125
+ 150
+ 175
+ 200
+ 225
+ 250
+√s=240 GeV
+dσ/dMinv (pb/GeV)
+Minv (GeV)
+signal Z(e-,e+)
+signal h(b, b-)
+background Z(e-,e+)
+background γ(e-,e+)
+background Z(J,J)
+10-8
+10-6
+10-4
+10-2
+100
+ 0
+ 25
+ 50
+ 75
+ 100
+ 125
+ 150
+ 175
+ 200
+ 225
+ 250
+√s=240 GeV
+(1/σ)dσ/dMinv (1/GeV)
+Minv (GeV)
+signal h(b,b-)
+background Z(J,J))
+Figure 8: Invariant mass plots for the signal 1 (left) and signal 2 (right) and the corresponding bacground
+processes in FCC-ee collider with 240 GeV center of mass energy.
+obtained data:
+S =
+�
+2((s + b) ln(1 + s/b) − s)
+(3)
+where s and b represent signal and background events, respectively [32]. Cross-sections, event rates, and
+significance values were calculated for the signal and background processes using the cut values in Table 2,
+Table 2: Cut values
+Emiss
+T
+(νl, νl) > 15 GeV
+PT (l, l) > 35 GeV
+PT (J) > 35 GeV
+-2.5 < η(l, l) < 2.5
+-2.5 < η(J) < 2.5
+80 GeV < Minv(l, l) < 100 GeV region is excluded
+115 GeV< Minv(J, J) < 135 GeV region is included
+∆R(l, J) > 0.5
+∆R(l, J) > 0.5
+∆R(J, J) > 0.5
+6
+
+Table 3: Cross sections, number of events and the significance values for CEPC.
+Colliders
+Processes
+Sg CS
+(pb)
+Bg CS
+(pb)
+L
+(pb−1)
+No. SgE
+No.BgE
+S
+CEPC
+(240 GeV)
+Signal 1
+9.91×10−5
+2.38×10−4
+6×105
+59.5
+142.8
+4.7
+Signal 1
+(with b-tagging)
+6.72×10−5
+3.68×10−5
+40.32
+22.08
+7
+Signal 2
+1.24×10−2
+5.22×10−1
+7440
+313200
+13.24
+Signal 2
+(with b-tagging)
+8.45×10−3
+7.17×10−2
+5070
+43020
+23.98
+b-tagging method, and nominal integrated luminosity given in Table 1. The event rates and significance
+values of the signals and corresponding backgrounds are obtained for four future lepton colliders. The
+numerical results are given in Table 3-6. The abbreviations used in the tables are: Sg CS (signal cross-
+section), Bg CS ( background cross-section),L (integrated luminosity), No. SgE (number of signal events)
+and No. BgE (number of background events).
+4
+Conclusion
+After the discovery of the Higgs particle, precise measurements of the Higgs properties became an im-
+portant step forward for future research in particle physics. Electron positron colliders to be installed
+for this purpose have unique capabilities for the measurement of Higgs boson parameters, including the
+Higgs total cross section of production processes, decay width, branching rates and determination of
+Higgs couplings. In this study, the Higgsstrahlung and W and Z fusion processes were examined, and the
+data obtained are presented in graphs and tables for four different electron-positron colliders. The pro-
+duction cross-sections for each process and additionally cross-sections for various final state backgrounds
+were calculated. In the calculations, we attempted to reduce the background by transverse momentum,
+pseudo rapidity, invariant mass, cone-angle constraints, and the b-tagging method. Significance calcula-
+tions were performed by determining the number of events related to the production processes and the
+background for each collider. The values are listed in Table 3-6.
+When the results are examined in Table 5 , it is seen that the desired significance value for Signal 1
+cannot be reached at the luminosity value given for ILC – 250 GeV. For Signal 1 processes to be observed
+in the ILC-250 GeV, the collider needs to accumulate data for a longer period of time. Again, at the end
+of one year, it was seen that the statistical significance value of 5σ would be reached after the b-tagging
+method for the Signal 1 processes in the CEPC collider. Therefore, the CEPC collider will enable the
+properties of the Higgs boson to be investigated precisely through Signal 1 processes. It is seen that at
+the end of 1 year in the FCC-ee collider, a significance value of 7.95 will be reached without b-tagging and
+a high significance value of 11.9 can be reached by using b-tagging. This shows that FCC-ee will be more
+advantageous than ILC-250 GeV and CEPC 250 GeV colliders for investigating Higgs boson properties
+through the Signal 1 group around these center of mass energies (240-250 GeV). In the ILC-500 GeV
+and CLIC-380-1500-3000 GeV colliders, results well above the desired significance value can be obtained
+for signal 1 processes, even without the b-tagging. Therefore, the properties of the Higgs boson through
+Signal 1 processes can be studied with precision in colliders other than the ILC-250 GeV collider. Since
+the results obtained for the Signal 2 process are greater than 5 significance values, the properties of the
+Table 4: Cross sections, number of events and the significance values for FCC-ee.
+Colliders
+Processes
+Sg CS
+(pb)
+Bg CS
+(pb)
+L
+(pb−1)
+No. SgE
+No.BgE
+S
+FCC-ee
+(240 GeV)
+Signal 1
+9.98×10−5
+2.36×10−4
+1.7×106
+169.7
+401.2
+7.95
+Signal 1
+(with b-tagging)
+6.79×10−5
+3.67×10−5
+115.4
+62.4
+11.9
+Signal 2
+1.25×10−2
+5.37×10−1
+21250
+912900
+22.15
+Signal 2
+(with b-tagging)
+8.53×10−3
+7.38×10−2
+14501
+125460
+40.18
+7
+
+Table 5: Cross sections, number of events and the significance values for ILC.
+Colliders
+Processes
+Sg CS
+(pb)
+Bg CS
+(pb)
+L
+(pb−1)
+No. SgE
+No.BgE
+S
+ILC
+(250 GeV)
+Signal 1
+1.33×10−4
+2.9×10−4
+1.35×105
+17.95
+39.15
+2.68
+Signal 1
+(with b-tagging)
+9.04×10−5
+4.34×10−5
+12.2
+5.86
+4.03
+Signal 2
+1.3×10−2
+5.33×10−1
+1755
+71955
+6.51
+Signal 2
+(with b-tagging)
+8.82×10−3
+7.32×10−2
+1190
+9882
+11.74
+ILC
+(500 GeV)
+Signal 1
+1.41×10−3
+1.7×10−3
+1.8×105
+253.8
+306
+12.98
+Signal 1
+(with b-tagging)
+9.57×10−4
+3.32×10−4
+172.3
+59.8
+16.88
+Signal 2
+2.86×10−2
+1.13×10−1
+5148
+20340
+34.71
+Signal 2
+(with b-tagging)
+1.94×10−2
+1.56×10−2
+3492
+2808
+56.54
+Table 6: Cross sections, number of events and the significance values for CLIC.
+Colliders
+Processes
+Sg CS
+(pb)
+Bg CS
+(pb)
+L
+(pb−1)
+No. SgE
+No.BgE
+S
+CLIC
+(380 GeV)
+Signal 1
+6.44×10−4
+1.08×10−3
+1.5×105
+96.6
+162
+6.98
+Signal 1
+(with b-tagging)
+4.38×10−4
+2.53×10−4
+65.7
+37.95
+8.76
+Signal 2
+1.7×10−2
+2.11×10−1
+2550
+31650
+14.14
+Signal 2
+(with b-tagging)
+1.16×10−2
+2.9×10−2
+1740
+4350
+24.86
+CLIC
+(1500 GeV)
+Signal 1
+1.95×10−3
+1.47×10−3
+3.7×105
+721.5
+543.9
+26.34
+Signal 1
+(with b-tagging)
+1.32×10−3
+2.34×10−4
+488.4
+86.58
+36.64
+Signal 2
+1.03×10−1
+9.22×10−3
+38110
+3411
+362
+Signal 2
+(with b-tagging)
+6.99×10−2
+1.27×10−3
+25863
+470
+400
+CLIC
+(3000 GeV)
+Signal 1
+6.01×10−4
+8.47×10−4
+5.9×105
+355
+499.7
+14.38
+Signal 1
+(with b-tagging)
+4.09×10−4
+1.32×10−4
+241
+77.8
+20.44
+Signal 2
+1.61×10−1
+2.72×10−3
+94990
+1605
+776
+Signal 2
+(with b-tagging)
+1.09×10−1
+3.74×10−4
+64310
+221
+777
+Higgs boson can be studied precisely for all colliders through this channel.
+As a result, in future lepton colliders, the Higgs boson can be observed with high event rates via
+Higgsstrahlung, W and Z fusion. Thus, electron–positron colliders can precisely measure the properties
+of the Higgs boson.
+Acknowledgment
+We would like to thank Professor Dr Inanc Sahin for his suggestions.
+8
+
+References
+[1] G. Aad et al. [ATLAS], “Observation of a new particle in the search for the Standard
+Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B 716 (2012) 1-29,
+arXiv:hep-ex/1207.7214.
+[2] S. Chatrchyan et al. [CMS], “Observation of a New Boson at a Mass of 125 GeV with the CMS
+Experiment at the LHC,” Phys. Lett. B 716(2012) 30-61, arXiv:hep-ex/1207.7235
+[3] P.
+W.
+Higgs,
+“Broken
+Symmetries
+and
+the
+Masses
+of
+Gauge
+Bosons,”
+Phys. Rev. Lett. 13 (1964) 508-509
+[4] P. W. Higgs, “Broken symmetries, massless particles and gauge fields,” Phys. Lett. 12(1964) 132-133
+[5] F. Englert and R. Brout,
+“Broken Symmetry and the Mass of Gauge Vector Mesons,”
+Phys. Rev. Lett. 13 (1964) 321-323
+[6] G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, “Global Conservation Laws and Massless Parti-
+cles,” Phys. Rev. Lett. 13 (1964) 585-587
+[7] S. Dittmaier et al. [LHC Higgs Cross Section Working Group], “Handbook of LHC Higgs Cross
+Sections: 1. Inclusive Observables,” ,arXiv:hep-ph/1101.0593.
+[8] S. Dittmaier, C. Mariotti, G. Passarino, R. Tanaka, S. Alekhin, J. Alwall, E. A. Bagnaschi, A. Banfi,
+J. Bl¨umlein and S. Bolognesi, et al. “Handbook of LHC Higgs Cross Sections: 2. Differential Distri-
+butions,” ,arXiv:hep-ph/1201.3084.
+[9] S. Heinemeyer et al. [LHC Higgs Cross Section Working Group], “Handbook of LHC Higgs Cross
+Sections: 3. Higgs Properties,” ,arXiv:hep-ph/1307.1347.
+[10] D. de Florian et al. [LHC Higgs Cross Section Working Group], “Handbook of LHC Higgs Cross
+Sections: 4. Deciphering the Nature of the Higgs Sector,” ,arXiv:hep-ph/1610.07922.
+[11] L. Evans et al. [Linear Collide], “The International Linear Collider Machine Staging Report 2017,”
+arXiv:physics.acc-ph/1711.00568 .
+[12] M. J. Boland et al. [CLIC and CLICdp], “Updated baseline for a staged Compact Linear Collider,”
+CERN 004 (2016) arXiv:physics.acc-ph/1608.07537 .
+[13] [CEPC
+Study
+Group],
+“CEPC
+Conceptual
+Design
+Report:
+Volume
+1
+-
+Accelerator,”
+arXiv:physics.acc-ph/1809.00285 .
+[14] A. Abada et al. [FCC], “FCC Physics Opportunities: Future Circular Collider Conceptual Design
+Report Volume 1,” Eur. Phys. J. C 79(2019) no.6, 474
+[15] D. R. T. Jones and S. T. Petcov, “Heavy Higgs Bosons at LEP,” Phys. Lett. B 84 (1979) 440-444,
+[16] D. A. Dicus and S. S. D. Willenbrock, “Higgs Bosons From Vector Boson Fusion in e+e−, ep and pp
+Collisions,” Phys. Rev. D 32 (1985) 1642,
+[17] G.
+Altarelli,
+B.
+Mele
+and
+F.
+Pitolli,
+“Heavy
+Higgs
+Production
+at
+Future
+Colliders,”
+Nucl. Phys. B 287 (1987) 205-224,
+[18] W. Kilian, M. Kramer and P. M. Zerwas, “Higgsstrahlung and W W fusion in e+ e- collisions,”
+Phys. Lett. B 373 (1996) 135-140, arXiv:hep-ph/9512355.
+[19] K. i. Hikasa, “Heavy Higgs Production in e+e− and e−e− Collisions,” Phys. Lett. B 164 (1985) 385
+[20] X. Mo, G. Li, M. Q. Ruan and X. C. Lou, “Physics cross sections and event generation of e+e−
+annihilations at the CEPC,” Chin. Phys. C 40 (2016), no.3, 033001, arXiv:hep-ex/1505.01008.
+[21] Y. Zhang, “WW fusion and Higgsstrahlung interplay in Higgs precision tests with the e-e+→νν¯h
+process,” Phys. Rev. D 99 (2019), no.3, 033011, arXiv:hep-ph/9704448.
+[22] A. Belyaev, N. D. Christensen and A. Pukhov, “CalcHEP 3.4 for collider physics within and beyond
+the Standard Model,” Comput. Phys. Commun. 184 (2013) 1729-1769, arXiv:hep-ph/1207.6082.
+[23] A. Pukhov, “CalcHEP 2.3: MSSM, structure functions, event generation, batchs, and generation of
+matrix elements for other packages,” arXiv:hep-ph/0412191.
+9
+
+[24] M. Skrzypek and S. Jadach, “Exact and approximate solutions for the electron nonsinglet structure
+function in QED,” Z. Phys. C 49 (1991) 557-584,
+[25] A. Pukhov, E. Boos, M. Dubinin, V. Edneral, V. Ilyin, D. Kovalenko, A. Kryukov, V. Savrin,
+S. Shichanin and A. Semenov, “CompHEP: A Package for evaluation of Feynman diagrams and
+integration over multiparticle phase space,” arXiv:hep-ph/9908288.
+[26] W. Kilian, M. Kramer and P. M. Zerwas, “Higgsstrahlung and W W fusion in e+ e- collisions,”
+Phys. Lett. B 373 (1996) 135-140, arXiv:hep-ph/9512355.
+[27] A. M. Sirunyan et al. [CMS], “Identification of heavy-flavour jets with the CMS detector in pp
+collisions at 13 TeV,” JINST 13 (2018) no.05, P05011, arXiv:physics.ins-det/1712.07158.
+[28] A. Djouadi, J. Kalinowski and M. Spira, “HDECAY: A Program for Higgs boson decays in the
+standard model and its supersymmetric extension,” Comput. Phys. Commun. 108 (1998) 59-74,
+arXiv:hep-ph/9704448.
+[29] M.
+Spira,
+“Higgs
+Boson
+Production
+and
+Decay
+at
+Hadron
+Colliders,”
+Prog. Part. Nucl. Phys. 95 (2017) 98-159, arXiv:hep-ph/1612.07651.
+[30] A. Djouadi, “The Anatomy of electro-weak symmetry breaking. I: The Higgs boson in the standard
+model,” Phys. Rept. 457 (2008) 1-216, arXiv:hep-ph/0503172.
+[31] S.
+Dawson,
+C.
+Englert
+and
+T.
+Plehn,
+“Higgs
+Physics:
+It
+ain’t
+over
+till
+it’s
+over,”
+Phys. Rept. 816 (2019) 1-85, [arXiv:1808.01324 [hep-ph]].
+[32] G. Cowan, K. Cranmer, E. Gross and O. Vitells, “Asymptotic formulae for likelihood-based tests of
+new physics,” Eur. Phys. J. C 71 (2011) 1554, arXiv:1007.1727 [physics.data-an].
+10
+

JNFRT4oBgHgl3EQfCTcy/content/tmp_files/2301.13468v1.pdf.txt

@@ -0,0 +1,593 @@
+Complete identification of spin-wave eigenmodes excited by parametric pumping in YIG microdisks
+T. Srivastava*,1, ∗ H. Merbouche*,2, † I. Ngouagnia Yemeli,1 N. Beaulieu,3 J. Ben Youssef,3 M.
+Mu˜noz,4 P. Che,5 P. Bortolotti,5 V. Cros,5 O. Klein,6 S. Sangiao,7 J. M. De Teresa,7 S. O.
+Demokritov,2 V. E. Demidov,2 A. Anane,5 C. Serpico,8 M. d’Aquino,8 and G. de Loubens1, ‡
+1SPEC, CEA, CNRS, Universit´e Paris-Saclay, Gif-sur-Yvette, France
+2Institute for Applied Physics, University of Muenster, Germany
+3LabSTICC, CNRS, Universit´e de Bretagne Occidentale, Brest, France
+4Instituto de Tecnolog´ıas F´ısicas y de la Informaci´on (CSIC), Madrid, Spain
+5Unit´e Mixte de Physique, CNRS, Thales, Universit´e Paris-Saclay, Palaiseau, France
+6Universit´e Grenoble Alpes, CEA, CNRS, Grenoble INP, Spintec, Grenoble, France
+7Instituto de Nanociencia y Materiales de Arag´on (INMA) and Laboratorio de
+Microscop´ıas Avanzadas (LMA), Universidad de Zaragoza, Zaragoza, Spain
+8Department of Electrical Engineering and ICT, University of Naples Federico II, Italy
+We present the parametric excitation of spin-wave modes in YIG microdisks via parallel pumping. Their
+spectroscopy is performed using magnetic resonance force microscopy (MRFM), while their spatial profiles
+are determined by micro-focus Brillouin light scattering (BLS). We observe that almost all the fundamental
+eigenmodes of an in-plane magnetized YIG microdisk, calculated using a micromagnetic eigenmode solver,
+can be excited using the parallel pumping scheme, as opposed to the transverse one. The comparison between
+the MRFM and BLS data on one side, and the simulations on the other side, provides the complete spectro-
+scopic labeling of over 40 parametrically excited modes. Our findings could be promising for spin-wave-based
+computation schemes, in which the amplitudes of a large number of spin-wave modes have to be controlled.
+I.
+INTRODUCTION
+Novel proposals for spin-wave-based computing schemes
+necessitate the generation and control of multiple spin-wave
+(SW) modes1–5. The most standard way to excite SW modes
+in a magnetic microstructure is by direct inductive coupling.
+There, the quasi-uniform microwave field, produced on the
+magnetic volume by an rf antenna, couples to the transverse
+dynamical component of the magnetization associated with
+the SW mode, with a maximal efficiency when the applied
+rf frequency coincides with the eigenfrequency of the mode.
+However, this method is not adapted to excite modes with anti-
+symmetric spatial profiles, as their overlap integral with the
+excitation field is zero6, nor short-wavelength modes, as their
+excitation efficiency quickly decreases with their wavevector.
+Yet, these two categories of modes make up a significant part
+of the SW k-space. In order to excite a large number of modes
+irrespective of their spatial profiles, parametric parallel pump-
+ing, which does not suffer from these limitations, becomes the
+ideal choice7. In this case, the microwave magnetic field cre-
+ated by the rf antenna is aligned parallel to the static field. As
+a result, it does not couple to the SW modes directly. Instead,
+it interacts with the dynamic component of magnetization os-
+cillating at 2ω in the static field direction, which arises due to
+the elliptical trajectory of magnetization precession at ω. An
+rf field at 2ω can therefore excite SW modes at ω. A quantum
+mechanical picture of this process is a photon generating two
+magnons of opposite momenta at half its frequency8. Since
+this is a nonlinear process, SWs are excited only if the am-
+plitude of the excitation field exceeds a parametric threshold,
+which depends on the mode relaxation, and on the mode ellip-
+ticity. The threshold power is lower for lower relaxation rates
+and higher ellipticities.
+Parallel pumping has been employed to generate SW modes
+in extended films9–13 and micro- and nano-waveguides14,15
+of yttrium iron garnet (YIG), as well as in magnetic
+nanocontacts16, magnetic tunnel junctions17, and micro- and
+nano-dots of Permalloy18–20. It has also been used for SW
+amplification21. All these studies have been limited to a hand-
+ful number of modes.
+The excitation and identification of
+many modes in an adequate system would pave the way to-
+wards simultaneous control and manipulation of a large num-
+ber of SW modes for different applications in magnonics.
+In this study, we present the excitation and identification of
+multiple SW modes in YIG microdisks via parametric pump-
+ing. The scheme of the experiments is shown in Fig. 1. The
+SW modes are excited in YIG disks of diameters 1 µm, 3 µm
+and 5 µm through an integrated rf antenna and detected us-
+ing a magnetic resonance force microscope (MRFM). Their
+spatial profiles can also be recorded using micro-focus Bril-
+louin light scattering spectroscopy (µ-BLS). We observe that
+almost all the SW eigenmodes are accessible by parametric
+pumping. As expected, these eigenmodes become fewer in
+number as the size of the disk decreases. For the 3 µm disk,
+we label over 40 eigenmodes by comparing its MRFM para-
+metric spectroscopy to micromagnetic simulations, and con-
+firm the identification of as many as 10 of them through their
+profiles thanks to µ-BLS. Our results could be instrumental in
+designing basic units for unconventional computing schemes
+like neuromorphic computing using hyperconnected popula-
+tions of a large number of eigen-excitations in a single mi-
+crostructure.
+II.
+RESULTS
+A.
+Sample
+We use 50 nm thick YIG grown on 0.5 mm thick GGG
+substrate by liquid phase epitaxy22.
+The characteristics of
+arXiv:2301.13468v1  [physics.app-ph]  31 Jan 2023
+
+2
+FIG. 1. Schematics of the experimental setup. Parallel pump-
+ing of a YIG microdisk using an rf antenna deposited on top. The
+spectroscopy of the parametrically excited modes is achieved using
+a magnetic resonance force microscope (MRFM) positioned above
+the sample. Their spatial profiles are measured by micro-focus Bril-
+louin light scattering (µ-BLS) using a separate experimental setup,
+the laser beam being focused to the bottom of the sample, through
+the transparent GGG substrate.
+the extended film are measured by standard magnetometry
+and broadband FMR techniques.
+These yield a saturation
+magnetization Ms = 140.7 kA/m, a gyromagnetic ratio γ =
+28.28 GHz/T, a Gilbert damping parameter α = 7.5 × 10−5,
+and a weak inhomogeneous broadening of the FMR linewidth,
+found to be 0.1 mT. These parameters are typical of the YIG
+material; the exchange constant, which has not been specifi-
+cally determined on this film, is assumed to be A = 3.7 pJ/m, a
+standard value from literature23. The YIG layer was patterned
+into disks of diameters 1 µm, 3 µm and 5 µm using e-beam
+lithography. A 220 nm thick Ti/Au antenna, of width equal to
+8 µm, was then deposited on top of the disks. Injecting an rf
+current in the antenna generates an rf in-plane magnetic field
+that is orthogonal to the long axis of the antenna.
+B.
+Parallel pumping spectroscopy
+The SW mode spectroscopy is done using MRFM. It em-
+ploys a very soft cantilever, at the end of which a submicronic
+magnetic spherical probe made of cobalt is attached24, to me-
+chanically detect the magnetization dynamics in the sample
+placed underneath25. When SWs are excited in the sample
+by the microwave field, the (static) longitudinal component of
+magnetization is reduced and so is the dipolar force on the
+MRFM probe, resulting in a displacement of the cantilever
+beam, which is detected optically. The rf excitation applied
+to the sample via the antenna is modulated at the mechanical
+resonance frequency of the cantilever to improve the quality
+factor and the signal-to-noise ratio.
+In these measurements, the dc magnetic field is applied in-
+plane at an angle of 45° with respect to the direction of the rf
+magnetic field, as displayed in Fig. 1. Therefore the rf field ex-
+citation has both a transverse and a parallel component relative
+to the magnetization direction. The parallel pumped SW spec-
+trum is studied for different-sized disks as a function of the
+applied microwave power. Figure 2 shows the results of the
+MRFM parametric spectroscopy performed at a constant mi-
+crowave frequency of 4 GHz for the three disks (color-coded
+intensity maps), together with the corresponding transverse
+excitation spectra measured at fixed frequency of 2 GHz and
+power of −5 dBm (continuous white curves). Only a few SW
+modes are detected in the latter regime. In contrast, we ob-
+serve that a large number of modes can be excited by parallel
+pumping at 4 GHz for all the disks, in the range of applied dc
+field corresponding to the direct excitation of modes at 2 GHz,
+because parametrically excited modes are generated at half the
+pumping frequency. As expected, this occurs only above a
+minimum power level, that ranges from about −4 dBm for the
+5 µm disk to −2 dBm for the 1 µm disk. The fact that the para-
+metric threshold increases and that the density of the excited
+modes decreases as the lateral size decreases can be explained
+by geometrical confinement effects, as reported earlier20.
+In the following, we will mainly focus on the 3 µm disk
+where the SW modes are quite abundant but at the same time
+discernible (not too closely spaced). We perform similar mea-
+surements on this disk, this time fixing the value of the dc field
+to 27 mT, and scanning the parallel pumping frequency as a
+function of the microwave power. Fig. 3b shows the intensity
+map of the parametrically excited modes in these conditions,
+as a function of half the pumping frequency fp/2 and the rf
+power P, varied along the horizontal and vertical axes, respec-
+tively. We note that the threshold power increases with the
+frequency in a non-monotonic way, which can be explained
+as follows. The threshold excitation field of each mode can
+indeed be computed as the ratio between the relaxation rate
+ωr(k) to a coupling coefficient V(k), that is related to the mode
+ellipticity7: the more elliptical a mode is, the larger its V(k)
+and the lower will its threshold be. Both the terms depend
+non-monotonously on the wavevector k and on the mode fre-
+quency. However, on a wide range, when k increases, so does
+the mode frequency and its relaxation rate, while its ellipticity
+and its coupling V(k) tends to decrease7,21. This leads to the
+clear but not monotonous increase of the experimental thresh-
+old power with frequency seen in Fig. 3b.
+C.
+Simulations
+In order to identify these parametrically excited modes, mi-
+cromagnetic simulations using the eigenmode solver imple-
+mented in the micromagnetic code MaGICo26 have been per-
+formed to calculate the SW spectrum. The magnetic ground
+state is first computed for the specific geometry and applied
+magnetic field. Once the magnetic ground state is known,
+the equation describing magnetization dynamics, the Landau-
+Lifshitz-Gilbert equation, is linearized around the ground state
+and small-amplitude spatial profiles of the modes are com-
+
+G
+rf antenna
+GGG3
+FIG. 2. MRFM parametric spectroscopy. Intensity maps of the parametrically excited modes in the field-power coordinates excited by the
+microwave field of frequency 4 GHz, measured by MRFM on the 5 µm (a), 3 µm (b) and 1 µm (c) diameter YIG disks. In each panel, the
+continuous white curve corresponds to the direct excitation spectrum at the frequency of 2 GHz. The rf pumping field hrf is at 45° from the dc
+field H.
+puted. This problem can be formulated as a generalized eigen-
+value problem as described in ref.27.
+The solution of the
+eigenvalue problem allows to the determination of the SW
+spectrum of the magnetic sample under investigation. Here,
+the geometry of the body, a 50 nm thick disk of 3 µm in di-
+ameter, was discretized using 300 × 300 × 5 cubic cells (mesh
+size of 10 × 10 × 10 nm3), and the values of the magnetic pa-
+rameters used in the simulation were those determined exper-
+imentally. As in the experimental case, the applied field lies
+in the plane of the disk and is set to 27 mT. The implemen-
+tation of suitable matrix-free large-scale methods described
+in ref.28 allowed the calculation of hundreds of eigenmodes
+for such an extended structure (353440 computational cells,
+eigenvalue problem size 706880 × 706880) in a few hours.
+Figure 3a displays the computed spatial profiles of the first
+100 eigenmodes. The 10 lowest frequency modes (first row)
+correspond to edge modes, where the precession of the mag-
+netization is strongly confined at the boundaries of the disk,
+in the (horizontal) direction of the applied dc field due to the
+demagnetizing field29,30. The following modes correspond to
+standing SW modes, which can be labelled by the number of
+precession lobes in the horizontal (nx) and vertical (ny) direc-
+tions. For instance, mode 20 (second row, last column) can be
+labelled by nx = 2 and ny = 1, i.e., it is the (2,1) mode. Mode
+40 (fourth row, last column) is the (11,3) mode. The most uni-
+form mode, usually referred as the FMR mode, is mode 29, or
+mode (1,1).
+Figure 3b presents the comparison between the experimen-
+tal spectroscopy and the computed eigenfrequencies of modes
+11 to 70, shown as red ticks on top of the intensity map of
+the parametrically excited modes. We observe a good agree-
+ment between the computed mode frequencies and the exper-
+imental mode frequencies (at half-pumping frequencies fp/2)
+observed at the bottom of the parametric instability regions
+(elongated yellow-green triangles extending downwards on
+the intensity map). From this comparison, it is possible to
+state that almost all, if not all SW eigenmodes, can be para-
+metrically excited, irrespective of their spatial profile. Due to
+the high density of modes in the investigated frequency range,
+we will only focus on a few modes, to emphasize the good
+agreement noted above. The lowest-lying computed modes in
+Fig. 3b are the pair of modes 11 and 12 with respective fre-
+quencies 1.938 and 1.9383 GHz, which correspond rather well
+to the measured parametric instability region with a threshold
+power of 2 dBm at around 1.95 GHz. The small disagree-
+ment of 10 MHz between the computed and measured fre-
+quencies is not unexpected, since these modes belong to the
+category of edge modes, whose characteristics are very sen-
+sitive to imperfections at the periphery of the disk30,31, which
+are not taken into account in the simulations. If we move to
+the next parametrically excited modes, which have the low-
+est power threshold and have frequencies around 1.975 GHz,
+the comparison with computed frequencies shows that they
+correspond to two pairs of modes: modes 13 and 14 with re-
+spective frequencies 1.973 and 1.9731 GHz, and modes 15
+and 16, at 1.9796 and 1.9809 GHz. The next excited modes
+in the experimental spectroscopy map are at around 2 GHz,
+and they correspond to mode 17 at 1.998 GHz and mode 18
+at 2.001 GHz. As a matter of fact, a detailed inspection of the
+data shows that indeed, the parametric instability region has
+two nearby minima with frequencies equally spaced around
+2 GHz. This good agreement between experimental and com-
+puted mode frequencies continues over the full range of inves-
+tigated frequencies. We note that among the 60 modes whose
+frequencies have been plotted in ig. 3b, only 44 modes have
+discernible frequencies and spatial profiles, a few of them be-
+ing pairs of modes with very similar characteristics (e.g., pairs
+of modes 11 and 12, 13 and 14, 15 and 16, 21 and 22, etc.).
+D.
+Spatial profiles with µ-BLS
+To push further the comparison between computed SW
+modes and experiments, it is possible to take advantage of
+µ-BLS, to map the spatial profiles of dynamic magnetization
+in micro-structures32. A probing laser light (λ = 473 nm and
+Plaser = 0.1 mW) is focused into a diffraction-limited spot on
+the surface of a similar 3 µm YIG disk (Fig. 1) and the mod-
+
+a
+C
+10.0
+10.0
+10.0
+7.5 
+7.5 
+7.5
+5.0 -
+5.0 
+5.0
+(dBm)
+2.5 
+(dBm)
+2.5
+(dBm)
+2.5
+0.0 -
+0.0 
+0.0
+P
+P
+P
+-2.5 -
+2.5
+2.5
+-5.0
+5.0 
+5.0
+7.5
+5
+wn
+7.5 
+μm
+7.5
+1 μm
+10.0-
+-10.0-
+10.0
+10
+15
+20
+25
+30
+35
+10
+15
+20
+25
+30
+35
+10
+15
+20
+25
+30
+35
+μoH (mT)
+μoH (mT)
+μoH (mT)4
+FIG. 3. Computed spin-wave eigenmodes. (a) Computed spatial
+profiles of the 100 lowest frequency modes of the 3 µm diameter YIG
+disk, in-plane magnetized by a field of 27 mT. The color code refers
+to the oscillation amplitude of the local magnetization, from blue
+(minimum) to red (maximum). (b) Comparison between parametric
+spectroscopy MRFM data (color-coded intensity map) and computed
+eigenfrequencies (red vertical ticks) of modes 11 to 70 (surrounded
+by the red rectangle in panel (a)).
+ulation of this probing light by the magnetization oscillations
+is analysed using a high-contrast optical spectrometer. The
+obtained signal – the BLS intensity – is proportional to the
+intensity of the magnetic oscillations at a given frequency. In
+this BLS measurement, the in-plane bias field is set at 30 mT.
+To compare the experimental mode profiles with the computed
+mode profiles, the micromagnetic simulations have therefore
+been repeated at 30 mT as well. To avoid nonlinear distor-
+tions of the mode profiles, known to occur when the mode
+amplitude increases too much, the BLS mapping of the mode
+profiles is performed at microwave power only slightly above
+threshold. By sweeping the laser spot position across the disk,
+a dozen of different modes are imaged, Fig. 4 presents the
+comparison between the experimental and computed profiles
+of 10 modes. Overall, the measured profiles are in good agree-
+ment with the computed ones, taking into account the exper-
+imental spatial resolution (≃ 250 nm) and the long duration
+of these measurements, which are subjected to experimental
+drifts. Similarly to the analysis performed in Fig. 3b, we ob-
+serve that the mode frequencies obtained by BLS correspond
+very well to the computed mode frequencies, with a mismatch
+that remains under 13 MHz for all modes. In particular, we
+observe well defined modes up to nx = 7 and ny = 3, which
+validates the agreement between experiment and simulations
+for a large number of modes.
+III.
+CONCLUSION
+Thanks to the comparison between parametric spectroscopy
+and mode imaging respectively performed by MRFM and
+BLS on one side, and micromagnetic simulations on the other
+side, we have successfully excited, detected and identified a
+large number (> 40) of SW eigenmodes in a 3 µm YIG disk,
+where the mode density is large due to the large lateral dimen-
+sions. The computed spatial profiles provide a direct way to
+label those modes, using the numbers of precession nodes in
+the directions parallel (nx) and transverse (ny) to the applied
+magnetic field. This study opens up the possibility to per-
+form experiments where many parametric modes are simulta-
+neously excited while using the normal mode approach33,34 to
+understand and harness the complex dynamics in the modal
+space of confined magnetic structures.
+ACKNOWLEDGEMENTS
+This work was supported by the Horizon2020 Research
+Framework Programme of the European Commission under
+grant no. 899646 (k-NET). It is also supported by a pub-
+lic grant overseen by the Agence Nationale de la Recherche
+as part of the “Investissements d’Avenir” program (Labex
+NanoSaclay, reference: ANR-10-LABX-0035).
+I.N.Y. ac-
+knowledges support from the ANR grant no. ANR-18-CE24-
+0021 (Maestro).
+
+0983GH2
+2.1275GHzf.
+n=2.1381GH
+Q0000
+2.1493 GH
+.1892GHzf
+.1992GHz
+2583GHz
+4GHz
+2GHzfa.2.3457GH2
+.3505GHz
+3519 GHz fgs =2.3557 GHz fa4
+9 GHz
+2.3774GHz fag-2.392GHzfgo-2.3949GHz5
+FIG. 4. BLS imaging of mode profiles. The central graph displays the BLS detected frequencies at 30 mT for 10 modes of the 3 µm disk
+(blue lines) and the corresponding computed frequencies of eigenmodes (red line). These frequencies are matched (dotted dark lines) by
+associating the mode profiles measured in the experiment (above the graph) to the ones computed in the simulation (below). The experimental
+and simulated mode frequencies (in GHz) and their difference (in MHz) are given in the table.
+∗ titiksha.srivastava@cea.fr
+† hugo.merbouche@uni-muenster.de
+‡ gregoire.deloubens@cea.fr
+1 T. Br¨acher and P. Pirro, “An analog magnon adder for all-
+magnonic neurons,” J. Appl. Phys. 124, 152119 (2018).
+2 R. Nakane, G. Tanaka, and A. Hirose, “Reservoir computing with
+spin waves excited in a garnet film,” IEEE Access 6, 4462–4469
+(2018).
+3 Tyler W. Hughes, Ian A. D. Williamson, Momchil Minkov, and
+Shanhui Fan, “Wave physics as an analog recurrent neural net-
+work,” Science Advances 5 (2019), 10.1126/sciadv.aay6946.
+4 ´Ad´am Papp, Wolfgang Porod,
+and Gyorgy Csaba, “Nanoscale
+neural network using non-linear spin-wave interference,” Nature
+Commun. 12, 1–8 (2021).
+5 Lukas K¨orber, Christopher Heins, Tobias Hula, Joo-Von Kim,
+Helmut Schultheiss, J¨urgen Fassbender, and Katrin Schultheiss,
+“Pattern recognition with a magnon-scattering reservoir,” arXiv
+preprint arXiv:2211.02328 (2022), 10.48550/arXiv.2211.02328.
+6 V. V. Naletov, G. de Loubens, G. Albuquerque, S. Borlenghi,
+V. Cros, G. Faini, J. Grollier, H. Hurdequint, N. Locatelli,
+B. Pigeau, A. N. Slavin, V. S. Tiberkevich, C. Ulysse, T. Valet,
+and O. Klein, “Identification and selection rules of the spin-wave
+eigenmodes in a normally magnetized nanopillar,” Phys. Rev. B
+84, 224423 (2011).
+7 A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and
+Waves (CRC Press, 1996).
+8 R. M. White and M. Sparks, “Ferromagnetic Relaxation. III. The-
+ory of instabilities,” Phys. Rev. 130, 632 (1963).
+9 H. Kurebayashi, O. Dzyapko, V. E. Demidov, D. Fang, A. J.
+Ferguson, and S. O. Demokritov, “Spin pumping by parametri-
+cally excited short-wavelength spin waves,” Appl. Phys. Lett. 99,
+162502 (2011).
+10 C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga, V. I.
+Vasyuchka, M. B. Jungfleisch, E. Saitoh,
+and B. Hillebrands,
+“Spin pumping by parametrically excited exchange magnons,”
+Phys. Rev. Lett. 106, 216601 (2011).
+11 A. A. Serga, C. W. Sandweg, V. I. Vasyuchka, M. B. Jungfleisch,
+B. Hillebrands, A. Kreisel, P. Kopietz,
+and M. P. Kostylev,
+“Brillouin light scattering spectroscopy of parametrically excited
+dipole-exchange magnons,” Phys. Rev. B 86, 134403 (2012).
+12 C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Naletov, and
+J. Ben Youssef, “Detection of microwave spin pumping using the
+inverse spin hall effect,” Phys. Rev. Lett. 111, 217204 (2013).
+13 V. Lauer, D. A. Bozhko, T. Br ˜A¤cher, P. Pirro, V. I. Vasyuchka,
+A. A. Serga, M. B. Jungfleisch, M. Agrawal, Yu. V. Kobljanskyj,
+G. A. Melkov, C. Dubs, B. Hillebrands, and A. V. Chumak, “Spin-
+transfer torque based damping control of parametrically excited
+spin waves in a magnetic insulator,” Appl. Phys. Lett. 108, 012402
+(2016).
+14 Morteza Mohseni, Martin Kewenig, Roman Verba, Qi Wang,
+Michael Schneider, Bj¨orn Heinz, Felix Kohl, Carsten Dubs, Bert
+L¨agel, Alexander A. Serga, Burkard Hillebrands, Andrii V. Chu-
+mak, and Philipp Pirro, “Parametric generation of propagating
+spin waves in ultrathin yttrium iron garnet waveguides,” Phys. Sta-
+tus Solidi RRL 14, 2000011 (2020).
+15 Bj¨orn Heinz, Morteza Mohseni, Akira Lentfert, Roman Verba,
+Michael Schneider, Bert L¨agel, Khrystyna Levchenko, Thomas
+Br¨acher, Carsten Dubs, Andrii V. Chumak,
+and Philipp Pirro,
+“Parametric generation of spin waves in nanoscaled magnonic
+conduits,” Phys. Rev. B 105, 144424 (2022).
+16 Sergei Urazhdin, Vasil Tiberkevich, and Andrei Slavin, “Paramet-
+ric excitation of a magnetic nanocontact by a microwave field,”
+Phys. Rev. Lett. 105, 237204 (2010).
+17 Yu-Jin Chen, Han Kyu Lee, Roman Verba, Jordan A. Katine, Igor
+Barsukov, Vasil Tiberkevich, John Q. Xiao, Andrei N. Slavin, and
+
+Profile
+00
+(experiment)
+2.10
+2.15
+Frequency,GHz
+2.20
+Profile
+00.00
+(simulation)
+fexp (GHz)
+2.098
+2.124
+2.141
+2.147
+2.162
+2.173
+2.192
+2.226
+2.236
+2.253
+fsimu (GHz)
+2.093
+2.117
+2.128
+2.145
+2.149
+2.173
+2.184
+2.217
+2.238
+2.252
+fexp - fsimu
+5
+7
+13
+2
+13
+0
+8
+9
+-2
+1
+(MHz)6
+Ilya N. Krivorotov, “Parametric resonance of magnetization ex-
+cited by electric field,” Nano Letters 17, 572–577 (2017).
+18 Henning Ulrichs, Vladislav E. Demidov, Sergej O. Demokritov,
+and Sergei Urazhdin, “Parametric excitation of eigenmodes in mi-
+croscopic magnetic dots,” Phys. Rev. B 84, 094401 (2011).
+19 E. R. J. Edwards, H. Ulrichs, V. E. Demidov, S. O. Demokritov,
+and S. Urazhdin, “Parametric excitation of magnetization oscilla-
+tions controlled by pure spin current,” Phys. Rev. B 86, 134420
+(2012).
+20 Feng Guo, L. M. Belova,
+and R. D. McMichael, “Parametric
+pumping of precession modes in ferromagnetic nanodisks,” Phys.
+Rev. B 89, 104422 (2014).
+21 T. Br¨acher, P. Pirro, and B. Hillebrands, “Parallel pumping for
+magnon spintronics: Amplification and manipulation of magnon
+spin currents on the micron-scale,” Physics Reports 699, 1–34
+(2017).
+22 N. Beaulieu, N. Kervarec, N. Thiery, O. Klein, V. Naletov, H. Hur-
+dequint, G. de Loubens, J. Ben Youssef, and N. Vukadinovic,
+“Temperature dependence of magnetic properties of a ultrathin
+yttrium-iron garnet film grown by liquid phase epitaxy: Effect of
+a pt overlayer,” IEEE Magnetics Letters 9, 1–5 (2018).
+23 Stefan Klingler, A V Chumak, Tim Mewes, Behrouz Khodadadi,
+Claudia Mewes, Carsten Dubs, Oleksii Surzhenko, Burkard Hille-
+brands, and Andr´es Conca, “Measurements of the exchange stiff-
+ness of yig films using broadband ferromagnetic resonance tech-
+niques,” J. Phys. D: Appl. Phys. 48, 015001 (2015).
+24 Soraya Sangiao, C´esar Mag´en, Darius Mofakhami, Gr´egoire
+de Loubens, and Jos´e Maria De Teresa, “Magnetic properties of
+optimized cobalt nanospheres grown by focused electron beam
+induced deposition (febid) on cantilever tips,” Beilstein J. Nan-
+otechnol. 8, 2106–2115 (2017).
+25 O. Klein, G. de Loubens, V. V. Naletov, F. Boust, T. Guillet,
+H. Hurdequint, A. Leksikov, A. N. Slavin, V. S. Tiberkevich, and
+N. Vukadinovic, “Ferromagnetic resonance force spectroscopy
+of individual submicron-size samples,” Phys. Rev. B 78, 144410
+(2008).
+26 http://wpage.unina.it/mdaquino/index file/MaGICo.html.
+27 Massimiliano d’Aquino, Claudio Serpico, Giovanni Miano, and
+Carlo Forestiere, “A novel formulation for the numerical com-
+putation of magnetization modes in complex micromagnetic sys-
+tems,” J. Comput. Phys. 228, 6130–6149 (2009).
+28 Massimiliano d’Aquino and Riccardo Hertel, “Micromagnetic
+frequency-domain simulation methods for magnonic systems,” J.
+Appl. Phys. 133, 033902 (2023).
+29 J. Jersch, V. E. Demidov, H. Fuchs, K. Rott, P. Krzysteczko,
+J. M¨unchenberger, G. Reiss, and S. O. Demokritov, “Mapping of
+localized spin-wave excitations by near-field brillouin light scat-
+tering,” Appl. Phys. Lett. 97, 152502 (2010).
+30 Feng Guo, L. M. Belova, and R. D. McMichael, “Spectroscopy
+and imaging of edge modes in permalloy nanodisks,” Phys. Rev.
+Lett. 110, 017601 (2013).
+31 H. T. Nembach, Justin M. Shaw, T. J. Silva, W. L. Johnson, S. A.
+Kim, R. D. McMichael,
+and P. Kabos, “Effects of shape dis-
+tortions and imperfections on mode frequencies and collective
+linewidths in nanomagnets,” Phys. Rev. B 83, 094427 (2011).
+32 Vladislav E. Demidov and Sergej O. Demokritov, “Magnonic
+waveguides studied by microfocus brillouin light scattering,”
+IEEE Trans. Magn. 51, 1–15 (2015).
+33 S. Perna, F. Bruckner, C. Serpico, D. Suess, and M. d’Aquino,
+“Computational micromagnetics based on normal modes: Bridg-
+ing the gap between macrospin and full spatial discretization,” J.
+Magn. Magn. Mater. 546, 168683 (2022).
+34 S. Perna, F. Bruckner, C. Serpico, D. Suess, and M. d’Aquino,
+“Normal modes description of nonlinear ferromagnetic resonance
+for magnetic nanodots,” AIP Adv. 12, 035244 (2022).
+

MtAyT4oBgHgl3EQf6vrQ/content/tmp_files/2301.00828v1.pdf.txt

@@ -0,0 +1,1347 @@
+MNRAS 000, 1–13 (2020)
+Preprint 4 January 2023
+Compiled using MNRAS LATEX style file v3.0
+From dark matter halos to pre-stellar cores: High resolution follow-up of
+cosmological Lyman-Werner simulations.
+Lewis R. Prole,1★ Anna T. P. Schauer,2 Paul C. Clark,1 Simon C. O. Glover,3 Felix D. Priestley,1 Ralf S. Klessen,3,4
+1Cardiff University School of Physics and Astronomy
+2Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA
+3Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Straße 2, D-69120 Heidelberg, Germany
+4Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+Molecular hydrogen allows cooling in primordial gas, facilitating its collapse into Population III stars within primordial halos.
+Lyman-Werner (LW) radiation from these stars can escape the halo and delay further star formation by destroying H2 in other
+halos. As cosmological simulations show that increasing the background LW field strength increases the average halo mass
+required for star formation, we perform follow-up simulations of selected halos to investigate the knock-on effects this has on
+the Population III IMF. We follow 5 halos for each of the 𝐽21 = 0, 0.01 and 0.1 LW field strengths, resolving the pre-stellar core
+density of 10−6 g cm−3 (1018 cm−3) before inserting sink particles and following the fragmentation behaviour for hundreds of
+years further. We find that the mass accreted onto sinks by the end of the simulations is proportional to the mass within the
+∼ 10−2 pc molecular core, which is not correlated to the initial mass of the halo. As such, the IMF shows little dependence on
+the LW strength. As the range of background LW field strengths tested here covers the most likely values from literature, we
+conclude that the IMF for so-called Pop III.2 stars is not significantly different from the initial population of Pop III.1 stars. The
+primordial IMF therefore likely remains unchanged until the formation of the next generation of Population II stars.
+Key words: stars: Population III – dark ages, reionization, first stars – hydrodynamics – stars: luminosity function, mass function
+– software: simulations
+1 INTRODUCTION
+This first stars are able to form because pristine baryonic gas can
+collapse within the gravitational potential well of dark matter (DM)
+halos (Couchman & Rees 1986; Haiman et al. 1996a; Tegmark et al.
+1997), heating it to ∼ 5000 K and facilitating the formation of H2
+(Bromm et al. 2002). H2 primarily forms via the radiative association
+reaction forming H−
+H + e− → H− + 𝛾,
+(1)
+followed by the fast associative detachment reaction forming H2
+H− + H → H2 + e−.
+(2)
+Radiative cooling from the molecular hydrogen renders the gas grav-
+itationally unstable, which allows it to decouple from the DM and
+collapse to form the first stars, known as Population III (Pop III)
+stars. The necessity of H2 renders any process of H2 destruction as a
+mechanism to delay or prevent Pop III star formation.
+While so-called Pop III.1 stars form from purely cosmological
+initial conditions, the radiation they produce affects Pop III.2 stars
+that form in its presence. As the masses of Pop III stars are predicted
+to be larger than present-day counterparts (due to the lack of cooling
+from dust and metals), they are expected to emit large amounts of
+★ E-mail: Prolel@cardiff.ac.uk
+ionizing radiation (e.g. Schaerer 2002). Ionizing photons above the
+Lyman limit (13.6 eV) create H ii regions around the stars up to the
+boundary of their Strömgren spheres (Whalen et al. 2004; Kitayama
+et al. 2004; Alvarez et al. 2006; Abel et al. 2007; Yoshida et al.
+2007a; Jaura et al. 2022) while photons below the Lyman limit are
+free to escape their Strömgren sphere. Lyman-Werner (LW) photons
+between 11.2 and 13.6 eV can dissociate H2 via the two-step Solomon
+process (Field et al. 1966; Stecher & Williams 1967)
+H2 + 𝛾 → H∗
+2 → 2H,
+(3)
+where H∗
+2 represents an electronically excited state of H2. Photons
+with energy above 0.76 eV can also photodissociate H− via
+H− + 𝛾 → H + e−,
+(4)
+reducing the H− abundance and hence the rate at which H2 can
+form via reaction 2 (e.g. Chuzhoy et al. 2007). This stellar feedback
+provides a potential obstacle for Pop III.2 stars to overcome during
+formation. Investigations into the effects of these far UV fields typ-
+ically categorise the field strength by the intensity in the LW band
+𝐽21, in units of 10−21 erg s−1 cm−2 Hz−1 sr−1.
+Calculations by Haiman et al. (1997) found that before the Ström-
+gren spheres of Pop III stars overlap, the UV background below the
+ionization threshold was able to penetrate large clouds and suppress
+their H2 abundance. They also found that the flux necessary for H2
+photodissociation is several orders of magnitude smaller than the flux
+© 2020 The Authors
+arXiv:2301.00828v1  [astro-ph.GA]  2 Jan 2023
+
+2
+L. R. Prole
+needed to reionize the universe. Haiman et al. (2000) showed that
+this photodissociation of H2 suppresses further Pop III star formation
+inside small halos and delays reionization until larger halos form.
+Collapse is not impossible without sufficient H2 for cooling.
+Omukai (2001) showed that if the LW field is sufficient to keep a
+halo free of molecular hydrogen, the gas can nevertheless collapse
+via atomic hydrogen line cooling if the halo has a virial temperature
+𝑇vir > 8000 K. The collapse occurs almost isothermally, possibly
+resulting in the formation of a direct collapse black hole (DCBH)
+(e.g. Bromm & Loeb 2003; Spaans & Silk 2006; Latif et al. 2013), a
+possible progenitor of the supermassive black holes observed at high
+redshifts (e.g. Mortlock et al. 2011; Matsuoka et al. 2019). However,
+for a T = 105 K blackbody spectrum expected from Pop III stars, a
+field strength of 𝐽21 ∼ 104 is required to keep the gas atomic during
+the collapse (Glover 2015; Agarwal & Khochfar 2015; Agarwal et al.
+2016; Sugimura et al. 2014), while the average exposure is expected
+to be 𝐽21 < 0.1 at 𝑧 ∼ 15 (Ahn et al. 2009; Trenti & Stiavelli 2009;
+Wise et al. 2012; Agarwal et al. 2012; Skinner & Wise 2020). Dijk-
+stra et al. (2008) showed that only a fraction of 10−8 − 10−6 of DM
+halos with virial temperatures > 104 K have a close luminous neigh-
+bour within < 10 kpc, and are exposed to an LW flux 𝐽21 > 103.
+The occurrence of atomically cooled halos is therefore expected to
+be rare.
+Studies have shown that values of 𝐽21 orders of magnitude lower
+than the critical intensity required to completely suppress H2 cooling
+in massive halos can still drastically affect halo collapse. Typically,
+the critical mass for efficient molecular hydrogen cooling and sub-
+sequent star formation increases with increasing 𝐽21 (e.g. Machacek
+et al. 2001; Yoshida et al. 2003; O’Shea & Norman 2008; Vis-
+bal et al. 2014; Schauer et al. 2021). Cosmological simulations by
+Yoshida et al. (2003) found gas cooling was suppressed for 𝐽21 = 0.1,
+leading them to predict that star formation would not occur in halos
+with 𝑇vir < 8000 K for LW field strengths greater than this. Con-
+versely, O’Shea & Norman (2008) found that for field strengths as
+high as 𝐽21 = 1, H2 cooling leads to collapse despite the depressed
+core molecular hydrogen fractions. They also noted that higher LW
+background fluxes lead to higher accretion rates. High resolution
+cosmological simulations by Schauer et al. (2021) (hereafter AS21)
+examined the impact of different values of the LW field strength on
+a large sample of minihalos. They showed that both Mav, the aver-
+age minihalo mass required for efficient H2 cooling (i.e. the mass
+above which more than 50% of minihalos of that mass can cool), and
+Mmin, the minimum minihalo mass required for efficient cooling, in-
+creased with increasing 𝐽21. An increase in the critical halo mass for
+star formation with increasing 𝐽21 was also found by Kulkarni et al.
+(2021), although they find a significant effect only for 𝐽21 > 1. In
+contrast, cosmological simulations by Skinner & Wise (2020) found
+no relationship between the LW intensity and host halo mass.
+The true average 𝐽21 intensity is expected to vary with redshift.
+Hirano et al. (2015) followed the formation and evolution of 1540 star-
+forming gas clouds. They found that in their models, the characteristic
+mass of Pop III stars shifted to lower masses with decreasing redshift
+due to the radiative feedback of previous generations of stars. For
+𝑧 > 20, half of the star-forming gas clouds were exposed to intense
+FUV radiation, with an average exposure of 𝐽21 ∼ 0.07. Due to
+smaller stellar masses and the expanding distance between stars, the
+FUV background became weaker at lower redshifts. For 15 < 𝑧 < 25,
+almost all the clouds had nonzero intensity 𝐽21 > 0.01. The average
+LW intensity in Skinner & Wise (2020) increased stochastically from
+10−3 at 𝑧 ∼ 25 to 10 at 𝑧 ∼ 10. For redshifts above ∼12, 𝐽21 remained
+> 0.1.
+Self-shielding is a process that occurs when large column den-
+sity of molecular hydrogen protects the inner regions against pho-
+todissociation because one photon can only photodissociate one H2
+molecule. This self-shielding allows further H2 production and H2
+cooling (Shang et al. 2010; Agarwal et al. 2014; Regan et al. 2014;
+Hartwig et al. 2015a). The large nonequilibrium abundance of elec-
+trons in gas cooling from above T > 104 K also boosts H2 formation
+(Oh & Haiman 2002). Early attempts to model self shielding (e.g.
+Shang et al. 2010) multiplied the intensity in the LW band by a self-
+shielding factor given by Draine & Bertoldi (1996). Wolcott-Green
+et al. (2011) showed that this method underestimated the numerically
+calculated self-shielding rate by more than an order of magnitude in
+low-density regions, by overestimating shielding by a large factor at
+temperatures above a few hundred kelvin. They modified the method
+of Draine & Bertoldi by estimating the shielding factor based on
+the Sobolev length, using local properties of the gas. This modifica-
+tion was computationally inexpensive and used in many subsequent
+investigations into the aforementioned critical intensity required to
+form atomic halos, typically producing values an order of magnitude
+lower than those using the original Draine & Bertoldi shielding (e.g.
+Glover 2015; Agarwal & Khochfar 2015; Agarwal et al. 2016). Clark
+et al. (2012) improved on this method further with their introduction
+of the TreeCol algorithm, which calculates maps of the column den-
+sity distribution seen by each computational element in a simulation
+in a computationally efficient fashion with the help of an oct-tree.
+Hartwig et al. (2015b) took this approach further by accounting for
+the relative velocities between different computational elements. This
+Doppler-shifts the spectral lines, reducing the effectiveness of self-
+shielding (since molecules shifted by more than the linewidth do not
+contribute to the effective column density).
+In addition to a background LW field, primordial star formation is
+complicated further by streaming velocities between the the gas and
+DM. Prior to recombination, baryons were tightly coupled to photons.
+As DM does not experience Thomson scattering, there should have
+been a relative velocity between the DM and baryons (e.g. Ma &
+Bertschinger 1995). At recombination, the relative velocity was ∼ 30
+km s−1 and was coherent over several comoving Mpc. Recombination
+resulted in a drop in the sound speed to ∼ 6 km s−1 as the gas
+transitioned from plasma to a neutral state, meaning the relative
+velocities were highly supersonic. Tseliakhovich & Hirata (2010)
+showed that the presence of these large-scale streaming velocities
+suppresses the abundance of the first bound objects by advecting
+small-scale perturbations near the baryonic Jeans scale. Moving-
+mesh calculations by Greif et al. (2011) found that the additional
+momentum and energy from the streaming velocities reduces the gas
+fractions and central densities of halos, increasing the typical virial
+mass required for efficient cooling by a factor of three. They also
+noted that the turbulent velocity dispersion increased in the presence
+of streaming velocities. The simulations of AS21 found that the
+increase in the average and minimum halo mass from increasing
+streaming velocities is additive on top of the effect of a LW field,
+with streaming velocities having the larger impact of the two.
+While it was initally believed that Pop III stars formed in isolation
+(Haiman et al. 1996b) and were massive (Abel et al. 2002; Bromm
+et al. 2002), more recent studies show that primordial gas fragments
+to give rise to a larger populations of lower mass stars (Clark et al.
+2011; Smith et al. 2011; Greif et al. 2012; Stacy & Bromm 2013;
+Machida & Doi 2013; Stacy et al. 2014; Susa 2019; Wollenberg
+et al. 2020). In Prole et al. (2022a) (hereafter LP22), we used high
+resolution simulations of idealised, purely hydrodynamical Pop III
+star formation to show that a number of cores are ejected from the
+system with masses capable of surviving until the present day. As
+small-scale primordial magnetic fields do not appear to prevent disc
+MNRAS 000, 1–13 (2020)
+
+Cosmological Lyman-Werner simulations.
+3
+fragmentation (Prole et al. 2022b) and accretion of metals onto the
+surface of these stars during their lifetime is unlikely (e.g. Johnson
+& Khochfar 2011; Tanaka et al. 2017), the question is raised about
+why these stars have not been found within archeological surveys
+(see e.g. Beers & Christlieb 2005; Frebel & Norris 2015; Starken-
+burg et al. 2017). Since most high resolution simulations of Pop III
+star formation have considered only the Pop III.1 case, one possible
+explanation could be that Pop III.2 star formation yields a different
+IMF, i.e. that Pop III stars forming in the presence of a LW back-
+ground have systematically larger masses than those forming in the
+absence of a background.
+In this paper, we aim to test this hypothesis by producing the
+most accurate prediction of the Pop III.1 and Pop III.2 initial mass
+functions (IMF) to date. We investigate how the increase in halo
+masses due to increasing LW field intensity changes star formation
+within them, by performing high resolution follow-up simulations
+of cosmological halos drawn from the simulations of AS21. The
+structure of our paper is as follows. In Section 2, we describe the
+cosmological simulations of AS21, our selection criteria for the halos
+chosen for follow-up simulations, the chemical model we use and our
+use of sink particles. In Section 3 we review the characteristics of the
+halos as they are taken from AS21, before presenting the results of the
+zoom-in simulations in Section 4, where we probe the density regime
+of the molecular core. In Section 5, we compare the fragmentation
+behaviour once sink particles have formed and present the IMFs at
+the end of the simulations. We discuss caveats in Section 6 before
+concluding in Section 7.
+2 NUMERICAL METHOD
+2.1 Arepo
+The simulations presented here were performed with the moving
+mesh code Arepo (Springel 2010) with a primordial chemistry set-
+up. Arepo combines the advantages of AMR and smoothed particle
+hydrodynamics (SPH: Monaghan 1992) with a mesh made up of a
+moving, unstructured, Voronoi tessellation of discrete points. Arepo
+solves hyperbolic conservation laws of ideal hydrodynamics with a
+finite volume approach, based on a second-order unsplit Godunov
+scheme with an exact Riemann solver. Automatic and continuous
+refinement overcome the challenge of structure growth associated
+with AMR (e.g. Heitmann et al. 2008).
+2.2 Cosmological simulations
+The cosmological simulations performed by AS21 assumed a ΛCDM
+cosmology with parameters ℎ = 0.6774, Ω0 = 0.3089, Ωb =
+0.04864, ΩΛ = 0.6911, 𝑛 = 0.96 and 𝜎8 = 0.8159 as derived by
+the Planck Collaboration et al. (2020). The simulations were ini-
+tialised at 𝑧 = 200 with an initial DM distribution created by MUSIC
+(Hahn & Abel 2011) using the transfer functions of Eisenstein &
+Hu (1998) and the gas distribution initially followed the DM. The
+DM was represented by 10243 particles and the gas was initially
+modelled with 10243 grid cells, all contained within a box with side
+length 1 ℎ−1 Mpc in comoving units. During the simulation, an ad-
+ditional Jeans refinement criterion was applied: cells were refined
+whenever necessary so as to ensure that the Jeans length was always
+resolved with at least 16 cells. This refinement was carried out until
+the gas reached a threshold density of ∼ 10−19 g cm−3. Above this
+threshold density, gravitationally bound and collapsing gas was con-
+verted into collisionless sink particles, as explained at greater length
+in AS21. The simulations were carried out with four different values
+of the baryonic streaming velocity (𝑣str = 0, 1, 2 and 3, in units of
+𝜎rms, the large-scale root mean squared value) and three different
+values for the LW field strength (𝐽21 = 0, 0.01 and 0.1). In this study,
+we make use of the three simulations with 𝑣str = 1𝜎rms because this
+is the most representative value available, as the volume fraction of
+streaming velocities peaks at 0.8𝜎rms (AS21).
+2.3 Halo selection
+Given snapshots at 𝑧 = 15 from the three simulations with 𝑣str = 1
+presented in AS21, we have selected 5 halos for each value of 𝐽21.
+Halo positions and masses were provided by the friends-of-friends
+(FoF) algorithm as described in that study. The selection criteria for
+the halos was as follows:
+• The halos identified by the FoF algorithm were sorted by their
+mass difference with Mav, the average minihalo mass above which
+minihalos become capable of cooling and forming stars. This average
+mass was defined in AS21, following Schauer et al. (2019), to be the
+minihalo mass above which more than 50% of minihalos can cool
+and form stars.1 AS21 report Mav at a range of redshifts between
+𝑧 = 22 and 𝑧 = 14; here, we adopt their values at 𝑧 = 15. By sorting
+the halos in this way, we ensure that the halos that we eventually
+select will have masses close to Mav, i.e. that they are representative
+of the common case at that redshift.
+• We considered a halo if it contained a cell denser than 𝜌th =
+10−22 g cm−3 within a search radius Rsearch = 1 kpc in physical
+units from the halo’s central coordinate. Note that checking the H2
+abundance is not necessary, as collapse to this density is not possible
+without a high H2 abundance outside of the atomically cooling halo
+scenario.
+• We check that the halo is sufficiently resolved i.e. that the 16
+cells per Jeans length criteria has not decreased as the cell sizes
+approach the maximum resolution of the cosmological simulations.
+This process begins when the density reaches ∼ 10−20 g cm−3, so
+we reject halos including a cell above this density. Note that this also
+ensures that a sink particle has not yet been formed within Rsearch.
+• Lastly, we removed the halo from consideration if there were
+other dense objects in the vicinity by checking there were no cells
+above 𝜌th within the radius Rsearch/2 < r < Rsearch.
+We performed this search throughout the simulation cube until 5
+suitable halos were selected for each 𝐽21 value. The selected halo
+masses from each simulation are shown in Table 1. Mass-weighted
+mean temperatures and H2 abundances are plotted as a function of
+density for each halo in Figure 1 and radial distributions of enclosed
+gas and DM along with velocity information are shown in Figure
+2. Projections of the initial density, temperature and H2 abundance
+distributions are visualised in Figures 3, 4 and 5. Note that all of
+these figures show the state of the halos at the point at which we
+extract them from the AS21 simulations, i.e. prior to our zoom-in
+calculation. Our choice of selection criteria means that the same
+halos are not chosen for direct comparison between each simulation.
+Rather, they are chosen to be most representative of the Universe at
+the given background LW field strength. Direct comparison also has
+the drawback that individual halos will collapse at different redshifts
+depending on the LW field strength, whereas we exclusively select
+halos that become able to form stars at 𝑧 = 15.
+1 Note that Mav is typically around a factor of three larger than 𝑀min, the
+mass of the least massive minihalo with cool gas.
+MNRAS 000, 1–13 (2020)
+
+4
+L. R. Prole
+10
+2
+10
+1
+100
+101
+102
+103
+n [cm
+3]
+101
+102
+103
+T [K]
+J21=0
+10
+2
+10
+1
+100
+101
+102
+103
+n [cm
+3]
+J21=0.01
+10
+2
+10
+1
+100
+101
+102
+103
+n [cm
+3]
+J21=0.1
+halo 1
+halo 2
+halo 3
+halo 4
+halo 5
+10
+26 10
+25 10
+24 10
+23 10
+22 10
+21
+ [g cm
+3]
+10
+12
+10
+10
+10
+8
+10
+6
+10
+4
+H2
+10
+26 10
+25 10
+24 10
+23 10
+22 10
+21
+ [g cm
+3]
+10
+26 10
+25 10
+24 10
+23 10
+22 10
+21
+ [g cm
+3]
+Figure 1. Summaries of the initial conditions for our simulations. Mass-weighted density profiles of temperature and H2 abundances.
+102
+104
+106
+Mgas [M
+]
+J21=0
+J21=0.01
+J21=0.1
+halo 1
+halo 2
+halo 3
+halo 4
+halo 5
+103
+105
+107
+MDM [M
+]
+0.25
+0.50
+0.75
+1.00
+v /v
+100
+101
+102
+103
+R [pc]
+100
+101
+v/cs
+100
+101
+102
+103
+R [pc]
+100
+101
+102
+103
+R [pc]
+Figure 2. Summaries of the initial conditions for our simulations. Cumulative radial profiles of gas and DM mass and mass-weighted radial profiles of the ratio
+of rotational to total velocity (note the dotted line represents the value above which rotational component dominates the velocity) and ratio of velocity to sound
+speed i.e. Mach number.
+2.4 Chemistry
+Collapse of primordial gas is closely linked to the chemistry involved
+(e.g. Glover et al. 2006; Yoshida et al. 2007b; Glover & Abel 2008;
+Turk et al. 2011). We therefore use a fully time-dependent chemical
+network to model the gas. We use the same chemistry and cooling
+as Wollenberg et al. (2020), which is described in the appendix of
+Clark et al. (2011), but with updated rate coefficients, as summarised
+in Schauer et al. (2017). The network has 45 chemical reactions
+to model primordial gas made up of 12 species: H, H+, H−, H+
+2 ,
+H2, He, He+, He++, D, D+, HD and free electrons. Included in the
+network are: H2 cooling (including an approximate treatment of the
+effects of opacity), collisionally-induced H2 emission, HD cooling,
+ionisation and recombination, heating and cooling from changes in
+the chemical make-up of the gas and from shocks, compression and
+expansion of the gas, three-body H2 formation and heating from
+accretion luminosity. For reasons of computational efficiency, the
+MNRAS 000, 1–13 (2020)
+
+Cosmological Lyman-Werner simulations.
+5
+500 pc
+Figure 3. Column-weighted density projections of the 2 kpc region around the 5 cosmological halos for each 𝐽21 value, taken from Schauer et al. (2021),
+which serve as the initial conditions of the high resolution follow-up simulations presented in this work. The halo numbers to compare with upcoming plots are
+indicated at the top of the figure.
+500 pc
+Figure 4. Column-weighted temperature projections of the 2 kpc region around the 5 cosmological halos for each 𝐽21 value, taken from Schauer et al. (2021),
+which serve as the initial conditions of the high resolution follow-up simulations presented in this work.
+network switches off tracking of deuterium chemistry2 at densities
+above 10−16 g cm−3, instead assuming that the ratio of HD to H2 at
+these densities is given by the cosmological D to H ratio of 2.6×10−5.
+The adiabatic index of the gas is computed as a function of chemical
+composition and temperature with the Arepo HLLD Riemann solver.
+As in Schauer et al. (2021), we use a radiation field expected from
+massive Pop III stars. We use a blackbody spectrum at temperature
+2 Note that HD cooling continues to be included in the model.
+105 K for energies below 13.6 eV. Above this energy, the flux is
+expected to drop due to absorption in the intergalactic medium, so
+we set the value of the radiation field to 0 above 13.6 eV. We model
+the effects of H2 self-shielding using the TreeCol algorithm (Clark
+et al. 2012).
+MNRAS 000, 1–13 (2020)
+
+6
+L. R. Prole
+500 pc
+Figure 5. Column-weighted H2 abundance projections of the 2 kpc region around the 5 cosmological halos for each 𝐽21 value, taken from Schauer et al. (2021),
+which serve as the initial conditions of the high resolution follow-up simulations presented in this work.
+Table 1. Total mass (DM and gas) as detected by the FoF algorithm within
+the selected halos from Schauer et al. (2021).
+𝐽21=0
+𝐽21=0.01
+𝐽21=0.1
+halo
+Mtot [106 M⊙]
+1
+3.50
+4.76
+8.70
+2
+3.65
+4.84
+8.30
+3
+2.81
+4.30
+6.70
+4
+2.76
+4.02
+4.15
+5
+2.96
+3.91
+3.90
+2.5 Sink particles
+If the local Jeans length falls below the minimum cell size of the
+mesh, artificial collapse occurs. We insert sink particles into the sim-
+ulations at a threshold density to prevent artificial collapse when the
+simulation reaches its maximum refinement level. Our sink particle
+implementation was introduced in Wollenberg et al. (2020) and Tress
+et al. (2020). A cell is converted into a sink particle if it satisfies three
+criteria: 1) it reaches a threshold density; 2) it is sufficiently far away
+from pre-existing sink particles so that their accretion radii do not
+overlap; 3) the gas occupying the region inside the sink is gravitation-
+ally bound and collapsing. Likewise, for the sink particle to accrete
+mass from surrounding cells it must meet two criteria: 1) the cell lies
+within the accretion radius; 2) it is gravitationally bound to the sink
+particle. A sink particle can accrete up to 90% of a cell’s mass, above
+which the cell is removed and the total cell mass is transferred to the
+sink.
+Increasing the threshold density for sink particle creation dras-
+tically increases the degree of fragmentation, reducing the masses
+of subsequent secondary protostars (LP22). Ideally, sink particles
+would be introduced when the gas becomes adiabatic at ∼ 10−4 g
+cm−3. This is currently computationally challenging. However, the
+zero metallicity protostellar model of Machida & Nakamura (2015)
+suggests that stellar feedback kicks in to halt collapse at ∼ 10−6 g
+cm−3 (1018 cm−3), so we choose this as our sink particle creation
+density.
+The initial accretion radius of a sink particle 𝑅sink is chosen to
+be the Jeans length 𝜆J corresponding to the sink particle creation
+density and corresponding temperature. At 10−6 g cm−3, we take
+the temperature value from LP22 of 4460 K to give a Jeans length
+of 1.67 × 1012 cm. We set the minimum cell length to fit 8 cells
+across the sink particle in compliance with the Truelove condition,
+by setting a minimum cell volume 𝑉min = (𝑅sink/4)3. The minimum
+gravitational softening length for cells and sink particles 𝐿soft is set
+to 𝑅sink/4.
+The increasing radius of a Pop III protostar is dependent on both
+its mass and accretion rate (e.g. Omukai & Palla 2003; Hosokawa
+& Omukai 2009; Hosokawa et al. 2012; Hirano et al. 2014). We
+allow the sink particle accretion radius 𝑅sink to vary throughout its
+accretion history, using on-the-fly calculations of the stellar radius
+using an approximate analytic formulae originally derrived by Stahler
+et al. (1986):
+𝑅sink = 26𝑅⊙
+� 𝑀
+𝑀⊙
+�0.27 �
+�𝑀
+10−3𝑀⊙yr−1
+�0.41
+,
+(5)
+where we smooth �𝑀 by taking the average over the time taken to
+accrete 0.1M⊙.
+The sink particle treatment also includes the accretion luminosity
+feedback from Smith et al. (2011), as implemented in Arepo by
+Wollenberg et al. (2020). Stellar internal luminosity is not included
+in this work because the Kelvin-Helmholtz times of the protostars
+formed in our simulations are much longer than the period simulated,
+meaning that none will have yet begun nuclear burning. We also
+include the treatment of sink particle mergers used in LP22.
+3 INITIAL HALO CHARACTERISTICS
+Figures 1 and 2 show some of the characteristics of the halos at the
+time when they were selected for zoom-in follow-up simulations.
+MNRAS 000, 1–13 (2020)
+
+Cosmological Lyman-Werner simulations.
+7
+The top panel of Figure 1 shows that the gas has already been shock
+heated as it fell into the gravitational potential well of the DM halo,
+allowing it to produce the necessary H2 to cool and collapse to higher
+densities. The bottom panel shows the destructive impact of the LW
+radiation field on the H2 abundance in the outer regions of the halo.
+The H2 abundance in the central regions reaches the same peak value
+of 4 × 10−4 due to self-shielding from the radiation field, however
+this happens at increasingly higher densities for larger 𝐽21 values.
+Figure 2 shows that within ∼ 100 pc, the gas velocity is dominated
+by its rotational component. The gas surrounding the halos is highly
+supersonic due to large-scale streaming, which cascades down to
+become subsonic at scales smaller than ∼ 10 pc, although we show
+in Section 4 that the velocities do not remain subsonic once the gas
+collapses further. From the density projections of Figure 3, halo sizes
+range from ∼ 100−200 pc, their shapes range from roughly spherical
+to structurally complex, and each is embedded within a network of
+filamentary structures. Figure 4 shows that these webs of filaments
+are a few hundred kelvin hotter than the ∼ 10 K gas that surrounds
+them, with the central halo reaching ∼ 1000 − 2000 K. Figure 5
+shows how the background H2 abundance is reduced drastically by
+the LW background, with significant levels of H2 only appearing in
+the central regions of the halo.
+4 FURTHER COLLAPSE
+We continue the collapse down to densities of 10−6 g cm−3 before
+inserting sink particles. Figure 6 shows temperature and chemical
+abundance profiles as a function of density, just before the formation
+of the first sink particle. At densities above 10−15 g cm−3, three-body
+H2 formation raises its abundance to over 0.1 within the molecular
+core. The abundance of free electrons falls off with density as the gas
+recombines. Figure 7 shows radial profiles of density and velocity.
+The rotational component of the gas velocity remains dominated by
+rotation only down to scales of ∼ 1 pc, below which infall begins to
+dominate. The velocities remain supersonic down to scales of 10−6
+pc (0.2 au). Halos experiencing a LW field are capable of achieving
+higher velocities, likely due to the higher halo mass.
+The left hand side of Figure 8 compares the temperature, density,
+accretion timescale and H2 abundance radial profiles for the differ-
+ent 𝐽21 values just before the formation of the first sink particle.
+Stronger LW fields require higher mass halos for star formation, as
+their stronger gravitational potential is capable of shock-heating the
+gas to higher temperatures, which increases the H2 formation rate
+enough to build up a high column density of H2 in order to self-shield
+the collapsing regions. Despite larger halo masses and shock-heating
+to higher initial temperatures, the density profile of the gas remains
+unaffected. Following Abel et al. (2002) and O’Shea & Norman
+(2008), we have also estimated the accretion timescale (𝑡acc = M/ �M)
+where
+�M = 4𝜋𝑅2𝜌(𝑅)𝑣rad(𝑅)
+(6)
+is our estimate of the mass inflow rate at radius 𝑅 and 𝜌(𝑅) and
+𝑣rad(𝑅) are the mass-weighted density and radial velocity within
+shells at radius 𝑅. For 𝑡acc > 104 yr, there is very little difference
+between the runs, suggesting that the accretion rate at early times is
+not influenced by the LW field. For larger 𝑡acc, we do see a difference
+between different runs, but this manifests as an increased scatter in
+𝑡acc at a given 𝑅 rather than any systematic dependence on the LW
+field strength.
+The right hand side of Figure 8 shows the gas as it transitions into a
+fully molecular state within the inner ∼ 10−2 pc core corresponding
+to the density regime above 10−15 g cm−3. Halos illuminated gy
+a LW background have higher gas kinetic energies owing to their
+larger masses, which promotes a larger molecular core. However,
+the increasing photodissociation rate with increasing 𝐽21 acts against
+this mechanism, reducing the H2 formation rate and shrinking the
+molecular core. This results in the 𝐽21 = 0.01 halos having molecular
+cores that extend to larger radii than the 𝐽21 = 0.1 halos, despite both
+having larger molecular cores than the 𝐽21 = 0 halos. As we only
+have access to these 3 values of 𝐽21, the value where the core size is
+maximised could lie anywhere between 0 < 𝐽21 < 0.1.
+The importance of the molecular core is shown in Figure 9, which
+shows the total mass in sink particles at the end of the simulations
+as functions of the halo mass, virial temperature and mass within
+different regimes of the collapse, just before the maximum density
+was reached. The mass in sink particles grows almost linearly with
+the mass within the inner molecular core with H2 abundances above
+10−1 as
+log10(Msinks) = (0.85±0.11)log10(MH2>10−1) + (0.14±0.24). (7)
+Due to competing effects between halo mass and H2 photodissoci-
+ation rate, the mass in sink particles is not correlated with the total
+halo mass or the subsequent mass that initially falls into its potential
+well with H2 abundances > 10−4. However, we have only followed
+the accretion for 300 yr, which corresponds to a free-fall time for gas
+at density 10−14 g cm−3. Gas below this density will not have been
+accreted within the simulation time, while gas above this density re-
+sides within the molecular core (see Figure 6). It is therefore unclear
+if the relationship between accretion and mass of the molecular core
+would remain if the simulations ran for a longer time. If the relation-
+ship does hold, we speculate that increasing the size of the streaming
+velocities between gas and DM may have a greater effect on the IMF,
+since this also increases halo masses without the counteracting ef-
+fects of H2 dissociation (Tseliakhovich & Hirata 2010; Greif et al.
+2011; Hirano et al. 2017; Schauer et al. 2019, AS21).
+5 FRAGMENTATION AND THE IMF
+The IMF of Pop III stars is determined by the fragmentation be-
+haviour of the disc around the initial central object and the subse-
+quent accretion onto fragments. Density projections of the inner 650
+AU of the halos are shown in Figure 10, while Figure 11 shows the
+evolution of the total number of sink particles formed, the total mass
+in sink particles and highest mass sink particle in each halo as a
+function of time. While the halos from the simulation without a LW
+background typically yield less fragmentation, the overall fragmen-
+tation behaviour is stochastic, as expected. The total mass accreted
+onto sinks is typically higher in the halos illuminated by a LW field
+due to their higher halo mass.
+Figure 12 shows the IMF at a time 300 yr after the formation
+of the first sink particle. The peak of the IMF positioned at ∼0.2-
+0.5 M⊙ shows little dependence on the LW strength, likely because
+the positive influence of larger halo masses on the molecular core
+is regulated by the increasing photodissociation rate. We also show
+the evolution of the cumulative IMFs in time, which converge by the
+end of the simulations for the 𝐽21=0.01 and 0.1 suites. The left side
+of Figure 13 compares the combined cumulative IMFs for different
+𝐽21 values. While the high mass end of the IMFs are nearly identical
+between the different 𝐽21 values, the low mass end of the IMFs show
+variance due to the random and stochastic nature of ejection events
+for low mass objects. Assuming these low mass objects (< 0.075M⊙)
+do not go on to accrete significant mass, they will remain as brown
+MNRAS 000, 1–13 (2020)
+
+8
+L. R. Prole
+105
+108
+1011
+1014
+1017
+n [cm
+3]
+103
+T [K]
+J21=0
+105
+108
+1011
+1014
+1017
+n [cm
+3]
+J21=0.01
+105
+108
+1011
+1014
+1017
+n [cm
+3]
+J21=0.1
+halo 1
+halo 2
+halo 3
+halo 4
+halo 5
+10
+3
+10
+2
+10
+1
+100
+H2
+10
+7
+10
+6
+10
+5
+HD
+10
+19
+10
+16
+10
+13
+10
+10
+10
+7
+10
+4
+ [g cm
+3]
+10
+11
+10
+9
+10
+7
+10
+5
+H +
+10
+19
+10
+16
+10
+13
+10
+10
+10
+7
+10
+4
+ [g cm
+3]
+10
+19
+10
+16
+10
+13
+10
+10
+10
+7
+10
+4
+ [g cm
+3]
+Figure 6. Mass-weighted temperature, H2, HD and H+ abundances versus density at a time shortly after the formation of the first sink particle. (Note that the
+HD abundance is only tracked self-consistently up to 𝜌 = 10−16 g cm−3; above this, we simply fix the HD/H2 ratio at 2.6×10−5, as explained in Section 2.4).
+10
+22
+10
+16
+10
+10
+10
+4
+ [g cm
+3]
+J21=0
+J21=0.01
+J21=0.1
+halo 1
+halo 2
+halo 3
+halo 4
+halo 5
+5
+10
+15
+|v| [km s
+1]
+10
+5
+0
+vr [km s
+1]
+0.50
+0.75
+1.00
+|v /v|
+10
+7
+10
+5
+10
+3
+10
+1
+101
+103
+R [pc]
+10
+1
+100
+101
+|v|/cs
+10
+7
+10
+5
+10
+3
+10
+1
+101
+103
+R [pc]
+10
+7
+10
+5
+10
+3
+10
+1
+101
+103
+R [pc]
+Figure 7. Radial distribution of cumulative gas mass and mass-weighted radial profiles of radial velocity, ratio of rotational to total velocity (note the dotted line
+represents the value above which rotational component dominates the velocity) and ratio of velocity to sound speed, taken at a time shortly after the formation
+of the first sink particle.
+MNRAS 000, 1–13 (2020)
+
+Cosmological Lyman-Werner simulations.
+9
+Figure 8. Left: Comparison of the mass-weighted temperature, density and and H2 abundance profiles between the 𝐽21 values. Right: Zoom-in of the transition
+to fully molecular core, showing the total kinetic energy within radial shells, mass weighted H2 photodissociation heating rate and H2 abundance.
+4
+6
+8
+Mhalo [M
+]
+1e6
+5
+10
+15
+20
+25
+30
+35
+40
+Msinks [M
+]
+2000
+3000
+4000
+Tvir [K]
+50000
+100000
+150000
+MH2 > 10
+4 [M
+]
+J21=0
+J21=0.01
+J21=0.1
+10
+20
+30
+40
+MH2 > 10
+1 [M
+]
+M0.85
+Figure 9. Total mass in sinks at the end of the simulations versus halo mass, virial temperature, mass within the halo with H2 abundance > 10−4 and mass
+within the molecular core with H2 abundance > 10−1. The size of the markers is scaled with the halo mass.
+dwarfs and never sustain nuclear fusion. The right side of the plot
+shows the cumulative IMFs if the brown dwarfs are ignored. Here,
+the IMFs fit well within each other’s regions of uncertainty, which
+are the standard deviations from the cumulative IMFs of the 5 halos
+individually. The overlap between the regions of uncertainty indicates
+that the LW strength does not significantly affect the primordial IMF.
+As the range of background LW field strengths tested here covers the
+most likely values from literature, we infer that the IMF for Pop III.2
+stars is not significantly different from the initial population of Pop
+III.1 stars.
+We also show the IMF from LP22 as a dotted line, which was
+produced from idealised halos as opposed the cosmological initial
+conditions. The cosmological halos produced a bi-modal distribution
+with a significant population of brown dwarfs that the idealised halos
+are missing. Even when ignoring the brown dwarfs, the cosmological
+initial conditions have yielded distributions tending to lower mass
+pre-stellar cores.
+MNRAS 000, 1–13 (2020)
+
+10
+L. R. Prole
+250 AU
+Figure 10. Column-weighted density projections of the inner 650 AU of the halos at a time 300 yr after the formation of the first sink particle. Sink particles are
+represented as red dots.
+0
+10
+20
+30
+Nsink
+J21=0
+J21=0.01
+J21=0.1
+halo 1
+halo 2
+halo 3
+halo 4
+halo 5
+100
+101
+Mtot [M
+]
+0
+100
+200
+300
+t [yr]
+10
+1
+100
+101
+Mmax [M
+]
+100
+200
+300
+t [yr]
+100
+200
+300
+t [yr]
+Figure 11. Number of sink particles formed, total mass in sinks, largest mass sink particle and median mass of sink particles as a function of time.
+6 CAVEATS
+Aside from the obvious uncertainties in the ΛCDM model on which
+this work and the work of AS21 are based on, there are a number of
+caveats to note.
+The LW fields we have used in this study assume a population
+of massive Pop III stars already exists. While previous studies do
+suggest that Pop III stars were massive, more recent work suggests
+that they may only grow to a few M⊙ (e.g. Stacy & Bromm 2013;
+Wollenberg et al. 2020; Prole et al. 2022a; Jaura et al. 2022). In this
+scenario, significantly less LW radiation would be produced, however
+radiation from solar mass stars can inhibit H2 formation through the
+destruction of H−. The fields used in this study therefore represent the
+maximum effect Pop III stars can have on the the Pop III.2 stars that
+follow. Since the effects of the LW fields appear to be insignificant, it
+is likely that this result also represents the outcome for weaker fields.
+We have assumed that the pre-stellar core radius grows as Equation
+5. This process begins immediately after sink particle formation.
+While this is predicted by stellar theory, it is unclear at what point
+the pre-stellar core would begin to expand, which may affect the
+accretion behaviour.
+MNRAS 000, 1–13 (2020)
+
+Cosmological Lyman-Werner simulations.
+11
+0
+1
+2
+3
+4
+J21=0
+0.5
+1.0
+0
+5
+10
+15
+20
+25
+Nsink
+J21=0.01
+0.5
+1.0
+M/Mtot
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+M [M
+]
+0
+2
+4
+6
+8
+J21=0.1
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+M [M
+]
+0.0
+0.5
+1.0
+10 yr
+50 yr
+100 yr
+200 yr
+300 yr
+Figure 12. Left: combined IMFs of sink particles for the different the 𝐽21 val-
+ues, Right: Cumulative IMFs of the combined sink particles for the different
+the 𝐽21 values.
+We have neglected the presence of primordial magnetic fields in
+this study. While the findings of Prole et al. (2022b) suggest that
+primordial magnetic fields make little difference to the primordial
+IMF due to their small-scale structure, the field is still active, with
+other recent studies finding that magnetic fields indeed lead to higher
+mass Pop III stars (e.g. Saad et al. 2022; Hirano & Machida 2022;
+Stacy et al. 2022).
+7 CONCLUSIONS
+The results of cosmological simulations by AS21 show that increas-
+ing the background LW field strength increases the average halo mass
+required for star formation. These simulations ran up to a maximum
+density of 10−19 g cm−3, hence the knock-on effects on the Pop III
+IMF were unclear. In this investigation, we have performed follow-up
+simulations of 5 halos for each of the 𝐽21 = 0, 0.01 and 0.1 LW field
+strengths, resolving the pre-stellar core density of 10−6 g cm−3 before
+inserting sink particles and following the fragmentation behaviour
+for hundreds of years further. We have found that the mass accreted
+onto sinks by the end of the simulations is proportional to the mass
+within the ∼ 10−2 pc molecular core, which is not correlated to the
+initial mass of the halo. As such, the IMF shows little dependence
+on the LW field strength. We also find no clear relationship between
+the estimated accretion time of gas lying further out within the halo
+and the LW field strength, suggesting that the LW field is unlikely
+to influence the development of the IMF at later times. As the range
+of background LW field strengths tested here covers the most likely
+values from literature, we conclude that the IMF for so-called Pop
+III.2 stars is not significantly different from that of the initial popu-
+lation of Pop III.1 stars, although we cannot rule out greater effects
+in the small subset of halos that are illuminated by LW fields much
+stronger than the average value (see e.g Latif et al. 2014). In a future
+paper, we will explore the effects of increasing the streaming veloci-
+ties between the gas and dark matter on the Pop III IMF, as this has
+been shown to increase halo masses through a different mechanism.
+ACKNOWLEDGEMENTS
+This work used the DiRAC@Durham facility managed by the
+Institute for Computational Cosmology on behalf of the STFC
+DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded
+by BEIS capital funding via STFC capital grants ST/P002293/1,
+ST/R002371/1 and ST/S002502/1, Durham University and STFC
+operations grant ST/R000832/1. DiRAC is part of the National e-
+Infrastructure.
+The authors gratefully acknowledge the Gauss Centre for Super-
+computing e.V. (www.gauss-centre.eu) for supporting this project by
+providing computing time on the GCS Supercomputer SuperMUC at
+Leibniz Supercomputing Centre (www.lrz.de) under project pr53ka.
+AS was partially supported by NSF grant AST-1752913.
+RSK and SCOG acknowledge computing resources provided by
+the Ministry of Science, Research and the Arts (MWK) of the State
+of Baden-Württemberg through bwHPC and the German Research
+Foundation (DFG) through grant INST 35/1134-1 FUGG and for
+data storage at SDS@hd through grant INST 35/1314-1 FUGG.
+RSK and SCOG acknowledge financial support from DFG via the
+collaborative research center (SFB 881, Project-ID 138713538) “The
+Milky Way System” (subprojects A1, B1, B2 and B8), from the Hei-
+delberg Cluster of Excellence “STRUCTURES” in the framework
+of Germany’s Excellence Strategy (grant EXC-2181/1, Project-ID
+390900948) and from the European Research Council (ERC) via the
+ERC Synergy Grant “ECOGAL” (grant 855130). RSK furthermore
+thanks the German Ministry for Economic Affairs and Climate Ac-
+tion for funding in the project “MAINN” (funding ID 50OO2206).
+We also acknowledge the support of the Supercomputing Wales
+project, which is part-funded by the European Regional Development
+Fund (ERDF) via Welsh Government.
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the corresponding author.
+REFERENCES
+Abel T., Bryan G. L., Norman M. L., 2002, Science (New York, N.Y.), 295,
+93
+Abel T., Wise J. H., Bryan G. L., 2007, The Astrophysical Journal, 659, L87
+Agarwal B., Khochfar S., 2015, Monthly Notices of the Royal Astronomical
+Society, 446, 160
+Agarwal B., Khochfar S., Johnson J. L., Neistein E., Dalla Vecchia C., Livio
+M., 2012, Monthly Notices of the Royal Astronomical Society, 425, 2854
+Agarwal B., Dalla Vecchia C., Johnson J. L., Khochfar S., Paardekooper J.-P.,
+2014, Monthly Notices of the Royal Astronomical Society, 443, 648
+Agarwal B., Smith B., Glover S., Natarajan P., Khochfar S., 2016, Monthly
+Notices of the Royal Astronomical Society, 459, 4209
+Ahn K., Shapiro P. R., Iliev I. T., Mellema G., Pen U.-L., 2009, The Astro-
+physical Journal, 695, 1430
+Alvarez M. A., Bromm V., Shapiro P. R., 2006, The Astrophysical Journal,
+639, 621
+Beers T. C., Christlieb N., 2005, Annual Review of Astronomy and Astro-
+physics, 43, 531
+MNRAS 000, 1–13 (2020)
+
+12
+L. R. Prole
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Msink [M
+]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+M/Mtot
+10
+1
+100
+101
+Msink [M
+]
+Msink > 0.075M
+J21=0
+J21=0.01
+J21=0.1
+Prole et al. 2022b
+Figure 13. Left: Comparison of the cumulative IMFs at 300 yr after the formation of the first sink particle. Right: Cumulative IMFs ignoring brown dwarfs
+(< 0.075M⊙).
+Bromm V., Loeb A., 2003, The Astrophysical Journal, 596, 34
+Bromm V., Coppi P. S., Larson R. B., 2002, The Astrophysical Journal, 564,
+23
+Chuzhoy L., Kuhlen M., Shapiro P. R., 2007, The Astrophysical Journal, 665,
+L85
+Clark P. C., Glover S. C. O., Smith R. J., Greif T. H., Klessen R. S., Bromm
+V., 2011, Science, 331, 1040
+Clark P. C., Glover S. C. O., Klessen R. S., 2012, Monthly Notices of the
+Royal Astronomical Society, 420, 745
+Collaboration P., et al., 2020, Astronomy and Astrophysics, 641, A6
+Couchman H. M. P., Rees M. J., 1986, Monthly Notices of the Royal Astro-
+nomical Society, 221, 53
+Dijkstra M., Haiman Z., Mesinger A., Wyithe J. S. B., 2008, Monthly Notices
+of the Royal Astronomical Society, 391, 1961
+Draine B. T., Bertoldi F., 1996, The Astrophysical Journal, 468, 269
+Eisenstein D. J., Hu W., 1998, The Astrophysical Journal, 496, 605
+Field G. B., Somerville W. B., Dressler K., 1966, Annual Review of Astron-
+omy and Astrophysics, 4, 207
+Frebel A., Norris J. E., 2015, Annual Review of Astronomy and Astrophysics,
+53, 631
+Glover S. C. O., 2015, Monthly Notices of the Royal Astronomical Society,
+451, 2082
+Glover S. C. O., Abel T., 2008, Monthly Notices of the Royal Astronomical
+Society, 388, 1627
+Glover S. C., Savin D. W., Jappsen A.-K., 2006, The Astrophysical Journal,
+640, 553
+Greif T. H., White S. D. M., Klessen R. S., Springel V., 2011, The Astrophys-
+ical Journal, 736, 147
+Greif T. H., Bromm V., Clark P. C., Glover S. C. O., Smith R. J., Klessen
+R. S., Yoshida N., Springel V., 2012, Monthly Notices of the Royal
+Astronomical Society, 424, 399
+Hahn O., Abel T., 2011, Monthly Notices of the Royal Astronomical Society,
+415, 2101
+Haiman Z., Thoul A. A., Loeb A., 1996a, The Astrophysical Journal, 464,
+523
+Haiman Z., Thoul A. A., Loeb A., 1996b, The Astrophysical Journal, 464,
+523
+Haiman Z., Rees M. J., Loeb A., 1997, The Astrophysical Journal, 476, 458
+Haiman Z., Abel T., Rees M. J., 2000, The Astrophysical Journal, 534, 11
+Hartwig T., Glover S. C. O., Klessen R. S., Latif M. A., Volonteri M., 2015a,
+Monthly Notices of the Royal Astronomical Society, 452, 1233
+Hartwig T., Clark P. C., Glover S. C. O., Klessen R. S., Sasaki M., 2015b,
+The Astrophysical Journal, 799, 114
+Heitmann K., et al., 2008, Computational Science and Discovery, 1, 015003
+Hirano S., Machida M. N., 2022, The Astrophysical Journal, 935, L16
+Hirano S., Hosokawa T., Yoshida N., Umeda H., Omukai K., Chiaki G., Yorke
+H. W., 2014, The Astrophysical Journal, 781, 60
+Hirano S., Hosokawa T., Yoshida N., Omukai K., Yorke H. W., 2015, Monthly
+Notices of the Royal Astronomical Society, 448, 568
+Hirano S., Hosokawa T., Yoshida N., Kuiper R., 2017, Science, 357, 1375
+Hosokawa T., Omukai K., 2009, The Astrophysical Journal, 691, 823
+Hosokawa T., Omukai K., Yorke H. W., 2012, The Astrophysical Journal,
+756, 93
+Jaura O., Glover S. C. O., Wollenberg K. M. J., Klessen R. S., Geen S., Haem-
+merlé L., 2022, Monthly Notices of the Royal Astronomical Society, 512,
+116
+Johnson J. L., Khochfar S., 2011, Monthly Notices of the Royal Astronomical
+Society, 413, 1184
+Kitayama T., Yoshida N., Susa H., Umemura M., 2004, The Astrophysical
+Journal, 613, 631
+Kulkarni M., Visbal E., Bryan G. L., 2021, The Astrophysical Journal, 917,
+40
+Latif M. A., Schleicher D. R. G., Schmidt W., Niemeyer J. C., 2013, Monthly
+Notices of the Royal Astronomical Society, 436, 2989
+Latif M. A., Schleicher D. R. G., Bovino S., Grassi T., Spaans M., 2014, The
+Astrophysical Journal, 792, 78
+Ma C.-P., Bertschinger E., 1995, The Astrophysical Journal, 455, 7
+Machacek M. E., Bryan G. L., Abel T., 2001, The Astrophysical Journal, 548,
+509
+Machida M. N., Doi K., 2013, Monthly Notices of the Royal Astronomical
+Society, 435, 3283
+Machida M. N., Nakamura T., 2015, Monthly Notices of the Royal Astro-
+nomical Society, 448, 1405
+Matsuoka Y., et al., 2019, The Astrophysical Journal, 872, L2
+Monaghan J. J., 1992, Annual Review of Astronomy and Astrophysics, 30,
+543
+Mortlock D. J., et al., 2011, Nature, 474, 616
+O’Shea B. W., Norman M. L., 2008, The Astrophysical Journal, 673, 14
+Oh S. P., Haiman Z., 2002, The Astrophysical Journal, 569, 558
+Omukai K., 2001, The Astrophysical Journal, 546, 635
+Omukai K., Palla F., 2003, The Astrophysical Journal, 589, 677
+Prole L. R., Clark P. C., Klessen R. S., Glover S. C. O., 2022a, Monthly
+Notices of the Royal Astronomical Society, 510, 4019
+Prole L. R., Clark P. C., Klessen R. S., Glover S. C. O., Pakmor R., 2022b,
+Monthly Notices of the Royal Astronomical Society, 516, 2223
+Regan J. A., Johansson P. H., Wise J. H., 2014, The Astrophysical Journal,
+795, 137
+Saad C. R., Bromm V., El Eid M., 2022, Monthly Notices of the Royal
+MNRAS 000, 1–13 (2020)
+
+Cosmological Lyman-Werner simulations.
+13
+Astronomical Society, 516, 3130
+Schaerer D., 2002, Astronomy & Astrophysics, 382, 28
+Schauer A. T. P., Regan J., Glover S. C. O., Klessen R. S., 2017, Monthly
+Notices of the Royal Astronomical Society, 471, 4878
+Schauer A. T. P., Glover S. C. O., Klessen R. S., Ceverino D., 2019, Monthly
+Notices of the Royal Astronomical Society, 484, 3510
+Schauer A. T. P., Glover S. C. O., Klessen R. S., Clark P., 2021, Monthly
+Notices of the Royal Astronomical Society, 507, 1775
+Shang C., Bryan G. L., Haiman Z., 2010, Monthly Notices of the Royal
+Astronomical Society, 402, 1249
+Skinner D., Wise J. H., 2020, Monthly Notices of the Royal Astronomical
+Society, 492, 4386
+Smith R. J., Glover S. C. O., Clark P. C., Greif T., Klessen R. S., 2011,
+Monthly Notices of the Royal Astronomical Society, 414, 3633
+Spaans M., Silk J., 2006, The Astrophysical Journal, 652, 902
+Springel V., 2010, Monthly Notices of the Royal Astronomical Society, 401,
+791
+Stacy A., Bromm V., 2013, Monthly Notices of the Royal Astronomical
+Society, 433, 1094
+Stacy A., Pawlik A. H., Bromm V., Loeb A., 2014, Monthly Notices of the
+Royal Astronomical Society, 441, 822
+Stacy A., McKee C. F., Lee A. T., Klein R. I., Li P. S., 2022, Monthly Notices
+of the Royal Astronomical Society, 511, 5042
+Stahler S., Palla F., Salpeter E., 1986, The Astrophysical Journal„ 302, 90
+Starkenburg E., et al., 2017, Monthly Notices of the Royal Astronomical
+Society, 471, 2587
+Stecher T. P., Williams D. A., 1967, The Astrophysical Journal, 149, L29
+Sugimura K., Omukai K., Inoue A. K., 2014, Monthly Notices of the Royal
+Astronomical Society, 445, 544
+Susa H., 2019, The Astrophysical Journal, 877, 99
+Tanaka S. J., Chiaki G., Tominaga N., Susa H., 2017, The Astrophysical
+Journal, 844, 137
+Tegmark M., Silk J., Rees M. J., Blanchard A., Abel T., Palla F., 1997, The
+Astrophysical Journal, 474, 1
+Trenti M., Stiavelli M., 2009, The Astrophysical Journal, 694, 879
+Tress R. G., Smith R. J., Sormani M. C., Glover S. C. O., Klessen R. S.,
+Mac Low M.-M., Clark P. C., 2020, Monthly Notices of the Royal Astro-
+nomical Society, 492, 2973
+Tseliakhovich D., Hirata C., 2010, Physical Review D, 82, 083520
+Turk M. J., Clark P., Glover S. C. O., Greif T. H., Abel T., Klessen R., Bromm
+V., 2011, Astrophys.J.726:55,2011, 726
+Visbal E., Haiman Z., Terrazas B., Bryan G. L., Barkana R., 2014, Monthly
+Notices of the Royal Astronomical Society, 445, 107
+Whalen D., Abel T., Norman M. L., 2004, The Astrophysical Journal, 610,
+14
+Wise J. H., Abel T., Turk M. J., Norman M. L., Smith B. D., 2012, Monthly
+Notices of the Royal Astronomical Society, 427, 311
+Wolcott-Green J., Haiman Z., Bryan G. L., 2011, Monthly Notices of the
+Royal Astronomical Society, 418, 838
+Wollenberg K. M. J., Glover S. C. O., Clark P. C., Klessen R. S., 2020,
+Monthly Notices of the Royal Astronomical Society, 494, 1871
+Yoshida N., Abel T., Hernquist L., Sugiyama N., 2003, The Astrophysical
+Journal, 592, 645
+Yoshida N., Oh S. P., Kitayama T., Hernquist L., 2007a, The Astrophysical
+Journal, 663, 687
+Yoshida N., Omukai K., Hernquist L., 2007b, The Astrophysical Journal, 667,
+L117
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–13 (2020)
+

MtFJT4oBgHgl3EQfzS0r/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4351c13dc146e6e995050b3a2d9240fda34239fa1fdf38fbcf595533bb10bc60
+size 114227

O9A0T4oBgHgl3EQfC_9-/content/tmp_files/2301.01997v1.pdf.txt

@@ -0,0 +1,1322 @@
+arXiv:2301.01997v1  [cs.LG]  5 Jan 2023
+1
+Data-Driven Inverse Reinforcement Learning for
+Expert-Learner Zero-Sum Games
+Wenqian Xue, Bosen Lian, Jialu Fan, Tianyou Chai, and Frank L. Lewis
+Abstract—In this paper, we formulate inverse reinforcement
+learning (IRL) as an expert-learner interaction whereby the
+optimal performance intent of an expert or target agent is
+unknown to a learner agent. The learner observes the states
+and controls of the expert and hence seeks to reconstruct the
+expert’s cost function intent and thus mimics the expert’s optimal
+response. Next, we add non-cooperative disturbances that seek to
+disrupt the learning and stability of the learner agent. This leads
+to the formulation of a new interaction we call zero-sum game
+IRL. We develop a framework to solve the zero-sum game IRL
+problem that is a modified extension of RL policy iteration (PI)
+to allow unknown expert performance intentions to be computed
+and non-cooperative disturbances to be rejected. The framework
+has two parts: a value function and control action update based
+on an extension of PI, and a cost function update based on
+standard inverse optimal control. Then, we eventually develop
+an off-policy IRL algorithm that does not require knowledge
+of the expert and learner agent dynamics and performs single-
+loop learning. Rigorous proofs and analyses are given. Finally,
+simulation experiments are presented to show the effectiveness
+of the new approach.
+I. INTRODUCTION
+For an agent or system suffering from disturbances, its con-
+trol input, as a defender, desires to complete a specified control
+mission by determining control policy to reject the influences
+of antagonistic input, i.e., non-cooperative disturbances that
+intend to disrupt the mission. This is known as zero-sum games
+or min-max problems [1], [2]. In real-world applications, agent
+dynamics may be unknown. In order to make such an agent
+perform in the target trajectories exhibited by a target agent
+with optimal policy, optimal control theory assumes that the
+performance cost function is known, and RL [3], [4] based
+optimal tracking control methods [5]–[7] compute optimal
+policy by observing states and control actions without knowing
+the system dynamics, where a standard iterative form for RL
+is known as policy iteration (PI) [3], [4], [8], [9]. However,
+in real interactions, operators may not know the appropriate
+specified cost functions, i.e., the weights on states and inputs.
+As a result, these optimal control methods may not obtain the
+expected control performance or even be used.
+Wenqian Xue, Jialu Fan, Tianyou Chai are with the State Key Laboratory of
+Synthetical Automation for Process Industries and International Joint Research
+Laboratory of Integrated Automation, Northeastern University, Shenyang
+110819, China. (e-mail: xuewenqian23@163.com, fanjialu@gmail.com, ty-
+chai@mail.neu.edu.cn).
+Bosen Lian and Frank L. Lewis are with the UTA Research Insti-
+tute, the University of Texas at Arlington, Texas 76118, USA. (email:
+bosen.lian@mavs.uta.edu; lewis@uta.edu).
+Instead of manually selecting cost function weights, many
+efforts have been made on constructing cost function weights.
+Inverse optimal control (IOC) and inverse RL (IRL) construct
+cost function weights given system control behaviors. Some-
+times they are referred to as the same thing [10]–[12], but they
+may differ in structure and how they are applied [13].
+Assuming a stable control system, IOC constructs a cost
+function concerning which the system behavior is optimal.
+The cost function is constructed in the framework of Lyapunov
+stability condition for continuous-time (CT) systems [14]–[17]
+and discrete-time (DT) systems [18]–[20] where [20] considers
+finite horizon. Online IOC methods to determine cost function
+in the infinite and finite horizon are studied in [21], [22]. IOC
+is also used to verify the effectiveness of the proposed control
+laws in [23], [24]. These works do not consider min-max
+or zero-sum games, but [25] does. They all require system
+dynamics, which cannot be applied directly to systems with
+unknown dynamics.
+IRL generally reconstructs reward and cost functions from
+expert demonstrations of the optimal policy. It is usually
+applied to apprenticeship learning and imitation learning prob-
+lems of Markov decision processes (MDPs) [12], [26]–[29]
+where a learner seeks to imitate the demonstrations by learning
+the unknown expert’s reward function from the observed
+demonstrations. IRL methods construct reward function since
+reward function is a more succinct, robust, and transferable
+definition for the task than the policy mapping from states to
+actions. Lyapunov stability is not necessarily considered here.
+IRL has also been developed for trajectory tracking and
+imitation problems of differential systems in [30], [31], where
+[30] uses a bilevel structure (also see [26], [27], [32]). That is,
+an optimal control problem is solved repeatedly in the inner
+loop. This two-loop iteration is computationally expensive. All
+of these works are model-based and do not consider min-max
+or zero-sum games. The work [33] makes an effort for data-
+driven control by estimating model parameters before adopt-
+ing the model-based IRL method. Unlike them, without the
+need for model identification, our previous studies [34], [35]
+propose completely model-free IRL methods that use merely
+system data, but not for zero-sum games. Our work [36]
+considers zero-sum games but propose an IRL method using
+a two-loop iteration structure and partial system dynamics.
+This paper considers an expert-learner zero-sum game, that
+is, a learner agent suffering from non-cooperative disturbances
+with unknown dynamics expects to mimic the behaviors of the
+expert agent of optimal policy. As the solution, we propose a
+new interaction called zero-sum game IRL, namely a novel
+data-driven off-policy IRL algorithm for expert-learner zero-
+
+2
+sum games of differential systems. It consists of a game
+solution correction modified from the standard RL and a cost
+function weight reconstruction using the standard IOC. Using
+only the behavior data of the expert and learner, a learner
+agent learns the unknown cost function objective and the
+optimal control policy to mimic the expert’s behavior. This
+algorithm does not need to know or identify system models
+and performs a single-loop learning procedure without solving
+optimal problems repeatedly in inner loops. Moreover, no
+initial stabilizing policy is needed to start the iteration. The
+properties and effectiveness of the proposed data-driven are
+well-analyzed.
+Notations. ∥·∥ is the Euclidean norm. In is n×n identity ma-
+trix, and diag{a,b,..} is diagonal matrix with a,b,.. in diagonal
+line. For a vector x = [x1, ··· , xn]T ∈ Rn, ˆx ≜ [x2
+1, 2x1x2, ···,
+2x1xn, x2
+2, ···, 2xn−1xn, x2
+n]T. For a matrix A = {aij} ∈ Rm×n,
+vec(A) ≜ [a11, a12, ···, a1n, a21, ···, am,n−1, amn]T.
+II. IRL PROBLEM FORMULATION
+We consider two dynamical agents. A target expert agent
+exhibits the demonstrations optimally associated with an ex-
+pert performance cost function. A learner agent attempts to
+determine the unknown cost function objective of the expert
+agent and mimic its behavior. The learner agent only knows
+the target agent’s control actions and state behavior but does
+not know its performance cost function and system dynamics.
+A. Target optimal control
+Consider a target expert agent
+˙xT = AxT + BuT + DdT,
+(1)
+where xT ∈ Rn is the target state, uT ∈ Rm is the target input
+and dT ∈ Rz is the non-cooperative disturbance. Matrices A, B,
+and D have appropriate dimensions. The pair (A,B) is assumed
+to be controllable.
+According to [2], the input of target (1) is uT = −KTxT
+that minimizes the following target performance cost function
+against dT
+VT(xT) =
+� ∞
+t (xT
+T QTxT + uT
+TRTuT − γ2
+TdT
+T dT)dτ,
+(2)
+where QT = QT
+T > 0 and RT = RT
+T > 0 are weights, γT > 0 is
+attenuation factor. The input uT and the worst disturbance d∗
+T
+are given by
+uT = −KTxT = −R−1
+T BTPTxT,
+(3a)
+d∗
+T = LTxT = 1
+γ2
+T
+DTPTxT,
+(3b)
+and form the Nash equilibrium
+VT(xT)∗ = min
+uT max
+dT
+� ∞
+t
+rTdτ = xT
+TPTxT
+(4)
+where rT = xT
+TQTxT + uT
+TRTuT − γ2
+TdT
+T dT, and PT = PT
+T > 0
+satisfies the target Bellman equation
+(AxT + BuT + DdT)TPTxT + xT
+TPT(AxT + BuT + DdT)
++ xT
+TQTxT + uT
+TRTuT − γ2
+TdT
+T dT = 0,
+(5)
+and the target game algebraic Riccati equation (GARE)
+ATPT + PTA+ QT − PTBR−1
+T BTPT + 1
+γ2
+T
+PTDDTPT = 0.
+(6)
+The GARE guarantees the uniqueness of stabilizing optimal
+control policy (3a) given the performance cost function (2).
+B. Learner dynamics and IRL control problem
+Consider a learner agent to be controlled
+˙x = Ax+ Bu + Dd,
+(7)
+where x ∈ Rn, u ∈ Rm, d ∈ Rz are the learner’s state, control
+input and disturbance, respectively.
+Assumption 1. The target cost function VT (2) and the target
+control policy in (3a) are unknown to the learner (7). That is,
+the weights QT, RT, γT in VT, the optimal strategy KT in (3a),
+the worst disturbance d∗
+T, LT and GARE solution PT are all
+unknown.
+Assumption 2. The learner knows the target behaviour data
+xT, uT and dT.
+Definition 1. Expert-Learner Zero-Sum Game. By using
+the behaviour data of the expert (1) and the learner itself (7),
+the learner desires to reconstruct the unknown performance
+cost function (2) to exhibit the same control actions uT in
+(3a) and states xT as the expert (1).
+□
+The learner (7) will be stabilized and perform the same way
+as the target (1) does if KT in (3a) is applied to the learner (7)
+with bounded dT = d. Hence, our control goal is to determine
+the unknown cost function objective (2) to produce the optimal
+control input u∗ = −K∗x with K∗ = KT using only target data
+of xT,uT,dT and learner data of x,u,d.
+III. MODEL-BASED IRL FRAMEWORK
+In this section, we develop a novel model-based IRL
+framework for learner (7) to determine the cost function (2)
+and use this knowledge to compute control input u(t), such
+that its behavior trajectories of u(t),x(t) mimic the observed
+target trajectories of uT(t),xT(t). In Section 5, we will finally
+propose a data-driven IRL algorithm that does not need any
+system dynamics.
+A. Learner optimal control
+Let us take a review of the zero-sum game [2] of the learner
+(7) with an arbitrary given cost function
+V(x) =
+� ∞
+t (xT Qx+ uTRu − γ2dTd)dτ,
+(8)
+where Q = QT > 0, R = RT > 0, and γ > 0. The optimal input
+uo and the worst dw are given by
+uo = −Kx = −R−1BTPx,
+(9)
+dw = Lx = 1
+γ2 DTPx,
+(10)
+
+3
+and (uo,dw) forms the Nash equilibrium
+V ∗(x) = min
+u max
+d
+� ∞
+t
+rdτ = xTPx,
+(11)
+where r = xTQx+uTRu−γ2dTd, and P = PT > 0 satisfies the
+following learner GARE
+ATP+ PA+ Q− PBR−1BTP+ 1
+γ2 PDDTP = 0.
+(12)
+B. Expert-learner zero-sum game solution
+We now present a theorem to show the conditions that the
+solution to the expert-learner zero-sum game must satisfy.
+Theorem 1. If the cost weights Q,R,γ and the solution P that
+satisfy the GARE (12) also satisfy the following equation
+(A− BKT)T P+P(A− BKT)
++Q+ 1
+γ2 PDDTP+ KT
+T RKT = 0,
+(13)
+then, the corresponding control strategy K in (9) equals the
+target strategy KT in (3a).
+Proof: Rewrite (12) with KT in (3a) and K in (9) as
+(A− BKT)TP+ P(A− BKT)+ Q+ 1
+γ2 PDDTP
++ KT
+T RK + KTRKT − KTRK = 0.
+(14)
+By subtracting (14) from (13), we have
+KT
+T RKT − KT
+T RK − KTRKT + KTRK
+= (KT − K)TR(KT − K) = 0.
+(15)
+Since R > 0, (15) concludes that K = KT .
+□
+C. Learning rules for cost function and game control policy
+To find the Q,R,γ,P satisfying Theorem 1, we select γ > 0
+and R > 0 and propose an iterative procedure based on
+Theorem 1 so as to learn the weight Q, the game control
+policy P, and consequently K in (9) and L in (10).
+First, we apply a modified PI to correct P using (13). Set
+current iteration step as i,i = 0,1,··· , and give the estimates Qi
+and Li. Then the iterative form of (13), i.e., (16) in Algorithm
+1 is presented below to obtain Pi. Then optimal control ((9)
+and (10)) is used to update the strategy Ki+1 and Li+1 based
+on the corrected Pi by (17a) and (17b), respectively.
+Now we must update the cost function weight estimate Qi+1
+based on the corrected Pi. By IOC [15], taking the iterative
+form of GARE (12) yields (18) in Algorithm 1 presented as
+follows.
+Remark 1. In Algorithm 1, step 4 is the standard IOC
+computation based on Lyapunov stability condition. Note also
+that if we set Qi = QT and KT = Ki in (16), then steps 2-3 are
+the standard RL PI. As such, Algorithm 1 combines modified
+PI with IOC to solve the IRL problem.
+Algorithm 1 Model-based IRL algorithm for expert-learner
+zero-sum games.
+Step 1: Initialize with R > 0 and γ > 0, Q0 > 0, and L0 = 0,
+and set i = 0.
+Step 2: (Game policy correction) Update policy Pi by
+(A− BKT)T Pi + Pi(A− BKT)
+= −Qi − KT
+T RKT − γ2(Li)TLi.
+(16)
+Step 3: (Input update) Update control input and disturbance
+gain based on (9) and (10)
+ui+1 = −Ki+1x = −R−1BTPix,
+(17a)
+di+1 = Li+1x = 1
+γ2 DTPix.
+(17b)
+Step 4: (Cost function weight construction) Update Qi+1 by
+Qi+1 =− ATPi − PiA
++ (Ki+1)TRKi+1 − γ2(Li+1)T Li+1.
+(18)
+Step 5: Stop if it converges. Otherwise, set i = i+1 and repeat
+steps 2 to 4.
+IV. ANALYSIS OF ALGORITHM 1
+The convergence, stability, and optimality of the proposed
+Algorithm 1 are analyzed here. It is also shown that Algorithm
+1 may not converge to a unique solution (Pi,Qi) even if all
+solutions give the correct target strategy Ki = KT .
+A. Convergence analysis
+Theorem 2. i). With an initial Q0 such that 0 < Q0 ≤ ˆQ and
+L0 = 0 where ˆQ > 0 is a solution to Theorem 1 associated with
+the R > 0 and γ > 0, Algorithm 1 converges. ii). As i → ∞, the
+solutions Qi,Pi,Ki,Li converge to Q∗,P∗,K∗,L∗ that satisfy
+ATP∗ + P∗A+ Q∗−P∗BR−1BTP∗+ 1
+γ2 P∗DDTP∗ = 0,
+(19a)
+K∗ = KT = R−1BTP∗,
+(19b)
+L∗ = 1
+γ2 DTP∗.
+(19c)
+iii). The solutions Q∗,P∗,K∗,L∗ satisfy Theorem 1.
+Proof: i). Convergence proof. First, we prove that Algorithm
+1 solves an increasing sequence Qi for all i = 0,1,···. Substi-
+tuting (17a) for Ki into (18) for Qi yields
+ATPi−1 + Pi−1A = (Ki)T RKi − Qi − γ2(Li)TLi,
+(20)
+which can be rewritten as
+(A− BKT)TPi−1 + Pi−1(A− BKT)
+= (Ki)TRKi − Qi − γ2(Li)T Li − (Ki)T RKT − KT
+T RKi.
+(21)
+Subtracting (16) from (21) gives
+(A− BKT)T (Pi−1 − Pi)+ (Pi−1 − Pi)(A− BKT)
+= (Ki − KT)TR(Ki − KT).
+(22)
+It follows from (Ki − KT )TR(Ki − KT ) ≥ 0 and Hurwitz
+A − BKT that Pi−1 ≤ Pi holds for all iterations. Each pair of
+
+4
+(Pi,Qi+1) satisfies (18), and they uniquely correspond to each
+other. This is followed by the known fact that Pi−1 ≤ Pi if
+Qi ≤ Qi+1 [37]. Therefore, Qi+1 ≥ Qi > 0 holds for i = 0,1,···,
+and Qi+1 = Qi holds if and only if KT = Ki+1. Note that
+achieving KT = Ki+1 is the goal of the algorithm.
+Now we show that Qi is bounded by an upper bound. Let
+ˆQ > 0 and ˆP > 0 be a group of solution to Theorem 1. That
+is
+AT ˆP+ ˆPA+ ˆQ− ˆPBR−1BT ˆP+ 1
+γ2 ˆPDDT ˆP = 0,
+(23a)
+KT = R−1BT ˆP, ˆL = 1
+γ2 DT ˆP.
+(23b)
+Rewriting (23a) using (23b) yields
+(A− BKT)T ˆP+ ˆP(A− BKT)+ ˆQ+ KT
+T RKT + γ2ˆLT ˆL = 0.
+(24)
+If Qi + γ2(Li)TLi ≤ ˆQ+ γ2 ˆLT ˆL holds, then (16) and (24) will
+solve 0 < Pi ≤ ˆP with Hurwitz A−BKT. With (3a), (17a) and
+(23b), AREs (18) and (23a) can be rewritten as
+(A− BKi+1)TPi + Pi(A− BKi+1)
+= −Qi+1 − (Ki+1)TRKi+1 − γ2(Li+1)T(Li+1),
+(25a)
+(A− BKi+1)T ˆP+ ˆP(A− BKi+1)
+= KT
+T RKT − ˆQ− γ2ˆLT ˆL− (Ki+1)TRKT − KT
+T RKi+1,
+(25b)
+respectively. Since (18) ensures Hurwitz A− BKi+1, subtract-
+ing (25a) from (25b) and using 0 < Pi ≤ ˆP obtains
+(A− BKi+1)T ( ˆP− Pi)+ ( ˆP− Pi)(A− BKi+1)
+= (Qi+1 + γ2(Li+1)TLi+1)− ( ˆQ+ γ2 ˆLT ˆL)
++ (Ki+1 − KT)TR(Ki+1 − KT) ≤ 0.
+(26)
+Therefore, Qi+1 + γ2(Li+1)T Li+1 ≤ ˆQ+ γ2ˆLT ˆL holds.
+By deduction, it is inferred that initializing Algorithm 1 with
+a Q0 such that 0 < Q0 ≤ ˆQ and L0 = 0, then Qi > 0,i = 0,1,···
+will be increasing with an upper bound. Therefore, Algorithm
+1 converges.
+ii). Converged solutions satisfy (19a), (19b), (19c).
+Substituting (18) into (16) yields
+ATPi + PiA− PiBR−1BTPi
+= ATPi+1 + Pi+1A− Pi+1BKT −KT
+T BTPi+1 + KT
+T RKT.
+(27)
+Taking Pi+1 = Pi = P∗ as converged value, (27) becomes
+(KT − K∗)R(KT − K∗) = 0,
+(28)
+where K∗ = R−1BTP∗. Since R > 0, (28) implies KT = K∗,
+which is exactly (19b). The converged P∗ produces the con-
+verged L∗ using (17b) as shown in (19c) and the converged
+Q∗ using (18) as shown in (19a).
+iii). Converged solutions satisfy Theorem 1.
+Rewriting (19a) with (19b) yields
+(A− BKT)T P∗ + P∗(A− BKT)+ Q∗
++ KT
+T RKT + 1
+γ2 P∗DDTP∗ = 0,
+(29)
+which is exactly (13). Obviously, (19a) is exactly (12). There-
+fore, Q∗,P∗,K∗ satisfy Theorem 1.
+□
+B. Stability and optimality analysis
+We now prove the stability of Algorithm 1 in Theorem 3,
+and optimality and Nash equilibrium in Theorem 4
+Theorem 3. Each iteration of Algorithm 1 exponentially
+stabilizes the learner agent (7) with d = 0.
+Proof: Rewriting (16) with (3a) and (17a) yields
+ATPi + PiA+ ˜Qi − PiBR−1BTPi + γ2(Li)TLi = 0.
+(30)
+where ˜Qi = Qi + (Ki+1 − KT)TR(Ki+1 − KT ). With (Ki+1 −
+KT)T R(Ki+1 −KT) ≥ 0 and Qi > 0, then ˜Qi > 0. It is obvious
+that Pi solved by (30) or equivalently (16) is a symmetrical
+positive definite matrix satisfying
+ATPi + PiA− PiBR−1BTPi + γ2(Li)T Li < 0,
+(31)
+and one has
+˙V i(x,Pi) = xT(A− BKi+1)TPix+ xTPi(A− BKi+1)x
+< −xTPiBR−1BTPix− γ2xT(Li)TLix < 0.
+(32)
+That is, Pi yields stabilizing Ki+1 by (17a) for learner (7) with
+d = 0.
+Considering (17a) and (31), Qi+1 in (18) satisfies
+Qi+1 + γ2(Li+1)T Li+1
+= −(ATPi + PiA− PiBR−1BTPi) > 0.
+(33)
+Using Qi+1 +γ2(Li+1)TLi+1 > 0 in (16) would still make (32)
+hold for the next iteration. Therefore, provided Q0 > 0 (32)
+will hold for all i = 0,1,···.
+□
+Before optimality analysis of Algorithm 1, we now give a
+lemma of importance which extends the idea of classic IOC
+[15] to two-player zero-sum games.
+Lemma 1. Consider the two-player learner agent (7) with
+x(t0) = x0, t ≥ t0 and (8) and (11). Assume there exists a
+positive definite symmetric matrix P ∈ Rn×n such that
+ATP+ PA− PBR−1BTP+ 1
+γ2 PDDTP < 0.
+(34)
+Then, with the optimal feedback control input uo and the worst
+disturbance dw such that
+uo = −R−1BTPx, dw = 1
+γ2 DTPx,
+(35)
+and the cost function weight
+Q = −(ATP+ PA− PBR−1BTP+ 1
+γ2 PDDTP),
+(36)
+the saddle point (uo,dw) makes the cost value function (8)
+reach the Nash equilibrium
+V(x0,uo,d) ≤ V(x0,uo,dw) ≤ V(x0,u,dw).
+(37)
+Proof: First, V(x) in (8) can be represented with P > 0 as
+V(x) = xTPx ≥ 0, V(0) = 0.
+(38)
+It follows from (35) and (36) that H(x,uo,dw) = 0 [2] where
+the Hamiltonian is
+H(x,u,d) =xTQx+ uTRu − γ2dTd + (Ax+ Bu + Dd)TPx
++ xTP(Ax+ Bu + Dd).
+(39)
+
+5
+One writes
+H(x,u,d) = H(x,u,d)− H(x,uo,dw)
+= (u − uo)R(u − uo)− γ2(d − dw)T (d − dw),
+(40)
+and hence
+H(x0,uo,d) ≤ H(x0,uo,dw) ≤ H(x0,u,dw).
+(41)
+Based on [2], we obtain the conclusion (37).
+□
+Theorem 4. The converged solutions Q∗,P∗,K∗,L∗ obtained
+by Algorithm 1 shown in Theorem 2 yield Nash equilibrium
+of the value function V(x) in (8) such that
+V(x0,u∗,d) ≤ V(x0,u∗,d∗) ≤ V(x0,u,d∗),
+(42)
+where u∗ = −K∗x and d∗ = L∗x.
+Proof: It follows from Theorem 1 that Qi > 0 holds for all
+i = 0,1,···. Thus one has the converged Q∗ > 0 and
+ATP∗ + P∗A− P∗BR−1BTP∗ + 1
+γ2 (P∗)TDDTP∗ < 0,
+(43)
+from (19a), which means that the converged P∗ satisfies (34)
+in Lemma 1. Also, the converged control strategy K∗ in (19b)
+and disturbance gain L∗ in (19c) satisfy (35) in Lemma 1. This
+indicates that (41) also holds for u∗ and d∗, namely the Nash
+equilibrium (42) holds.
+□
+C. Non-uniqueness of solution
+In fact, the Q∗,R,γ,P∗ satisfying (19a)-(19c) that explain
+the same strategy K∗ = KT may not be unique and can be
+different from the actual target values QT,RT,γT,PT shown
+in (3b) and (6). This multi-solution phenomenon is known
+as the ill-posedness property, which is well-analyzed for DT
+ARE in [38] and coupled ARE in [39]. In the next result,
+we characterize the relationship between QT,RT,γT ,PT, and
+Q∗,R,γ,P∗ for CT GARE and show the conditions for the
+occurrence of this phenomenon.
+Theorem 5. Recall QT, RT, γT, PT satisfying (6) and (3b),
+and let Qo, Ro, Po satisfy
+BTPo = RoR−1
+T BTPT,
+(44a)
+Qo + ATPo + PoA− KT
+T RoKT + 1
+γ2
+T
+PTDDTPT − 1
+γ2 P∗DDTP∗
+= 0,
+(44b)
+where Ro = RT −R. Then any Q∗ = QT −Qo and P∗ = PT −Po
+satisfy (19a)-(19c).
+Proof: Subtracting (44b) from (6) and using (3b) yields
+AT(PT − Po)+ (PT − Po)A+ (QT − Qo)
+− KT
+T (RT − Ro)KT + 1
+γ2 P∗DDTP∗ = 0.
+(45)
+Using P∗ = PT − Po, Ro = RT − R, and R > 0 in (44a) gives
+K∗ = R−1BTP∗ = R−1
+T BTPT = KT,
+(46)
+which is (19b). Substituting it into (45) yields (19a) and (19c).
+This proves the relationship between the obtained solution
+Q∗,R,γ,P∗ and the expert’s QT,RT,γT ,PT. We observe that
+Po, Ro, Qo satisfying (44a) and (44b) can be nonzero. That
+is, Q∗,R,γ,P∗ associate optimally with the same strategy as
+QT,RT,γT,PT, i.e., KT = K∗, but QT ̸= Q∗, RT ̸= R, γT ̸= γ.
+Therefore, there could be multiple solutions to (19a) to gen-
+erate a K∗ in (19b) equal to the target KT in (3a).
+□
+The following corollary shows a special case of the Q∗,R,γ
+of Theorem 5 which gives V(x)∗ = cVT(xT)∗.
+corollary 1. With scalar c > 0, any Q∗ = cQT, R = cRT, γ =
+√cγT would yield V(x)∗ = cVT(xT)∗ in (4) and (11), and they
+optimally associate with the same K∗ as the expert such that
+K∗ = KT in (19b) and (3a).
+Proof: Bring such Q∗,R,γ into (19a), since QT,RT,γT satisfy
+(6), then one has P∗ = cPT and K∗ = KT . Then, using this
+result in (11) for V(x)∗ and comparing it with VT(xT)∗ in (4)
+shows that V(x)∗ = cVT(xT)∗.
+□
+V. DATA-DRIVEN OFF-POLICY IRL ALGORITHM
+Algorithm 1 relies on the system dynamics A,B,D and the
+target strategy KT . To remove this requirement, we develop
+here a data-driven IRL algorithm for expert-learner zero-sum
+games based on Algorithm 1, which only requires the data
+xT,uT,dT of the target agent (1) and x,u,d of the learner
+agent (7). To accomplish this, we use two techniques similar
+to integral RL [6], [8] and off-policy RL [6], [40]. The end
+result is Algorithm 2.
+A. Data-driven game policy correction
+In order to update Pi, Ki+1 and Li+1 in (16)-(17b) using
+only target data xT,uT,dT, inspired by the idea of off-policy
+integral RL technique [6], [40], rewrite (1) as
+˙xT = AxT − BKixT + DdT + B(uT + KixT).
+(47)
+Using (47) and (16) one writes
+˙xT
+T PixT + xT
+TPi ˙xT
+= (AxT − BKixT)T PixT + xT
+TPi(AxT − BKixT)
++ 2(uT + KixT)TBTPixT + 2dT
+T DTPixT
+= −xT
+TQixT − xT
+TKT
+T RKTxT − γ2xT
+T(Li)T LixT
++ 2xT
+TKT
+T BTPixT + 2uT
+TBTPixT + 2dT
+T DTPixT.
+(48)
+Using (17a), (17b), (3a) in (48) and integrating both sides from
+t to t + T, where T > 0 is the integral time period, obtains
+(50) in Algorithm 2 to be presented, by which Pi, Ki+1 and
+Li+1 are updated simultaneously. Similar to [6], probing noise
+e is added to uT, i.e., uT = −KTxT + e, for the persistence
+of excitation condition in merely learning process. It is not
+needed anymore when solutions converge. Unlike (16)-(17b),
+(50) does not need the knowledge of agent dynamics A,B,D
+or the strategy KT in (3a).
+
+6
+B. Data-driven cost function weight reconstruction
+In order to update Qi+1 in (18) using only data x,u,d,
+inspired by the integral RL technique [8], multiplying both
+sides of (18) by x and adding and subtracting terms uTBTPix
+and dTDTPix, (18) can be rewritten as
+xTQi+1x =− [(Ax+ Bu + Dd)TPix+ xTPi(Ax+ Bu + Dd)
+− xT(Ki+1)TRKi+1x+ γ2xT(Li+1)T Li+1x
+− 2uTBTPix− 2dTDTPix],
+(49)
+where u can be generated by any stabilizing policy and d
+can be random and different from dT in the learning process.
+Substituting (7), (17a) and (17b) into (49) and integrating it
+gives (51) in Algorithm 2 below. Using Pi, Ki+1 and Li+1
+obtained by (50), (51) equivalently replaces (18) in Algorithm
+1 to calculate Qi+1 without knowing any system dynamics.
+Algorithm 2 Data-driven off-policy IRL algorithm for expert-
+learner zero-sum games.
+Step 1: Initialize with R > 0, γ > 0, Q0 > 0, and L0 = 0,
+and collect system data generated by any stabilizing
+control input u. Set i = 0.
+Step 2: (Game policy correction) Update policy Pi, control
+strategy Ki+1 and disturbance gain Li+1 by
+xT(t + T)TPixT(t + T)− xT(t)TPixT(t)
+− 2
+� t+T
+t
+eTRKi+1xTdτ − 2γ2
+� t+T
+t
+dT
+T Li+1xTdτ
+= −
+� t+T
+t
+�
+xT
+TQixT + (uT − e)TR(uT − e)
++ γ2xT
+T (Li)TLixT
+�
+dτ.
+(50)
+Step 3: (Cost function weight construction) Update cost
+function weight Qi+1 by
+� t+T
+t
+xT Qi+1xdτ
+= −[x(t + T)TPix(t + T)− x(t)TPix(t)
+−
+� t+T
+t
+(2uTRKi+1x+ xT(Ki+1)TRKi+1x)dτ
+−γ2
+� t+T
+t
+(2dTLi+1x− xT(Li+1)TLi+1x)dτ].
+(51)
+Step 4: Stop if it converges. Otherwise, set i = i+1 and repeat
+steps 2 to 4, .
+Remark 2. Algorithm 2 does not need system dynamics.
+Moreover, it iterates in single loop indicated by i, no inner-
+loop iteration is needed.
+C. Implementation and Analysis of Algorithm 2
+In order to show how to implement data-driven IRL Al-
+gorithm 2 using only data, first, consider Kronecker product
+aTWb = (bT ⊗ aT)vec(W) for (50) and define the following
+operators,
+OxT xT = [x2
+T1,2xT1xT2,...,x2
+T2,2xT2xT3,...,x2
+Tn]T;
+dxT xT = [OxT xT (t + T)− OxT xT (t),...,
+OxT xT (t + lT)− OxT xT (t + (l − 1)T)]T;
+IxT xT = [
+� t+T
+t
+OxT xT dτ,...,
+� t+lT
+t+(l−1)T OxT xT dτ]T;
+IxT uT = [
+� t+T
+t
+xT ⊗ uTdτ,...,
+� t+lT
+t+(l−1)T xT ⊗ uTdτ]T ;
+IxT dT = [
+� t+T
+t
+xT ⊗ dTdτ,...,
+� t+lT
+t+(l−1)T xT ⊗ dTdτ]T ;
+IxT e = [
+� t+T
+t
+xT ⊗ edτ,...,
+� t+lT
+t+(l−1)T xT ⊗ edτ]T;
+Φp = [dxT xT ,−2IxT e(In ⊗ R),−2γ2IxT dT ]T;
+ri = xT
+TQixT + (uT − e)TR(uT − e)+ γ2xT
+T(Li)T LixT;
+Ψi
+p = −[
+� t+T
+t
+ridτ,...,
+� t+lT
+t+(l−1)T ridτ]T;
+ˆPi = [Pi
+11,Pi
+12,...,Pi
+22,Pi
+23,...,Pi
+nn]T,
+(52)
+where l is the group number of collected data and should be
+l ≥ n(n+1)
+2
++ nm + nz. Using batch least squares method [8],
+[40], [41], ˆPi, Ki+1, Li+1 can be calculated by
+�
+( ˆPi)T ,vec(Ki+1)T,vec(Li+1)T �T=(ΦT
+pΦp)−1ΦT
+pΨi
+p.
+(53)
+Similarly, for (51), we define
+Oxx = [x2
+1,2x1x2,...,x2
+2,2x2x3,...,x2
+n]T;
+dxx = [Oxx(t + T)− Oxx(t),...,
+Oxx(t + kT)− Oxx(t + (k − 1)T)]T ;
+ˆqi = [Qi
+11,Qi
+12,...,Qi
+22,Qi
+23,...,Qi
+nn]T;
+Φq = Ixx = [
+� t+T
+t
+Oxxdτ,...,
+� t+kT
+t+(k−1)T Oxxdτ]T ;
+Ii+1
+q
+(t) =
+� t+T
+t
+(2uTRKi+1x+ xT(Ki+1)TRKi+1x)dτ
++ γ2
+� t+T
+t
+(2dTLi+1x− xT(Li+1)TLi+1x)dτ
+Ψi+1
+q
+= −dxx ˆPi + [Ii+1
+q
+(t),...,Ii+1
+q
+(t + (k − 1)T)]T ,
+(54)
+where k is the group number of collected data and should be
+k ≥ n(n+1)
+2
+. Then, Qi+1 can be uniquely solved by
+ˆqi+1 = (ΦT
+q Φq)−1ΦT
+q Ψi+1
+q
+.
+(55)
+By using (53) and (55), we solve Pi, Qi+1, Ki+1, Li+1 in a
+data-driven mode.
+Each step of Algorithm 1 yields a unique solution. Algo-
+rithm 2 is developed based on Algorithm 1. To illustrate that
+the solution obtained by Algorithm 2 estimates the solution
+obtained by Algorithm 1, we show that equations (53) and
+(55) yield unique solutions in the next result.
+
+7
+Theorem 6. If there exist lo > 0, ko > 0, for all l ≥ lo and
+k ≥ ko,
+rank([IxT xT ,IxT uT ,IxT dT ]) = n(n + 1)
+2
++ nm+ nz,
+(56a)
+rank(Ixx) = n(n + 1)
+2
+,
+(56b)
+then, (53) and (55) solve unique solution, respectively.
+Proof: First, we show that (53) solves unique solution. This
+is aiming to show that
+ΦpΩ = 0
+(57)
+has only the trivial solution Ω = 0. Now, we prove Ω = 0
+by contradiction. Assume Ω = [XT
+v ,Y T
+v ,ZT
+v ]T ∈ R
+n(n+1)
+2
++nm+nz
+is a nonzero solution of (57), where Xv ∈ R
+n(n+1)
+2
+, Yv ∈ Rmn,
+Zv ∈ Rnz. Then, Xv,Yv,Zv uniquely determine matrices X,Y,Z
+by Xv = ˆX, Yv = vec(Y) and Zv = vec(Z), respectively, where
+X is a symmetrical matrix.
+Define
+Inx = [
+� t+T
+t
+xT ⊗ xTdτ,...,
+� t+kT
+t+(k−1)T xT ⊗ xTdτ]T.
+(58)
+Integrating (48) from t to t + T gives
+ΦpΩ = Inxvec(E)+ 2IxT uT vec(G)+ 2IxTdT vec(F)= 0, (59)
+where
+E = ATX + XA− KT
+T RY −YT RKT,
+(60a)
+G = BTX − RY,
+(60b)
+F = DTX − γ2Z.
+(60c)
+Since E is a symmetrical matrix, one has Inxvec(E) = IxT xT ˆE.
+Using this in (59) yields
+ΦpΩ = [IxT xT ,2IxT uT ,2IxT dT ]
+
+
+ˆE
+vec(G)
+vec(F)
+
+ = 0.
+(61)
+Under (56a), we know that [Ixx,Ixu,Ixd] has full column rank,
+and thus (61) has only the solution ˆE = 0, vec(G) = 0 and
+vec(F) = 0. That is,
+ATX + XA− KT
+T RY −Y TRKT = 0,
+(62a)
+BTX = RY,
+(62b)
+DTX = γ2Z.
+(62c)
+Since A − BKT is Hurwitz, substituting (62b) and (62c) into
+(62a) gives X = 0. This implies that Y = 0 and Z = 0 due to
+R > 0 and γ2 > 0. In summary, we have Ω = 0. However, this
+conflicts with the assumption that Ω is nonzero. Therefore, it
+concludes that under (56a), (55) solves unique solution.
+Second, from the integral RL work [8], we conclude that
+when (56b) holds for (51), Qi+1 in (51) can be uniquely
+determined by (55) with collected data.
+□
+VI. SIMULATION
+We show three simulation experiments, a first one of the
+data-driven Algorithm 2 to show its performance, a com-
+parison simulation with the bilevel IRL method in [36] to
+show the reduction of iteration steps of Algorithm 2, and
+a second comparison simulation with the RL-based optimal
+tracking control method in [6] to show the improvement of
+control performance with the cost function weights correction
+of Algorithm 2.
+A. Simulation result of Algorithm 2
+The system dynamics information of the target (1) and the
+learner (7) for simulation is
+A =
+�
+−1
+2
+2.2
+1.7
+�
+,B =
+�
+0
+3
+�
+,D =
+�
+1
+0
+�
+.
+(63)
+For the target agent (1), the actual target cost function objec-
+tive (2) consists of QT = diag{8,12},RT = 2I1,γT = 3. The
+disturbance is dT = 0.003rand(1). The expert’s LT, KT and PT
+are
+KT = [1.9869,3.5779],LT = [0.4162,0.1472],
+PT =
+� 3.7459
+1.3246
+1.3246
+2.3853
+�
+.
+(64)
+For the learner, the behaviour strategy to generate data
+is Kb = [1.2129,
+2.2812], and the disturbance is d =
+0.003rand(1). To start Algorithm 2, the initial weights for cost
+function, initial disturbance gain L0, and integral time period
+T are given by
+Q0 = diag{1,0.5},R = I1,γ = 40,L0 = [0, 0],T = 0.008s.
+(65)
+The Fig. 1(a) captures the iteration process from the initial
+spot to the spot on ∥Ki+1−KT∥ ≤ 0.01. The final values of Ki,
+Qi, Li, and Pi are K∗, Q∗, L∗, and P∗, respectively, as follows
+K∗ =
+� 1.9827
+3.5839 �
+,
+Q∗ =
+�
+2.2796
+2.6670
+2.6670
+6.0151
+�
+,
+L∗ = 10−3 ×
+�
+0.4021
+0.4131
+�
+,
+P∗ =
+� 0.6441
+0.6622
+0.6622
+1.1968
+�
+,
+(66)
+where K∗ closely approximates the target KT in (64) with
+∥K∗ − KT∥ = 0.0073, while Q∗, L∗ and P∗ are not equal to
+QT,LT and PT in (64), respectively. This is the multiple-
+solution phenomenon. In Fig. 1(b), the learner’s state x can
+mimic the trajectories of the target xT very well under the
+learned K∗. Therefore, the proposed Algorithm 2 can learn an
+appropriate cost function and optimal policy for the learner to
+mimic the target trajectories.
+B. Comparison simulation case 1
+This subsection shows the simulation results of the bilevel
+IRL method in [36] that iterates in two-loop to show the
+reduction of computational complexity in terms of iteration
+steps. The same expert-learner system, initial parameters for
+
+8
+0
+10
+20
+30
+40
+50
+0
+1
+2
+||Ki − KT||
+0
+10
+20
+30
+40
+50
+9
+10
+11
+12
+||Qi − QT||
+0
+10
+20
+30
+40
+50
+0.44095
+0.441
+0.44105
+0.4411
+||Li − LT||
+0
+10
+20
+30
+40
+50
+Update steps i
+(a)
+3.4
+3.6
+||P i − PT||
+0
+0.5
+1
+1.5
+2
+Time(s)
+(b)
+-20
+-10
+0
+10
+20
+x1
+x2
+xT1
+xT2
+Fig. 1. Convergence and imitation performance using Algorithm 2
+Algorithm 2 in (63)-(65) are used for this comparison method
+1.
+Fig. 2 also captures the iteration process from the initial
+spot to the spot on ∥Ki+1 −KT∥ ≤ 0.01 as Fig. 1 to show the
+difference in iteration steps of the two methods. The inner-
+loop iteration figures are omitted since they are too many to
+put here. In Fig. 2, the final values of K j,Qj,Lj, and P j of
+the outer-loop iterations are
+K∗ =
+�
+1.9822
+3.5691
+�
+,
+Q∗ =
+� 2.3186
+2.6974
+2.6974
+6.0506
+�
+,
+L∗ = 10−3 ×
+� 0.4053
+0.4130 �
+,
+P∗ =
+� 0.6486
+0.6607
+0.6607
+1.1897
+�
+.
+(67)
+where K∗ approximates the target KT in (64) with ∥K∗−KT∥ =
+0.01. Table I shows that the total iteration steps of the method
+is 3370, including 587 outer-loop updates (See Fig. 2) and
+2783 inner-loop updates, while Algorithm 2 iterates 51 steps
+in total (See Fig. 1(a)). The time of the learning process of
+Algorithm 2 is 4.08s, while that of the comparison method 1 is
+169.936s. It is proportional to the amount of utilized collected
+data. Algorithm 2 uses 510 groups of data, and comparison
+method 1 uses 21242 groups of data. Therefore, Algorithm 2
+costs much fewer data and time than the bilevel comparison
+method 1.
+0
+100
+200
+300
+400
+500
+600
+0
+1
+2
+||Kj − KT||
+0
+100
+200
+300
+400
+500
+600
+8.5
+9
+9.5
+10
+||Qj − QT||
+0
+100
+200
+300
+400
+500
+600
+0.441
+0.4411
+||Lj − LT||
+0
+100
+200
+300
+400
+500
+600
+Outer-loop update steps j
+3.4
+3.6
+||P j − PT||
+Fig. 2. Convergence of K j,Qj,Lj, and Pj using the comparison method 1
+TABLE I
+ITERATION STEPS AND LEARNING TIME OF ALGORITHM 2 AND THE
+COMPARISON METHOD 1
+Methods
+Total updates
+Learning time
+Algorithm 2
+51
+4.08s
+Comparison method 1
+3370
+169.936s
+C. Comparison simulation case 2
+In this subsection, the typical RL-based optimal tracking
+control method [6] for linear disturbed systems, which com-
+putes optimal control policy given cost function weights, is
+simulated to show the advantage of Algorithm 2 in control
+performance by computing both optimal control policy and
+cost function weights.
+The same system and cost weights shown in (63)-(65) and
+discount factor α = 0.9 are used. The obtained optimal control
+law is u∗ = −[1.1760 1.9139 1.0044 1.7639][xT,rT]T, and the
+corresponding imitation performance in Fig. 3 is not as good
+as that of Algorithm 2 in Fig. 1(b). By evenly sampling the
+trajectory data, the imitation performance of the two methods
+is quantified by the error index defined as follows
+Te = 1
+n
+n
+∑
+i=1
+�
+1
+a
+a
+∑
+k=1
+|xi(kT)− xTi(kT)|2
+where n = 2, a = 250, and T = 0.008s. As shown in Table II,
+Te of Algorithm 2 is much smaller than that of the comparison
+method 2. The reason is that Algorithm 2 can correct the
+given cost function weights when it is inappropriate, but the
+comparison method 2 cannot. Algorithm 2 thus obtains much
+better imitation performance.
+TABLE II
+IMITATION INDEX OF ALGORITHM 2 AND THE COMPARISON METHOD 2
+Methods
+Error index Te
+Algorithm 2
+0.0162
+Comparison method 2
+1.4461
+
+9
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+-20
+-10
+0
+10
+20
+Fig. 3. Imitation performance of learner’s x to target xT using the comparison
+method 2
+VII. CONCLUSION
+This paper proposes a novel data-driven off-policy IRL
+approach to determine both cost function and optimal con-
+trol policy to stabilize a learner agent suffering from non-
+cooperative disturbances by mimicking a target agent’s tra-
+jectories using data of both agents. The proposed approach
+does not need any system dynamics and guarantees stability,
+Nash optimality, and imitation performance with single-loop
+iteration. The rigorous theoretical proofs and simulation ex-
+periments verify its effectiveness.
+REFERENCES
+[1] T. Bas¸ar and P. Bernhard, H-infinity optimal control and related minimax
+design problems: a dynamic game approach.
+Springer Science &
+Business Media, 2008.
+[2] F. L. Lewis, D. Vrabie, and V. Syrmos, Optimal Control.
+John Wiley
+& Sons, Inc., Hoboken, NJ, USA., pages 444–453, 2012.
+[3] S. R. S. and B. A. G., Reinforcement learning: An introduction.
+MIT
+Press, Cambridge, USA., pages 73–88, 1998.
+[4] F. L. Lewis and D. Vrabie, “Reinforcement learning and adaptive
+dynamic programming for feedback control,” IEEE Circ. Syst. Mag.,
+vol. 9, no. 3, pp. 32–50, 2009.
+[5] L. R. G. Carrillo and K. Vamvoudakis, “Deep-learning tracking for
+autonomous flying systems under adversarial inputs,” IEEE Trans.
+Aerosp. Electron. Syst., vol. 56, no. 2, pp. 1444–1459, 2019.
+[6] H. Modares, F. L. Lewis, and Z. Jiang, “H∞ tracking control of com-
+pletely unknown continuous-time systems via off-policy reinforcement
+learning,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 10, pp.
+2550–2562, 2015.
+[7] B. Kiumarsi, F. L. Lewis, and Z. Jiang, “H∞ control of linear discrete-
+time systems: Off-policy reinforcement learning,” Automatica, vol. 78,
+pp. 144–152, 2017.
+[8] H. Modares and F. L. Lewis, “Linear quadratic tracking control of
+partially-unknown continuous-time systems using reinforcement learn-
+ing,” IEEE Trans. Autom. Control, vol. 59, no. 11, pp. 3051–3056, 2014.
+[9] D. Kleinman, “On an iterative technique for riccati equation computa-
+tions,” IEEE Trans. Autom. Control, vol. 13, no. 1, pp. 114–115, 1968.
+[10] X. Chen, M. Monfort, B. D. Ziebart, and P. Carr, “Adversarial inverse
+optimal control for general imitation learning losses and embodiment
+transfer.” in Conf. Uncertain. Artif. Intell., 2016, pp. 1–10.
+[11] R. Self, K. Coleman, H. Bai, and R. Kamalapurkar, “Online observer-
+based inverse reinforcement learning,” arXiv:2011.02057., 2020.
+[12] S. Levine and V. Koltun, “Continuous inverse optimal control with
+locally optimal examples,” arXiv preprint arXiv:1206.4617, 2012.
+[13] N. Ab Azar, A. Shahmansoorian, and M. Davoudi, “From inverse
+optimal control to inverse reinforcement learning: A historical review.”
+Annu. Rev. Control, vol. 50, pp. 119–138, 2020.
+[14] R. E. Kalman, “When is a linear control system optimal?” J. Basic Eng.,
+vol. 86, no. 1, pp. 51–60, 1964.
+[15] W. M. Haddad and V. Chellaboina, Nonlinear dynamical systems and
+control: A Lyapunov-based approach.
+Princeton University Press,
+Princeton, USA, pages 519–540, 2011.
+[16] M. Johnson, N. Aghasadeghi, and T. Bretl, “Inverse optimal control for
+deterministic continuous-time nonlinear systems,” in Proc. 52nd IEEE
+CDC, 2013, pp. 2906–2913.
+[17] X. Cai and Z. Han, “Inverse optimal control of nonlinear systems with
+structural uncertainty,” IEE Proc-C, vol. 152, no. 1, pp. 79–83, 2005.
+[18] F. Ornelas, E. N. Sanchez, and A. G. Loukianov, “Discrete-time non-
+linear systems inverse optimal control: A control lyapunov function
+approach,” in Proc. IEEE Int. Conf. Control Autom., 2011, pp. 1431–
+1436.
+[19] E. N. Sanchez and F. Ornelas-Tellez, Discrete-time inverse optimal
+control for nonlinear systems.
+CRC Press, Boca Raton, USA., pages
+67–107, 2017.
+[20] T. L. Molloy, J. J. Ford, and T. Perez, “Finite-horizon inverse optimal
+control for discrete-time nonlinear systems,” Automatica, vol. 87, pp.
+442–446, 2018.
+[21] ——, “Online inverse optimal control on infinite horizons,” in Proc.
+57th IEEE CDC, 2018, pp. 1663–1668.
+[22] ——, “Online inverse optimal control for control-constrained discrete-
+time systems on finite and infinite horizons,” Automatica, vol. 87, pp.
+1–8, 2020.
+[23] Y. Park, “Inverse optimal and robust nonlinear attitude control of rigid
+spacecraft,” Aerosp. Sci. Technol., vol. 28, no. 1, pp. 257–265, 2013.
+[24] W. Luo, Y. Chu, and K. V. Ling, “Inverse optimal adaptive control for
+attitude tracking of spacecraft,” IEEE Trans. Autom. Contr., vol. 50,
+no. 11, pp. 1639–1654, 2005.
+[25] D. Tsai, T. L. Molloy, and T. Perez, “Inverse two-player zero-sum
+dynamic games,” in Pro. 2016 AUCC, 2016, pp. 192–196.
+[26] P. Abbeel and A. Y. Ng, “Apprenticeship learning via inverse reinforce-
+ment learning,” in Proc. 21st. Conf. Mach. Learn. ICML, 2004, pp. 1–8.
+[27] M. Wulfmeier, P. Ondruska, and I. Posner, “Maximum entropy deep
+inverse reinforcement learning,” arXiv preprint arXiv:1507.04888, 2015.
+[28] G. Neu and C. Szepesv´ari, “Apprenticeship learning using inverse
+reinforcement learning and gradient methods,” pp. 295–302, 2012.
+[29] B. D. Ziebart, A. Maas, J. Andrew Bagnell, and A. K. Dey, “Maximum
+entropy inverse reinforcement learning,” in Pro. 2008 AAAI., 2008, pp.
+1433–1438.
+[30] S. Choi, S. Kim, and H. Jin Kim, “Inverse reinforcement learning control
+for trajectory tracking of a multirotor uav,” Int. J. Control Autom. Syst.,
+vol. 15, no. 4, pp. 1826–1834, 2017.
+[31] R. Self, M. Abudia, and R. Kamalapurkar, “Online inverse reinforcement
+learning for systems with disturbances,” in Proc. 2020 ACC., 2020, pp.
+1118–1123.
+[32] K. Mombaur, A. Truong, and J. P. Laumond, “From human to humanoid
+locomotion: An inverse optimal control approach,” Auton. Robots.,
+vol. 28, no. 3, pp. 369–383, 2010.
+[33] R. Kamalapurkar, “Linear inverse reinforcement learning in continuous
+time and space.” in Proc. 2018 ACC, 2018, pp. 1683–1688.
+[34] W. Xue, K. P., J. Fan, L. B., T. Chai, and F. L. Lewis, “Inverse rein-
+forcement learning in tracking control based on inverse optimal control,”
+IEEE Trans. Cybern., doi: 10.1109/TCYB.2021.3061894, 2021.
+[35] W. Xue, L. B., J. Fan, T. Chai, and F. L. Lewis, “Inverse reinforcement
+learning in tracking control based on inverse optimal control,” IEEE
+Trans. Neural Netw. Learn., doi: 10.1109/TNNLS.2021.3106635, 2021.
+[36] B. Lian, W. Xue, F. L. Lewis, and T. Chai, “Online inverse reinforcement
+learning for nonlinear systems with adversarial attacks,” Int. J. Robust
+Nonlinear Control., vol. 31, no. 14, pp. 6646–6667, 2021.
+[37] P. Lancaster and L. Rodman, Algebraic Riccati equations.
+Clarendon
+Press, Oxford, UK., pages: 231–243, 1995.
+[38] M. Menner and M. N. Zeilinger, “Convex formulations and algebraic
+solutions for linear quadratic inverse optimal control problems,” in Proc.
+17th ECC, 2018, pp. 2107–2112.
+[39] J. Inga, E. Bischoff, T. L. Molloy, M. Flad, and S. Hohmann, “Solution
+sets for inverse non-cooperative linear-quadratic differential games.”
+IEEE Control Syst. Lett., vol. 3, no. 4, pp. 871–876, 2019.
+[40] Y. Jiang and Z. Jiang, “Computational adaptive optimal control for
+continuous-time linear systems with completely unknown dynamics,”
+Automatica, vol. 48, no. 10, pp. 2699–2704, 2012.
+[41] Y. Jiang, J. Fan, T. Chai, J. Li, and F. L. Lewis, “Data-driven flotation
+industrial process operational optimal control based on reinforcement
+learning,” IEEE Trans. Ind. Informa., vol. 14, no. 5, pp. 1974–1989,
+2018.
+

PdE2T4oBgHgl3EQfrgiA/content/tmp_files/2301.04050v1.pdf.txt

@@ -0,0 +1,1332 @@
+arXiv:2301.04050v1  [cs.RO]  10 Jan 2023
+1
+Design, Modeling and Control of a Quadruped Robot SPIDAR:
+Spherically Vectorable and Distributed Rotors Assisted
+Air-Ground Amphibious Quadruped Robot
+Moju Zhao1, Tomoki Anzai2, Takuzumi Nishio2
+Abstract—Multimodal locomotion capability is an emerging
+topic in robotics field, and various novel mobile robots have
+been developed to enable the maneuvering in both terrestrial
+and aerial domains. Among these hybrid robots, several state-of-
+the-art bipedal robots enable the complex walking motion which
+is interlaced with flying. These robots are also desired to have the
+manipulation ability; however, it is difficult for the current forms
+to keep stability with the joint motion in midair due to the cen-
+tralized rotor arrangement. Therefore, in this work, we develop
+a novel air-ground amphibious quadruped robot called SPIDAR
+which is assisted by spherically vectorable rotors distributed in
+each link to enable both walking motion and transformable flight.
+First, we present a unique mechanical design for quadruped robot
+that enables terrestrial and aerial locomotion. We then reveal the
+modeling method for this hybrid robot platform, and further
+develop an integrated control strategy for both walking and
+flying with joint motion. Finally, we demonstrate the feasibility of
+the proposed hybrid quadruped robot by performing a seamless
+motion that involves static walking and subsequent flight. To
+the best of our knowledge, this work is the first to achieve a
+quadruped robot with multimodal locomotion capability, which
+also shows the potential of manipulation in multiple domains.
+Index Terms—Legged Robots; Aerial Systems: Mechanics and
+Control; Motion Control
+I. INTRODUCTION
+During the last decade, robots with multimodal locomotion
+capability have undergone intensive development, and demon-
+strated the versatile maneuvering in multiple domains (i.e.,
+terrestrial, aerial, and aquatic) [1]–[5] which can benefit the
+performance in various situations, such as disaster response
+and industrial surveillance. Among various forms for mul-
+timodal locomotion, the legged type has the advantage in
+unstructured terrains, and several bipedal models have been
+developed to demonstrate the promising walking that is inter-
+laced with flying motion [6], [7]. The legged robots are desired
+to provide not only the advanced locomotion capability, but
+also the manipulation ability by the limb end-effector. Then,
+multilegged (multilimbed) is considered effective from the
+aspect of both the stability in the terrestrial locomotion and
+the freedom in manipulation. In spite of the achievement
+of a simple arm motion in midair by the flying humanoid
+robot in [7], the centralized rotor arrangement in most of the
+existing robot platforms can hardly handle the large change of
+Center-of-Gravity (CoG) caused by the joint motion in midair.
+Besides, the external force at the limb end during the aerial
+manipulation is also difficult to compensate by the centralized
+1 Department of Mechanical Engineering, The University of Tokyo, 2
+Department of Mechano-Infomatics, The University of Tokyo, 7-3-1 Hongo,
+Bunkyo-ku, Tokyo 113-8656, Japan. chou@hnl.t.u-tokyo.ac.jp
+Digital Object Identifier (DOI): see top of this page.
+(A)
+(B)
+Fig. 1.
+Air-ground amphibious quadruped robot SPIDAR: SPherIcally
+vectorable and Distributed rotors assisted Air-ground amphibious quadruped
+Robot. (A) walk on the ground like an usual quadruped robot, but with the
+assistance of thrust force; (B) fly and transform in midair.
+rotors due to the large moment arm. Hence, a distributed rotor
+arrangement is necessary for aerial manipulation. Then, we
+propose a novel quadruped robot platform in this work, which
+can both walk and fly by using the spherically vectorable rotors
+distributed in each link unit as illustrated in Fig. 1.
+One of the efficient type of air-ground hybrid robot plat-
+forms equips an ordinary multirotor for flying, and then de-
+ploys rolling cage or wheels to perform terrestrial locomotion
+[2], [8], [9]. Although the rolling mechanism can achieve the
+stable terrestrial locomotion without any complex control, it is
+relatively difficult for this type to handle the unstructured ter-
+rains (e.g., very few foothold). Then, legged model is proposed
+in several researches [6], [7], [10]–[12] to solve this issue by
+the walking motion. Most of these robots are bipedal model
+which attaches the multirotor unit in their torso for aerial
+maneuvering. Compared with the bipedal model, it is relatively
+easier to achieve the walking stability by the multilegged
+model because of the larger support polygon. Besides, the legs
+can be considered as the limbs for manipulation in midair, and
+thus more limbs can provide more end-effectors for complex
+manipulation task. Therefore, we choose the quadruped model
+that can offer both the stable terrestrial locomotion and the
+potential of aerial manipulation.
+For most of the quadruped robot, the form is bio-inspired
+and specialized for terrestrial locomotion [13]–[15]. However,
+in our work, the robot is also desired to perform manipulation
+task in multiple domains. Several ape-like quadruped robots
+show the manipulation ability by the hands attached on the leg
+ends [16], [17]. The point symmetry is the crucial difference of
+these robots from the skeleton design for common quadruped
+robots. In our work, we also adopt the point symmetry for
+our hybrid quadruped robot to enable the omni-directional
+maneuvering and manipulation in both terrestrial and aerial
+
+2
+domains. Regarding the rotor arrangement, it is difficult for
+rotors centralized in the robot torso like [6] to handle the
+change of CoG caused by the joint motion. Besides, the
+external force acted at the end-effectors can also induce a
+large rotational load for the centralized rotors due to the
+large moment arm. To ensure a sufficient control margin for
+the stable joint motion in midair, [11] proposes a distributed
+rotor arrangement that deploys the thrust units at each limb
+end. However, this rotor arrangement deprives the robot of
+manipulation ability. Therefore, a fully-distributed rotor design
+proposed by [18] is applied in this work. In this design,
+spherically vectorable rotor apparatus is embedded in each link
+and thus can generate individual three-dimensional thrust force
+for the promising maneuvering and manipulation in midair as
+presented in [19].
+For the stable flying motion, the whole platform should be
+lightweight, which leads a relatively compact and weak joint
+actuator for legs. Hence, the thrust force is also required to
+assist the walking motion by reducing the load from gravity.
+For model with centralized rotor arrangement, both the model-
+based [6] and the policy-based [20] methods are developed to
+obtain the assistive thrust. However, these methods are not
+suitable for the model with rotors distributed in all links.
+Then, [21] proposes an assistive thrust control method for a
+bipedal robot with the vectorable rotor attached at each foot,
+whereas [19] presents a control method for a fully-distributed
+rotor model in the aerial domains. Based on these methods, a
+comprehensive investigation of the modeling and control for
+the multilegged model is performed in this work to handle
+multiple types of torques and forces (i.e., the joint torque, the
+contact force on each limb end, the gravity of each link, and
+the thrust force from each vectorable rotor).
+The main contributions of this work can be summarized as
+follows:
+1) We propose a unique mechanical design for air-ground
+quadruped robot with the spherically vectorable rotors
+distributed in all links.
+2) We present a modeling and control methods for this
+multilegged platform for hybrid locomotion in both
+terrestrial and aerial domains.
+3) We achieve the seamless and stable motion that involves
+walking, flying and joint motion in midair by the proto-
+type of quadruped robot.
+The remainder of this paper is organized as follows. The
+mechanical design for this unique quadruped robot is intro-
+duced in Sec. II. The modeling of our robot is presented in
+Sec. III, followed by the integrated control method for hybrid
+locomotion in Sec. IV. We then show the experimental results
+in Sec. V before concluding in Sec. VI.
+II. DESIGN
+In this section, we present the mechanical design for the
+quadruped robot that is capable of terrestrial/aerial hybrid
+locomotion. The key of the whole structure is the spherically
+vectorable rotor embedded in each link unit, along with the
+unique quadruped shape that differs from the common bio-
+inspired type.
+Fig. 2. Mechanical design for the air-ground hybrid quadruped robot.
+(A) skeleton model for common bio-inspired mammal-type quadruped robot.
+(B) proposed point-symmetric skeleton model for hybrid quadruped model.
+(C1)/(C2) two-DoF joint module for each limb, where the yaw axis qi yaw
+comes first followed by the pitch axis qi pitch. (D) spherically vectorable
+rotor apparatus with two vectoring angles (φ, θ), and a combined thrust force
+λi from the counter rotating dual rotors. There is a small offset between two
+vectoring axes because two rods cannot intersect with each other.
+A. Skeleton Model
+As shown in Fig. 2(A), the common skeleton for quadruped
+robot is mammal-type, which puts a priority on the forward
+motion. Thus, the model is plane symmetric, and each leg has
+three DoF (two in the hip, and one in knee). However, our
+robot is desired to enable not only the terrestrial/aerial hybrid
+locomotion, but also the manipulation in midair. Therefore, the
+omni-directional movement is a critical feature for the skeleton
+design. Then, a point symmetric structure is introduced as
+depicted in Fig. 2(B). This is similar to the sprawling-type
+quadruped design proposed by [22], which can provide s wider
+supporting polygon and also a lower CoG than the mammal-
+type. According to this design concept, each limb consists
+of two links that have the same length, and is connected to
+the center torso with a joint module that has two Degree-of-
+Freedom (DoF). For this joint module, the yaw axis qi yaw
+comes first, which is followed by the pitch axis qi pitch
+to allow a larger swinging range for walking as shown in
+Fig. 2(C1)/(C2). The crucial difference from the ordinary
+sprawling-type is that we also introduce an identical two-
+DoF joint module to connect neighboring links, which can
+provide a four-DoF manipulation capability by each limb end
+without the help of the torso motion. Eventually, this robot
+is composed of 8 links with 16 joints for walking and flying.
+Given that the lightweight design is significantly important for
+the flight performance, we deploy a compact servo motor to
+individually actuate each joint at the expense of the torque
+power. Nevertheless, the shortage of the joint torque can be
+compensated by the rotor thrust in our robot.
+B. Spherically Vectorable Rotor
+Rotors embedded in links are used to achieve flight with
+arbitrary joint motion in midair. Besides, it is also necessary
+
+(B)
+(A)
+qi+1_yaw
+i+1_pitch
+yaw
+qi_pitch
+(D)
+C1 top view
+(C2) side view3
+to use the rotor thrust to assist leg lifts for walking, because
+the joint actuator is weak due to the lightweight design as
+mentioned in Sec. II-A. Therefore, the rotor is required to
+point arbitrary direction to handle the change in link ori-
+entation. In other words, it is required to generate a three-
+dimensional thrust force by each rotor module to interact
+with not only the gravity and also the external force (i.e.,
+the contact force on each foot). Then a spherically vectorable
+apparatus proposed by [18] is equipped in each link as depicted
+in Fig. 2(D). To achieve the spherical vectoring around the
+link unit, two rotation axes is necessary. We first introduce
+a roll axis φi around the link rod. Then, we need the second
+orthogonal vectoring axis. If we use a single rotor, the collision
+between the propeller and the link rod will be inevitable
+while performing the second vectoring. Therefore, we apply
+the counter-rotating dual-rotor module to avoid the collision
+as shown in Fig. 2(D). In addition, this dual-rotor can also
+counteract the drag moment and gyroscopic moment. Then,
+we define the pitch axis across dual rotors as the second
+vectoring axis θi. Each vectoring axis is also actuated by an
+individual compact servo motor. Regarding the thrust force,
+since we assume that a pair of rotors rotate with the same
+speed, it is possible to introduced a combined thrust λi for
+each spherically vectorable rotor module. Eventually, there are
+three control input (two vectoring angles φi, θi, and one thrust
+λi) for each vectorable rotor module, and totally 8 sets are used
+to control the whole quadruped model.
+III. MODELING
+In this section, we describe the modeling for this robot
+which can be divided into two parts: the thrust model and
+the whole dynamics model.
+A. Spherically Vectorable Thrust
+Based on the kinematic model depicted in Fig. 3, the three-
+dimensional force fi generated by the i-th rotor module can
+contact
+contact
+contact
+free
+Υ
+Œ•>Ú
+“••
+Œ‰•
+Ε
+Ε>Ú
+Ε>Û
+Ε>Ü
+“•>Ú•
+<(Ü=
+<%K)=
+<.Ü=
+Fig. 3.
+Dynamics model of the proposed quadruped robot. The entire
+dynamics involves the joint torque τj, the contact force at each limb end fci,
+the gravity of each link mig, and the thrust force from each vectorable rotor
+fi. {Li} and {Fi} denote the frame for the i-th link and rotor respectively,
+whereas {CoG} is the CoG frame for the whole model. For the free leg
+during walking, the contact force fci at the limb end disappears.
+be written as:
+fi = λiui,
+(1)
+ui = CoGRFi(q, φi, θi)
+�0
+0
+1�T ,
+(2)
+where CoGRFi denotes a rotation matrix of the rotor frame
+{Fi} w.r.t. the frame {CoG}. For this robot, we define the
+frame {CoG} to have an origin at the CoG point as depicted in
+Fig. 3, and a xyz coordinate that is identical to the baselink at
+the center torso. ui denotes the unit vector for the spherically
+vectorable mechanism that is effected by two vectoring angles
+φi and θi. Besides, this vector also depends on the joint angles
+q ∈ RNJ , NJ is the number of joints.
+Then the total wrench in the frame {CoG} can be given by
+�
+fλ
+τλ
+�
+=
+
+
+Nr
+�
+i=1
+fi
+Nr
+�
+i=1
+pi × fi
+
+
+= Qλ,
+(3)
+Q =
+�
+u1
+u2
+· · ·
+uNr
+p1 × u1
+p2 × u2
+· · ·
+pNr × uNr
+�
+,
+(4)
+λ =
+�λ1
+λ2
+· · ·
+λNr
+�T ,
+where pi is the position of the frame {Fi} origin from the
+frame {CoG} that is influenced by the joint angles q and the
+first vectoring angle φi for the i-th rotor. Nr is the number of
+rotors.
+B. Dynamics of Multilinked Model
+The whole dynamic model can be written as follows:
+˙PΣ =Rcfλ − mΣg +
+Nc
+�
+i=1
+fci,
+(5)
+˙LΣ =τλ +
+Nc
+�
+i=1
+pci × RT
+c fci,
+(6)
+MJ(q)¨q + c(q, ˙q) =τq +
+Nc
+�
+i=1
+JT
+cifci
++
+Nr
+�
+i=1
+JT
+rifi +
+Ns
+�
+i=1
+JT
+simsig.
+(7)
+(5) and (6) denote the centroidal dynamics for the whole
+multibody model. PΣ and LΣ are the entire linear and
+rotational momentum described in the inertial frame {W}
+and the frame {CoG}, respectively. These momentum are
+both affected by the joint angles, vectoring angles, and their
+velocities (i.e., q, ˙q, φ, ˙φ, θ, ˙θ). Rc is the orientatin of the
+frame {CoG} w.r.t. the frame {W}, and is identical to Rb
+that is the orientatin of the baselink. fλ and τλ corresponds
+to the total wrench described in (3). fci is the contact force at
+the i-th limb end (foot) w.r.t. the frame {W}, whereas pci is
+the position of this contact point from the frame {CoG} which
+is also influenced by the joint angles q. Nc is the number of
+contact points (i.e., standing legs). mΣ is the total mass, and
+g is a three-dimensional vector expressing gravity.
+
+4
+(7) corresponds to the joint motion. MJ(q) denotes the
+inertial matrix, whereas c(q, ˙q) is the term related to the
+centrifugal and Coriolis forces in joint motion. J∗i ∈ R3×NJ
+is the Jacobian matrix for the frame of the i-th contact point
+(∗ → c), the i-th rotor (∗ → r), and the i-th segment’s CoG
+(∗ → s), respectively. τq ∈ RNJ is the vector of joint torque
+and fi is the vectoring thrust force corresponding to (1).
+(5) ∼ (7) are highly complex due to the joint motion. Then
+the realtime feedback control based on these nonlinear equa-
+tions is significantly difficult for an onboard computational
+resource. Therefore, for the joint motion, a crucial assumption
+is introduced in our work to simplify the dynamics, i.e., all
+the joints are actuated slowly by individual servo motors.
+This is also called the quasi-static assumption that allows
+˙q ≈ 0; ¨q ≈ 0 during the joint motion.
+Under this assumption, the original dynamic model can be
+approximated as follows:
+mΣ¨rc(q) = Rcfλ − mΣg +
+Nc
+�
+i=1
+fci,
+(8)
+IΣ(q) ˙ω + ω × IΣ(q)ω = τλ +
+Nc
+�
+i=1
+pci × fci,
+(9)
+0 = τq +
+Nc
+�
+i=1
+JT
+cifci +
+Nr
+�
+i=1
+JT
+rifi +
+Ns
+�
+i=1
+JT
+simsig,
+(10)
+where rc is the position of the frame {CoG} w.r.t the frame
+{W}, which can be calculated using the forward-kinematics
+from the baselink states with the joint angles q. ωc is the
+angular velocity of the frame {CoG} w.r.t the frame of
+{CoG}, and is identical to ωb that is the angular velocity
+of the baselink. IΣ(q) is the total inertia tensor that is also
+influenced by the joint angles q.
+(8) and (9) show the property of a single rigid body, whereas
+(10) indicates the equilibrium between the forces and torques
+for the joint motion. Given that we apply the quasi-static
+assumption for joint motion, only the slow terrestrial motion,
+such as the static walk gait, is allowed.
+IV. CONTROL
+In this section, we first describe a unified control framework
+as depicted in Fig. 4, and then present the modification for the
+aerial and terrestrial locomotion, respectively.
+Control
+Allocation
+Centroidal
+Motion 
+Control
+Robot
+Dynamics
+Model
+Approximation
+4Ö×
+˜Ö×
+6˜Ö×
+Ó‰×
+�×
+Ø× Â× Å×
+˜Õ
+4Õ
+6˜Õ
+ÓÕ
+—
+—×
+IŠ +Š
+Vectorable Rotor Control
+4Ö ÓÖ
+˜Ö
+6˜Ö
+—
+Joint
+PD Control
+Îä×
+Fig. 4.
+Unified control framework for terrestrial/aerial locomotion.
+“Model approximation” presented in Sec. III-B is followed by the vectorable
+rotor control based on the centroidal motion. The joint control is performed
+independently.
+A. Centroidal Motion Control
+For the approximated dynamics of (8) and (9), the position
+feedback control based on an ordinary PID control is given by
+f d
+λ = mΣRT
+c (Kf,per + Kf,i
+�
+er + Kf,d ˙er)
++ RT
+c (mΣg −
+Nc
+�
+i=1
+fci),
+(11)
+where er = rd
+c − rc, and Kf,∗ are the PID gain diagonal
+matrices.
+The attitude control follows the SO(3) control method
+proposed by [23]:
+τ d
+λ = IΣ(Kτ,peR + Kτ,i
+�
+eR + Kτ,deω)
++ ωc × IΣωc −
+Nc
+�
+i=1
+pci × RT
+c fci,
+(12)
+eR = 1
+2
+�
+RT
+c Rd
+c − RdT
+c Rc
+�∨ ,
+(13)
+eω = RT
+c Rd
+cωd
+c − ωc,
+(14)
+where [⋆]∨ is the inverse of a skew map.
+Then, the desired wrench w.r.t the frame {CoG} can be
+summarized as follows:
+wd =
+�
+f d
+λ
+τ d
+λ
+�T .
+(15)
+B. Control Allocation
+The goal of vectorable rotor control is to obtain the control
+input (the desired thrust λd and the desired vectoring angles
+φd, θd) from the desired wrench wd. Meanwhile, it is also
+important to suppress the rotor output and the joint load from
+the aspect of the energy consumption. Then, an optimization
+problem should be design to obtain the desired control input.
+Since the vectoring angles φ and θ demand the trigono-
+metric function, nonlinear constraints would appear in the
+optimization problem and thus lead a complex computation.
+To decrease the computational load during the realtime control
+loop, we introduce an alternative three-dimensional forces f
+′
+i
+that is the vectorable thrust w.r.t, the related link frame {Li}:
+f
+′
+i = LiRFi(φi, θi)
+�0
+0
+λi
+�T. The definition of the frames
+of {Li} and {Fi} can be found in Fig. 3. Then the above
+optimization problem can be modified as follows:
+min
+f ′
+i ,τq,fci
+w1
+Nr
+�
+i=1
+∥f
+′
+i∥2 + w2∥τq∥2,
+(16)
+s.t.
+wd =
+Nr
+�
+i=1
+Qif
+′
+i ,
+Qi =
+�E3×3
+[pi×]
+�
+CoGRLi,
+(17)
+τq = −
+Nc
+�
+i=1
+JT
+cifci −
+Nr
+�
+i=1
+JT
+rifi −
+Ns
+�
+i=1
+JT
+simsig,
+(18)
+0 < λi < ¯λ,
+(19)
+− ¯τq < τqi < ¯τq,
+(20)
+0 < fci(2),
+(21)
+
+5
+where w1 and w2 in (16) are the weights for the cost of rotor
+thrust and joint torque, respectively. (17) is the modified form
+of wrench allocation from (3) by using the alternative variable
+f
+′
+i . pi is defined in (3), whereas CoGRLi is the orientation of
+the frame {Li} w.r.t. the frame {CoG}. E3×3 is a 3 × 3
+identity matrix and [·×] denotes the skew symmetric matrix
+of a three dimensional vector. (18) denotes the equilibrium
+between the joint torque τq, the contact force fci, the thrust
+force fi, and the segment gravity msig to satisfy the joint
+quasi-static assumption. (19) and (20) denote the bounds for
+the rotor thrust and joint torque, respectively. The contact force
+fci is also considered as the searching variable, and the z
+element fci(2) should be always non-negative as shown in
+(21).
+Given that all constraints (17) ∼ (21) are linear, an ordinary
+algorithm for quadratic problem can be applied. Once the
+optimized thrust force ˜f
+′
+i is calculated, the true control input
+for the spherically vector rotor apparatus can be obtained as
+follows:
+λi = ∥f
+′
+i∥,
+(22)
+φi = tan−1(−f
+′
+i(1)
+f
+′
+i(2) ),
+(23)
+θi = tan−1(
+f
+′
+i (0)
+−f
+′
+i(1)sin(φi) + f
+′
+i(2)cos(φi)),
+(24)
+where f
+′
+i (0), f
+′
+i(1), and f
+′
+i(2) are the x, y, and z element of
+the vector.
+As a unique mechanical feature of the spherically vectorable
+apparatus depicted in Fig. 2(D), the result of vectoring angles
+φ and θ from (23) and (24) will deviate the position pi in
+(17) because of the small offset between two vectoring axes
+as depicted in Fig. 2(D). Then, the results of (22) ∼ (24) will
+no longer satisfy the constraint (17) because Qi has changed.
+To solve this problem, we apply the iteration process that is
+based on the gradient of a residual term ǫ := wd − Q(θ, φ)λ,
+and finally we can obtain the convergent values of φd, θd and
+λd. The detail can be found in [19].
+C. Joint Control
+The proposed optimization problem of (16) can also pro-
+vides the joint torque that however only satisfies the quasi-
+static assumption for joint motion. In addition, the measure-
+ment bias and noisy from the joint encoders along with the
+slight deformation of the link and joint structure can also
+induce the model error. To handle this model error, it is
+necessary to apply a feed-back control to track the desired
+position for joints. Therefore, a simple PD control for joint
+position is introduced for each joint:
+τ d
+i = kj,p(qd
+i − qi) − kj,d ˙qi,
+(25)
+where qd
+i is the desired joint angle from the walking gait or
+the aerial transformation planning. kj,p and kj,d are the P and
+D control gains. It is also notable that we also used the same
+PD control for the rotor vectoring angles φi and θi.
+D. Aerial Locomotion
+The control mode for aerial locomotion follows the flow
+shown in Fig. 4 but without the contact force fci. Then the
+constraint of (21) and the first term �Nc
+i=1 JT
+cifci at the right
+side of (18) can be omitted. The joint control is executed
+independently to follow the trajectory given by other task
+planing.
+E. Terrestrial Locomotion
+1) Torso altitude control: The terrestrial locomotion is
+totally based on the quasi-static joint motion. Therefore the
+centroidal motion should be also assumed to be static, which
+results in a desired wrench only handling gravity (wd =
+�0
+0
+−mΣg
+0
+0
+0�
+) for (17). Despite of the joint
+position control proposed in (25), a small error regarding
+the torso (i.e., baselink) pose, particularly along the altitude
+direction, would still remain mainly due to the influence of
+gravity. Therefore, we apply a feedback control using the rotor
+thrust for the torso altitude. Instead of the PID position control
+for the centroidal motion as proposed in (11), a truncated
+feedback control for the torso altitude is introduced as follows:
+f d
+z = kb(zd
+b − zb),
+(26)
+where kb is the P gain, and zb is the torso altitude. We assume
+this altitude control is for the “floating” baselink even in the
+terrestrial locomotion mode. Therefore, instead of considering
+f d
+z in (17) and (18), we introduce another independent control
+allocation to obtain the additional thrust force as follows:
+∆wd =
+Nr
+2
+�
+i=1
+Q2i∆f
+′
+2i,
+(27)
+where ∆wd =
+�
+0
+0
+f d
+z
+0
+0
+0
+�T. It is notable that we
+only choose the rotors in the inner link of each leg to suppress
+the influence on the joint quasi-static motion as presented in
+(18). Then ∆f
+′
+2i can be given by
+∆f
+′ = ˜Q#∆wd,
+(28)
+˜Q =
+�Q0
+Q2
+· · ·
+QNr
+�
+,
+∆f
+′ =
+�
+∆f
+′
+0
+∆f
+′
+2
+· · ·
+∆f
+′
+Nr
+�T ,
+where ˜Q# is the psuedo-inverse matrix of ˜Q. Finally, f
+′
+2i →
+f
+′
+2i + ∆f
+′
+2i is performed before substituting it into (22)∼(24).
+2) Static walking gait: In this work, we only focus on the
+static walking gait. Hence only one leg is allowed to lift during
+walking. As the update of the foot step for the lifting leg,
+we analytically solve the inverse-kinematics for the related
+three joint angles: qi yaw, qi pitch, and qi+1 pitch as depicted
+in Fig. 2, which can be uniquely determined. Regarding the
+gait for linear movement, we design a creeping gait that lifts
+the front-left, front-right, rear-right, and rear-left legs in order
+for one gait cycle, and also solely moves the torso in standing
+mode just after the two front legs have moved to the new
+position. To enable the repetition of the gait cycle, the stride
+length of all feet is set equal to the moving distance of torso.
+We further assume the robot only walks on a flat floor, and
+thus the height of feet should be always zero. Then, we first
+
+6
+set an intermediate target position right above the new foot
+step with a small height offset. Thus qi yaw and qi+1 pitch are
+identical to the final target, whereas qi pitch is smaller than
+the final value. Once the lifting leg moves to this intermediate
+pose, the robot starts lowering the leg to reach the new foot
+step only by changing qi pitch. Given that there is no tactile
+sensor on the foot, we introduce a threshold ∆qc for the joint
+angle error of qi pitch to detect touchdown. That is, if qd
+i pitch−
+qi pitch < ∆qc, then switch the lifting leg to the standing
+mode, and thus the number of the contact force fci changes
+from three to four.
+V. EXPERIMENT
+A. Robot Platform
+In this work, we developed a prototype of SIDAR as shown
+in Fig. 5, and the basic specification is summarized in Tab. I.
+Given the lightweight design, we employed CFRP material for
+link rod where cables can pass through. For the joint module,
+we used the Aluminum sheet to connect links, whereas the
+joint servo was Dynamixel XH430-V350R of which the torque
+was enhanced by pulley made from PLA. The range of joint
+angle was [−90◦ 90◦]. For the vectorable rotor module, a
+pair of counter-rotating plastic propellers were enclosed by
+ducts with the aim of safety and increase of thrust, whereas
+Dynamixel XL430-W250T was used for the rotor vectoring.
+Batteries are distributed in each link unit in parallel as shown
+in Fig. 5(G) which can provide a flight duration up to 9 min
+and a longer walk duration up to 20 min. A hemisphere foot
+with anti-slip tape was equipped to ensure the stable point
+contact during the terrestrial locomotion.
+On the center torso as shown in Fig. 5(A), NVIDIA Jetson
+TX2 and an original MCU board called Spinal were deployed
+to perform the realtime control framework as presented in
+Fig. 4. For each link unit, there was a distributed MCU
+board called Neuron that served as relay node between Spinal
+(C1)
+(A)
+(B)
+(D)
+(C2)
+(G)
+(F)
+(A)
+(B)
+(C1)
+(C2)
+(D)
+(E)
+(E)
+(F)
+(G)
+àÜ
+öÜ
+1.1m
+MÜ4w_ê
+MÜ>54w_ê
+MÜ4ngraf
+MÜ>54ngraf
+CAN
+Fig. 5. Prototype of SPIDAR: (A) center torso that employed an original
+red MCU called Spinal and a high level processor (Nvidia Jetson TX2); (B)
+spherically vectorable dual-rotor module; (C1)(C2) two-DoF joint module
+for the “hip” and the “knee”, respectively; (D) single leg (limb) that had the
+maximum length of 1.1 m; (E) small relay board called “Neuron” for each
+link unit that was connected with “Spinal” via CAN; (F) hemisphere foot
+with anti-slip tape; (G) distributed battery attached at each link unit.
+TABLE I
+PROTOTYPE SPECIFICATIONS
+1. Main Feature
+3. Vectorable Rotor
+Attribute
+Value
+Attribute
+Value
+total mass
+15.2 kg
+rotor KV
+1550
+max size (dia.)
+2.7 m
+propeller diameter
+5 inch
+max flight time
+9 min
+max thrust (¯λ)
+42 N
+max walk time
+20 min
+pulley ratio
+1:1.5
+max vectoring torque
+1.5 Nm
+2. Link and Joint
+max vectoring speed
+4.2 rad/s
+Attribute
+Value
+link length
+0.54 m
+4. Lipo Battery
+joint pulley ratio
+1:2
+Attribute
+Value
+max joint speed
+0.34 rad/s
+capacity
+6S 3Ah
+max torque (¯τq)
+6.5 Nm
+amount
+8
+and each actuator. Neurons and Spinal were connected by
+CAN cable. The detail of the onboard communication can be
+found in [18]. Besides, an external motion capture system was
+applied in our experiment to obtain the state of the baselink
+(i.e., rb, ˙rb, Rb, and ωb), which were used to calculate the
+state of centroidal motion based on forward-kinematics.
+B. Basic Experimental Evaluation
+1) Aerial transformation: A unique feature of SPIDAR is
+the aerial maneuvering with joint motion (i.e., aerial transfor-
+mation). To validate the stability during flight, simple transfor-
+mation as shown in Fig. 6 was performed. All limbs changed
+their joints with the same trajectories as plotted in Fig. 7(C).
+(A)
+(B)
+Fig. 6. Stable joint motion in midair: (A) extended pose that has diameter
+of 2.6 m; (B) standing pose, implying the feasibility to takeoff directly from
+the terrestrial mode.
+32
+34
+36
+38
+40
+42
+44 [s]
+(B)
+(A)
+(D)
+0.05
+-0.05
+0
+Š Aåã
+Š Aåä
+Š Aåå
+Š Aåâßß Š AãÜçÖÛ Š AìÔê
+[m]
+1.5
+-1.5
+-1.0
+-0.5
+0.5
+1.0
+0
+[;]
+6
+4
+2
+0
+-2
+-4
+[Nm]
+32
+34
+36
+38
+40
+42
+44 [s]
+(C)
+80
+60
+40
+20
+0
+-20
+[;]
+Š M54ìÔê
+Š M54ãÜçÖÛ
+Š M64ìÔê
+Š M64ãÜçÖÛ
+Š ì54ìÔê
+Š ì54ãÜçÖÛ
+Š ì64ìÔê
+Š ì64ãÜçÖÛ
+Fig. 7. Plots related to Fig. 6 : (A) positional errors of {CoG}; (B) rotational
+errors of {CoG} described in XYZ Euler angles; (C) joint trajectories for
+leg1 (q1 yaw, q1 pitch, q2 yaw, and q2 pitch), other legs followed the same
+joint trajectories; (D) torques for those joints.
+
+MM
+MwM
+M装配件7
+(A)
+(B)
+Fig. 8. lifting a single leg from standing mode: (A) standing mode where
+all feet have contact with the ground; (B) lifting single leg and keeping the
+raised pose with the assistance of rotor thrust.
+For the control gains in (11) and (12), we set Kf,p, Kf,i, Kf,d,
+Kτ,p, Kτ,i, and Kτ,d as D(3.6, 3.6, 2.8), D(0.03, 0.03, 1.2),
+D(4, 4, 2.8), D(15, 15, 10), D(0.3, 0.3, 0.1), and D(5, 5, 5),
+where D(∗, ∗, ∗) ∈ R3×3 is a diagonal matrix. For the
+optimization problem of (16), we omitted the second term
+(i.e., w2 = 0) to put a priority on the minimization of the
+thrust force. Fig. 7(A) and (B) plotted the positional and
+rotational errors during the flight and transformation, and the
+RMS of those errors were [0.014, 0.023, 0.038] m and [0.81,
+0.69, 0.92]◦. The altitude error erz indicated a relatively large
+deviation during the joint motion, which was caused by the
+violation of the quasi-static assumption. Nevertheless, this
+deviation rapidly decreased once the joint motion finished.
+Fig. 7(C) and (D) showed that all joint were well controlled by
+the PD control as presented in (25). Eventually, these results
+demonstrated the stability of both the baselink pose and the
+joint motion in aerial locomotion.
+2) Leg lifting: The key to achieve walking by legged robot
+is the stability while lifting the leg. Then, we evaluated the
+proposed control method by performing a long-term single
+leg lifting as shown in Fig. 8. The cost weights in (16) were
+set as w1 = 1, w2 = 1, and the bound for joint torque ¯τq
+was decreased to 1.5 Nm to ensure sufficient margin for joint
+control. Besides, the gain kb in (26) was set as 25. Leg1
+was lifted by changing q1 pitch from −16 ◦ to −28 ◦, and the
+lifting motion lasted around 30 s as shown in Fig. 9(A). Other
+joints were kept constant in the whole motion as shown in
+Fig. 9(A) and (C), and their torques were within the bounds as
+depicted in Fig. 9(B) and (D). These results demonstrated the
+stability of joint motion against the influence of thrust force.
+Besides the stability of the baselink pose can be confirmed
+in Fig. 9(E) and (F), where both the positional and rotational
+errors converged to the sufficiently small value (i.e., 0.01 m
+and 0.5 ◦). Fig. 9(G) showed the large increase of the thrust
+forces in the lifting leg, whereas Fig. 9(H) showed small
+changes in other standing legs. In addition, these plots also
+confirmed the stable transition between standing mode and
+leg lifting mode. In particular, the shift back to the standing
+model around 40 s demonstrated the smooth touchdown, which
+indicates the promising terrestrial locomotion.
+C. Seamless Terrestrial/Aerial Hybrid Locomotion
+We further evaluated the feasibility of seamless locomotion
+transition as shown in Fig. 10. Fig. 11(A) and (C) demon-
+strated the baselink pose trajectory during walking with five
+15
+20
+25
+30
+35
+40
+[s]
+45
+15
+20
+25
+30
+35
+40
+[s]
+45
+15
+20
+10
+5
+10
+8
+6
+4
+2
+[N]
+[N]
+(F)
+(E)
+(G)
+(H)
+0.02
+-0.02
+0
+[m]
+-0.01
+0.01
+1.5
+-1.5
+-1.0
+-0.5
+0.5
+1.0
+0
+[;]
+Š Aåã Š Aåä Š Aåå
+Š Aåâßß Š AãÜçÖÛ Š AìÔê
+Š ã5 Š ã6
+Š ã7 Š ã8 Š ã9 Š ã:
+(A)
+(B)
+(C)
+(D)
+2
+1
+0
+-1
+-2
+6
+5
+[Nm]
+4
+2
+1
+0
+-1
+3
+-30
+-25
+-20
+-15
+[;]
+80
+[;]
+86
+84
+82
+Š M54ãÜçÖÛ Š M74ãÜçÖÛ Š M94ãÜçÖÛ
+Š ì54ãÜçÖÛ Š ì74ãÜçÖÛ Š ì94ãÜçÖÛ
+Š M64ãÜçÖÛ Š M84ãÜçÖÛ Š M:4ãÜçÖÛ
+Š ì64ãÜçÖÛ Š ì84ãÜçÖÛ Š ì:4ãÜçÖÛ
+Fig. 9. Plots related to Fig. 8: (A) trajectories for hip pitch joints. q7 pitch
+was omitted due to the symmetric pose of leg4 related to leg2; (B) torques
+of joints in (A); (C) trajectories for knee pitch joints; (D) torques of joints in
+(C); (E) positional errors of baselink ; (F) rotational errors of baselink; (G)
+thrust forces in leg1; (H) thrust forces in other legs.
+gait cycles. We observed that the translational drift along the
+walking direction (x axis) and the orthogonal direction (y axis)
+finally grew to 0.18 m and 0.10 m, whereas the rotational drift
+along the yaw axis also increased to 9 ◦. These drifts can be
+attributed to the feed-froward gait planing where the target
+baselink pose was updated based on the last target values
+�
+ 
+!
+"
+0.2m
+0.2m
+0.2m
+5 gait 
+cycles
+gait cycle
+Fig. 10. Seamless Terrestrial/Aerial Hybrid Locomotion: 1⃝ ∼ 3⃝ shows
+the representative phases (moving the front-left leg, the torso, and rear-left
+leg) in one creeping gait cycle. After five gait cycles, robot switched to the
+aerial locomotion directly from the terrestrial pose as shown in 4⃝.
+
+82prMN8
+gait 
+cycle1
+gait 
+cycle2
+gait 
+cycle3
+gait 
+cycle4
+gait 
+cycle5
+stand
+sprawl
+Š NKHH Š LEP?D Š U=S
+--- NKHH×, LEP?D×, U=S×
+-6
+4
+2
+0
+-2
+-4
+-8
+-10
+takeoff
+land
+(B)
+(A)
+(C)
+(D)
+1.2
+0.8
+1.0
+[;]
+-6
+0
+-2
+-4
+-8
+-10
+[;]
+0.6
+0.4
+0.2
+0
+-0.2
+0.8
+0.6
+0.4
+0.2
+0
+-0.2
+[m]
+[m]
+Š Në Š Nì Š Ní
+--- Në× --- Në× --- Në×
+--- U=S×
+20
+40
+60
+80
+100
+120 [s]
+120
+125
+130
+135
+[s]
+Fig. 11. Plots related Fig. 10.(A)/(B) trajectories of baselink position during
+the terrestrial locomotion and the aerial locomotion, respectively; (C)/(D)
+trajectories of baselink orientation during the terrestrial locomotion and the
+aerial locomotion, respectively.
+but not the actual values. Nevertheless, these drifts can be
+considered relatively small compared to the total displacement,
+and are possible to be suppressed by adding a feed-back loop
+in planning as a future work. Furthermore, the deviations
+regarding the z, roll, and pitch axes were sufficiently small,
+which demonstrated the efficiency of the proposed control
+method presented in Sec. IV.
+As shown in Fig. 11(B) and (D), the transition to the
+aerial locomotion was smooth and stable, and the stability in
+midair was also confirmed, Thus, these results demonstrated
+the feasibility of the mechanical design, modeling and control
+methods for the terrestrial/aerial hybrid quadruped platform.
+VI. CONCLUSION
+In this paper, we presented the achievement of the terres-
+trial/aerial hybrid locomotion by the quadruped robot SPIDAR
+that were equipped with the vectorable rotors distributed in all
+links. We first proposed the mechanical design for this unique
+quadruped platform, and then developed the modeling and
+control methods to enable static walking and transformable
+flight. The feasibility of the above methods were verified by
+the experiment of seamless terrestrial/aerial hybrid locomotion
+with the prototype of SPIDAR.
+A crucial issue remained in this work is the oscillation
+and deviation of the baselink pose and joint angles during
+walking. To improve the stability, the rotor thrust should be
+directly used in the joint position control to replace the current
+simple PD control. Furthermore, the gait planning should be
+also robust against the drift by adding a feed-back loop as
+discussed in Sec. V-C. Last but not least, the dynamic walking
+and the aerial manipulation will be investigated to enhance the
+versatility of this robot in both maneuvering and manipulation.
+REFERENCES
+[1] Koji Kawasaki, Moju Zhao, Kei Okada, and Masayuki Inaba. MUWA:
+Multi-field universal wheel for air-land vehicle with quad variable-pitch
+propellers. In 2013 IEEE/RSJ International Conference on Intelligent
+Robots and Systems, pp. 1880–1885, 2013.
+[2] Arash Kalantari and Matthew Spenko.
+Modeling and performance
+assessment of the HyTAQ, a hybrid terrestrial/aerial quadrotor. IEEE
+Transactions on Robotics, Vol. 30, No. 5, pp. 1278–1285, 2014.
+[3] Hiroya Yamada, et al.
+Development of amphibious snake-like robot
+ACM-R5. In the 36th International Symposium on Robotics (ISR), 2005.
+[4] Auke Jan Ijspeert, Alessandro Crespi, Dimitri Ryczko, and Jean-Marie
+Cabelguen. From swimming to walking with a salamander robot driven
+by a spinal cord model. Science, Vol. 315, No. 5817, pp. 1416–1420,
+2007.
+[5] Hamzeh Alzu’bi, Iyad Mansour, and Osamah Rawashdeh. Loon Copter:
+Implementation of a hybrid unmanned aquatic–aerial quadcopter with
+active buoyancy control. Journal of Field Robotics, Vol. 35, No. 5, pp.
+764–778, 2018.
+[6] Kyunam Kim, Patrick Spieler, Elena-Sorina Lupu, Alireza Ramezani,
+and Soon-Jo Chung. A bipedal walking robot that can fly, slackline,
+and skateboard. Science Robotics, Vol. 6, No. 59, p. eabf8136, 2021.
+[7] Tomoki Anzai, Yuta Kojio, Tasuku Makabe, Kei Okada, and Masayuki
+Inaba. Design and development of a flying humanoid robot platform with
+bi-copter flight unit. In 2020 IEEE-RAS 20th International Conference
+on Humanoid Robots (Humanoids), pp. 69–75, 2021.
+[8] Stella Latscha, et al. Design of a hybrid exploration robot for air and land
+deployment (H.E.R.A.L.D) for urban search and rescue applications.
+In 2014 IEEE/RSJ International Conference on Intelligent Robots and
+Systems, pp. 1868–1873, 2014.
+[9] Nitzan Ben David and David Zarrouk. Design and analysis of FCSTAR,
+a hybrid flying and climbing sprawl tuned robot. IEEE Robotics and
+Automation Letters, Vol. 6, No. 4, pp. 6188–6195, 2021.
+[10] Azumi Maekawa, Ryuma Niiyama, and Shunji Yamanaka.
+Pseudo-
+locomotion design with a quadrotor-assisted biped robot. In 2018 IEEE
+International Conference on Robotics and Biomimetics (ROBIO), pp.
+2462–2466, 2018.
+[11] Daniele Pucci, Silvio Traversaro, and Francesco Nori.
+Momentum
+control of an underactuated flying humanoid robot. IEEE Robotics and
+Automation Letters, Vol. 3, No. 1, pp. 195–202, 2018.
+[12] Yuhang Li, et al. Jet-HR2: A flying bipedal robot based on thrust vector
+control. IEEE Robotics and Automation Letters, Vol. 7, No. 2, pp. 4590–
+4597, 2022.
+[13] Michael P. Murphy, Aaron Saunders, Cassie Moreira, Alfred A. Rizzi,
+and Marc Raibert. The LittleDog robot. The International Journal of
+Robotics Research, Vol. 30, No. 2, pp. 145–149, 2011.
+[14] Marco Hutter, et al.
+ANYmal - a highly mobile and dynamic
+quadrupedal robot.
+In 2016 IEEE/RSJ International Conference on
+Intelligent Robots and Systems (IROS), pp. 38–44, 2016.
+[15] Benjamin Katz, Jared Di Carlo, and Sangbae Kim.
+Mini Cheetah:
+A platform for pushing the limits of dynamic quadruped control.
+In
+2019 International Conference on Robotics and Automation (ICRA), pp.
+6295–6301, 2019.
+[16] Paul Hebert, et al. Mobile manipulation and mobility as manipulation-
+design and algorithm RoboSimian. Journal of Field Robotics, Vol. 32,
+No. 2, pp. 255–274, 2015.
+[17] Kenji Hashimoto, et al.
+WAREC-1 - a four-limbed robot having
+high locomotion ability with versatility in locomotion styles. In 2017
+IEEE International Symposium on Safety, Security and Rescue Robotics
+(SSRR), pp. 172–178, 2017.
+[18] Moju Zhao, et al.
+Design, modeling, and control of an aerial robot
+DRAGON: A dual-rotor-embedded multilink robot with the ability
+of multi-degree-of-freedom aerial transformation. IEEE Robotics and
+Automation Letters, Vol. 3, No. 2, pp. 1176–1183, April 2018.
+[19] Moju Zhao, Kei Okada, and Masayuki Inaba. Versatile articulated aerial
+robot DRAGON: Aerial manipulation and grasping by vectorable thrust
+control. The International Journal of Robotics Research, 2022.
+[20] Fan Shi, Tomoki Anzai, Yuta Kojio, Kei Okada, and Masayuki In-
+aba. Learning agile hybrid whole-body motor skills for thruster-aided
+humanoid robots.
+In 2022 IEEE/RSJ International Conference on
+Intelligent Robots and Systems (IROS), pp. 12986–12993, 2022.
+[21] Zhifeng Huang, et al. Three-dimensional posture optimization for biped
+robot stepping over large ditch based on a ducted-fan propulsion system.
+In 2020 IEEE/RSJ International Conference on Intelligent Robots and
+Systems (IROS), pp. 3591–3597, 2020.
+[22] Satoshi Kitano, Shigeo Hirose, Gen Endo, and Edwardo F. Fukushima.
+Development of lightweight sprawling-type quadruped robot TITAN-
+XIII and its dynamic walking. In 2013 IEEE/RSJ International Confer-
+ence on Intelligent Robots and Systems, pp. 6025–6030, 2013.
+[23] T. Lee, M. Leok, and N. H. McClamroch. Geometric tracking control
+of a quadrotor UAV on SE(3). In 49th IEEE Conference on Decision
+and Control (CDC), pp. 5420–5425, 2010.
+
+E

QdAyT4oBgHgl3EQf7vpN/content/tmp_files/2301.00843v1.pdf.txt

@@ -0,0 +1,1564 @@
+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+1
+Explicitly Solvable Continuous-time Inference for
+Partially Observed Markov Processes
+Daniel Chen, Alexander G. Strang, Andrew W. Eckford Senior Member, IEEE, Peter J. Thomas
+Abstract—Many natural and engineered systems can be mod-
+eled as discrete state Markov processes. Often, only a subset
+of states are directly observable. Inferring the conditional prob-
+ability that a system occupies a particular hidden state, given
+the partial observation, is a problem with broad application.
+In this paper, we introduce a continuous-time formulation of
+the sum-product algorithm, which is a well-known discrete-time
+method for finding the hidden states’ conditional probabilities,
+given a set of finite, discrete-time observations. From our new
+formulation, we can explicitly solve for the conditional probability
+of occupying any state, given the transition rates and observations
+within a finite time window. We apply our algorithm to a
+realistic model of the cystic fibrosis transmembrane conductance
+regulator (CFTR) protein for exact inference of the conditional
+occupancy probability, given a finite time series of partial
+observations.
+I. INTRODUCTION
+Markov processes—dynamic processes whose future be-
+havior depends only on their present state—approximate a
+wide variety of natural and engineered systems. Despite rapid
+advances in high-throughput data acquisition and data pro-
+cessing, many systems of interest contain important degrees
+of freedom that cannot be directly observed. Inferring the
+conditional probability that such a partially observed Markov
+process occupies specific hidden states, given the available ob-
+servations, is a ubiquitous problem in science and engineering.
+Examples appear in robotics [1], ecology [2], neuroscience [3],
+and algorithmic text analysis [4].
+We are motivated by biological examples in the present
+paper. Ion channels in excitable membranes, such as the
+sodium (Na+) and potassium (K+) channels described in
+Hodgkin and Huxley’s quantitative model for action potential
+generation [5], [6], [7], provide an early example. Discrete
+state Markov models based on Hodgkin and Huxley’s K+
+channel contain five states, only one of which conducts an
+ionic current; the other states are “silent” and cannot be distin-
+guished by direct electrophysiological observation. Similarly,
+This work was supported in part by National Institutes of Health BRAIN
+Initiative grant R01 NS118606 and a National Science Foundation grant
+DMS-2052109 to PJT, as well as research support from the Oberlin College
+Libraries, and an NSERC Discovery grant to AWE.
+D. Chen and P. J. Thomas are with the Department of Mathematics, Applied
+Mathematics, and Statistics; Department of Electrical, Control and Systems
+Engineering; Department of Computer and Data Science; Department of
+Biology, Case Western Reserve University, Cleveland, OH 44106 USA (e-
+mail: txc461/pjthomas@case.edu).
+A. G. Strang is with Department of Statistics, University of Chicago,
+Chicago, IL 60637 USA (email: alexstrang@uchicago.edu).
+A. W. Eckford is with Department of Electrical Engineering and Computer
+Science, York University, Toronto, ON M3J 1P3 Canada (e-mail: aeck-
+ford@yorku.ca).
+the Na+ channel has eight states: seven with zero conductance
+and one with nonzero conductance. Colquhoun and Hawkes
+introduced maximum likelihood methods for inferring the rate
+constants of a partially observed Markov process representing
+the nicotinic Acetylcholine receptor [8], [9], [10], [11], but
+did not address the question of inferring microscopic state
+occupancy from observable conductance time series. More
+recently, research into the molecular biology of cystic fibrosis
+(CF) has focused on the CF transmembrane conductance
+regulator (CFTR), which can be modeled as a 7-state system
+with two conducting states and five nonconducting states
+[12] (detailed below in Section V). Beyond these biological
+examples, problems of inferring or estimating hidden states
+from incomplete observations are widely studied in the signal
+processing literature [13].
+The literature contains several approaches to approximating
+the behavior of hidden states of partially observed Markov
+processes. Sampling provides one common technique for
+approximate inference [14]. As an example, recent work by
+Fang et al. demonstrated an efficient algorithm for simulating
+stochastic reaction networks with multiple separated time
+scales using particle filters [15]. In general, Markov Chain
+Monte Carlo is a widely employed sampling technique used
+to infer hidden states [16], [17], that has also been applied to
+ion channels [18]. For partially observed Bayesian networks
+operating in discrete time, message passing algorithms on
+factor graphs provide an efficient and exact inference method
+[19]. The factor-graph formalism is highly flexible. Algorithms
+based on the message passing concept have been extended to
+applications in localization [20], compressed sensing [21], and
+decision fusion [22]. Factor graphs are not limited to models
+with a discrete number of variables. For example, Gaussian
+message passing in linear Gaussian models (e.g. Kalman
+filtering and smoothing) has been developed for continuous-
+time models with discrete-time observations [23], [24], [25].
+However, the state reconstruction problem for continuous-time
+finite-state hidden Markov models has not been addressed in
+the literature, to be best of our knowledge.
+In this work, we extend the message-passing algorithm in
+order to analytically interpolate state-occupancy probabilities
+of a continuous-time system, given a discretely sampled time
+series. That is, we show how to infer the time-dependent
+conditional probabilities of latent states for continuous-time
+discrete-state homogeneous Markov processes given a set of
+partial observations over a finite time window. We derive
+an equivalent formulation of the sum-product algorithm in
+continuous time that allows one to find an explicit ana-
+lytic solution for the state occupancy probability. Having
+arXiv:2301.00843v1  [eess.SP]  2 Jan 2023
+
+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+2
+explicit solutions lowers the computational cost and, unlike
+sampling-based approximate methods, does not sacrifice ac-
+curacy. Furthermore, the continuous-time formalism leads to
+elegant simplification of the analytic solutions. For certain
+systems—like the three-state systems shown in Figure 2—the
+conditional probability obeys a second-order inhomogeneous
+linear ordinary differential equation. Finally, we demonstrate
+the practical functionality of the algorithm with the 7-state
+model for CFTR using simulated data.
+The paper is organized as follows. Section II reviews the
+message-passing algorithm for Markov processes with binary
+observations. Section III displays the main result of the paper:
+a continuous-time formulation of the sum-product algorithm.
+We present the derivation of the conditional probability using
+the message-passing approach in continuous time, and the
+corresponding sum-product algorithm. Section IV discusses
+the implications of the continuous-time formulation further.
+We give examples of small systems to display how analytic
+solutions may be found. In Section V we demonstrate the value
+of the algorithm for a larger, realistic system.
+II. THEORY OF THE SUM-PRODUCT ALGORITHM
+The sum-product algorithm can be used generally for in-
+ference on probabilistic models that can be written as factor
+graphs [19]. There are many variants of the sum-product
+algorithm, each suitable for accomplishing a different task.
+For our purposes, we will focus on the forward/backward
+algorithm for inference on hidden Markov models.
+We consider a continuous-time, discrete-state homogeneous
+Markov process on a finite state space Ω. Given a discrete,
+uniformly spaced sampling interval, the continuous-time pro-
+cess induces a discrete-time Markov process specified by some
+column-stochastic transition matrix P that is invariant in time.
+Let S ⊂ Ω be a subset of states, and let St ∈ Ω be the state
+of the system at time t. We assume an observer can only
+see whether St is in S or not. Accordingly, let Yt = m(St)
+represent the observable where m(s) : Ω → {0, 1} is the
+indicator function for the set S. The goal of the algorithm
+is to infer the conditional probability of being at a particular
+state, i ∈ Ω, given the binary observation.
+The algorithm involves three vector-valued quantities: a for-
+ward message αt, a backward message βt, and the observation
+message χt. One can interpret the forward message as the
+probability of arriving at a certain state from time 0 to time
+t and the backward message as the likelihood of occupying
+a certain state at time t conditioned on ending up in a given
+target state, or a given target set of states, at the end of the
+measurement T. On the other hand, the observation message
+χt is an indicator function of the possible states given the
+observation. For instance, if at time t, the observation of the
+total system were Yt = 1, then χt would be a vector with 1’s
+on the states in the observation set S and 0 otherwise. Using
+superscripts to denote vector indices, i.e. v(i) denotes the i-
+th element of vector v, we present pseudo-code for the sum-
+product algorithm in Algorithm 1. Note that, the normalization
+constant Z = �
+k α(k)
+t
+β(k)
+t
+in Line 10 of Algorithm 1 is time
+invariant [26].
+Algorithm 1 Forward/Backward Algorithm
+Input: The transition matrix P ∈ Rn×n, the observation
+message χt for t ∈ {1, . . . , T}.
+Output: The inferred probability pt for t ∈ {1, . . . , T}.
+1: Initialize the forward message α0 = π, the stationary
+distribution of P
+2: Initialize the backward message β(i)
+T
+= 1, for all i
+3: for t from 2, . . . , T do
+4:
+αt = diag(χt)Pαt−1
+5: end for
+6: for s from T − 1, . . . , 1 do
+7:
+βs = P ⊺diag(χs+1)βs+1
+8: end for
+9: for t from 1, . . . , T do
+10:
+pt =
+αt⊙βt
+�
+k α(k)
+t
+β(k)
+t
+11: end for
+Fig. 1 illustrates the sum-product algorithm’s application to
+a time series of discretely sampled observations. Consider a
+three-state-chain with symmetric transition rates as shown in
+the top-left of Figure 1. If the system takes states 1 or 2,
+“0” will be observed, and “1” will be observed otherwise.
+Using the sum-product algorithm, we can find the probability
+of the system occupying each state given the discrete-time
+observations. As shown in the bottom row, as the sampling
+time step decreases, the conditional probabilities appear to
+converge to a smooth curve within each interval with a fixed
+observation (either “0” or “1”). Intuitively, there should exist
+a continuous, perhaps piecewise differentiable, representation
+of the conditional probability of a partially observed process.
+We formalize this intuition below.
+III. CONTINUOUS-TIME MESSAGE PASSING
+In this section, we present the main result, namely the
+derivation of the continuous-time message passing algorithm.
+In this new formalism, messages are passed in the form
+of linear differential equations on possible states given the
+observable system. In order to guarantee the existence of the
+continuous sum-product algorithm, we assume the following
+conditions.
+Assumptions:
+A1 The continuous-time process {S(t); t ∈ [0, T]} takes
+values in a finite state space Ω = {1, 2, . . . , N}.
+A2 S(t) has the Markov property, and has exponentially
+distributed waiting times parameterized by a rate matrix
+W, with wji specifying the transition rate from state i to
+state j. Note that W is constant in the interval [0, T].
+A3 There is a distinguished subset S ⊂ Ω such that the
+observable process Y(t) satisfies
+Y(t) =
+�
+1
+if S(t) ∈ S,
+0
+otherwise .
+Further details appear in Appendix A.
+Under these assumptions, we obtain a continuous-time ver-
+sion of the sum-product algorithm by executing the following
+steps (made rigorous in the proof of Theorem 1 below). Write
+
+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+3
+Fig. 1: Illustrating convergence of conditional state occupancy probabilities to a differentiable function for a three-state model.
+(Top left) The state diagram. (Top middle) True simulated states. (Top right) Binary observation derived from true states.
+(Bottom row) Inference of hidden states via the sum-product algorithm with time steps 2.5 sec (Bottom Left), 1.0 sec (Bottom
+Middle) and 0.5 sec (Bottom Right).
+out the matrix multiplication of the discrete-time algorithm
+element-wise. Focus on one sojourn where the observation
+doesn’t change. Within that time interval, take the limit as the
+time step goes to zero to derive the continuous-time dynamics
+of the conditional probabilities. Extend the solution to the
+full time interval via appropriate boundary conditions at the
+transition between each sojourn. The main result is stated in
+the theorem below.
+Theorem 1. Suppose processes S(t) and the associated
+process Y(t) satisfies assumptions A1, A2, and A3 above.
+Then, given a realization of the process Y(t), the conditional
+probability p(t) = Pr[S(t)|Y(t)] exists, is piecewise smooth
+(C∞), and is C∞ on all intervals where Y(t) is constant. In
+particular, p(t) takes the form
+p(t) =
+ρ(t)
+�
+k ρ(k)(t) =
+α(t) ⊙ β(t)
+�
+k α(k)(t) ⊙ β(k)(t)
+(1)
+where ⊙ denotes the element-wise product, and the quantities
+α(t) and β(t) are functions of time that follow the linear
+ordinary differential equations
+dα(i)(t)
+dt
+=
+�
+k∈S\{i}
+wkiα(k)(t) −
+�
+l̸=i
+wilα(i)(t),
+(2)
+dβ(j)(t)
+dt
+= −
+�
+k∈S\{j}
+wjkβ(k)(t) +
+�
+l̸=j
+wjlβ(j)(t).
+(3)
+Proof. Without loss of generality, focus on the case where
+Y(0) = Y(T) = 0, and Y(t) = 1 for 0 < t < T. We use
+q(t) to denote a quantity, q, evolving in continuous time on
+the interval [0, T], and qt to denote the same process sampled
+at discrete times.
+Let the time interval [0, T] be discretized with a step size
+∆t = T/n for some integer n ≫ 1. At each time step,
+S(t) is sampled. Then, the sum-product algorithm can be used
+to solve for pt. Writing the matrix multiplication out yields
+the following set of equations for the forward and backward
+messages in discrete time:
+α(i)
+t+∆t =
+�
+k
+Pr[st+∆t = i|st = k]α(k)
+t
+χ(i)
+t ,
+(4)
+β(i)
+t
+=
+�
+k
+Pr[st+∆t = k|st = i]β(k)
+t+∆tχ(k)
+t+∆t.
+(5)
+We neglect states not in S because only the probability
+conditioned on the observations is of interest. Then, for any
+state i ∈ S, we argue that the corresponding forward message,
+α(i)(t), and backward message, β(i)(t) in continuous time can
+be written as solutions of systems of differential equations,
+upon taking limits as ∆t → 0. For notational simplicity, for
+
+true states
+observation
+3
+W2
+2
+W/2
+W23
+0
+0
+50
+100
+0
+50
+100
+sample every 2.5 seconds
+sample every 1 seconds
+sample every 0.5 seconds
+1
+0.8
+0.8
+0.8
+0.6
+0.6
+0.6
+0.4
+0.4
+0.4
+0.2
+0.2
+0.2
+0
+0
+0
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+4
+t > τ we let P(i,j)
+t,τ
+= Pr[S(t) = i|S(τ) = j].
+dα(i)(t)
+dt
+= lim
+∆t→0
+α(i)(t + ∆t) − α(i)(t)
+∆t
+(6)
+= lim
+∆t→0
+1
+∆t
+� �
+k
+P(i,k)
+t+∆t,tα(k)(t)χ(i)(t) − α(i)(t)
+�
+(7)
+= lim
+∆t→0
+1
+∆t
+�
+�
+k∈S\{i}
+(wki∆t + o(∆t))α(k)(t) . . .
+−
+�
+l̸=i
+(wil∆t + o(t))α(i)(t)
+�
+(8)
+=
+�
+k∈S\{i}
+wkiα(k)(t) −
+�
+l̸=i
+wilα(i)(t)
+(9)
+dβ(i)(t)
+dt
+= lim
+∆t→0
+β(i)(t + ∆t) − β(i)(t)
+∆t
+(10)
+= lim
+∆t→0
+1
+∆t
+�
+β(i)(t + ∆t) . . .
+−
+�
+k
+P(k,i)
+t+∆t,tβ(k)(t + ∆t)χ(k)(t + ∆t)
+�
+(11)
+= lim
+∆t→0
+1
+∆t
+�
+−
+�
+k∈S\{i}
+(wik∆t + o(∆t))β(k)(t + ∆t)
++
+�
+l̸=i
+(wil∆t + o(t))β(i)(t + ∆t)
+�
+(12)
+= −
+�
+k∈S\{i}
+wikβ(k)(t) +
+�
+l̸=i
+wilβ(i)(t)
+(13)
+where lim∆t→0 o(∆t)/∆t = 0. Readers could refer to Ap-
+pendix A or consult existing literature such as [27] for the
+relationship between transition probabilities and transition
+rates of a Markov jump process.
+The conditional probability can be found by solving the
+differential equations, taking the component-wise product of
+α(t) and β(t) for every t ∈ [0, T], and normalizing so as to
+obtain a valid probability distribution.
+The differential equation formulation is only applicable for
+the time intervals where the observation Y(t) is constant.
+When the observable Y(t) changes (when the systems transi-
+tions from a state in S to a state not in S), the probability with
+respect to time might not be differentiable; in some cases, it
+is not even continuous. Therefore, we must specify boundary
+conditions to connect the probabilities from one sojourn to
+the next. The observable may change either by the system
+entering S or else leaving S. Suppose a transition occurred
+within time (t∗ −∆t, t∗] such that, for t < t∗ −∆t, S(t) ∈ S,
+and S(t) ̸∈ S for t ≥ t∗. Call this event E. Then, we obtain
+the following transition rule for the forward message.
+Pr[S(t∗) = j|E]
+=
+�
+i∈S Pj,i
+t∗,t∗−∆tPr[S(t∗ − ∆t) = i]
+�
+k̸∈S
+�
+i∈S Pk,i
+t∗,t∗−∆tPr[S(t∗ − ∆t) = i]
+(14)
+=
+�
+i∈S(wji∆t + o(∆t))Pr[S(t∗ − ∆t) = i]
+�
+k̸∈S
+�
+i∈S(wki∆t + o(∆t))Pr[S(t∗ − ∆t) = i]
+(15)
+→
+�
+i∈S wjiPr[S(t−
+∗ ) = i]
+�
+k̸∈S
+�
+i∈S wkiPr[S(t−
+∗ ) = i]
+(16)
+as ∆t → 0. Here Pr[S(t−
+∗ ) = i] is the probability of occupying
+state i the instant before the transition, which can be found by
+solving the differential equations introduced above.
+We handle the boundary conditions at state transitions for
+the backward message similarly. Define E as above and let s∗
+be a particular goal state. Then:
+Pr[S(T) = s∗|S(t∗ − ∆t) = j, E]
+=
+�
+i̸∈S
+Pr[S(T) = s∗, S(t∗) = i|S(t∗ − ∆t) = j, E]
+=
+�
+i̸∈S Pr[S(T) = s∗|S(t∗) = i]Pi,j
+t∗,t∗−∆t
+�
+k∈S
+�
+i̸∈S Ps∗,i
+T,t∗Pi,k
+t∗,t∗−∆t
+(17)
+=
+�
+i̸∈S Pr[S(T) = s∗|S(t∗) = i](wij∆t + o(∆t))
+�
+k∈S
+�
+i̸∈S Ps∗,i
+T,t∗(wik∆t + o(∆t))
+(18)
+=
+�
+i̸∈S Pr[S(T) = s∗|S(t∗) = i](wij + o(∆t)
+∆t )
+�
+k∈S
+�
+i̸∈S Ps∗,i
+T,t∗(wik + o(∆t)
+∆t )
+(19)
+→
+�
+i̸∈S wijPr[S(T) = s∗|S(t∗) = i]
+�
+k∈S
+�
+i̸∈S wikPr[S(T) = s∗|S(t∗) = i]
+(20)
+as ∆t → 0. Here, Pr[S(T) = s∗|S(t∗) = i] is a hitting
+probability associated with the backward message at time
+t∗, which can be found by solving the backward-message
+differential equation. These boundary conditions, together with
+the differential equations (2)-(3), give the continuous-time
+evolution of the conditional probability for any finite-length
+observations.
+Note that the equations (2)-(3) extend to the case where
+Y(0) = Y(T) = 1 and Y(t) = 0 for 0 < t < T by viewing
+S ← Ω \ S. So, given a time series observation Y(t) where
+observation (the value of Y(t)) changes at time 0 < t1 < t2 <
+. . . tm, we can solve for the analytic solution at any interval
+with consistent observation (ti, ti+1] using equation (2) and
+(3). Then, use the result to compute the initial condition for the
+next interval — namely, (ti+1, ti+2] for the forward message
+and (ti−1, ti] for the backward message — as specified in (16)
+and (20). Thus, the statement of Theorem 1 holds.
+A general expression for the conditional probability can be
+obtained, but it is not of great utility in most systems. Yet,
+there are certain special cases that yield elegant solutions; we
+introduce several examples in Section IV.
+The continuous-time sum-product algorithm follows directly
+from the derivation above, and is outlined in Algorithm 2.
+
+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+5
+The discrete algorithm passes information through matrix
+multiplication of a truncated transition matrix; the continuous-
+time algorithm does the same by solving a system of linear
+differential equations using the truncated rate matrix. Since the
+final conditional probability is only piecewise differentiable,
+the boundary condition must be applied whenever a transition
+in or out of the observable set S occurs.
+Algorithm 2 Continuous-time Forward/Backward Algorithm
+Input: The rate transition matrix W ∈ Rn×n, the observed
+process Y(t).
+Output: The inferred probability p(t) for t ∈ [0, T].
+1: Let [t1, t2, . . . , tm] be a list of times where transitions
+occur
+2: τ ← 0
+3: α∗ ← π, the stationary distribution
+4: for j from 1, . . . , m do
+5:
+αj ← solution to the forward message differential
+equation (Equation 2) from τ to tj with initial condition
+α∗
+6:
+τ ← tj
+7:
+α∗ ← distribution specified according to Equation 16
+8: end for
+9: τ ← T
+10: β∗ ← the uniform distribution
+11: for j from m, . . . , 1 do
+12:
+βj ← solution to the backward message differential
+equation (Equation 3) from τ to tj with initial condition
+β∗
+13:
+τ ← tj
+14:
+β∗ ← distribution specified according to Equation 20
+15: end for
+16: α(t), β(t) ← concatenation αj’s and βj’s
+17: Compute ρ(t) = α(t)⊙β(t), the component-wise product
+between α(t) and β(t) pointwise with respect to t
+18: Compute the conditional probability p(t) =
+ρ(t)
+�
+k ρ(k)(t)
+From a practical perspective, having the ability to solve
+for the conditional probabilities exactly through differential
+equations drastically lowers the computational cost of the
+forward/backward algorithm. Traditionally, the discrete-time
+algorithm propagates the forward and backward messages
+through matrix operations at each time step. For long time-
+series and/or high-dimensional systems, this is computation-
+ally prohibitive. Through our continuous-time formalism, we
+solve the differential equations analytically, which is an op-
+eration that is independent of the length of the time-series,
+to find the forward or backward message at any time point.
+This difference effectively reduces the asymptotic scaling
+from O(∆t−1) to O(1), with the later scaling only in the
+number of observable transitions. In scenarios where finding
+the appropriate boundary condition would require an iterative
+procedure of solving the forward and backward messages
+multiple times, our continuous time approach should be much
+more efficient than the traditional discrete-time method. We
+discuss the performance of the continuous-time message-
+Fig. 2: State diagrams of two systems for which the
+continuous-time message passing algorithm exhibit analytic
+simplifications. (Left) 3-state chain with symmetric rates
+w12 = w21. (Right) Irreversible 3-state loop. States marked
+in red return Y(t) = 1 and blue return Y(t) = 0.
+passing algorithm further in Section V.
+IV. ANALYTIC SOLUTION
+Theorem 1 in the previous section shows that the conditional
+probability is always available analytically upon normalizing
+the component-wise product of the forward and backward
+messages,
+i.e. p(t) = ρ(t)/Z where Z = �
+k ρ(k)(t).
+As
+in the discrete-time case, the normalizing term Z is time-
+invariant in the continuous-time case as well. See Appendix
+B. The conditional probability may therefore be expressed in
+a particularly elegant form in certain cases, namely as the
+solution of a linear nonhomogeneous second-order differential
+equation. We begin this section by considering two examples.
+Following the examples, we consider extensions to higher
+dimensions.
+A. Symmetric 3-State Chain
+Consider the three-state chain depicted in the left panel of
+Figure 2, where states 1 and 2 are hidden. Assume that the per-
+capita transition rates within the hidden block are symmetric,
+i.e. w12 = w21 > 0, and assume w13 = w31 = 0. The rates
+w23 > 0 and w32 > 0 may be arbitrary. These assumptions
+result in the following rate matrix:
+W =
+�
+�
+−w21
+w12
+0
+w21
+−(w12 + w32)
+w23
+0
+w32
+−w23
+�
+� .
+(21)
+In this case, let S = {3}, the singleton set of state 3. When
+Y(t) = 1 the inference problem is trivial since the system
+takes state 3 with probability one. Thus, we emphasize the
+intervals when Y(t) = 0.
+First, consider the forward message given by the following
+system of differential equations
+dα
+dt =
+�−w21
+w12
+w21
+−(w12 + w32)
+�
+α .
+(22)
+Note that the matrix defining the system of equations corre-
+sponds to the upper left block of W. To simplify notation, let
+w21 = w12 = a and w32 = b. Then, the submatrix reduces to
+the following form:
+dα
+dt =
+�−a
+a
+a
+−a − b
+�
+α .
+(23)
+
+W
+13
+W
+W
+21
+32
+2
+W
+32
+W
+W
+12
+23
+W
+21
+2CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+6
+This is a linear system of differential equations that can
+be solved exactly. The sub-matrix is real-symmetric, so is
+diagonalizable. Therefore, the solution will be of the form:
+α(t) = Aeλ1tv1 + Beλ2tv2.
+(24)
+where λi is an eigenvalue of the rate submatrix and vi is the
+corresponding eigenvector. The eigenvalues and vectors are:
+λ1/2 = a(−1 ±
+�
+1 + γ2) − γ,
+(25)
+v1/2 =
+�
+γ ±
+�
+1 + γ2
+1
+�
+,
+(26)
+where γ = b/2a. Constants A and B are found through
+the initial condition given below. In the previous section, the
+initial forward message was set to the equilibrium distribution.
+However, the chain structure leaves no ambiguity in the state
+occupied when the observation changes from 1 to 0. So, the
+initial forward message is the delta distribution on state 2:
+α(0) =
+�0
+1
+�
+.
+(27)
+After some time T, the system re-enters the visible state,
+namely, state 3 again. By the same reasoning, we also have an
+unambiguous boundary condition for the backward message:
+β(T) =
+�0
+1
+�
+.
+(28)
+Also, by the symmetry in the rates, the backward message
+evolves according to the same equations as the forward mes-
+sage, but backward in time. Thus, β(t) = α(T − t),
+β(t) = Aeλ1(T −t)v1 + Beλ2(T −t)v2
+(29)
+where λi’s, vi’s, constants A and B all remain the same due
+to symmetry.
+The
+conditional
+probability
+is
+proportional
+to
+the
+component-wise product ρ = α ⊙ β, which yields:
+ρ(t) = A
+�
+eλ1(T −t)+λ2t + eλ1t+λ2(T −t)�
++ B,
+(30)
+where
+A = AB(v1 ⊙ v2) =
+�−AB
+AB�⊺,
+(31)
+B = A2eλ1T (v1 ⊙ v1) + B2eλ2T (v2 ⊙ v2).
+(32)
+To recover the conditional probability, we must normalize such
+that the component sum evaluates to 1. In this case, since the
+submatrix in Equation 22 is symmetric, the eigenvectors of the
+submatrix are orthogonal to each other, which means the sum
+of the components of v1⊙v2, or the inner-product between v1
+and v2, evaluates to 0. Then, the component-wise sum of A is
+zero, so the normalizing constant equals the component-wise
+sum of B.
+Write p(t) to represent the conditional probability. We write
+Z = �
+k B(k) to represent the normalizing constant. The final
+equation describing the conditional probability can thus be
+rewritten in the following form:
+p(t) = 1
+Z
+�
+Ae(λ1−λ2)teλ2T + Ae(λ2−λ1)teλ1T + B
+�
+. (33)
+The components of p(t) may also be expressed as the solutions
+of a second-order ordinary differential equation, cf. (51)-(52).
+B. Irreversible 3-State Loop
+Light-gated Channelrhodopsin-2 (ChR2) receptors can be
+modeled with a 3-state chain where each vertex has out-degree
+1 and forms a cycle, as depicted in Figure 2, right panel [28],
+[29]. Let state 1 be open, and states 2 and 3 be closed. That
+is, S = {1}. The open/closed status of the channel is observed
+through voltage recordings: high conductance indicates that the
+channel is in the open state, and low conductance implies the
+closed state. The rate matrix can be written as the following:
+W =
+�
+�
+−w21
+0
+w13
+w21
+−w32
+0
+0
+w32
+−w13
+�
+�
+(34)
+where wji is the transition rate from state i to state j.
+Conditioning on the channel being closed, the message-passing
+algorithm takes the lower right 2 × 2 block as the transition
+matrix.
+When transitioning from S to a state not in S, the system
+must enter state 2 first and exit through state 3. Thus the
+boundary conditions for the forward and backward message
+are
+α(0) =
+�1
+0
+�
+,
+β(T) =
+�0
+1
+�
+(35)
+where the first component corresponds to state 2 and second
+state 3. Suppose w32 ̸= w13 and let γ =
+w32
+w13−w32 . Then, the
+solution of the message-passing differential equations with the
+initial condition enforced satisfies:
+α(t) = e−w32t
+�1
+γ
+�
+− γe−w31t
+�0
+1
+�
+,
+(36)
+β(t) = e−w32(T −t)
+�
+1
+0
+�
++ 1
+γ e−w13(T −t)
+�
+−γ
+1
+�
+.
+(37)
+To get the conditional probability, first take the component-
+wise product between the forward and backward message at
+time t.
+ρ(t) = e−w32t−w13(T −t)
+�
+−1
+1
+�
++
+� e−w32T
+−e−w13T
+�
+(38)
+As in the previous example, the normalizing constant is
+invariant in time. In this case, Z = e−w32T − e−w13T . Thus,
+the time evolution of the conditional probability can be written
+p(t) = ρ(t)/Z.
+(39)
+Next, we consider a case in which the submatrix is not diag-
+onalizable. We set w = w32 = w13, so that the corresponding
+submatrix:
+U =
+�−w
+0
+w
+−w
+�
+(40)
+admits the following Jordan normal form:
+U =
+�0
+w
+1
+0
+� �−w
+1
+0
+−w
+� � 0
+1
+1
+w
+0
+�
+.
+(41)
+Using the same initial condition as Equation (35) to solve the
+system of differential equations yields the following forward
+message.
+α(t) =
+� e−wt
+te−wt
+�
+(42)
+
+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+7
+We solve for the backward message by undergoing a similar
+procedure and yield
+β(t) =
+�(T − t)e−w(T −t)
+e−w(T −t)
+�
+.
+(43)
+Upon taking the component-wise product, we observe that the
+evolution of conditional probability is independent of time:
+ρ(t) = e−wT
+�
+T − t
+t
+�
+,
+Z =
+1
+Te−wT .
+(44)
+Since both the time entering and exiting the hidden states are
+fixed, we can see that the probability flows linearly from one
+state to the other at equal rates.
+Time-invariant normalization is a fundamental property of
+the sum-product algorithm [26]. In Appendix B we give an
+elementary demonstration of this property for continuous-time
+systems with time-homogeneous transition rates, as illustrated
+by the two cases presented above.
+C. Generalization
+We show that under certain circumstances, the conditional
+probability follows a second-order nonhomogeneous linear
+ordinary differential equation.
+Corollary 1. For 3-state systems where the truncated subma-
+trix U has distinct eigenvalues, the conditional probability can
+be written as the solution of a nonhomogeneous second-order
+linear differential equation with constant coefficients.
+Proof. Let (λi, vi) be eigenpairs for U and let (λi, wi) be
+eigenpairs for U ⊺. Then the forward and backward messages
+are given by the expressions:
+α(t) = Aeλ1tv1 + Beλ2tv2,
+(45)
+β(t) = Ceλ1(T −t)w1 + Deλ2(T −t)w2.
+(46)
+Taking the component-wise product yields the form:
+ρ(t) = A
+�
+eλ1t+λ2(T −t)�
++ B
+�
+eλ1(T −t)+λ2t�
++ C,
+(47)
+= A
+�
+e(λ1−λ2)teλ2T �
++ B
+�
+e(λ2−λ1)teλ1T �
++ C,
+(48)
+where
+A = AD(v1 ⊙ w2),
+B = BC(v2 ⊙ w1),
+(49)
+C = ACeλ1T (v1 ⊙ w1) + BDeλ2T (v2 ⊙ w2).
+(50)
+Since the left and right eigenvectors corresponding to dif-
+ferent eigenvalues are orthogonal, all time-dependent terms
+cancel when normalizing. Thus the normalization constant
+Z = �
+k C(k). Upon differentiating p(t) = ρ(t)/Z twice with
+respect to time, we arrive at the second-order linear differential
+equation that describes the time evolution of the condition
+probability:
+d2p
+dt2 = (λ1 − λ2)2
+Z
+�
+Ae(λ1−λ2)teλ2T + Be(λ2−λ1)teλ1T �
+(51)
+= (λ1 − λ2)2
+Z
+(p − C).
+(52)
+This second order equation comes equipped with two bound-
+ary conditions set by fixing α(0) and β(T).
+Fig. 3: State diagram for CFTR. When the protein enters one of
+the red states, the ion channels will open and conduct a current.
+On the other hand, when CFTR is in one of the blue states,
+then the ion channels are closed and conducts no current. For
+this system S = {4, 5}.
+Corollary 2. For a system of arbitrary dimension, for which
+the truncated submatrix U has exactly two distinct eigenvalues
+and is diagonalizable, the conditional probability can be
+written as a second-order linear differential equation of the
+same form as Equation 52.
+Proof. Let λ1 and λ2 be the two distinct eigenvalues. Let vi be
+the sum of all right eigenvectors corresponding to eigenvalue
+λi, and let wi be the sum of all left eigenvectors corresponding
+to eigenvalue value λi. Then by diagonalizability, the proof for
+this corollary follows the proof of Corollary 1.
+This result extends Corollary 1 to higher dimensional sys-
+tems, under special conditions. These two corollaries match
+intuition: we are concerned with the state occupancy prob-
+ability conditioned on both the entrance and exit times of
+an observable state, so the probability flow obeys a second-
+order differential equation, which also requires specifying two
+boundary conditions. We note, however, that if the submatrix
+has three or more distinct eigenvalues, we do not expect a
+similar result to hold. Moreover, generic matrices typically
+have as many distinct eigenvalues as their dimension, so we
+do not expect this scenario to occur outside of special cases
+when the rate submatrix is highly regular. In addition, it would
+be difficult to check if a given rate submatrix satisfies these
+conditions through numerical solvers, as equality of repeated
+eigenvalues might be obscured by floating point arithmetic.
+Therefore, as much as the second-order differential equations
+interpretation is intuitively appealing, it is likely not a generic
+phenomenon that we should expect for arbitrary rate matrices.
+V. BIOLOGICAL EXAMPLE: CFTR
+Here, we show that our algorithm applies to a higher
+dimensional system, namely the 7-state model of the cystic
+fibrosis transmembrane conductance regulator (CFTR) protein.
+CF is a common life-threatening genetic disorder. CFTR is
+an important protein that regulates the opening and closing
+of ion channels. Loss of CFTR function causes pancreatic
+insufficiency as well as airway infection due to excessive
+mucus, which in turn can cause a variety of complications
+such as impaired innate immunity and respiratory failure [30].
+Mathematically, the behavior of CFTR can be captured
+with a 7-state hidden Markov process. Figure 3 illustrates
+
+2
+3
+4
+5
+7
+6CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+8
+Fig. 4: The inference result of the continuous-time algorithm (top) and the discrete-time algorithm (middle) for the true state
+occupancy simulated from the Gillespie algorithm (bottom). Sampling time step ∆t = 10−4.
+the state diagram [12]. The ion channel opens (conducts an
+ionic current) when CFTR is in state four or five (marked in
+red). When in the nonconducting states (marked in blue) the
+ion channel is closed. Transmembrane conductance recordings
+report whether CFTR is in a conducting or a nonconducting
+state, but do not directly indicate which of the possible states
+is occupied. Using the numbering shown in Figure 3, the rate
+matrix is defined as follows:
+W =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−9.9
+5.0
+0
+0
+0
+0
+1.7
+9.9
+−12.7
+5.8
+0
+0
+0
+0
+0
+7.7
+−10.7
+10.0
+0
+0
+0
+0
+0
+4.9
+−17.1
+0
+0
+0
+0
+0
+0
+7.1
+−3.0
+7.0
+0
+0
+0
+0
+0
+3.0
+−13.0
+12.8
+0
+0
+0
+0
+0
+6.0
+−14.5
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(53)
+We used Gillespie’s exact stochastic simulation algorithm
+[31], [32], [33] to generate sample traces of the ion channel
+states and recordings. The simulation produces discrete state,
+continuous time trajectories. We introduced a finite sampling
+time step to discretize the simulated data along the time axis,
+consistent with data obtained through experimental recordings.
+Then, we place the generated data into discretized time bins
+where the size of each bin (one can think of this as the sam-
+pling time step) is a parameter that can be altered. We solved
+the differential equations giving the forward and backward
+messages exactly via function handles in Matlab. Figure 4
+compares the traces produced by the classical discrete-time
+algorithm, our continuous-time algorithm, and the true states,
+and shows excellent agreement among the respective curves.
+With a sufficiently small time step (∆t = 10−4), the contin-
+uous and discrete-time algorithms show no visible discrepancy,
+as expected. Figure 5 shows that the maximum discrepancy
+Fig. 5: First-order convergence of the message passing al-
+gorithms as a function of sampling time step. The mean
+maximum difference was plotted along with one standard
+deviation about the mean as the error bar for an ensemble
+of forty trajectories.
+(ℓ∞ norm) between the conditional probabilities generated by
+the continuous time and discrete time algorithms decreases
+linearly with the sampling time step ∆t. The figure shows
+the results from an ensemble of forty independent repeated
+trials for each sample size. Again, we emphasize that while
+the discrete time algorithm requires iterating over all samples,
+which scales O(∆t−1), the continuous time algorithm outputs
+a description of the conditional probability — that is, a
+function that returns a value at an exact time point queried
+— in O(1) time, scaling only with the number of transitions.
+
+inferred probability of states: continuous
+1234567
+0.9
+0.8
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+×104
+0.7
+inferred probability of states: discrete
+1234567
+0.6
+0.5
+0.4
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+×104
+0.3
+true states
+0.2 
+134567
+0.1
+0.5
+1.5
+2
+2.5
+3.5
+4.5
+0
+3
+4
+5
+time
+×10410~2
+maximum difference 
+10~3
+10~5
+10~6
+10-7
+10-7
+10-6
+10-5
+10-4
+10-3
+size of sampling intervalCHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+9
+VI. DISCUSSION & CONCLUSIONS
+This paper presents an algorithm for continuous-time infer-
+ence on partially observable Markov processes with discrete
+state spaces. We show that the well-known sum-product al-
+gorithm can be extended to the continuous-time domain via
+two sets of differential equations and pointwise normalization.
+In the continuous time setting, we were able to solve the
+trajectory of conditional probabilities exactly given a finite
+time-series. Moveover, we find that the dynamics of the state
+occupancy probabilities can be reduced to second-order dif-
+ferential equations under special circumstances. These results
+are valuable not only for their mathematical interest, but
+also because they have the potential to reduce the inference
+problem to solving systems of linear differential equations,
+with a potentially significant reduction in computational com-
+plexity for long time series. Numerically, the continuous-time
+algorithm is consistent with the discrete-time algorithm in the
+limit of small time step, but executes in approximately constant
+time rather than linearly in the number of time steps.
+Our formalism extends naturally to non-binary observations.
+Briefly, let S1, S2, . . . , Sm be a partition over the sample space
+Ω, and suppose the observable process Y is given as:
+Y(t) = i if S(t) ∈ Si.
+(54)
+For a sojourn with observation i, we can apply the forward
+and backward message-passing scheme as introduced for the
+binary case, viewing the observation as either in Si or Ω\Si. At
+the boundaries, the same technique (as in the proof of Theorem
+1) can be used for finding the update rule by noting the
+possible transitions from Si to Sj, for all j ̸= i. This approach
+can finally be extended to an arbitrary collection of subsets
+of Ω where the elements are not necessarily disjoint. One
+may accomplish this extension by expanding all unions and
+intersections as disjoint sets possibly with the same observable.
+We defer detailed investigations in this direction to future
+work.
+Extending the continuous-time message passing to the case
+with inhomogeneous transition rates remains an open problem.
+We expect a derivation similar to the proof of the main theorem
+would be applicable, possibly with smoothness constraints
+on the transition rates. While the normalization constant will
+remain invariant in time, we do not expect a result such as
+Corollary 1 to hold beyond constant transition rates.
+Finally, we note the relationship between conditional prob-
+ability and a second-order differential equation is intuitively
+satisfying: the entry and exit times act as two boundary
+conditions that fix the endpoints of the evolution, whereas
+the unconstrained forward evolution equation, a first-order
+differential equation, requires only the starting condition. It
+is an interesting question for future work to investigate under
+which assumptions a similar result as Corollary 1 would hold
+for systems with more than two distinct eigenvalues.
+APPENDIX A
+NOTATION AND PRELIMINARIES
+For completeness, we define a continuous-time discrete-
+space Markov process with exponential waiting times.
+Definition 1 (Markov Process). Let {S(t); t ∈ [0, T]} be
+a discrete-space, continuous-time stochastic process where
+for each t, S(t) is a random variable with state space
+Ω = {1, 2, . . . , N}, N < ∞. Then, the process S(t) has the
+Markov Property if for any s1, s2, . . . , sm ∈ Ω, 0 < τ1 <
+τ2 < · · · < τm−1 < t,
+Pr[S(t) = sm|S(τ1) = s1, S(τ2) = s2, . . . , S(τm−1) = sm−1]
+= Pr[S(t) = sm|S(τm−1) = sm−1].
+(55)
+Furthermore, the process has exponential waiting times if for
+any i, j ∈ Ω, i ̸= j, there is a constant rate 0 ≤ wji < ∞
+such that
+�
+Pr[S(t + ∆t) = i|S(t) = i] = 1 − �
+k̸=i wki∆t + o(∆t)
+Pr[S(t + ∆t) = j|S(t) = i] = wji∆t + o(∆t)
+Pr[S(t + ∆t) = j, S(t + 2∆t) = k|S(t) = i] = o(∆t),
+∀k
+for sufficiently small ∆t.
+The following table lists notation used in the paper.
+Symbol
+Meaning
+St
+discrete-time process with (integer) time index t
+S(t)
+continuous-time process with time index t
+Ω
+sample space of a process at fixed time
+S
+states that give observable “1”
+Y(t)
+the observed process (indicating S(t) ∈ S)
+P
+transition matrix of a Markov chain
+W
+transition rate matrix of Markov process
+αt; α(t)
+forward message
+βt; β(t)
+backward message
+χt; χ(t)
+observation message
+ρt; ρ(t)
+unnormalized conditional state-occupancy probability
+Z(t)
+normalizing constant for ρ(t)
+pt; p(t)
+conditional state-occupancy probability
+∆t
+sampling time step
+APPENDIX B
+INVARIANT NORMALIZATION IN CONTINUOUS TIME
+Time-invariant invariant normalization is a fundamental
+property of the sum-product algorithm [26]. In the continuous
+time case, we observed that this property can be easily con-
+firmed when the submatrices are diagonalizable, because the
+left and right eigenvectors corresponding to different eigenval-
+ues are orthogonal. When the submatrix has nontrivial Jordan
+blocks, the time-dependent term cancels in less obvious ways.
+Here we show that time-independent normalization holds in
+general, using only elementary calculus without invoking the
+machinery of factor-graphs.
+Recall
+that
+the
+normalizing
+constant
+is
+Z(t)
+=
+�
+i α(i)(t)β(i)(t). Using Equation 2 and 3, we can obtain the
+
+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+10
+following series of expressions.
+dZ(t)
+dt
+= d
+dt
+�
+i∈S
+α(i)(t)β(i)(t)
+(56)
+=
+�
+i∈S
+dα(i)(t)
+dt
+β(i)(t) + α(i)(t)dβ(i)(t)
+dt
+(57)
+=
+�
+i∈S
+�
+� �
+k∈S\{i}
+wkiα(k) −
+�
+l̸=i
+α(i)
+�
+� β(i) . . .
++ α(i)
+�
+�−
+�
+k∈S\{i}
+wikβ(k) +
+�
+l̸=i
+β(i)
+�
+�
+(58)
+=
+�
+i∈S
+�
+k∈S\{i}
+wkiα(k)β(i) − wikα(i)β(k)
+(59)
+= 0
+(60)
+Since the rate of change of Z(t) is zero, the normalizing
+constant is invariant of time.
+REFERENCES
+[1] G. Grisettiyz, C. Stachniss, and W. Burgard, “Improving grid-based
+slam with Rao-Blackwellized particle filters by adaptive proposals and
+selective resampling,” in Proceedings of the 2005 IEEE International
+Conference on Robotics and Automation.
+IEEE, 2005, pp. 2432–2437.
+[2] L. E. Baum and J. A. Eagon, “An inequality with applications to
+statistical estimation for probabilistic functions of Markov processes and
+to a model for ecology,” Bulletin of the American Mathematical Society,
+vol. 73, no. 3, pp. 360–363, 1967.
+[3] D. F. Anderson, B. Ermentrout, and P. J. Thomas, “Stochastic repre-
+sentations of ion channel kinetics and exact stochastic simulation of
+neuronal dynamics,” Journal of Computational Neuroscience, vol. 38,
+no. 1, pp. 67–82, 2015.
+[4] D. M. Blei, A. Y. Ng, and M. I. Jordan, “Latent Dirichlet allocation,”
+Journal of Machine Learning research, vol. 3, pp. 993–1022, 2003.
+[5] A. L. Hodgkin and A. F. Huxley, “A quantitative description of mem-
+brane current and its application to conduction and excitation in nerve,”
+The Journal of Physiology, vol. 117, no. 4, p. 500, 1952.
+[6] E. Skaugen and L. Walløe, “Firing behaviour in a stochastic nerve
+membrane model based upon the Hodgkin—Huxley equations,” Acta
+Physiologica Scandinavica, vol. 107, no. 4, pp. 343–363, 1979.
+[7] B. Hille, “Ionic channels in excitable membranes. current problems and
+biophysical approaches,” Biophysical Journal, vol. 22, no. 2, pp. 283–
+294, 1978.
+[8] D. Colquhoun and A. G. Hawkes, “Relaxation and fluctuations of mem-
+brane currents that flow through drug-operated channels,” Proceedings
+of the Royal Society of London. Series B. Biological Sciences, vol. 199,
+no. 1135, pp. 231–262, 1977.
+[9] D. Colquhoun and A. Hawkes, “On the stochastic properties of single
+ion channels,” Proceedings of the Royal Society of London. Series B.
+Biological Sciences, vol. 211, no. 1183, pp. 205–235, 1981.
+[10] D. Colquhoun and A. G. Hawkes, “On the stochastic properties of bursts
+of single ion channel openings and of clusters of bursts,” Philosophical
+Transactions of the Royal Society of London. Series B, Biological
+Sciences, pp. 1–59, 1982.
+[11] F. Qin, A. Auerbach, and F. Sachs, “Maximum likelihood estimation
+of aggregated Markov processes,” Proceedings of the Royal Society of
+London. Series B: Biological Sciences, vol. 264, no. 1380, pp. 375–383,
+1997.
+[12] M. D. Fuller, Z.-R. Zhang, G. Cui, and N. A. McCarty, “The block
+of CFTR by scorpion venom is state-dependent,” Biophysical Journal,
+vol. 89, no. 6, pp. 3960–3975, 2005.
+[13] H.-A. Loeliger, J. Dauwels, J. Hu, S. Korl, L. Ping, and F. R. Kschis-
+chang, “The factor graph approach to model-based signal processing,”
+Proceedings of the IEEE, vol. 95, no. 6, pp. 1295–1322, 2007.
+[14] D. Koller and N. Friedman, Particle-based Approximate Inference. MIT
+Press, 2012.
+[15] Z. Fang, A. Gupta, and M. Khammash, “Stochastic filtering for mul-
+tiscale stochastic reaction networks based on hybrid approximations,”
+Journal of Computational Physics, vol. 467, p. 111441, 2022.
+[16] O. Capp´e, C. P. Robert, and T. Ryd´en, “Reversible jump, birth-and-
+death and more general continuous time Markov chain Monte Carlo
+samplers,” Journal of the Royal Statistical Society: Series B (Statistical
+Methodology), vol. 65, no. 3, pp. 679–700, 2003.
+[17] P. M. Djuric and J.-H. Chun, “Estimation of nonstationary hidden
+Markov models by MCMC sampling,” in 1999 IEEE International
+Conference on Acoustics, Speech, and Signal Processing. Proceedings.
+ICASSP99 (Cat. No. 99CH36258), vol. 3.
+IEEE, 1999, pp. 1737–1740.
+[18] R. Rosales, J. A. Stark, W. J. Fitzgerald, and S. B. Hladky, “Bayesian
+restoration of ion channel records using hidden Markov models,” Bio-
+physical Journal, vol. 80, no. 3, pp. 1088–1103, 2001.
+[19] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and
+the sum-product algorithm,” IEEE Transactions on Information Theory,
+vol. 47, no. 2, pp. 498–519, 2001.
+[20] D. Jin, F. Yin, C. Fritsche, F. Gustafsson, and A. M. Zoubir, “Bayesian
+cooperative localization using received signal strength with unknown
+path loss exponent: Message passing approaches,” IEEE Transactions
+on Signal Processing, vol. 68, pp. 1120–1135, 2020.
+[21] S. Som and P. Schniter, “Compressive imaging using approximate
+message passing and a Markov-tree prior,” IEEE Transactions on Signal
+Processing, vol. 60, no. 7, pp. 3439–3448, 2012.
+[22] A. Abrardo, M. Barni, K. Kallas, and B. Tondi, “A message passing
+approach for decision fusion of hidden-Markov observations in the
+presence of synchronized attacks,” in Int. Conf. Advances in Multimedia
+(MMEDIA), Special Track on Models and Algorithms for Spatially and
+Temporally Correlated Data (STCD), Venice, Italy, 2017.
+[23] L. Bolliger, Digital Estimation of Continuous-Time Signals Using Factor
+Graphs, Hartung-Gorre Verlag, Konstanz, Series in Signal and Informa-
+tion Processing, vol. 22, 2012.
+[24] L. Bolliger, H.-A. Loeliger, and C. Vogel, “LMMSE estimation and
+interpolation of continuous-time signals from discrete-time samples
+using factor graphs,” arXiv preprint arXiv:1301.4793, 2013.
+[25] L. Bruderer and H.-A. Loeliger, “Estimation of sensor input signals that
+are neither bandlimited nor sparse,” in 2014 Information Theory and
+Applications Workshop (ITA).
+IEEE, 2014, pp. 1–5.
+[26] G. D. Forney Jr and P. O. Vontobel, “Partition functions of normal factor
+graphs,” arXiv preprint arXiv:1102.0316, 2011.
+[27] R. G. Gallager, Stochastic Processes: Theory for Applications.
+Cam-
+bridge University Press, 2013.
+[28] A. W. Eckford and P. J. Thomas, “The channel capacity of Channel-
+rhodopsin and other intensity-driven signal transduction receptors,” IEEE
+Transactions on Molecular, Biological and Multi-Scale Communications,
+vol. 4, no. 1, pp. 27–38, 2018.
+[29] G. Nagel, T. Szellas, W. Huhn, S. Kateriya, N. Adeishvili, P. Berthold,
+D. Ollig, P. Hegemann, and E. Bamberg, “Channelrhodopsin-2, a directly
+light-gated cation-selective membrane channel,” Proceedings of the
+National Academy of Sciences, vol. 100, no. 24, pp. 13 940–13 945,
+2003.
+[30] D. Goetz and C. L. Ren, “Review of cystic fibrosis,” Pediatric Annals,
+vol. 48, no. 4, pp. e154–e161, 2019.
+[31] D. T. Gillespie, “Exact stochastic simulation of coupled chemical
+reactions,” The Journal of Physical Chemistry, vol. 81, no. 25, pp. 2340–
+2361, 1977.
+[32] ——, “Stochastic simulation of chemical kinetics,” Annual Review of
+Physical Chemistry, vol. 58, pp. 35–55, 2007.
+[33] D. J. Wilkinson, Stochastic Modelling for Systems Biology.
+Chapman
+and Hall/CRC, 2018.
+

SNE0T4oBgHgl3EQf1wKp/content/tmp_files/2301.02704v1.pdf.txt

@@ -0,0 +1,704 @@
+Revista Mexicana de Física 3 040909 (2022) 1–4
+September 2022
+Rivet and the analysis preservation in heavy-ion collisions experiments
+Antonio Carlos Oliveira da Silva (for the ALICE Collaboration)
+University of Tennessee, Knoxville, 1408 Circle Drive, Knoxville TN 37996-1200
+Received 3 July 2022; accepted 15 September 2022
+The comparison of experimental data and theoretical predictions is important for our understanding of the mechanisms for interactions and
+particle production in hadron collisions, both at the Large Hadron Collider and at the Relativistic Heavy-Ion Collider experiments. Several
+tools were ideated to help with that. Rivet (Robust Independent Validation of Experiment and Theory) is a framework that facilitates the
+comparison between measurements from high-energy physics experiments and Monte Carlo event generators able to produce outputs using
+the HepMC package. Rivet contains a repository with analysis algorithms developed by experiments, providing analysis documentation and
+preservation.
+The recent developments for the implementation of centrality and multiplicity classes in Rivet are presented in this contribution.
+Keywords:
+1
+Introduction
+Currently, the data and analysis preservation in high-energy
+physics experiments is becoming a common concern. Previ-
+ous experiments and collaborations are losing the power of
+reproducing their measurements since the data are not prop-
+erly kept in a accessible way.
+The old code, which con-
+tains crucial and detailed information like detector accep-
+tance, particle and event selections, and corrections, is no
+longer maintained and it is very difficult, if possible, to be
+run again. Comparisons of previous measurements with new
+models is, therefore, very challenging.
+Robust Independent Validation of Experiment and The-
+ory (Rivet) [1] is a framework that aims to facilitate the com-
+parison between data and Monte Carlo (MC) event genera-
+tors.
+2
+Rivet framework
+Rivet analyses are written in C++ and it currently contains
+more than 1000 analyses from several high-energy physics
+collaborations. The data, when available, are downloaded di-
+rectly from HepData [2]. Any model that is incorporated in
+an event generator able to produce output that complies with
+HepMC framework [3] can be used by Rivet for the compar-
+ison with data. The integration of Rivet with HepMC and
+HepData is pictured in the scheme presented in Fig. 1.
+The references for the event generators in Fig.
+1 can
+be found in [4–12]. Not all of them provide output using
+HepMC standards.
+In principle, an article presenting a measurement should
+present enough information to make the measurement able
+to be reproduced by another experiment or theoretician inter-
+ested in comparing the data with a model. However, some
+subtle details about detector acceptances, particle selections,
+trigger conditions, etc, could be missing or not clearly de-
+scribed. This can be the case even in internal notes in large
+collaborations.
+FIGURE 1. Schematic diagram showing how Rivet is integrated
+with HepMC and HepData.
+Rivet aims at preserving all the analysis details. Further-
+more, it should reproduce the methods used in a measurement
+as close as possible. One of the pillars of the Rivet philoso-
+phy is that we should treat Monte Carlo simulations in the
+same way as data.
+In order to assure maximum fidelity to the original ac-
+ceptances, selections and methods used in a measurement,
+whenever possible, the rivet analysis should be implemented
+by the collaboration that published the article containing the
+measurement. Currently, ALICE has an official internal pro-
+cedure for Rivet analysis approvals.
+3
+Centrality and multiplicity determination
+Measurements in heavy-ion collisions are commonly differ-
+ential in centrality intervals to study different physical phe-
+nomena. Therefore, centrality determination is a crucial fea-
+ture Rivet has to provide in order to reproduce measurements
+in heavy-ion collisions.
+Centrality determination in ALICE is commonly pro-
+vided by the V0 detector [13], which consists of two ar-
+rays, V0-A and V0-C covering the pseudorapidity ranges 2.8
+< η < 5.1 and 3.7 < η < 1.7 respectively.
+Figure 2 presents the distribution of the total energy de-
+arXiv:2301.02704v1  [physics.data-an]  6 Jan 2023
+
+AMPT
+FXR
+HIJING
+JETSCAPE
+PYTHIA
+Monte Carlo Mode
+JEWEL
+EPOS
+PHSI
+smash
+HepMC
+HEpData
+Rivet2
+ANTONIO CARLOS OLIVEIRA DA SILVA
+posited in the V0 scintillators (amplitude). The most cen-
+tral collisions are associated with those event with highest
+V0 amplitude. The details of the centrality determination are
+described in [14].
+FIGURE 2. Distribution of the total amplitude in the V0 scintilla-
+tors in black points. The data are fitted using a Negative Binomial
+Distribution (NBD) using parameters from the Glauber model.
+b
+b
+b
+b
+b b
+b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b
+b b b b b b b b b b
+b b b b
+b
+b b b b b b b b b b b b b b
+b b b b b b
+b b b b b b
+b b
+b b b b
+b b
+b b b b b b b b b b b b b b b b b b b b b b b b
+b b b b b
+b b b b b b b b b b b b b b b b b b b b b b b b b b b b
+b
+b b b b
+b
+b
+b
+b b b b
+b b b b b b b b
+b b
+b
+b b b b
+b b
+b b b b b b b b b b
+b
+b b b b b b b
+b b b b b b b b b b b b b b b
+b
+b b b
+b
+b b b b b
+b b b
+b b b
+b b
+b
+b b b
+b b b
+b b
+b
+b
+b b b b b b b b b b b
+b
+b b
+b b b b b
+b b
+b b
+b b
+b
+b
+b b
+b b b
+b
+b b b b b
+b
+b
+b
+b b
+b b
+b b
+b
+b
+b
+b b
+b
+b
+b b
+b
+b b
+b
+b
+b
+b
+b b b
+b
+b b b b
+b
+Data
+calibration ALICE PbPb2760GeV
+0
+5000
+1.0·104
+1.5·104
+2.0·104
+2.5·104
+3.0·104
+3.5·104
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+1
+VZERO amplitude (a.u.) or ch. particle multiplicity
+Events
+FIGURE 3. V0 amplitude in arbitrary units (black markers) mea-
+sured by ALICE and the charged particle multiplicity in the V0 ac-
+ceptance calculated with Rivet from Pb–Pb collisions at 2.76 TeV
+events simulated with PYTHIA 8 Angantyr (red line).
+The centrality determination in Rivet uses the multiplic-
+ity of charged particles in the acceptance of the V0. Since
+the multiplicity of particles in heavy-ion collision events in
+Monte Carlo event generators is model dependent, it is neces-
+sary to create a calibration file that depends on collisions sys-
+tem, energy and event generator. Figure 3 shows the V0 am-
+plitude distribution measured by ALICE and the multiplicity
+of charged particles in Pb–Pb collisions at √sNN = 2.76 TeV
+generated with PYTHIA 8 Angantyr [6].
+Rivet divides the charged particle multiplicity presented
+in figure 3 (red line) in centrality percentiles.
+So the
+most central events are associated to the highest multiplic-
+ity events. When running the Rivet analysis that requires
+centrality determination, this calibration has to be provided.
+A similar strategy is used for multiplicity determination in
+pp and p–Pb collisions. Currently, ALICE is developing the
+possibility to characterize the event using the self-normalized
+charged particle multiplicity distribution. This development
+is presented in sections 4 and 5.
+4
+Self-normalized multiplicity
+Forward-rapidity multiplicity classes can be defined in
+ALICE using the V0 detector. Figure 4 shows the distribution
+of the V0M amplitude, which is proportional to the number
+of charged particles passing through the V0A and V0C de-
+tectors, scaled by its average value ⟨V0M⟩ [15].
+FIGURE 4. Distribution of the V0M amplitude scaled by its aver-
+age value ⟨V0M⟩ used to determined forward-rapidity multiplicity
+classes in pp collisions at √s = 13 TeV.
+The Silicon Pixel Detector (SPD) [16, 17] is the closest
+detector to the interaction point in ALICE. The SPD provides
+mid-rapidity multiplicity classes determination using the re-
+constructed tracklets, which are track segments that connects
+hits in the two SPD layers pointing to the primary vertex. The
+self-normalized estimator is obtained with the distribution of
+SPD tracklets NSPD tracklets in -2 < η < 2 scaled by the av-
+erage of its value ⟨NSPD tracklets⟩. The self-normalized SPD
+tracklets distribution is presented in Fig. 5.
+Supl. Rev. Mex. Fis. 3 040909
+
+Yield (a.u.)
+10-3
+10-3
++
+Data
+Glauberfit
+80-90%
+70-80%
+60-70%
+10-4
+10-4
+500
+1000
+10-5
+50-60%
+40-50%
+0-40%
+20-30%
+10-20%
+10-6
+5-10%
+-5%
+8
+0
+4000
+8000
+12000
+16000
+20000
+VZERO Amplitude (a.u.)
+ALI-PUB-89941103
+ALICE
+Forward Multiplicity Classes
+Min. Bias data (%)
+102
+Normalised counts
+pp, Vs = 13 TeV
+0-1
+1-5
+5-10
+10-15
+10
+15-20
+20-30
+30-40
+40-50
+50-70
+70-100
+High-Mult data (%)
+10-
+0-0.01
+0.01-0.1
+10-3
+10
+10
+5
+0
+2
+4
+6
+8
+10
+VOM/VOM)
+ALI-PUB-499489A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF
+3
+FIGURE 5. Distribution of the number of SPD tracklets scaled by its
+average value ⟨NSPD tracklets⟩ used to determine midrapidity mul-
+tiplicity classes in pp collisions at √s = 13 TeV.
+The event characterization using self-normalized multi-
+plicity estimators in Rivet is under development in ALICE.
+Similar to what was discussed in the previous section, instead
+of using the V0M amplitude, Rivet uses the charged particle
+multiplicity in the V0 acceptance. In particular, the forward-
+rapidity self-normalized estimator uses the multiplicity of
+charged particles in the V0 acceptance scaled by its average
+value. Figure 6 shows the V0M/⟨V0M⟩ distribution calcu-
+lated with Rivet using pp collisions at 13 TeV generated with
+PYTHIA 8 Monash 2013 tune [4,18].
+Rivet pp13TeV
+0
+2
+4
+6
+8
+10
+10−5
+10−4
+10−3
+10−2
+10−1
+1
+V0M/⟨V0M⟩
+Normalized counts
+FIGURE 6. Self-normalized multiplicity distribution of charged par-
+ticles in the acceptance of the V0 detector in pp collisions at √s =
+13 TeV generated with PYTHIA 8 Monash 2013.
+Similarly to what is done for the V0M, the self-
+normalized multiplicity estimator at mid-rapidity uses the
+number of charged particles in the acceptance of the SPD.
+Figure 7 shows the distribution of the charged particles in the
+SPD acceptance scaled by its average value in pp collisions
+at 13 TeV generated with PYTHIA 8 Monash 2013.
+Rivet pp13TeV
+0
+2
+4
+6
+8
+10
+10−4
+10−3
+10−2
+10−1
+1
+NSPD tracklets
+SPD tracklets
+SPD tracklets/⟨NSPD tracklets
+SPD tracklets
+SPD tracklets⟩
+Normalized counts
+FIGURE 7. Self-normalized multiplicity distribution of charged par-
+ticles in the acceptance of the SPD detector in pp collisions at
+√s = 13 TeV generated with PYTHIA 8 Monash 2013.
+5
+Results using the self-normalized multiplic-
+ity estimators
+The self-normalized multiplicity estimators framework in
+Rivet is currently work in progress and being tested using ar-
+ticles published by ALICE that use such estimators. The first
+measurement used to test the V0M/⟨V0M⟩ estimator was the
+transverse momentum (pT) of jets in different multiplicity in-
+tervals in pp collisions at √s = 13 TeV [19].
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+Data
+Rivet [cent=GEN]
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+1/Nevent d2N/dpTd
+event d2N/dpTd
+event d2N/dpTdηηη (GeV/ccc)))−1
+−1
+−1
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0.5
+0.6
+0.7
+0.8
+0.91
+1.1
+1.2
+1.3
+1.4
+jet pT
+pT
+pT (GeV/ccc)))
+MC/Data
+FIGURE 8. Charged-particle jet transverse momentum distribution
+in pp collisions at √s = 5.02 TeV for the 0-1% multiplicity
+class corresponding to the self-normalized V0M-based multiplic-
+ity estimator. Jets were reconstructed using jet resolution parame-
+ter R = 0.2. Data (black markers) are compared with PYTHIA 8
+Monash 2013 (red line).
+Supl. Rev. Mex. Fis. 3 040909
+
+103
+ALICE
+Central Multiplicity Classes
+Min. Bias data (%)
+102
+pp, Vs = 13 TeV
+Normalised counts
+0-1
+1-5
+5-10
+10-15
+10
+15-20
+20-30
+30-40
+40-50
+50-70
+70-100
+10
+10
+3
+10
+10
+5
+0
+2
+4
+6
+8
+10
+KN
+SPD Tracklet4
+ANTONIO CARLOS OLIVEIRA DA SILVA
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+Data
+Rivet SPD pp5TeV [cent=GEN]
+0
+5
+10
+15
+20
+25
+dN/dη
+dN/dη
+dN/dη
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+0.5
+0.6
+0.7
+0.8
+0.91
+1.1
+1.2
+1.3
+1.4
+ηηη
+MC/Data
+FIGURE 9. Charged particle pseudorapidity distribution in pp colli-
+sions at √s = 5.02 TeV for the 0-1% multiplicity class correspond-
+ing to the self-normalized SPD-based multiplicity estimator. Data
+(black markers) are compared with PYTHIA 8 Monash 2013 (red
+line).
+The transverse momentum of jets reconstructed with
+FastJet anti-kT algorithm [20] and resolution parameter R =
+0.2 in the multiplicity class 0-1% in pp collisions at √s =
+13 TeV is presented in figure 8. The measurement was com-
+pared with PYTHIA 8 Monash 2013 using Rivet and the self-
+normalized V0M multiplicity framework. The agreement of
+the model to data is a positive indication that the framework
+can reproduce the multiplicity determination method used by
+ALICE. Other multiplicity classes presented a similar perfor-
+mance.
+The self-normalized estimator using the SPD was also
+tested using the measurements in [15].
+Figure 9 presents
+the charged particle pseudorapidity distribution in pp colli-
+sions at √s = 5.02 TeV. The ALICE data are compared
+with PYTHIA 8 Monash 2013 using Rivet and the SPD self-
+normalized framework. The results fairly reproduce the com-
+parisons to MC presented in the cited article.
+6
+Summary
+Rivet is a valuable tool for analysis preservation and compar-
+ison of data to Monte Carlo event generators. The develop-
+ment of additional tools to facilitate the implementation of
+Rivet analyses is an important contribution to the framework
+and can be of benefit both for the experiment and the theory
+side. The self-normalized multiplicity estimators are provid-
+ing consistent results between MC curves in Rivet and those
+provided by experiments. The final goal is to make this mul-
+tiplicity framework available soon in the Rivet official frame-
+work.
+1. A. Buckley, et al., Rivet user manual (2010).
+2. E. Maguire, L. Heinrich, and G. Watt, HEPData: a repository
+for high energy physics data, Journal of Physics: Conference
+Series 898 (2017) 102006,
+10.1088/1742-6596/898/
+10/102006
+3. M. Dobbs and J. B. Hansen, The HepMC C++ Monte Carlo
+Event Record for High Energy Physics,
+Tech. rep., CERN,
+Geneva (2000), URL https://cds.cern.ch/record/
+684090, Revised version number 1 submitted on 2001-02-27
+09:54:32.
+4. T. Sjöstrand, S. Mrenna, and P. Z. Skands, A Brief Introduc-
+tion to PYTHIA 8.1, Comput. Phys. Commun. 178 (2008) 852,
+10.1016/j.cpc.2008.01.036
+5. Z.-W. Lin, et al., Multiphase transport model for relativistic
+heavy ion collisions, Physical Review C 72 (2005), 10.1103/
+physrevc.72.064901
+6. C. Bierlich, et al., The Angantyr model for Heavy-Ion Col-
+lisions in PYTHIA8,
+JHEP 10 (2018) 134,
+10.1007/
+JHEP10(2018)134
+7. J. H. Putschke, et al., The JETSCAPE framework (2019), 10.
+48550/ARXIV.1903.07706,
+URL https://arxiv.
+org/abs/1903.07706.
+8. J. Weil, et al., Particle production and equilibrium properties
+within a new hadron transport approach for heavy-ion colli-
+sions, Physical Review C 94 (2016), 10.1103/physrevc.
+94.054905
+9. T. Pierog, et al.,
+EPOS LHC: Test of collective hadroniza-
+tion with data measured at the CERN Large Hadron Collider,
+Phys. Rev. C 92 (2015) 034906, 10.1103/PhysRevC.92.
+034906
+10. K. Zapp, et al., A Monte Carlo model for ‘jet quenching’, The
+European Physical Journal C 60 (2009), 10.1140/epjc/
+s10052-009-0941-2
+11. X.-N. Wang and M. Gyulassy, hijing: A Monte Carlo model
+for multiple jet production in pp, pA, and AA collisions, Phys.
+Rev. D 44 (1991) 3501, 10.1103/PhysRevD.44.3501
+12. W. Cassing and E. Bratkovskaya,
+Parton–hadron–string dy-
+namics: An off-shell transport approach for relativistic ener-
+gies,
+Nuclear Physics A 831 (2009) 215,
+10.1016/j.
+nuclphysa.2009.09.007
+13. T. A. Collaboration, Performance of the ALICE VZERO sys-
+tem, Journal of Instrumentation 8 (2013) P10016, 10.1088/
+1748-0221/8/10/p10016
+14. B. Abelev, J. Adam, and D. Adamov,
+Centrality determina-
+tion of Pb-Pb collisions at √sNN = 2.76 TeV with ALICE 88
+(2013), 10.1103/physrevc.88.044909
+15. J. Adam et al.,
+Pseudorapidity distributions of charged par-
+ticles as a function of mid- and forward rapidity multiplic-
+Supl. Rev. Mex. Fis. 3 040909
+
+A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF
+5
+ities in pp collisions at √s = 5.02, 7 and 13 TeV (2021),
+10.1140/epjc/s10052-021-09349-5
+16. ALICE Inner Tracking System (ITS): Technical Design Report,
+Technical design report. ALICE (CERN, Geneva, 1999), URL
+http://cds.cern.ch/record/391175.
+17. R. Santoro, et al., The ALICE Silicon Pixel Detector: readiness
+for the first proton beam, Journal of Instrumentation 4 (2009)
+P03023, 10.1088/1748-0221/4/03/p03023
+18. P. Skands, S. Carrazza, and J. Rojo,
+Tuning PYTHIA 8.1:
+the Monash 2013 tune, The European Physical Journal C 74
+(2014), 10.1140/epjc/s10052-014-3024-y
+19. ALICE Collaboration,
+Multiplicity dependence of charged-
+particle jet production in pp collisions at √s = 13 TeV
+(2022), 10.48550/ARXIV.2202.01548, URL https:
+//arxiv.org/abs/2202.01548.
+20. M. Cacciari, G. P. Salam, and G. Soyez,
+FastJet user man-
+ual, The European Physical Journal C 72 (2012), 10.1140/
+epjc/s10052-012-1896-2
+Supl. Rev. Mex. Fis. 3 040909
+

SNE0T4oBgHgl3EQf1wKp/content/tmp_files/load_file.txt

@@ -0,0 +1,382 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf,len=381
+page_content='Revista Mexicana de Física 3 040909 (2022) 1–4  September 2022  Rivet and the analysis preservation in heavy-ion collisions experiments  Antonio Carlos Oliveira da Silva (for the ALICE Collaboration)  University of Tennessee, Knoxville, 1408 Circle Drive, Knoxville TN 37996-1200  Received 3 July 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' accepted 15 September 2022  The comparison of experimental data and theoretical predictions is important for our understanding of the mechanisms for interactions and  particle production in hadron collisions, both at the Large Hadron Collider and at the Relativistic Heavy-Ion Collider experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Several  tools were ideated to help with that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rivet (Robust Independent Validation of Experiment and Theory) is a framework that facilitates the  comparison between measurements from high-energy physics experiments and Monte Carlo event generators able to produce outputs using  the HepMC package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rivet contains a repository with analysis algorithms developed by experiments, providing analysis documentation and  preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The recent developments for the implementation of centrality and multiplicity classes in Rivet are presented in this contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Keywords:  1  Introduction  Currently, the data and analysis preservation in high-energy  physics experiments is becoming a common concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Previ-  ous experiments and collaborations are losing the power of  reproducing their measurements since the data are not prop-  erly kept in a accessible way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The old code, which con-  tains crucial and detailed information like detector accep-  tance, particle and event selections, and corrections, is no  longer maintained and it is very difficult, if possible, to be  run again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Comparisons of previous measurements with new  models is, therefore, very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Robust Independent Validation of Experiment and The-  ory (Rivet) [1] is a framework that aims to facilitate the com-  parison between data and Monte Carlo (MC) event genera-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  2  Rivet framework  Rivet analyses are written in C++ and it currently contains  more than 1000 analyses from several high-energy physics  collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The data, when available, are downloaded di-  rectly from HepData [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Any model that is incorporated in  an event generator able to produce output that complies with  HepMC framework [3] can be used by Rivet for the compar-  ison with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The integration of Rivet with HepMC and  HepData is pictured in the scheme presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The references for the event generators in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  1 can  be found in [4–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Not all of them provide output using  HepMC standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  In principle, an article presenting a measurement should  present enough information to make the measurement able  to be reproduced by another experiment or theoretician inter-  ested in comparing the data with a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' However, some  subtle details about detector acceptances, particle selections,  trigger conditions, etc, could be missing or not clearly de-  scribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' This can be the case even in internal notes in large  collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Schematic diagram showing how Rivet is integrated  with HepMC and HepData.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Rivet aims at preserving all the analysis details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Further-  more, it should reproduce the methods used in a measurement  as close as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' One of the pillars of the Rivet philoso-  phy is that we should treat Monte Carlo simulations in the  same way as data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  In order to assure maximum fidelity to the original ac-  ceptances, selections and methods used in a measurement,  whenever possible, the rivet analysis should be implemented  by the collaboration that published the article containing the  measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Currently, ALICE has an official internal pro-  cedure for Rivet analysis approvals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  3  Centrality and multiplicity determination  Measurements in heavy-ion collisions are commonly differ-  ential in centrality intervals to study different physical phe-  nomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Therefore, centrality determination is a crucial fea-  ture Rivet has to provide in order to reproduce measurements  in heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Centrality determination in ALICE is commonly pro-  vided by the V0 detector [13], which consists of two ar-  rays, V0-A and V0-C covering the pseudorapidity ranges 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='8  < η < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='7 < η < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Figure 2 presents the distribution of the total energy de-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='02704v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='data-an]  6 Jan 2023  AMPT  FXR  HIJING  JETSCAPE  PYTHIA  Monte Carlo Mode  JEWEL  EPOS  PHSI  smash  HepMC  HEpData  Rivet2  ANTONIO CARLOS OLIVEIRA DA SILVA  posited in the V0 scintillators (amplitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The most cen-  tral collisions are associated with those event with highest  V0 amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The details of the centrality determination are  described in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Distribution of the total amplitude in the V0 scintilla-  tors in black points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The data are fitted using a Negative Binomial  Distribution (NBD) using parameters from the Glauber model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b b b b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b b b b b b b b b b b b b b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b b b b b b b b b b b b b b b b b b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b b b b b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b b b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b b b b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='calibration ALICE PbPb2760GeV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='0·104  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5·104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='0·104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5·104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='0·104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5·104  10−6  10−5  10−4  10−3  10−2  10−1  1  VZERO amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=') or ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' particle multiplicity  Events  FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' V0 amplitude in arbitrary units (black markers) mea-  sured by ALICE and the charged particle multiplicity in the V0 ac-  ceptance calculated with Rivet from Pb–Pb collisions at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='76 TeV  events simulated with PYTHIA 8 Angantyr (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The centrality determination in Rivet uses the multiplic-  ity of charged particles in the acceptance of the V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Since  the multiplicity of particles in heavy-ion collision events in  Monte Carlo event generators is model dependent, it is neces-  sary to create a calibration file that depends on collisions sys-  tem, energy and event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Figure 3 shows the V0 am-  plitude distribution measured by ALICE and the multiplicity  of charged particles in Pb–Pb collisions at √sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='76 TeV  generated with PYTHIA 8 Angantyr [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Rivet divides the charged particle multiplicity presented  in figure 3 (red line) in centrality percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  So the  most central events are associated to the highest multiplic-  ity events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' When running the Rivet analysis that requires  centrality determination, this calibration has to be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  A similar strategy is used for multiplicity determination in  pp and p–Pb collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Currently, ALICE is developing the  possibility to characterize the event using the self-normalized  charged particle multiplicity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' This development  is presented in sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  4  Self-normalized multiplicity  Forward-rapidity multiplicity classes can be defined in  ALICE using the V0 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Figure 4 shows the distribution  of the V0M amplitude, which is proportional to the number  of charged particles passing through the V0A and V0C de-  tectors, scaled by its average value ⟨V0M⟩ [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  FIGURE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Distribution of the V0M amplitude scaled by its aver-  age value ⟨V0M⟩ used to determined forward-rapidity multiplicity  classes in pp collisions at √s = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The Silicon Pixel Detector (SPD) [16, 17] is the closest  detector to the interaction point in ALICE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The SPD provides  mid-rapidity multiplicity classes determination using the re-  constructed tracklets, which are track segments that connects  hits in the two SPD layers pointing to the primary vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The  self-normalized estimator is obtained with the distribution of  SPD tracklets NSPD tracklets in -2 < η < 2 scaled by the av-  erage of its value ⟨NSPD tracklets⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The self-normalized SPD  tracklets distribution is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Supl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 3 040909  Yield (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=')  10-3  10-3  +  Data  Glauberfit  80-90%  70-80%  60-70%  10-4  10-4  500  1000  10-5  50-60%  40-50%  0-40%  20-30%  10-20%  10-6  5-10%  5%  8  0  4000  8000  12000  16000  20000  VZERO Amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=')  ALI-PUB-89941103  ALICE  Forward Multiplicity Classes  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Bias data (%)  102  Normalised counts  pp, Vs = 13 TeV  0-1  1-5  5-10  10-15  10  15-20  20-30  30-40  40-50  50-70  70-100  High-Mult data (%)  10-  0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='01-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1  10-3  10  10  5  0  2  4  6  8  10  VOM/VOM)  ALI-PUB-499489A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF  3  FIGURE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Distribution of the number of SPD tracklets scaled by its  average value ⟨NSPD tracklets⟩ used to determine midrapidity mul-  tiplicity classes in pp collisions at √s = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The event characterization using self-normalized multi-  plicity estimators in Rivet is under development in ALICE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Similar to what was discussed in the previous section, instead  of using the V0M amplitude, Rivet uses the charged particle  multiplicity in the V0 acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' In particular, the forward-  rapidity self-normalized estimator uses the multiplicity of  charged particles in the V0 acceptance scaled by its average  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Figure 6 shows the V0M/⟨V0M⟩ distribution calcu-  lated with Rivet using pp collisions at 13 TeV generated with  PYTHIA 8 Monash 2013 tune [4,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Rivet pp13TeV  0  2  4  6  8  10  10−5  10−4  10−3  10−2  10−1  1  V0M/⟨V0M⟩  Normalized counts  FIGURE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Self-normalized multiplicity distribution of charged par-  ticles in the acceptance of the V0 detector in pp collisions at √s =  13 TeV generated with PYTHIA 8 Monash 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Similarly to what is done for the V0M, the self-  normalized multiplicity estimator at mid-rapidity uses the  number of charged particles in the acceptance of the SPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Figure 7 shows the distribution of the charged particles in the  SPD acceptance scaled by its average value in pp collisions  at 13 TeV generated with PYTHIA 8 Monash 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Rivet pp13TeV  0  2  4  6  8  10  10−4  10−3  10−2  10−1  1  NSPD tracklets  SPD tracklets  SPD tracklets/⟨NSPD tracklets  SPD tracklets  SPD tracklets⟩  Normalized counts  FIGURE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Self-normalized multiplicity distribution of charged par-  ticles in the acceptance of the SPD detector in pp collisions at  √s = 13 TeV generated with PYTHIA 8 Monash 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  5  Results using the self-normalized multiplic-  ity estimators  The self-normalized multiplicity estimators framework in  Rivet is currently work in progress and being tested using ar-  ticles published by ALICE that use such estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The first  measurement used to test the V0M/⟨V0M⟩ estimator was the  transverse momentum (pT) of jets in different multiplicity in-  tervals in pp collisions at √s = 13 TeV [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  Data  Rivet [cent=GEN]  10−6  10−5  10−4  10−3  10−2  10−1  1/Nevent d2N/dpTd  event d2N/dpTd  event d2N/dpTdηηη (GeV/ccc)))−1  −1  −1  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  10  20  30  40  50  60  70  80  90  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='4  jet pT  pT  pT (GeV/ccc)))  MC/Data  FIGURE 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Charged-particle jet transverse momentum distribution  in pp collisions at √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='02 TeV for the 0-1% multiplicity  class corresponding to the self-normalized V0M-based multiplic-  ity estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Jets were reconstructed using jet resolution parame-  ter R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Data (black markers) are compared with PYTHIA 8  Monash 2013 (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Supl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 3 040909  103  ALICE  Central Multiplicity Classes  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Bias data (%)  102  pp, Vs = 13 TeV  Normalised counts  0-1  1-5  5-10  10-15  10  15-20  20-30  30-40  40-50  50-70  70-100  10  10  3  10  10  5  0  2  4  6  8  10  KN  SPD Tracklet4  ANTONIO CARLOS OLIVEIRA DA SILVA  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  Data  Rivet SPD pp5TeV [cent=GEN]  0  5  10  15  20  25  dN/dη  dN/dη  dN/dη  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='4  ηηη  MC/Data  FIGURE 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Charged particle pseudorapidity distribution in pp colli-  sions at √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='02 TeV for the 0-1% multiplicity class correspond-  ing to the self-normalized SPD-based multiplicity estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Data  (black markers) are compared with PYTHIA 8 Monash 2013 (red  line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The transverse momentum of jets reconstructed with  FastJet anti-kT algorithm [20] and resolution parameter R =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2 in the multiplicity class 0-1% in pp collisions at √s =  13 TeV is presented in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The measurement was com-  pared with PYTHIA 8 Monash 2013 using Rivet and the self-  normalized V0M multiplicity framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The agreement of  the model to data is a positive indication that the framework  can reproduce the multiplicity determination method used by  ALICE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Other multiplicity classes presented a similar perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The self-normalized estimator using the SPD was also  tested using the measurements in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Figure 9 presents  the charged particle pseudorapidity distribution in pp colli-  sions at √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='02 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The ALICE data are compared  with PYTHIA 8 Monash 2013 using Rivet and the SPD self-  normalized framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The results fairly reproduce the com-  parisons to MC presented in the cited article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  6  Summary  Rivet is a valuable tool for analysis preservation and compar-  ison of data to Monte Carlo event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The develop-  ment of additional tools to facilitate the implementation of  Rivet analyses is an important contribution to the framework  and can be of benefit both for the experiment and the theory  side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The self-normalized multiplicity estimators are provid-  ing consistent results between MC curves in Rivet and those  provided by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The final goal is to make this mul-  tiplicity framework available soon in the Rivet official frame-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Buckley, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=', Rivet user manual (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Maguire, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Heinrich, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Watt, HEPData: a repository  for high energy physics data, Journal of Physics: Conference  Series 898 (2017) 102006,  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1088/1742-6596/898/  10/102006  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Dobbs and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Hansen, The HepMC C++ Monte Carlo  Event Record for High Energy Physics,  Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=', CERN,  Geneva (2000), URL https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='ch/record/  684090, Revised version number 1 submitted on 2001-02-27  09:54:32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Sjöstrand, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Mrenna, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Skands, A Brief Introduc-  tion to PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 178 (2008) 852,  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='cpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='036  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Lin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=', Multiphase transport model for relativistic  heavy ion collisions, Physical Review C 72 (2005), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1103/  physrevc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='064901  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Bierlich, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=', The Angantyr model for Heavy-Ion Col-  lisions in PYTHIA8,  JHEP 10 (2018) 134,  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1007/  JHEP10(2018)134  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Putschke, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=', The JETSCAPE framework (2019), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='07706,  URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  org/abs/1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='07706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Weil, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=', Particle production and equilibrium properties  within a new hadron transport approach for heavy-ion colli-  sions, Physical Review C 94 (2016), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1103/physrevc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='054905  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Pierog, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=',  EPOS LHC: Test of collective hadroniza-  tion with data measured at the CERN Large Hadron Collider,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' C 92 (2015) 034906, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  034906  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Zapp, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=', A Monte Carlo model for ‘jet quenching’, The  European Physical Journal C 60 (2009), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1140/epjc/  s10052-009-0941-2  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Wang and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Gyulassy, hijing: A Monte Carlo model  for multiple jet production in pp, pA, and AA collisions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' D 44 (1991) 3501, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='3501  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Cassing and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Bratkovskaya,  Parton–hadron–string dy-  namics: An off-shell transport approach for relativistic ener-  gies,  Nuclear Physics A 831 (2009) 215,  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  nuclphysa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='007  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Collaboration, Performance of the ALICE VZERO sys-  tem, Journal of Instrumentation 8 (2013) P10016, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1088/  1748-0221/8/10/p10016  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Abelev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Adam, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Adamov,  Centrality determina-  tion of Pb-Pb collisions at √sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='76 TeV with ALICE 88  (2013), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1103/physrevc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='044909  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Adam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=',  Pseudorapidity distributions of charged par-  ticles as a function of mid- and forward rapidity multiplic-  Supl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 3 040909  A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF  5  ities in pp collisions at √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='02, 7 and 13 TeV (2021),  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1140/epjc/s10052-021-09349-5  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' ALICE Inner Tracking System (ITS): Technical Design Report,  Technical design report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' ALICE (CERN, Geneva, 1999), URL  http://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='ch/record/391175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Santoro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=', The ALICE Silicon Pixel Detector: readiness  for the first proton beam, Journal of Instrumentation 4 (2009)  P03023, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1088/1748-0221/4/03/p03023  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Skands, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Carrazza, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rojo,  Tuning PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1:  the Monash 2013 tune, The European Physical Journal C 74  (2014), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1140/epjc/s10052-014-3024-y  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' ALICE Collaboration,  Multiplicity dependence of charged-  particle jet production in pp collisions at √s = 13 TeV  (2022), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='01548, URL https:  //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='org/abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='01548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Cacciari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Salam, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Soyez,  FastJet user man-  ual, The European Physical Journal C 72 (2012), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1140/  epjc/s10052-012-1896-2  Supl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 3 040909' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}

TtE4T4oBgHgl3EQfLwyC/content/tmp_files/2301.04941v1.pdf.txt

@@ -0,0 +1,491 @@
+arXiv:2301.04941v1  [math.RT]  12 Jan 2023
+RIGID INTEGRAL REPRESENTATIONS OF QUIVERS REVISITED
+WILLIAM CRAWLEY-BOEVEY
+Abstract. In earlier work, the author classified rigid representations of a quiver by
+means of finitely generated free modules over a principal ideal ring.
+We show that
+the classification of exceptional pointwise free lattices can be extended to arbitrary
+commutative rings and that the classification of rigid lattices can be extended to reduced
+commutative rings.
+1. Introduction
+In [4] we studied rigid representations of a quiver over a principal ideal domain. Here
+we generalize the results to more general commutative base rings.
+Let R be a commutative ring (always non-zero). Given an R-module X and a homo-
+morphism R → S we write XS for the induced S-module S ⊗R X. Recall that a finitely
+generated projective R-module P has constant rank n if Pp is a free Rp-module of rank
+n for each prime ideal p of R, or equivalently if dimK P K = n for all homomorphisms
+R → K with K a field, which we may take to be algebraically closed.
+Let Q = (Q0, Q1, h, t) be a finite quiver and let RQ be the path algebra of Q over R,
+with a trivial path ei for each vertex i ∈ Q0. By an RQ-lattice we mean an RQ-module
+which is finitely generated projective as an R-module. We say that an RQ-module X
+is pointwise free if eiX is a free R-module for all i. We say that an RQ-lattice X has
+rank vector α ∈ NQ0 if eiX has constant rank αi for each i, and that X has pointwise
+constant rank if it has rank vector α for some α. If R → S is a homomorphism, then XS
+is naturally an SQ-module.
+We say that an RQ-module X is rigid if Ext1
+RQ(X, X) = 0, and exceptional if also the
+natural map R → EndRQ(X) is an isomorphism. If K is an algebraically closed field, then
+by definition the possible dimension vectors of exceptional KQ-modules are the real Schur
+roots for Q. We call the possible dimension vectors of rigid KQ-modules rigid dimension
+vectors. By results of [2] or [3] these do not depend on the field K. If X is a rigid
+(respectively exceptional) RQ-lattice of rank vector α, then XK is a rigid (respectively
+exceptional) KQ-module of dimension vector α for any homomorphism R → K with K
+a field (see Lemmas 3.1,3.2), and so α must be a rigid dimension vector (respectively real
+Schur root).
+Theorem A. For any commutative ring R and real Schur root α for Q, there is a unique
+rigid pointwise free RQ-lattice with rank vector α and it is exceptional.
+The existence follows easily from [4], and the uniqueness follows from the next result.
+As mentioned, the case when R is a principal ideal domain is in [4]. The case when R is
+a truncated polynomial ring K[ǫ]/(ǫn) is a special case of [6, Theorem 1.2]. Recall that a
+commutative ring R is reduced if it has no nilpotent elements.
+2020 Mathematics Subject Classification. Primary 16G20; Secondary 16G30,16H20,13C10.
+Key words and phrases. Quiver representations, Rigid representations, Lattices over orders.
+Partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
+SFB-TRR 358/1 2023 – 491392403.
+1
+
+Theorem B. If R is reduced, then any rigid RQ-lattice X of pointwise constant rank
+has a decomposition
+X ∼= (X1 ⊗R P1) ⊕ · · · ⊕ (Xr ⊗R Pr)
+where the Xi are pairwise non-isomorphic exceptional pointwise free RQ-lattices satisfying
+Ext1
+RQ(Xi, Xj) = 0 for all i, j and the Pi are finitely generated projective R-modules of
+constant rank. Moreover this decomposition is unique up to isomorphism and reordering.
+In particular any exceptional RQ-lattice is of the form Y ⊗R P with Y an exceptional
+pointwise free RQ-module and P a projective R-module of constant rank 1.
+Note that in general one doesn’t have uniqueness for rigid pointwise free lattices of a
+given rank vector. For example let Q be the quiver 1 → 2 and R a reduced ring with a
+stably free projective module P which is not free, say P ⊕Rn ∼= Rm. Then (RQe2⊗RP)⊕
+(RQe1 ⊗R Rn) is pointwise free, but not isomorphic to (RQe2 ⊗R Rm−n) ⊕(RQe1 ⊗R Rn).
+2. Commutative rings
+Concerning our definitions and results, note that every finitely generated projective
+R-module has constant rank if and only if Spec R is connected, or equivalently R has no
+idempotents other than 0 and 1, see for example Exercise 20.12 in [5].
+Lemma 2.1. Let R be a commutative ring.
+(i) If X is a finitely generated R-module, then X = 0 if and only if XK = 0 for all
+homomorphisms R → K with K an algebraically closed field.
+(ii) If θ : X → Y is a homomorphism of R-modules and Y is finitely generated, then
+θ is onto if and only if θK is onto for all R → K with K an algebraically closed
+field.
+(iii) If R is reduced and X is a finitely generated R-module, then X is projective of
+constant rank n if and only if dimK XK = n for all homomorphisms R → K with
+K an algebraically closed field.
+(iv) If R is reduced and θ : X → Y is a homomorphism with X, Y finitely generated
+projective of constant rank, then θ is a split monomorphism if and only if θK is a
+monomorphism for all R → K with K an algebraically closed field.
+Proof. (i) Since X is finitely generated, if non-zero it has a map onto some R/m, and
+taking K to be the algebraic closure of this field gives a surjective map XK → R/m⊗RK ∼=
+K.
+(ii) Clear.
+(iii) See Exercise 20.13 in [5].
+(iv) If θK is a monomorphism for all R → K, then Coker θK has constant dimension,
+so Coker θ is projective of constant rank. Then Im θ is a summand of Y , so projective.
+But then 0 → Ker θ → X → Im θ → 0 splits, so is exact after inducing to K. Thus
+Ker θK = 0. Also Ker θ is summand of X, so finitely generated. Thus Ker θ is zero.
+□
+Lemma 2.2. Any commutative ring R can be written as a quotient R ∼= ˜R/I with ˜R an
+integral domain and I is an ideal contained in the Jacobson radical of ˜R.
+Proof. Let A = Z[xr : r ∈ R] be the polynomial ring over Z with indeterminates indexed
+by the elements of R. There is a surjective homomorphism θ : A → R sending each xr
+to r and the set
+S = {a ∈ A : θ(a) is invertible}
+is a multiplicative subset. We set ˜R to be the localization AS, an integral domain. The
+homomorphism θ extends to a surjective homomorphism ˜θ : ˜R → R.
+2
+
+Let x ∈ Ker ˜θ. If y ∈ ˜R, then xy ∈ Ker ˜θ. Write xy = as−1 with a ∈ A and s ∈ S.
+Then θ(a) = 0, so s + a ∈ S. Thus 1 + xy = (s + a)s−1 is invertible in ˜R. Since this is
+true for all y, it follows that x is in the Jacobson radical of ˜R.
+□
+3. Lattices
+Let R be a commutative ring and Q a quiver. Lemma 1 of [4] generalizes immediately
+to this setting.
+Lemma 3.1. If X is an RQ-lattice, then proj. dim X ≤ 1 and
+Ext1
+RQ(X, Y )S ∼= Ext1
+SQ(XS, Y S)
+for any homomorphism R → S and any RQ-module Y . In particular if X is a rigid
+RQ-lattice, then XS is a rigid SQ-lattrice for all homomorphisms R → S. Conversely, if
+X is an RQ-lattice and XK rigid for all homomorphisms R → K with K an algebraically
+closed field, then X is rigid.
+Proof. Letting S be the R-subalgebra of RQ with basis the trivial paths ei and B be the
+free R-submodule of RQ with basis the arrows, the path algebra RQ is isomorphic to the
+tensor algebra of B over S, so there is a standard resolution
+0 → RQ ⊗S B ⊗S RQ → RQ ⊗S RQ → RQ → 0,
+or equivalently
+0 →
+�
+a∈Q1
+RQeh(a) ⊗R et(a)RQ →
+�
+i∈Q0
+RQei ⊗R eiR → RQ → 0.
+Tensoring with the RQ-module X gives
+0 = TorRQ
+1 (RQ, X) →
+�
+a∈Q1
+RQeh(a) ⊗R et(a)X →
+�
+i∈Q0
+RQei ⊗R eiX → X → 0.
+and this is a resolution of X by finitely generated projective RQ-modules. We denote it
+by 0 → P1 → P0 → X → 0. Since X is projective over R, the induced sequence
+0 → P S
+1 → P S
+0 → XS → 0
+is exact, so a resolution of XS by projective SQ-modules. Now if P is a finitely generated
+projective RQ-module, then
+HomRQ(P, Y )S ∼= HomSQ(P S, Y S).
+Thus we get a commutative diagram with exact rows
+HomRQ(P0, Y )S −−−→ HomRQ(P1, Y )S −−−→ Ext1
+RQ(X, Y )S
+−−−→ 0
+�
+�
+HomSQ(P S
+0 , Y S) −−−→ HomSQ(P S
+1 , Y S) −−−→ Ext1
+SQ(XS, Y S) −−−→ 0
+in which the vertical maps are isomorphisms, and hence an isomorphism
+Ext1
+RQ(X, Y )S ∼= Ext1
+SQ(XS, Y S).
+Now Ext1
+RQ(X, X) is a finitely generated R-module, so if Ext1
+RQ(X, X)K = 0 for all
+homomorphisms from R to an algebraically closed field K then Ext1
+RQ(X, X) = 0, that
+is, X is rigid.
+□
+3
+
+Over an algebraically closed field K, the isomorphism classes of representations X of
+Q of dimension vector α correspond to orbits OX of a group GL(α) acting on an affine
+space Rep(Q, α). Moreover X is rigid if and only if OX is open orbit. Since the affine
+space it irreducible, it follows that there is at most one rigid module of dimension α, up
+to isomorphism.
+The general dimension of Hom(X, Y ) and Ext1(X, Y ) for X of dimension α and Y
+of dimension β is denoted hom(α, β) and ext(α, β). In particular this is these are the
+dimensions for X and Y rigid. Based on work of Schofield [11], it is proved in [3] that
+hom(α, β) and ext(α, β) do not depend on the algebraically closed field K.
+Lemma 3.2. If R is reduced, and if X, Y are rigid RQ-lattices of pointwise constant
+rank α, β, then
+HomRQ(X, Y )S ∼= HomSQ(XS, Y S)
+for all homomorphisms R → S. Moreover Ext1
+RQ(X, Y ) and HomRQ(X, Y ) are finitely
+generated projective R-modules of constant rank ext(α, β) and hom(α, β).
+Proof. For any R → K with K an algebraically closed field, Ext1
+RQ(X, Y )K has constant
+dimension ext(α, β), so Ext1
+RQ(X, Y ) is projective over R of this rank by Lemma 2.1(iii).
+Now the projective resolution 0 → P1 → P0 → X → 0 of X gives an exact sequence
+0 → HomRQ(X, Y ) → HomRQ(P0, Y ) → HomRQ(P1, Y ) → Ext1
+RQ(X, Y ) → 0.
+The last three are projective R-modules of constant rank, so this sequence splits. Thus
+the first module is also projective over R. Moreover the sequence remains exact under
+induction using a homomorphism R → S. It follows that
+HomRQ(X, Y )S ∼= HomSQ(XS, Y S).
+Considering the case when S is a field, we see that HomRQ(X, Y ) has constant rank
+hom(α, β).
+□
+The following result is well known.
+Lemma 3.3. The full subquiver of Q on the support of any rigid dimension vector α has
+no oriented cycles.
+Proof. If not, one can easily construct representations of Q of dimension α over an alge-
+braically closed field K on which the trace of the oriented cycle is arbitrary, but the variety
+of representations has a dense orbit, so the trace must be constant. (This argument is
+already used in [4, Theorem 1] for real Schur roots.)
+□
+4. Mutations
+Lemma 4.1. If R is a commutative ring and α is real Schur root for Q, then there exists
+an exceptional pointwise free RQ-lattice of dimension vector α.
+Proof. By [4, Theorem 2], there is a (unique) exceptional ZQ-lattice X0 of rank vector
+α. Now we obtain an RQ-lattice XR = X ⊗Z R, and it is exceptional by Lemmas 3.1
+and 3.2.
+□
+The cited theorem [4, Theorem 2] depends on a braid group action for exceptional
+sequences of lattices, defined using mutations of exceptional pairs.
+The result about
+mutations in turn depends on the construction of mutations of pairs of representations of
+KQ where K is a field, for which [4] cites [10] and [2] cites [7]. We take this opportunity
+to give a direct proof, based on the following generalization of the Happel-Ringel Lemma
+[8, Lemma 4.1].
+4
+
+Lemma 4.2. Let θ : X → Y be a homomorphism between finite-dimensional A-modules
+for a hereditary algebra A and suppose that Ext1
+A(Y, X) = 0. If Y is indecomposable, then
+either θ is an epimorphism or the map X → Im(θ) is a split epimorphism.
+Proof. The map θ gives exact sequences
+ξ : 0 → Im(θ)
+α−→ Y → Coker(θ) → 0
+and
+η : 0 → Ker(θ) → X
+β−→ Im(θ) → 0.
+From Ext1
+A(Coker(θ), η) we get an exact sequence
+· · · → Ext1
+A(Coker(θ), X)
+f−→ Ext1
+A(Coker(θ), Im(θ)) → Ext2
+A(Coker(θ), Ker(θ)) = 0
+so ξ = f(ζ) for some ζ. Thus there is a commutative diagram
+ζ : 0 −−−→
+X
+δ
+−−−→ Z −−−→ Coker(θ) −−−→ 0
+β
+�
+γ
+�
+���
+ξ : 0 −−−→ Im(θ)
+α
+−−−→ Y −−−→ Coker(θ) −−−→ 0
+Now the sequence
+0 → X
+
+δ
+β
+
+
+−−−→ Z ⊕ Im(θ)
+�
+γ
+−α
+�
+−−−−−−→ Y → 0
+is exact, so splits since Ext1
+A(Y, X) = 0. Thus there are maps
+�p
+q�
+: Z ⊕ Im(θ) → X
+and
+�
+r
+s
+�
+: Y → Z ⊕ Im(θ)
+with
+�
+1Z
+0
+0
+1Im(θ)
+�
+=
+�
+δ
+β
+� �p
+q�
++
+�
+r
+s
+� �γ
+−α�
+.
+Thus 1Im(θ) = βq − sα. Now if θ is not an epimorphism, then αs cannot be an auto-
+morphism of Y . Since Y is indecomposable, αs is nilpotent. This implies that βq is
+invertible, and hence that β is a split epimorphism.
+□
+If X is also indecomposable, one recovers the Happel-Ringel Lemma: if θ is non-
+zero and not an epimorphism, then X → Im(θ) must be an isomorphism, so θ is a
+monomorphism. Dually we have:
+Lemma 4.3. Let θ : X → Y be a homomorphism between finite-dimensional A-modules
+for a hereditary algebra A and suppose that Ext1
+A(Y, X) = 0. If X is indecomposable,
+then either θ is a monomorphism or the map Im(θ) → Y is a split monomorphism.
+A sequence of exceptional RQ-lattices (X1, . . . , Xr) is called an exceptional sequence
+provided that HomRQ(Xi, Xj) = Ext1
+RQ(Xi, Xj) = 0 for all i > j. The result about
+mutations is as follows.
+Theorem 4.4. Suppose R is reduced. If (X, Y ) is an exceptional pair of RQ-lattices, then
+there are exceptional pairs (LXY, X) and (Y, RY X) given as follows. If HomRQ(X, Y ) = 0
+then LXY and RY X are given by the universal exact sequences
+0 → Y → LXY → X ⊗R Ext1
+RQ(X, Y ) → 0,
+0 → Y ⊗R D Ext1
+RQ(X, Y ) → RY X → X → 0.
+where D = HomR(−, R). If HomRQ(X, Y ) ̸= 0, then the universal map
+f : X ⊗R HomRQ(X, Y ) → Y
+5
+
+is an epimorphism or a monomorphism, and LXY is its kernel or cokernel, and the
+universal map
+g : X → HomR(HomRQ(X, Y ), Y ) ∼= Y ⊗R D HomRQ(X, Y )
+is an epimorphism or a monomorphism and RY X is its kernel or cokernel.
+Proof. Since R is reduced and the lattices are rigid, the Hom and Ext spaces are projective
+over R of constant rank, independent of R. Moreover using Lemma 2.1 we can, as in [4,
+Lemma 4], reduce to the case when R = K is an algebraically closed field.
+In this setting, the claim follows easily once one knows that if HomKQ(X, Y ) ̸= 0 then
+Ext1
+KQ(X, Y ) = 0, and that f and g are either epimorphisms or monomorphisms.
+Now if θ ∈ HomKQ(X, Y ) is a non-zero map, then by the Happel-Ringel Lemma θ is
+either an epimorphism or a monomorphism. In the first case we have a exact sequence
+· · · → Ext1
+KQ(X, X) → Ext1
+KQ(X, Y ) → Ext2
+KQ(X, Ker(θ)) = 0
+and in the second case we have
+· · · → Ext1
+KQ(Y, Y ) → Ext1
+KQ(X, Y ) → Ext2
+KQ(Coker(θ), Y ) = 0.
+Since Ext1
+KQ(X, X) = Ext1
+KQ(Y, Y ) = 0, in both cases we get Ext1
+KQ(X, Y ) = 0.
+If the map f : X ⊗K HomKQ(X, Y ) → Y is not an epimorphism, then the induced
+map f ′ : X ⊗K HomKQ(X, Y ) → Im(f) is a split epimorphism by Lemma 4.2. Since
+EndKQ(X) = K, the map f can be identified with the universal map from a direct sum
+of copies of X to Y given by a basis of HomKQ(X, Y ), from which it is easy to see
+that f is right minimal [1, §I.2]. Thus by [1, Corollary I.2.3] there is no non-zero direct
+summand of the domain on which f vanishes. Thus f ′ is an isomorphism, and hence f
+is a monomorphism. The argument for g is dual.
+□
+5. Classification of rigids
+We prove the following result. Once Theorem A is proved, it gives Theorem B. The
+proof of the theorem is the same as Theorem 2(ii) of [4].
+Theorem 5.1. Let R be reduced. For each real Schur root α, we choose an exceptional
+pointwise free RQ-lattice X(α) of rank vector α. Any rigid RQ-lattice X of pointwise
+constant rank has a decomposition
+X ∼= (X(α1) ⊗R P1) ⊕ · · · ⊕ (X(αr) ⊗R Pr)
+where the α1, . . . , αr are distinct real Schur roots, ext(αi, αj) = 0 for all i, j and the Pi
+are finitely generated projective R-modules of constant rank. Moreover this decomposition
+is unique up to isomorphism and reordering. In particular any exceptional RQ-lattice of
+rank vector α is of the form X(α) ⊗R P with X exceptional pointwise free and P a
+projective R-module of constant rank 1.
+Proof. By Lemma 3.3, we may assume that Q has no oriented cycles. We fix temporarily
+a homomorphism R → K with K an algebraically closed field. Then XK is rigid, so
+decomposes as direct sum
+XK ∼= Mm1
+1
+⊕ · · · ⊕ Mmr
+r
+.
+with the Mi pairwise non-isomorphic exceptional KQ-modules, Ext1
+KQ(Mi, Mj) = 0 for
+all i, j and all mi > 0. By [8, Corollary 4.2], we can order the Mi so that (M1, . . . , Mr) is
+an exceptional sequence, so Hom(Mi, Mj) = 0 for i > j. Let Mi have dimension vector αi.
+We have Hom(X(αi), X(αj)) = 0 for i > j since dim Hom(Mi, Mj) = hom(αi, αj).
+6
+
+Now Pr = HomRQ(X(αr), X) is a finitely generated projective R-module of constant
+rank. Consider the evaluation map θ : X(αr) ⊗R Pr → X and let C be its cokernel. For
+an arbitrary homomorphism R → K with K an algebraically closed field (no longer the
+one fixed above), using Lemma 3.2, we can identify θK with the evaluation map
+X(αr)K ⊗K HomKQ(X(αr)K, XK) → X
+but we know that
+XK ∼= (X(α1)K)m1 ⊕ · · · ⊕ (X(αr)K)mr
+since both sides are rigid KQ-modules of the same dimension vector. Thus θK is the
+inclusion of (X(αr)K)mr as a direct summand of XK. Thus by Lemma 2.1, θ is a split
+monomorphism of R-modules, so C is projective over R. Also
+CK ∼= Coker(θK) ∼= (X(α1)K)m1 ⊕ · · · ⊕ (X(αr−1)K)mr−1
+so Ext1
+KQ(CK, CK) = 0 and Ext1
+KQ(CK, X(αr)K) = 0. Thus C is rigid and Ext1(C, Xr) =
+0. Thus X ∼= X(αr) ⊗R Pr ⊕C. Now the result follows by induction (on r or on the total
+rank of X).
+For uniqueness, note that if we have another decomposition
+X = (X(β1) ⊗ P ′
+1) ⊕ · · · ⊕ (X(βs) ⊗ P ′
+s)
+with the βi being distinct real Schur roots and the P ′
+i projective R-modules of constant
+rank, then using a fixed homomorphism R → K and the uniqueness of rigid KQ-modules
+of a given dimension vector, we see that s = r and that the βi are a permutation of the
+αi. Thus we reduce to the case that βi = αi. But then
+P ′
+r ∼= HomRQ(X(αr), X) ∼= Pr
+and then the corresponding cokernels of the evaluation maps are isomorphic, so by in-
+duction P ′
+i ∼= Pi for all i.
+The special case of exceptional lattices follows.
+□
+6. Uniqueness for exceptional pointwise frees
+We now prove Theorem A. The existence part was Lemma 4.1. Suppose X is a point-
+wise free lattice for RQ of rank vector α, a real Schur root. By Lemma 2.2 we can write
+R ∼= ˜R/I with ˜R reduced and I contained in the Jacobson radical of ˜R.
+We define a pointwise free ˜RQ-lattice ˜X by giving it as a representation of Q by free
+˜R-modules as follows. For each vertex i we take ei ˜X to be a free ˜R-module of rank αi,
+and we fix an isomorphism
+(ei ˜X)/I(ei ˜X) ∼= (ei ˜X)R ∼= eiX.
+For each arrow a ∈ Q1, multiplication by a induces an R-linear map et(a)X → eh(a)X,
+and we lift this to an ˜R-linear map et(a) ˜X → eh(a) ˜X. This defines ˜X as a representation
+of Q and clearly we have ˜XR ∼= X.
+Let E = Ext1
+˜RQ( ˜X, ˜X), a finitely generated ˜R-module. Now ER ∼= Ext1
+RQ(X, X) = 0
+by Lemma 3.1, so E = IE, so by Nakayama’s Lemma E = 0. Thus ˜X is rigid. (This
+idea of lifting X to a rigid representation ˜X is taken from [6].)
+By Theorem B we have ˜X ∼= X(α)⊗ ˜R ˜P, where X(α) is a chosen exceptional pointwise
+free ˜RQ-lattice of rank vector α and ˜P is a finitely generated projective ˜R-module of
+constant rank 1.
+7
+
+Tensoring with R, we get X ∼= Y ⊗R P where Y = X(α)R is an exceptional pointwise
+free RQ-lattice and P = ˜P R is a finitely generated projective R-module of constant
+rank 1.
+Now since X and Y are pointwise free of rank vector α, we have
+Rαi ∼= eiX ∼= eiY ⊗R P ∼= Rαi ⊗R P ∼= P αi
+for all i. Since α is a real root for Q, it is indivisible, that is, its components are coprime.
+Thus we can find ai, bi ∈ N such that
+1 +
+�
+i
+aiαi =
+�
+i
+biαi.
+Then
+P ⊕ R
+�
+i aiαi ∼= P ⊕
+�
+i
+(Rαi)ai ∼= P ⊕
+�
+i
+(P αi)ai ∼=
+�
+i
+(P αi)bi ∼=
+�
+i
+(Rαi)bi ∼= R
+�
+i biαi.
+Thus P is stably free. Now any stably free projective module of constant rank 1 for a
+commutative ring is free, see for example [9, Theorem 4.11], so P ∼= R. Thus X ∼= Y .
+We have already observed that Y is exceptional, hence so is X.
+References
+[1] M. Auslander, I. Reiten and S. O. Smalø, Representation theory of Artin algebras, Cambridge
+Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1995.
+[2] W. Crawley-Boevey, Exceptional sequences of representations of quivers, in ‘Representations of
+algebras’ (Ottawa, ON, 1992), 117-–124, CMS Conf. Proc., 14, Amer. Math. Soc., Providence, RI,
+1993.
+[3] W. Crawley-Boevey, Subrepresentations of general representations of quivers, Bull. London Math.
+Soc. 28 (1996), 363—366.
+[4] W. Crawley-Boevey, Rigid integral representations of quivers, in ‘Representation theory of
+algebras’ (Cocoyoc, 1994), 155-–163, CMS Conf. Proc., 18, Amer. Math. Soc., Providence, RI,
+1996.
+[5] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Graduate Texts in
+Mathematics, 150. Springer-Verlag, New York, 1995.
+[6] C. Geiß, B. Leclerc and J. Schr¨oer, Rigid modules and Schur roots. Math. Z. 295 (2020), no. 3-4,
+1245–1277.
+[7] A. L. Gorodentsev, Exceptional bundles on surfaces with a moving anticanonical class (Russian),
+Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 740-–757, 895; translation in Math. USSR-Izv. 33
+(1989), 67-–83.
+[8] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399-–443.
+[9] T. Y. Lam, Serre’s problem on projective modules, Springer Monographs in Mathematics.
+Springer-Verlag, Berlin, 2006.
+[10] A. N. Rudakov, Exceptional collections, mutations and helices, in ‘Helices and vector bundles’,
+1-–6, London Math. Soc. Lecture Note Ser., 148, Cambridge Univ. Press, Cambridge, 1990.
+[11] A. Schofield, General representations of quivers, Proc. London Math. Soc. (3) 65 (1992), 46-–64.
+Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, 33501 Bielefeld, Germany
+Email address: wcrawley@math.uni-bielefeld.de
+8
+

TtE4T4oBgHgl3EQfLwyC/content/tmp_files/load_file.txt

@@ -0,0 +1,326 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf,len=325
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='04941v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='RT]  12 Jan 2023  RIGID INTEGRAL REPRESENTATIONS OF QUIVERS REVISITED  WILLIAM CRAWLEY-BOEVEY  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' In earlier work, the author classified rigid representations of a quiver by  means of finitely generated free modules over a principal ideal ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  We show that  the classification of exceptional pointwise free lattices can be extended to arbitrary  commutative rings and that the classification of rigid lattices can be extended to reduced  commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Introduction  In [4] we studied rigid representations of a quiver over a principal ideal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Here  we generalize the results to more general commutative base rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Let R be a commutative ring (always non-zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Given an R-module X and a homo-  morphism R → S we write XS for the induced S-module S ⊗R X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Recall that a finitely  generated projective R-module P has constant rank n if Pp is a free Rp-module of rank  n for each prime ideal p of R, or equivalently if dimK P K = n for all homomorphisms  R → K with K a field, which we may take to be algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Let Q = (Q0, Q1, h, t) be a finite quiver and let RQ be the path algebra of Q over R,  with a trivial path ei for each vertex i ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By an RQ-lattice we mean an RQ-module  which is finitely generated projective as an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We say that an RQ-module X  is pointwise free if eiX is a free R-module for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We say that an RQ-lattice X has  rank vector α ∈ NQ0 if eiX has constant rank αi for each i, and that X has pointwise  constant rank if it has rank vector α for some α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If R → S is a homomorphism, then XS  is naturally an SQ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  We say that an RQ-module X is rigid if Ext1  RQ(X, X) = 0, and exceptional if also the  natural map R → EndRQ(X) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If K is an algebraically closed field, then  by definition the possible dimension vectors of exceptional KQ-modules are the real Schur  roots for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We call the possible dimension vectors of rigid KQ-modules rigid dimension  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By results of [2] or [3] these do not depend on the field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If X is a rigid  (respectively exceptional) RQ-lattice of rank vector α, then XK is a rigid (respectively  exceptional) KQ-module of dimension vector α for any homomorphism R → K with K  a field (see Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2), and so α must be a rigid dimension vector (respectively real  Schur root).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For any commutative ring R and real Schur root α for Q, there is a unique  rigid pointwise free RQ-lattice with rank vector α and it is exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  The existence follows easily from [4], and the uniqueness follows from the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  As mentioned, the case when R is a principal ideal domain is in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The case when R is  a truncated polynomial ring K[ǫ]/(ǫn) is a special case of [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Recall that a  commutative ring R is reduced if it has no nilpotent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Primary 16G20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Secondary 16G30,16H20,13C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Quiver representations, Rigid representations, Lattices over orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –  SFB-TRR 358/1 2023 – 491392403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  1  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If R is reduced, then any rigid RQ-lattice X of pointwise constant rank  has a decomposition  X ∼= (X1 ⊗R P1) ⊕ · · · ⊕ (Xr ⊗R Pr)  where the Xi are pairwise non-isomorphic exceptional pointwise free RQ-lattices satisfying  Ext1  RQ(Xi, Xj) = 0 for all i, j and the Pi are finitely generated projective R-modules of  constant rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover this decomposition is unique up to isomorphism and reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  In particular any exceptional RQ-lattice is of the form Y ⊗R P with Y an exceptional  pointwise free RQ-module and P a projective R-module of constant rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Note that in general one doesn’t have uniqueness for rigid pointwise free lattices of a  given rank vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For example let Q be the quiver 1 → 2 and R a reduced ring with a  stably free projective module P which is not free, say P ⊕Rn ∼= Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Then (RQe2⊗RP)⊕  (RQe1 ⊗R Rn) is pointwise free, but not isomorphic to (RQe2 ⊗R Rm−n) ⊕(RQe1 ⊗R Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Commutative rings  Concerning our definitions and results, note that every finitely generated projective  R-module has constant rank if and only if Spec R is connected, or equivalently R has no  idempotents other than 0 and 1, see for example Exercise 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='12 in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let R be a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (i) If X is a finitely generated R-module, then X = 0 if and only if XK = 0 for all  homomorphisms R → K with K an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (ii) If θ : X → Y is a homomorphism of R-modules and Y is finitely generated, then  θ is onto if and only if θK is onto for all R → K with K an algebraically closed  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (iii) If R is reduced and X is a finitely generated R-module, then X is projective of  constant rank n if and only if dimK XK = n for all homomorphisms R → K with  K an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (iv) If R is reduced and θ : X → Y is a homomorphism with X, Y finitely generated  projective of constant rank, then θ is a split monomorphism if and only if θK is a  monomorphism for all R → K with K an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' (i) Since X is finitely generated, if non-zero it has a map onto some R/m, and  taking K to be the algebraic closure of this field gives a surjective map XK → R/m⊗RK ∼=  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (ii) Clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (iii) See Exercise 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='13 in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (iv) If θK is a monomorphism for all R → K, then Coker θK has constant dimension,  so Coker θ is projective of constant rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Then Im θ is a summand of Y , so projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  But then 0 → Ker θ → X → Im θ → 0 splits, so is exact after inducing to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus  Ker θK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Also Ker θ is summand of X, so finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus Ker θ is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Any commutative ring R can be written as a quotient R ∼= ˜R/I with ˜R an  integral domain and I is an ideal contained in the Jacobson radical of ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let A = Z[xr : r ∈ R] be the polynomial ring over Z with indeterminates indexed  by the elements of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' There is a surjective homomorphism θ : A → R sending each xr  to r and the set  S = {a ∈ A : θ(a) is invertible}  is a multiplicative subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We set ˜R to be the localization AS, an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The  homomorphism θ extends to a surjective homomorphism ˜θ : ˜R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  2  Let x ∈ Ker ˜θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If y ∈ ˜R, then xy ∈ Ker ˜θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Write xy = as−1 with a ∈ A and s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Then θ(a) = 0, so s + a ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus 1 + xy = (s + a)s−1 is invertible in ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since this is  true for all y, it follows that x is in the Jacobson radical of ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Lattices  Let R be a commutative ring and Q a quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Lemma 1 of [4] generalizes immediately  to this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If X is an RQ-lattice, then proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' dim X ≤ 1 and  Ext1  RQ(X, Y )S ∼= Ext1  SQ(XS, Y S)  for any homomorphism R → S and any RQ-module Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' In particular if X is a rigid  RQ-lattice, then XS is a rigid SQ-lattrice for all homomorphisms R → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Conversely, if  X is an RQ-lattice and XK rigid for all homomorphisms R → K with K an algebraically  closed field, then X is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Letting S be the R-subalgebra of RQ with basis the trivial paths ei and B be the  free R-submodule of RQ with basis the arrows, the path algebra RQ is isomorphic to the  tensor algebra of B over S, so there is a standard resolution  0 → RQ ⊗S B ⊗S RQ → RQ ⊗S RQ → RQ → 0,  or equivalently  0 →  �  a∈Q1  RQeh(a) ⊗R et(a)RQ →  �  i∈Q0  RQei ⊗R eiR → RQ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Tensoring with the RQ-module X gives  0 = TorRQ  1 (RQ, X) →  �  a∈Q1  RQeh(a) ⊗R et(a)X →  �  i∈Q0  RQei ⊗R eiX → X → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  and this is a resolution of X by finitely generated projective RQ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We denote it  by 0 → P1 → P0 → X → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since X is projective over R, the induced sequence  0 → P S  1 → P S  0 → XS → 0  is exact, so a resolution of XS by projective SQ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now if P is a finitely generated  projective RQ-module, then  HomRQ(P, Y )S ∼= HomSQ(P S, Y S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Thus we get a commutative diagram with exact rows  HomRQ(P0, Y )S −−−→ HomRQ(P1, Y )S −−−→ Ext1  RQ(X, Y )S  −−−→ 0  \uf8e6\uf8e6�  \uf8e6\uf8e6�  HomSQ(P S  0 , Y S) −−−→ HomSQ(P S  1 , Y S) −−−→ Ext1  SQ(XS, Y S) −−−→ 0  in which the vertical maps are isomorphisms, and hence an isomorphism  Ext1  RQ(X, Y )S ∼= Ext1  SQ(XS, Y S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Now Ext1  RQ(X, X) is a finitely generated R-module, so if Ext1  RQ(X, X)K = 0 for all  homomorphisms from R to an algebraically closed field K then Ext1  RQ(X, X) = 0, that  is, X is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  3  Over an algebraically closed field K, the isomorphism classes of representations X of  Q of dimension vector α correspond to orbits OX of a group GL(α) acting on an affine  space Rep(Q, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover X is rigid if and only if OX is open orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since the affine  space it irreducible, it follows that there is at most one rigid module of dimension α, up  to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  The general dimension of Hom(X, Y ) and Ext1(X, Y ) for X of dimension α and Y  of dimension β is denoted hom(α, β) and ext(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' In particular this is these are the  dimensions for X and Y rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Based on work of Schofield [11], it is proved in [3] that  hom(α, β) and ext(α, β) do not depend on the algebraically closed field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If R is reduced, and if X, Y are rigid RQ-lattices of pointwise constant  rank α, β, then  HomRQ(X, Y )S ∼= HomSQ(XS, Y S)  for all homomorphisms R → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover Ext1  RQ(X, Y ) and HomRQ(X, Y ) are finitely  generated projective R-modules of constant rank ext(α, β) and hom(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For any R → K with K an algebraically closed field, Ext1  RQ(X, Y )K has constant  dimension ext(α, β), so Ext1  RQ(X, Y ) is projective over R of this rank by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Now the projective resolution 0 → P1 → P0 → X → 0 of X gives an exact sequence  0 → HomRQ(X, Y ) → HomRQ(P0, Y ) → HomRQ(P1, Y ) → Ext1  RQ(X, Y ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  The last three are projective R-modules of constant rank, so this sequence splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus  the first module is also projective over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover the sequence remains exact under  induction using a homomorphism R → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' It follows that  HomRQ(X, Y )S ∼= HomSQ(XS, Y S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Considering the case when S is a field, we see that HomRQ(X, Y ) has constant rank  hom(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  The following result is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The full subquiver of Q on the support of any rigid dimension vector α has  no oriented cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If not, one can easily construct representations of Q of dimension α over an alge-  braically closed field K on which the trace of the oriented cycle is arbitrary, but the variety  of representations has a dense orbit, so the trace must be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' (This argument is  already used in [4, Theorem 1] for real Schur roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=')  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Mutations  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If R is a commutative ring and α is real Schur root for Q, then there exists  an exceptional pointwise free RQ-lattice of dimension vector α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By [4, Theorem 2], there is a (unique) exceptional ZQ-lattice X0 of rank vector  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now we obtain an RQ-lattice XR = X ⊗Z R, and it is exceptional by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  The cited theorem [4, Theorem 2] depends on a braid group action for exceptional  sequences of lattices, defined using mutations of exceptional pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  The result about  mutations in turn depends on the construction of mutations of pairs of representations of  KQ where K is a field, for which [4] cites [10] and [2] cites [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We take this opportunity  to give a direct proof, based on the following generalization of the Happel-Ringel Lemma  [8, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  4  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let θ : X → Y be a homomorphism between finite-dimensional A-modules  for a hereditary algebra A and suppose that Ext1  A(Y, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If Y is indecomposable, then  either θ is an epimorphism or the map X → Im(θ) is a split epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The map θ gives exact sequences  ξ : 0 → Im(θ)  α−→ Y → Coker(θ) → 0  and  η : 0 → Ker(θ) → X  β−→ Im(θ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  From Ext1  A(Coker(θ), η) we get an exact sequence  · · → Ext1  A(Coker(θ), X)  f−→ Ext1  A(Coker(θ), Im(θ)) → Ext2  A(Coker(θ), Ker(θ)) = 0  so ξ = f(ζ) for some ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus there is a commutative diagram  ζ : 0 −−−→  X  δ  −−−→ Z −−−→ Coker(θ) −−−→ 0  β  \uf8e6\uf8e6�  γ  \uf8e6\uf8e6�  ���  ξ : 0 −−−→ Im(θ)  α  −−−→ Y −−−→ Coker(θ) −−−→ 0  Now the sequence  0 → X  \uf8eb  \uf8edδ  β  \uf8f6  \uf8f8  −−−→ Z ⊕ Im(θ)  �  γ  −α  �  −−−−−−→ Y → 0  is exact, so splits since Ext1  A(Y, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus there are maps  �p  q�  : Z ⊕ Im(θ) → X  and  �  r  s  �  : Y → Z ⊕ Im(θ)  with  �  1Z  0  0  1Im(θ)  �  =  �  δ  β  � �p  q�  +  �  r  s  � �γ  −α�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Thus 1Im(θ) = βq − sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now if θ is not an epimorphism, then αs cannot be an auto-  morphism of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since Y is indecomposable, αs is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' This implies that βq is  invertible, and hence that β is a split epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  If X is also indecomposable, one recovers the Happel-Ringel Lemma: if θ is non-  zero and not an epimorphism, then X → Im(θ) must be an isomorphism, so θ is a  monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Dually we have:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let θ : X → Y be a homomorphism between finite-dimensional A-modules  for a hereditary algebra A and suppose that Ext1  A(Y, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If X is indecomposable,  then either θ is a monomorphism or the map Im(θ) → Y is a split monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  A sequence of exceptional RQ-lattices (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' , Xr) is called an exceptional sequence  provided that HomRQ(Xi, Xj) = Ext1  RQ(Xi, Xj) = 0 for all i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The result about  mutations is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Suppose R is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If (X, Y ) is an exceptional pair of RQ-lattices, then  there are exceptional pairs (LXY, X) and (Y, RY X) given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If HomRQ(X, Y ) = 0  then LXY and RY X are given by the universal exact sequences  0 → Y → LXY → X ⊗R Ext1  RQ(X, Y ) → 0,  0 → Y ⊗R D Ext1  RQ(X, Y ) → RY X → X → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  where D = HomR(−, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If HomRQ(X, Y ) ̸= 0, then the universal map  f : X ⊗R HomRQ(X, Y ) → Y  5  is an epimorphism or a monomorphism, and LXY is its kernel or cokernel, and the  universal map  g : X → HomR(HomRQ(X, Y ), Y ) ∼= Y ⊗R D HomRQ(X, Y )  is an epimorphism or a monomorphism and RY X is its kernel or cokernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since R is reduced and the lattices are rigid, the Hom and Ext spaces are projective  over R of constant rank, independent of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1 we can, as in [4,  Lemma 4], reduce to the case when R = K is an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  In this setting, the claim follows easily once one knows that if HomKQ(X, Y ) ̸= 0 then  Ext1  KQ(X, Y ) = 0, and that f and g are either epimorphisms or monomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Now if θ ∈ HomKQ(X, Y ) is a non-zero map, then by the Happel-Ringel Lemma θ is  either an epimorphism or a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' In the first case we have a exact sequence  · · → Ext1  KQ(X, X) → Ext1  KQ(X, Y ) → Ext2  KQ(X, Ker(θ)) = 0  and in the second case we have  · · → Ext1  KQ(Y, Y ) → Ext1  KQ(X, Y ) → Ext2  KQ(Coker(θ), Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Since Ext1  KQ(X, X) = Ext1  KQ(Y, Y ) = 0, in both cases we get Ext1  KQ(X, Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  If the map f : X ⊗K HomKQ(X, Y ) → Y is not an epimorphism, then the induced  map f ′ : X ⊗K HomKQ(X, Y ) → Im(f) is a split epimorphism by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since  EndKQ(X) = K, the map f can be identified with the universal map from a direct sum  of copies of X to Y given by a basis of HomKQ(X, Y ), from which it is easy to see  that f is right minimal [1, §I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus by [1, Corollary I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='3] there is no non-zero direct  summand of the domain on which f vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus f ′ is an isomorphism, and hence f  is a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The argument for g is dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Classification of rigids  We prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Once Theorem A is proved, it gives Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The  proof of the theorem is the same as Theorem 2(ii) of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let R be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For each real Schur root α, we choose an exceptional  pointwise free RQ-lattice X(α) of rank vector α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Any rigid RQ-lattice X of pointwise  constant rank has a decomposition  X ∼= (X(α1) ⊗R P1) ⊕ · · · ⊕ (X(αr) ⊗R Pr)  where the α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' , αr are distinct real Schur roots, ext(αi, αj) = 0 for all i, j and the Pi  are finitely generated projective R-modules of constant rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover this decomposition  is unique up to isomorphism and reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' In particular any exceptional RQ-lattice of  rank vector α is of the form X(α) ⊗R P with X exceptional pointwise free and P a  projective R-module of constant rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='3, we may assume that Q has no oriented cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We fix temporarily  a homomorphism R → K with K an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Then XK is rigid, so  decomposes as direct sum  XK ∼= Mm1  1  ⊕ · · · ⊕ Mmr  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  with the Mi pairwise non-isomorphic exceptional KQ-modules, Ext1  KQ(Mi, Mj) = 0 for  all i, j and all mi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By [8, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2], we can order the Mi so that (M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' , Mr) is  an exceptional sequence, so Hom(Mi, Mj) = 0 for i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let Mi have dimension vector αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  We have Hom(X(αi), X(αj)) = 0 for i > j since dim Hom(Mi, Mj) = hom(αi, αj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  6  Now Pr = HomRQ(X(αr), X) is a finitely generated projective R-module of constant  rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Consider the evaluation map θ : X(αr) ⊗R Pr → X and let C be its cokernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For  an arbitrary homomorphism R → K with K an algebraically closed field (no longer the  one fixed above), using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2, we can identify θK with the evaluation map  X(αr)K ⊗K HomKQ(X(αr)K, XK) → X  but we know that  XK ∼= (X(α1)K)m1 ⊕ · · · ⊕ (X(αr)K)mr  since both sides are rigid KQ-modules of the same dimension vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus θK is the  inclusion of (X(αr)K)mr as a direct summand of XK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1, θ is a split  monomorphism of R-modules, so C is projective over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Also  CK ∼= Coker(θK) ∼= (X(α1)K)m1 ⊕ · · · ⊕ (X(αr−1)K)mr−1  so Ext1  KQ(CK, CK) = 0 and Ext1  KQ(CK, X(αr)K) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus C is rigid and Ext1(C, Xr) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus X ∼= X(αr) ⊗R Pr ⊕C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now the result follows by induction (on r or on the total  rank of X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  For uniqueness, note that if we have another decomposition  X = (X(β1) ⊗ P ′  1) ⊕ · · · ⊕ (X(βs) ⊗ P ′  s)  with the βi being distinct real Schur roots and the P ′  i projective R-modules of constant  rank, then using a fixed homomorphism R → K and the uniqueness of rigid KQ-modules  of a given dimension vector, we see that s = r and that the βi are a permutation of the  αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus we reduce to the case that βi = αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' But then  P ′  r ∼= HomRQ(X(αr), X) ∼= Pr  and then the corresponding cokernels of the evaluation maps are isomorphic, so by in-  duction P ′  i ∼= Pi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  The special case of exceptional lattices follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Uniqueness for exceptional pointwise frees  We now prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The existence part was Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Suppose X is a point-  wise free lattice for RQ of rank vector α, a real Schur root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2 we can write  R ∼= ˜R/I with ˜R reduced and I contained in the Jacobson radical of ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  We define a pointwise free ˜RQ-lattice ˜X by giving it as a representation of Q by free  ˜R-modules as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For each vertex i we take ei ˜X to be a free ˜R-module of rank αi,  and we fix an isomorphism  (ei ˜X)/I(ei ˜X) ∼= (ei ˜X)R ∼= eiX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  For each arrow a ∈ Q1, multiplication by a induces an R-linear map et(a)X → eh(a)X,  and we lift this to an ˜R-linear map et(a) ˜X → eh(a) ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' This defines ˜X as a representation  of Q and clearly we have ˜XR ∼= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Let E = Ext1  ˜RQ( ˜X, ˜X), a finitely generated ˜R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now ER ∼= Ext1  RQ(X, X) = 0  by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1, so E = IE, so by Nakayama’s Lemma E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus ˜X is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' (This  idea of lifting X to a rigid representation ˜X is taken from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=')  By Theorem B we have ˜X ∼= X(α)⊗ ˜R ˜P, where X(α) is a chosen exceptional pointwise  free ˜RQ-lattice of rank vector α and ˜P is a finitely generated projective ˜R-module of  constant rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  7  Tensoring with R, we get X ∼= Y ⊗R P where Y = X(α)R is an exceptional pointwise  free RQ-lattice and P = ˜P R is a finitely generated projective R-module of constant  rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Now since X and Y are pointwise free of rank vector α, we have  Rαi ∼= eiX ∼= eiY ⊗R P ∼= Rαi ⊗R P ∼= P αi  for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since α is a real root for Q, it is indivisible, that is, its components are coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Thus we can find ai, bi ∈ N such that  1 +  �  i  aiαi =  �  i  biαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Then  P ⊕ R  �  i aiαi ∼= P ⊕  �  i  (Rαi)ai ∼= P ⊕  �  i  (P αi)ai ∼=  �  i  (P αi)bi ∼=  �  i  (Rαi)bi ∼= R  �  i biαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Thus P is stably free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now any stably free projective module of constant rank 1 for a  commutative ring is free, see for example [9, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='11], so P ∼= R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus X ∼= Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  We have already observed that Y is exceptional, hence so is X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  References  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Auslander, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Reiten and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Smalø, Representation theory of Artin algebras, Cambridge  Studies in Advanced Mathematics, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Cambridge University Press, Cambridge, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Crawley-Boevey, Exceptional sequences of representations of quivers, in ‘Representations of  algebras’ (Ottawa, ON, 1992), 117-–124, CMS Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=', 14, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=', Providence, RI,  1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [3] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Crawley-Boevey, Subrepresentations of general representations of quivers, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' 28 (1996), 363—366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [4] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Crawley-Boevey, Rigid integral representations of quivers, in ‘Representation theory of  algebras’ (Cocoyoc, 1994), 155-–163, CMS Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=', 18, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=', Providence, RI,  1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Eisenbud, Commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' With a view toward algebraic geometry, Graduate Texts in  Mathematics, 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Springer-Verlag, New York, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Geiß, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Leclerc and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Schr¨oer, Rigid modules and Schur roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' 295 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' 3-4,  1245–1277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Gorodentsev, Exceptional bundles on surfaces with a moving anticanonical class (Russian),  Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Nauk SSSR Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' 52 (1988), 740-–757, 895;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' translation in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' USSR-Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' 33  (1989), 67-–83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Happel and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Ringel, Tilted algebras, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' 274 (1982), 399-–443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Lam, Serre’s problem on projective modules, Springer Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Springer-Verlag, Berlin, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Rudakov, Exceptional collections, mutations and helices, in ‘Helices and vector bundles’,  1-–6, London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Lecture Note Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=', 148, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Press, Cambridge, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Schofield, General representations of quivers, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' (3) 65 (1992), 46-–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, 33501 Bielefeld, Germany  Email address: wcrawley@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='uni-bielefeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='de  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}

U9E2T4oBgHgl3EQftwgF/content/tmp_files/2301.04072v1.pdf.txt

@@ -0,0 +1,993 @@
+The spectral reconstruction of inclusive rates
+John Bulava 𝑎,∗
+𝑎Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany
+E-mail: john.bulava@desy.de
+A recently re-discovered variant of the Backus-Gilbert algorithm for spectral reconstruction en-
+ables the controlled determination of smeared spectral densities from lattice field theory correlation
+functions. A particular advantage of this approach is the a priori specification of the kernel with
+which the underlying spectral density is smeared, allowing for variation of its peak position,
+smearing width, and functional form. If the unsmeared spectral density is sufficiently smooth
+in the neighborhood of a particular energy, it can be obtained from an extrapolation to zero
+smearing-kernel width at fixed peak position. A natural application for this approach is scattering
+processes summed over all hadronic final states. As a proof-of-principle test, an inclusive rate
+is computed in the two-dimensional O(3) sigma model from a two-point correlation function of
+conserved currents. The results at finite and zero smearing radius are in good agreement with
+the known analytic form up to energies at which 40-particle states contribute, and are sensitive to
+the 4-particle contribution to the inclusive rate. The straight-forward adaptation to compute the
+𝑅-ratio in lattice QCD from two-point functions of the electromagnetic current is briefly discussed.
+The 39th International Symposium on Lattice Field Theory,
+8th-13th August, 2022,
+Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.04072v1  [hep-lat]  10 Jan 2023
+
+The spectral reconstruction of inclusive rates
+John Bulava
+1.
+Introduction
+Lattice QCD simulations proceed by computing 𝑛-point euclidean correlation functions of
+(quasi-) local interpolating operators. Single-hadron states and finite-volume few-hadron states are
+isolated from correlation functions in the asymptotic large euclidean time limit. However, some
+hadronic phenomena are best studied by other means. As an example, this work considers inclusive
+rates defined as a sum over all hadronic final states produced by an external current. At large
+center-of-mass energies, a finite-volume approach to such a process is impractical since it requires
+the isolation of all individual finite-volume levels with arbitrarily many particles. Such processes
+are a cornerstone of QCD and connect the low-energy hadronic and high-energy perturbative
+regimes [1], serving as a manifestation of ‘quark-hadron duality’ [2] whereby perturbative QCD in
+terms of quarks and gluons becomes increasingly effective at computing inclusive rates summed
+over final states consisting entirely of hadrons.
+For concreteness, consider the QCD part of the process 𝑒+𝑒− → hadrons
+𝜌(𝑠) = 𝑅(𝑠)
+12𝜋2 ,
+𝑅(𝑠) = 𝜎 [𝑒+𝑒− → hadrons] (𝑠)
+4𝜋𝛼em(𝑠)2/(3𝑠)
+,
+(1)
+𝜌𝜇𝜈(𝑘) = 1
+2𝜋
+ˆ
+𝑑4𝑥 𝑒−𝑖𝑘·𝑥⟨Ω| ˆ𝑗em
+𝜇 (𝑥) ˆ𝑗em
+𝜈 (0)|Ω⟩ = (𝑔𝜇𝜈𝑘2 − 𝑘𝜇𝑘𝜈) 𝜌(𝑘2),
+(2)
+where ˆ𝑗em
+𝜇
+is the quark-level electromagnetic current. The desired inclusive rate is given by the
+spectral density 𝜌(𝑠), which is also present in the analogous infinite-volume Euclidean correlator
+𝐶(𝑡) =
+ˆ
+𝑑3𝒙 ⟨Ω| ˆ𝑗em
+𝑧 (𝒙) 𝑒− ˆ𝐻𝑡 ˆ𝑗em
+𝑧 (0)†|Ω⟩ =
+ˆ ∞
+0
+𝑑𝜔 𝜔2𝜌(𝜔2) 𝑒−𝜔𝑡 .
+(3)
+The direct determination of 𝜌(𝑠) in lattice QCD is not straightforward, however. First, the inversion
+of integral equations like Eq. 3 using 𝐶(𝑡) evaluated at a finite number of discrete times with
+statistical errors is notoriously ill-posed.
+Furthermore, the finite volume introduces additional
+complication. Even if the inverse problem were solved successfully and the finite-volume euclidean
+correlator 𝐶𝐿(𝑡) used to determine its spectral density 𝜌𝐿(𝑠), it differs qualitatively from its infinite-
+volume counterpart 𝜌(𝑠). While 𝜌𝐿(𝑠) is a sum over Dirac 𝛿-functions for each finite-volume state,
+𝜌(𝑠) is smooth apart from non-analyticities due to the opening of thresholds. In no way does 𝜌𝐿(𝑠)
+‘approach’ 𝜌(𝑠) as 𝐿 → ∞.
+The bridge between finite and infinite volume is made more effectively using the smeared
+spectral density
+𝜌𝜖 (𝐸) =
+ˆ ∞
+0
+𝑑𝜔 𝛿𝜖 (𝐸 − 𝜔) 𝜌(𝜔) ,
+(4)
+where lim𝜖 →0 𝛿𝜖 (𝑥) = 𝛿(𝑥), 𝛿(𝑥) is the Dirac-delta function, and
+´ ∞
+−∞ 𝑑𝑥 𝛿𝜖 (𝑥) = 1. This solves
+(in principle) both of the difficulties mentioned above: the inverse problem can be made arbitrarily
+mild by increasing in the smearing width 𝜖 and 𝜌𝐿,𝜖 (𝐸) approaches its infinite-volume counterpart
+in a well-defined manner. The goal is now to take the ordered double limit [3]
+𝜌(𝐸) = lim
+𝜖 →0+ lim
+𝐿→∞ 𝜌𝐿,𝜖 (𝐸),
+(5)
+2
+
+The spectral reconstruction of inclusive rates
+John Bulava
+the asymptotic corrections to which are discussed in Sec. 21.
+Although spectral reconstruction has a long history in lattice QCD, particularly at finite temper-
+ature [4], the treatment of the inverse problem in Eq. 3 demands special care. In order to define the
+result of the spectral reconstruction procedure, precise knowledge of the smearing kernel in Eq. 4
+is required. As detailed in Sec. 2, the Backus-Gilbert approach [5, 6] is suitable in this respect2.
+Since the estimator for the smeared spectral density ˆ𝜌𝜖 (𝐸) is simply a linear combination of the
+input correlator data ˆ𝜌𝜖 (𝐸) = �
+𝑡 𝑔𝑡 (𝜖, 𝐸)𝐶(𝑡), the resultant smearing kernel is given by the same
+linear combination of decaying exponentials in Eq. 3
+ˆ𝜌𝜖 (𝐸) =
+ˆ
+𝑑𝜔 ˆ𝛿𝜖 (𝐸, 𝜔) 𝜌(𝜔),
+ˆ𝛿𝜖 (𝐸, 𝜔) =
+∑︁
+𝑡
+𝑔𝑡 (𝜖, 𝐸)𝜔2𝑒−𝜔𝑡.
+(6)
+Explicit knowledge of ˆ𝛿𝜖 (𝐸, 𝜔) is a minimum requirement for a well-defined spectral reconstruction
+procedure.
+Naively the Backus-Gilbert approach provides knowledge of the kernel in Eq. 6 only a posteriori
+for a given choice of coefficients. However, the coefficients themselves can be chosen to approximate
+a particular smearing kernel specified a priori [9]. This important innovation is employed here
+and was applied to lattice field theory for the first time in Ref. [10]. In order to understand this
+reconstruction algorithm, and in particular demonstrate control over the systematic errors, a test in a
+controlled context is warranted. Such a test has been performed in Ref. [11] for the two-dimensional
+O(3) sigma model together with attempts at saturating the ordered double limit of Eq. 5.
+The remainder of this work is organized as follows. The spectral reconstruction method is
+presented in Sec. 2 in the context of the O(3) model test mentioned above. Prospects for adapting
+the method to current correlators in lattice QCD is discussed in Sec. 3 and Sec. 4 concludes.
+2.
+O(3) model test
+This section reviews a spectral reconstruction test which was recently published in Ref. [11].
+It employs the spectral reconstruction procedure of Ref. [10] in the two-dimensional O(3) sigma
+model. Consider the standard lattice discretization
+𝑆[𝜎] = 𝛽
+2
+∑︁
+𝑥∈Λ
+𝑎2 ∑︁
+𝜇
+ˆ𝜕𝜇𝜎(𝑥) · ˆ𝜕𝜇𝜎(𝑥) = 𝛽
+∑︁
+𝑥∈Λ
+∑︁
+𝜇
+[1 − 𝜎(𝑥) · 𝜎(𝑥 + 𝑎 ˆ𝜇)] ,
+(7)
+where 𝜎(𝑥) ∈ R3, |𝜎(𝑥)| = 1, and ˆ𝜕𝜇 𝑓 (𝑥) = 1
+𝑎 [ 𝑓 (𝑥 + 𝑎 ˆ𝜇) − 𝑓 (𝑥)]. This model has a conserved
+current
+𝑗 𝑎
+𝜇(𝑥) = 𝛽𝜖 𝑎𝑏𝑐𝜎𝑏(𝑥) ˆ𝜕𝜇𝜎𝑐(𝑥)
+(8)
+at finite lattice spacing, and possesses a dynamically-generated mass gap 𝑚. The total zero spatial
+momentum euclidean current-current correlation function analogous to Eq. 3 (but without the factor
+1Finite lattice spacing effects must also be removed by taking the continuum limit, which is here performed at fixed
+𝜖 and 𝐸.
+2Another spectral reconstruction algorithm for which the smearing kernel is formally known a posteriori is the
+Chebyshev polynomial approach of Ref. [7]. Ref. [8] compares that approach to the one employed here.
+3
+
+The spectral reconstruction of inclusive rates
+John Bulava
+n=2
+n=4
+10
+20
+30
+40
+50
+μ/m
+0.5
+1.0
+1.5
+2.0
+2.5
+ρ(n)(μ)
+Figure 1:
+Left: exactly known 𝑛 = 2 and 𝑛 = 4 particle contributions to the (continuum, infinite-
+volume) spectral density associated with the conserved current in the two-dimensional O(3) sigma model.
+Contributions from states with more particles are insignificant in the energy range shown here. Right: the
+four smearing kernels 𝛿x
+𝜖 (𝑥), where x = {g, c0, c1, c2}, defined in Eq. 9 plotted against 𝑥 with 𝜖 = 1 .
+of 𝜔2) is computed by numerical simulations using the single-cluster algorithm of Ref. [12]. A
+variety of ensembles are generated, with 𝑚𝐿 ≈ 30 − 60 and 𝑎𝑚 ∈ [0.01, 0.04] to assess finite
+volume effects and take the continuum limit. In the continuum the contributions to the associated
+spectral density 𝜌(𝜔) from each fixed-particle number sector can be computed exactly [13]. Below
+energies 𝐸 < 50𝑚, only the 𝑛 = 2 and 𝑛 = 4 particle contributions are significant and are shown in
+Fig.1.
+In order to demonstrate the a priori specification of the smearing kernel, consider four kernels
+with different profiles as a function of 𝑥 = 𝐸 − 𝜔:
+𝛿g
+𝜖 (𝑥) =
+1
+√
+2𝜋𝜖
+exp
+�
+− 𝑥2
+2𝜖2
+�
+,
+𝛿c0
+𝜖 (𝑥) = 1
+𝜋
+𝜖
+𝑥2 + 𝜖2 ,
+(9)
+𝛿c1
+𝜖 (𝑥) = 2
+𝜋
+𝜖3
+(𝑥2 + 𝜖2)2 ,
+𝛿c2
+𝜖 (𝑥) = 8
+3𝜋
+𝜖5
+(𝑥2 + 𝜖2)3 ,
+(10)
+including the gaussian (denoted ‘g’) and three Cauchy-like kernels denoted ‘c𝑛’, for which 𝑛 = 0, 1, 2
+distinguishes the power of the pole. These kernels are depicted in Fig. 1.
+The method advocated in Ref. [10] to reconstruct the smeared spectral density 𝜌x
+𝜖 (𝐸), where
+x = {g, c0, c1, c2}, is based on two criteria. First, the reconstructed smearing kernel ˆ𝛿x
+𝜖 (𝐸, 𝜔)
+should be close to the desired one 𝛿x
+𝜖 (𝐸 − 𝜔). Second, the coefficients {𝑔𝑡 (𝜖, 𝐸)} in Eq. 6 should
+not induce a large statistical variance on the estimator ˆ𝜌x
+𝜖 (𝐸). These two considerations are encoded
+in the functionals
+𝐴[𝑔] =
+ˆ ∞
+𝐸0
+𝑑𝜔
+�
+𝛿x
+𝜖 (𝐸 − 𝜔) − ˆ𝛿x
+𝜖 (𝐸, 𝜔)
+�2 ,
+(11)
+𝐵[𝑔] = Var[ ˆ𝜌x
+𝜖 (𝐸)] =
+∑︁
+𝑡𝑡′
+𝑔𝑡 (𝜖, 𝐸) 𝑔𝑡′(𝜖, 𝐸) Cov[𝐶(𝑡), 𝐶(𝑡′)]
+(12)
+respectively. The coefficients are then chosen to minimize the combination functional 𝐺𝜆[𝑔] = (1−
+𝜆)𝐴[𝑔]/𝐴[0] + 𝜆𝐵[𝑔], where the ‘trade-off’ parameter 𝜆 is introduced. For small 𝜆, the ‘accuracy’
+functional 𝐴[𝑔] takes preference over the ‘precision’ one 𝐵[𝑔] resulting in small systematic but large
+4
+
+The spectral reconstruction of inclusive rates
+John Bulava
+3
+−
+10
+2
+−
+10
+1
+−
+10
+1
+0.66
+0.67
+0.68
+0.69
+0.7
+0.71
+0.72
+0.73
+(E)
+ερ
+ =  0.20
+c
+λ
+ = 0.75m, gauss, 
+ε
+E = 3.0m, 
+A[q]/A[0]
+(E)
+ερ
+1.5
+−
+1
+−
+0.5
+−
+0
+0.5
+1
+1.5
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+ε
+)/
+ω
+(E-
+ε
+λ
+δ
+ε
+ε
+λ
+δ
+ε
+Figure 2: Left: indicative illustration of the trade-off between statistical and systematic errors for a particular
+choice of 𝐸 and 𝜖 on a single ensemble. Each point corresponds to a different 𝜆 and the horizontal band
+indicates the chosen reconstruction (with 𝜆 = 𝜆c = 0.20) for which statistical errors dominate systematic ones.
+Right: for this same setup and 𝜆 = 𝜆c, the reconstructed kernel ˆ𝛿g
+𝜖 (𝐸, 𝜔), together with the desired kernel
+𝛿g
+𝜖 (𝐸 − 𝜔) shown as a solid line. All reconstructions employ the correlator timeslices 𝑡 = 1𝑎, . . . , 160𝑎.
+statistical errors. By contrast, large 𝜆 results in small statistical errors but a reconstructed smearing
+kernel which does not resemble the desired one. The choice of the parameter 𝜆 is performed
+automatically and results in an approximate balance of these two criteria. The effect of varying
+𝜆 is illustrated in Fig. 2 together with the reconstructed kernel ˆ𝛿g
+𝜖 (𝐸, 𝜔) for a sample setup. The
+trade-off between statistical and systematic errors is familiar to lattice field theorists and the left
+panel of Fig. 2 resembles the identification of a plateau in effective mass plots.
+The procedure described above is performed for a variety of 𝐸 and 𝜖, and for all four smearing
+kernels. Next, finite volume effects must be assessed and the continuum limit taken independently
+for each 𝐸, 𝜖, and kernel. Finite-volume effects are assessed at a single lattice spacing by simulating
+two additional ensembles with doubled spatial and temporal extents, respectively. The differences
+Δ𝐿,𝑇 between the spectral reconstruction on the doubled lattices divided by the statistical error are
+shown in Fig. 3, which show (at most) moderately significant hints for finite-𝐿 effects at energies
+near the two-particle production threshold.
+With the finite-volume effects demonstrably controlled for the (𝐸, 𝜖) values and the kernels in
+question, the continuum limit can be investigated. Cutoff effects for ‘on-shell’ quantities in the two-
+dimensional O(3) sigma model have a long history, due to their apparently linear behaviour which
+is caused by large logarithmic corrections [14]. Unfortunately, the analysis there is incomplete
+for the ‘off-shell’ smeared spectral density considered here. In order to explore the influence of
+logarithmic cutoff effects, the fit forms
+𝑄(𝑎) = 𝑄(0) + 𝐶𝑎2𝛽𝑟,
+𝑟 = 0, 3, 6
+(13)
+are explored to extrapolate the smeared spectral densities to the continuum limit. A comparison
+of the extrapolation forms is shown in Fig. 4. The continuum limits are generally mild and well-
+constrained by the data, although the slope becomes steeper for increasing 𝐸.
+Given the assessment of systematic errors due to spectral reconstruction, finite 𝐿 and 𝑇 effects,
+and finite lattice spacing, it is time to confront the computations of 𝜌x
+𝜖 (𝐸) with the exact spectral
+density 𝜌(𝜔) (comprised of the two-, four-, and six-particle contributions) smeared with the exact
+5
+
+The spectral reconstruction of inclusive rates
+John Bulava
+Figure 3: The difference Δ𝐿,𝑇 between spectral reconstruction on ensembles using 𝐿 and 2𝐿 (top row), and
+using ensembles with 𝑇 and 2𝑇 in the bottom row. In both cases Δ𝐿,𝑇 is divided by the statistical error on
+the smaller ensemble. Perhaps some marginally significant hints for finite-𝐿 effects are observed at small 𝐸
+near the two-particle production threshold.
+Figure 4: The continuum limit for a single 𝐸, 𝜖, and smearing kernel using the ansatz of Eq. 13. The shaded
+band indicates the fit, and the horizontal dotted region the extrapolated value for 𝑟 = 3, which is taken as the
+final result.
+6
+
+The spectral reconstruction of inclusive rates
+John Bulava
+Figure 5: Lattice results for 𝜌x
+𝜖 (𝐸) in the continuum limit (the data points shown in the legend) compared
+against the exact spectral density including two-, four-, and six-particle contributions smeared with the exact
+kernel 𝛿x
+𝜖 (𝐸 − 𝜔). The exact results are shown as lines in the top row, and the bottom row shows the ‘pull’
+between the numerical data and the exact result, divided by the statistical and systematic error combined
+in quadrature. A naive histogram of the differences, which ignores correlations among the data, is shown
+horizontally in the bottom row and approximately resembles the unit gaussian.
+smearing kernel 𝛿x
+𝜖 (𝐸 − 𝜔).
+Such a comparison is performed in Fig. 5 where the numerical
+computations are demonstrably consistent with the exact results, within the quoted errors. These
+errors take into account the statistical errors due to the reconstruction, with any residual systematic
+errors from finite-𝐿,𝑇 effects and the continuum limit added in quadrature.
+At this point the verification of the spectral reconstruction approach of Ref. [10] is complete.
+Smeared spectral densities in the two-dimensional O(3) sigma model have been reconstructed with
+smearing kernels specified a priori, which are consistent with the exact result after the continuum
+limit has been taken. Consider now the 𝜖 → 0 limit in Eq. 5. For this, an important property of the
+unsmeared spectral density evident in Fig. 1 is required, namely that it varies increasingly slowly
+with increasing 𝐸. This circumvents the limitations in reconstructing kernels with a fixed 𝜖 and
+increasing 𝐸 evident in Fig. 5: larger smearing widths are sufficient at larger 𝐸. To this end, the
+smearing width is scaled 𝜖 ∝ (𝐸 −2𝑚). Also, rather than using a single smearing kernel, 𝜌x
+𝜖 (𝐸) for
+all kernels are used to perform constrained extrapolations. For this the small-𝜖 expansion is useful
+𝜌x
+𝜖 (𝐸) ≡
+ˆ ∞
+0
+𝑑𝜔 𝛿x
+𝜖 (𝐸 − 𝜔) 𝜌(𝜔) = 𝜌(𝐸) +
+∞
+∑︁
+𝑘=1
+𝑤x
+𝑘𝑎𝑘(𝐸)𝜖 𝑘 ,
+(14)
+7
+
+The spectral reconstruction of inclusive rates
+John Bulava
+x
+𝑤x
+𝑘, even 𝑘
+𝑤x
+𝑘, odd 𝑘
+𝑤x
+1
+𝑤x
+2
+𝑤x
+3
+𝑤x
+4
+g
+𝑘!
+(−2)𝑘/2(𝑘/2)!
+0
+0
+−1
+0
+3
+c0
+1
+1
+1
+1
+1
+1
+c1
+(1 − 𝑘)
+(1 − 𝑘)
+0
+−1
+−2
+−3
+c2
+1
+3 (𝑘 − 3)(𝑘 − 1)
+1
+3 (𝑘 − 3)(𝑘 − 1)
+0
+−1/3
+0
+1
+Table 1: The kernel-dependent coefficients 𝑤x
+𝑘 appearing in the small-𝜖 expansion of Eq. (14). For the c1
+and c2 kernels, 𝑤c1
+3 and 𝑤c2
+5 (respectively) are the non-zero coefficients with lowest odd order.
+where the contribution at the 𝑘th order in 𝜖 is the product of a kernel-independent factor
+𝑎𝑘(𝐸) =
+������
+������
+(−1)𝑘/2
+𝑘!
+�
+𝑑
+𝑑𝐸
+� 𝑘
+𝜌(𝐸) ,
+𝑘 even
+lim𝜂→0+ (−1) (𝑘−1)/2
+2𝜋
+´ ∞
+−∞ d𝜔 𝜌(𝐸+𝜔)+𝜌(𝐸−𝜔)
+(𝜔+𝑖𝜂)𝑘+1
+,
+𝑘 odd
+.
+(15)
+which depends on the unsmeared spectral density, and a kernel-independent piece 𝑤(x)
+𝑘
+which is
+however independent of 𝜌(𝜔). The 𝑤(x)
+𝑘
+for the kernels used here are given for all orders in Tab. 1.
+The c0 kernel is however not practically useful in such extrapolations due to the O(𝜖) term.
+A representative constrained extrapolation, in which all kernels (apart from c0) are used to fit
+for 𝜌(𝐸) and the 𝑎𝑘(𝐸) up to a certain order, is shown in Fig. 6. A final estimate for 𝜌(𝐸) is chosen
+with a statistical error larger than the variation between different extrapolation orders and ranges.
+Repeating this procedure for all values of 𝐸 yields the final results for the spectral density 𝜌(𝐸)
+shown in Fig. 10. Not only do the numerical results agree with the exact spectral density including
+two-, four-, and six-particle contributions, but differ significantly from the two-particle contribution
+alone, indicating the sensitivity to four-particle states. Furthermore, the largest energy of 𝐸 = 40𝑚
+is statistically consistent with the two-loop perturbative result, demonstrating that 𝜌(𝐸) has been
+computed up to the onset of the perturbative regime.
+3.
+Prospects for QCD
+It is in principle straightforward to adopt the analysis of the O(3) sigma model in Sec. 2 to the
+lattice QCD computation of current spectral densities. However, while it is difficult to compare
+the density of finite-volume states in one and three spatial dimensions, the O(3) model setup with
+𝑚𝐿 ≈ 30 may be difficult to achieve in QCD. Fortunately, the masterfield paradigm [15–17] offers
+the possibility of large lattice volumes by accumulating statistics from widely-separated space-time
+regions rather than widely-separated Markov chain elements.
+Work in this direction has been detailed at this conference in talks by M. Cè and P. Fritzsch.
+This section describes preliminary work toward the spectral reconstruction of the isovector vector
+8
+
+The spectral reconstruction of inclusive rates
+John Bulava
+Figure 6: Left: a sample constrained extrapolation using the known coefficients in Tab. 1 up to and including
+O(𝜖4) terms for a fixed energy 𝐸 = 14𝑚. The relative fit ranges of the different smearing kernels are adjusted
+so that each kernel has an equal amount of support between the two-particle threshold 2𝑚 and 𝐸. Right:
+variation of the extrapolated value of 𝜌(𝐸) for different extrapolation ranges and orders. The final result is
+conservatively taken as the horizontal shaded region.
+Figure 7: Left: a selection of some of values of 𝜖 (given in the legend) used in the 𝜖 → 0 extrapolation,
+together with the exact smeared spectral density shown as solid lines for the gaussian kernel. Right: the
+final extrapolated results for 𝜌(𝐸) together with the exact two-particle contribution to the spectral density
+and the sum of the two-, four-, and six-particle contributions. The two-loop perturbative spectral density is
+also shown.
+current spectral density with the collaborators and setup mentioned in those talks. Using 𝑁f = 2+1
+dynamical flavors of stabilized Wilson fermions [17] at 𝑎 = 0.09 fm, two ensembles were generated
+with (𝐿/𝑎)4 = 964 and 1924. The analysis described below is based on two and five thermalized,
+widely separated configuations on the 𝐿/𝑎 = 96 and 192 ensembles, respectively. Details about the
+construction of the correlators and the estimation of the statistical errors were given by M. Cè. For
+the data presented here, a variant of the bootstrap procedure is employed.
+As suggested in Sec. 1, the prototypical QCD analogue of Sec. 2 is the hadronic component
+to 𝑒+𝑒− → hadrons, which can be obtained by solving the inverse problem of the euclidean
+current-current correlator projected onto zero spatial momentum in Eq. 3. However, including both
+the isoscalar and isovector components of the electromagnetic current requires valence quark-line
+9
+
+The spectral reconstruction of inclusive rates
+John Bulava
+disconnected Wick contractions, incurring additional computational cost and statistical variance.
+Consider then the simpler case of the isovector-vector correlator. Phenomenologically this spectral
+density can be accessed directly from hadronic decays of the tau lepton [18]. A state-of-the-art
+phenomenological determination of the isovector-vector spectral density is performed in Ref. [19].
+The spectral reconstruction approach of Sec. 2 is adopted nearly identically here, apart from
+some key differences.
+First, the basis functions provided by the correlator data in Sec. 2 are
+𝑏𝑡 (𝜔) = e−𝜔𝑡 + e−𝜔(𝑇 −𝑡), but those employed in this analysis from Eq. 3 are 𝑏𝑡 (𝜔) = 𝜔2 e−𝜔𝑡.
+The flexibility of the formalism of Sec. 2 to handle these different basis functions is an advantage
+over the Chebyshev approach of Ref. [7]. Also, for these large lattices the finite temporal extent
+can be demonstrably ignored. For a first test of the approach in QCD, only the gaussian smearing
+kernel from Eq. 9 is considered. All correlator timeslices from 𝑡min = 𝑎 to 𝑡max = 35𝑎 are used in
+the reconstruction, and all arithmetic operations are performed with 400 bits of computer precision
+using the Arb library [20].
+Another innovation for this analysis compared to Sec. 2 is the procedure for choosing the 𝜆 at
+which statistical errors dominate the systematic errors. As suggested by the left panel in Fig. 2, the
+procedure in Sec. 2 which balances the two functionals 𝐴[𝑔]/𝐴[0] and 𝐵[𝑔] from Eq. 11 is perhaps
+over-conservative and somewhat arbitrary. The alternative approach employed here makes use of
+one of the possible constraints introduced in Ref. [11]. By the addition of a lagrange multiplier, it is
+possible to enforce constraints on the reconstructed smearing kernel ˆ𝛿g
+𝜖 (𝐸, 𝜔). Ref. [11] describes
+how to impose the coincidence of the reconstructed and desired kernels at a particular point
+ˆ𝛿g
+𝜖 (𝐸, 𝜔∗) = 𝛿g
+𝜖 (𝐸 − 𝜔∗).
+(16)
+Although Ref. [11] only considers 𝜔 = 𝐸, the generalization to arbitrary 𝜔∗, even outside the
+interval [𝐸0, ∞), is straightforward.
+Using this ‘equal value’ constraint on the reconstructed kernel, it is possible to estimate how
+small 𝐴[𝑔]/𝐴[0] must be for the statistical errors to dominate. An ‘ensemble’ of reconstructions
+are performed with different values of 𝜔∗, in addition to the unconstrained one. The systematic error
+estimate is then obtained from the variation of ˆ𝜌g
+𝜖 (𝐸) among this ensemble at similar 𝐴[𝑔]/𝐴[0].
+The point at which this variation is smaller than the statistical error on the unconstrained result is
+taken as the optimal reconstruction. Of course this procedure depends on the ensemble of constraint
+points {𝜔∗} which are considered. However, it is sensitive the unsmeared spectral density 𝜌(𝜔), in
+contrast to the approach of Ref. [11]. If additional values of 𝜔∗ are added for which 𝜌(𝜔∗) has little
+support, these will likely differ little from the unconstrained case, apart from possible variations in
+ˆ𝛿g
+𝜖 (𝐸, 𝜔) away from 𝜔∗ induced by the constraint at 𝜔∗. An illustration of this procedure is given
+in Fig. 8.
+After applying the procedure discussed above for a variety of 𝜖 and 𝐸 for the gaussian kernel
+on each of the 𝐿 = 9 fm and 18 fm ensembles, finite volume effects can be examined. This is done
+in Fig. 9, using 𝑣1(𝑠) = 2𝜋2𝜌(𝑠) for a variety of energies at two different values of the smearing
+width 𝜖. While there are possibly hints of finite-volume effects at the one-to-few sigma level at both
+𝜖, these effects are generally under control. Additional volumes will however elucidate the situation
+in the future.
+We finally turn to a comparison of the reconstructed isovector vector spectral density with
+experiment [21]. For this a preliminary value of the vector current renormalization factor 𝑍𝑉 is
+10
+
+The spectral reconstruction of inclusive rates
+John Bulava
+3
+−
+10
+2
+−
+10
+1
+−
+10
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+0.11
+0.12
+0.13
+A[g]/A[0]
+(E)
+ε
+g
+ρ
+π
+ = 0.5m
+ε
+, 
+π
+E = 2.5m
+none
+π
+2.5m
+π
+2m
+π
+3m
+π
+4m
+π
+5m
+π
+6m
+π
+7m
+π
+8m
+π
+9m
+π
+10m
+π
+11m
+π
+12m
+2
+−
+0
+2
+4
+6
+8
+10
+12
+0.05
+−
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+ε
+ - E)/
+ω
+(
+εδ
+ε
+ε
+g
+δ
+ε
+none
+π
+2.5m
+π
+2m
+π
+3m
+π
+4m
+π
+5m
+π
+6m
+π
+7m
+π
+8m
+π
+9m
+π
+10m
+π
+11m
+π
+12m
+Figure 8: Illustration of the method for choosing the optimal tradeoff parameter 𝜆 described in the text for the
+gaussian reconstruction on the 𝐿 = 18 fm ensemble with 𝜖 = 0.5𝑚 𝜋 and 𝐸 = 2.5𝑚 𝜋. Left: different values of
+𝜆 for the unconstrained reconstruction and reconstructed kernels constrained to agree with 𝛿g
+𝜖 (𝐸 − 𝜔∗) at the
+various values of 𝜔∗ indicated in the legend. The horizontal band indicates the chosen estimate for which the
+statistical error on the unconstrained reconstruction covers the spread given by the ensemble of constraints.
+For comparison, the method for balancing statistical and systematic errors of Sec. 2 (and Ref. [11]) chooses
+the unconstrained point with 𝐴[𝑔]/𝐴[0] ≈ 0.0016. Right: the reconstructed smearing kernel compared
+to the desired gaussian (solid line) for each member of the constraint ensemble near the chosen value of
+𝐴[𝑔]/𝐴[0] indicated by the horizontal band in the left plot. The residual variation between the different
+constraints is evidently smaller than the statistical error on the constrained reconstruction, although perhaps
+additional values of 𝜔∗ near 𝜔∗ − 𝐸 ≈ 2𝜖 should be added in the future.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+)
+2
+s (GeV
+(s)
+ε
+1,
+g
+v
+π
+ = 0.5m
+ε
+L = 9 fm
+L = 18 fm
+0
+0.5
+1
+1.5
+2
+2.5
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+)
+2
+s (GeV
+(s)
+ε
+1,
+g
+v
+π
+ = 1.0m
+ε
+L = 9 fm
+L = 18 fm 
+Figure 9: Finite volume effects in the reconstructed vector isovector spectral density on the two masterfield
+ensembles described in the text. Gaussian smearing is used for a variety of energies at smearing width
+𝜖 = 0.5𝑚 𝜋, shown on the left, and 𝜖 = 𝑚 𝜋, shown on the right. These effects are generally small apart from
+some mild discrepancies near 𝑠 = 0.4 GeV2 for 𝜖 = 0.5𝑚 𝜋 and 𝑠 = 0.75GeV2 for 𝜖 = 𝑚 𝜋. Additional smaller
+lattice volumes could further examine these potential finite volume effects.
+11
+
+The spectral reconstruction of inclusive rates
+John Bulava
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+0
+0.5
+1
+1.5
+2
+2.5
+3
+)
+2
+s (GeV
+(s)
+ε
+1,
+g
+v
+ = 265 MeV, a = 0.09 fm
+π
+L = 18 fm, m
+π
+ = 0.5m
+ε
+π
+ = 0.75m
+ε
+π
+ = 1.0m
+ε
+π
+ = 1.25m
+ε
+0
+0.5
+1
+1.5
+2
+2.5
+3
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+τ– → V–ντ
+π–π0
+π–3π0, 2π–π+π0, (6π)–
+ωπ–, ηπ–π0, (KK
+–(π))–
+QCD prediction
+parton model
+s   (GeV2)
+v1(s)
+ALEPH
+Figure 10:
+Comparison of the lattice QCD results for the isovector vector spectral density on the larger
+𝐿/𝑎 = 192 master field ensemble discussed in the text (shown on the left), with experimental results for
+hadronic 𝜏-decays on the right. Statistical errors due to the scale setting and the renormalization of the vector
+current are not yet taken into account.
+employed, which was presented at this conference by J. Kuhlmann. The statistical error on 𝑍𝑉
+is ignored in these preliminary results, as is the error on the lattice scale, which is crudely set
+by assuming 𝑚 𝜋 = 265 MeV. The results are summarized in Fig. 10, and broadly resemble the
+experimental plot, with a narrow peak likely due to the 𝜌(770) vector resonance followed by a slow
+rise due to four-particle states. Particularly interesting is the mild indication of this rise in the lattice
+QCD data, which (like in the O(3) model) show the effects of four-pion states. It should be noted
+that the current state-of-the-art for the finite-volume approach to lattice QCD scattering amplitudes
+is the numerical computation of (exclusive rather than inclusive) three-pion scattering amplitudes3.
+4.
+Conclusions
+Alternative techniques are required to compute phenomena arising from many hadronic states.
+The spectral reconstruction of smeared spectral densities from euclidean correlator data not only
+bridges the gap between finite and infinite volume, but also helps to regulate the ill-posed nature
+of the problem. The application discussed here is the computation of inclusive rates summed over
+all final states produced by an external current. In the two-dimensional O(3) model, after taking
+the continuum limit, the algorithm presented in Sec. 2 (first proposed for lattice field theory in
+Ref. [10]) results in smeared spectral densities consistent with known analytic results. Spectral
+reconstruction algorithms based on the Backus-Gilbert approach [5, 6] enable a precise definition
+of the smeared spectral density that has been computed, while the modification of Ref. [9] further
+allows the a priori specification of a desired smearing kernel. The simple linear ansatz on which
+these approaches are based enables the direct expression for the smearing kernel given in Eq. 6.
+Smeared spectral densities are useful not only for inclusive decay rates. An incomplete list
+of recent applications of the Backus-Gilbert approach includes the nucleon hadronic tensor [24],
+3For a review of the current status of computations of three-particle scattering amplitudes using the finite-volume
+approach, see the presentation by F. Romero-López and the recent reviews in Refs. [22, 23].
+12
+
+The spectral reconstruction of inclusive rates
+John Bulava
+the determination of PDFs from Ioffe time data [25], and the photon emissivity of the quark-
+gluon plasma [26]. These applications do not employ the algorithmic variant enabling a priori
+specification of the smearing kernel, but could perhaps benefit from it in the future. This a priori
+specification of the kernel enabled in Refs. [9, 10] is also present in the Chebyshev approach of
+Ref. [7], but the stabilizing effect of the functional 𝐵[𝑔] in Eq. 11 is naively not present. The work
+of Ref. [8] is a first step towards comparing the two approaches.
+The advantages of the a priori approach are leveraged in the two-dimensional O(3) model to
+perform joint constrained 𝜖 → 0 extrapolations with several different kernels. The presence of the
+narrow 𝜌(770) peak in the isovector vector spectral density in QCD discussed in Sec. 3 complicates
+such an extrapolation and more work is required toward an implementation. A similar approach has
+been employed to compute inclusive decay rates in Refs. [27, 28], and taken up by additional groups
+in Refs. [8, 29, 30]. Work towards computing the 𝑅-ratio was reported in this conference [31], as
+well as a similar analyses of the total hadronic tau decay rate [32, 33], albeit with a wider gaussian
+smearing radius than employed here. The interplay between the spatial extent and the smallest
+achievable smearing width requires further study. Furthermore, the a priori approach of Ref. [7]
+led to the direct computation of the Borel transform of a current-current correlator required for
+the Shifman-Vainshtein-Zakharov sum rule in Ref. [34], possibly opening the door for additional
+interaction between lattice QCD and QCD sum rules. Another interesting application is pursued in
+Ref. [35] in which fits to smeared spectral densities are considered as an alternative to ‘standard’
+spectroscopy. Additional applications could appear in the future. The a priori approach in principle
+enables the computation of exclusive scattering amplitudes using Refs. [36, 37], while the formalism
+for inclusive rates was developed already in Refs. [3, 38].
+References
+[1] A. Pich, Precision physics with inclusive QCD processes, Prog. Part. Nucl. Phys. 117 (2021)
+103846, [arXiv:2012.04716].
+[2] E. C. Poggio, H. R. Quinn, and S. Weinberg, Smearing the Quark Model, Phys. Rev. D 13
+(1976) 1958.
+[3] M. T. Hansen, H. B. Meyer, and D. Robaina, From deep inelastic scattering to heavy-flavor
+semileptonic decays: Total rates into multihadron final states from lattice QCD, Phys. Rev. D
+96 (2017), no. 9 094513, [arXiv:1704.08993].
+[4] O. Kaczmarek and H.-T. Shu, Spectral and Transport Properties from Lattice QCD, Lect.
+Notes Phys. 999 (2022) 307–345, [arXiv:2206.14676].
+[5] G. Backus and F. Gilbert, The resolving power of gross earth data, Geophysical Journal
+International 16 (1968), no. 2 169–205.
+[6] G. Backus and F. Gilbert, Uniqueness in the inversion of inaccurate gross earth data,
+Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and
+Engineering Sciences 266 (1970), no. 1173 123–192,
+[http://rsta.royalsocietypublishing.org/content/266/1173/123.full.pdf].
+13
+
+The spectral reconstruction of inclusive rates
+John Bulava
+[7] G. Bailas, S. Hashimoto, and T. Ishikawa, Reconstruction of smeared spectral function from
+Euclidean correlation functions, PTEP 2020 (2020), no. 4 043B07, [arXiv:2001.11779].
+[8] A. Barone, S. Hashimoto, A. Jüttner, T. Kaneko, and R. Kellermann, Inclusive semi-leptonic
+𝐵(𝑠) mesons decay at the physical 𝑏 quark mass, in 39th International Symposium on Lattice
+Field Theory, 11, 2022. arXiv:2211.15623.
+[9] F. Pijpers and M. Thompson, Faster formulations of the optimally localized averages method
+for helioseismic inversions, Astronomy and Astrophysics 262 (08, 1992) L33–L36.
+[10] M. Hansen, A. Lupo, and N. Tantalo, Extraction of spectral densities from lattice correlators,
+Phys. Rev. D 99 (2019), no. 9 094508, [arXiv:1903.06476].
+[11] J. Bulava, M. T. Hansen, M. W. Hansen, A. Patella, and N. Tantalo, Inclusive rates from
+smeared spectral densities in the two-dimensional O(3) non-linear 𝜎-model, JHEP 07
+(2022) 034, [arXiv:2111.12774].
+[12] M. Luscher and U. Wolff, How to Calculate the Elastic Scattering Matrix in Two-dimensional
+Quantum Field Theories by Numerical Simulation, Nucl. Phys. B339 (1990) 222–252.
+[13] J. Balog and M. Niedermaier, Off-shell dynamics of the O(3) NLS model beyond Monte
+Carlo and perturbation theory, Nucl. Phys. B 500 (1997) 421–461, [hep-th/9612039].
+[14] J. Balog, F. Niedermayer, and P. Weisz, The Puzzle of apparent linear lattice artifacts in the
+2d non-linear sigma-model and Symanzik’s solution, Nucl. Phys. B 824 (2010) 563–615,
+[arXiv:0905.1730].
+[15] M. Lüscher, Stochastic locality and master-field simulations of very large lattices, EPJ Web
+Conf. 175 (2018) 01002, [arXiv:1707.09758].
+[16] L. Giusti and M. Lüscher, Topological susceptibility at 𝑇 > 𝑇c from master-field simulations
+of the SU(3) gauge theory, Eur. Phys. J. C 79 (2019), no. 3 207, [arXiv:1812.02062].
+[17] A. Francis, P. Fritzsch, M. Lüscher, and A. Rago, Master-field simulations of O(𝑎)-improved
+lattice QCD: Algorithms, stability and exactness, Comput. Phys. Commun. 255 (2020)
+107355, [arXiv:1911.04533].
+[18] M. Davier, A. Hocker, and Z. Zhang, The Physics of Hadronic Tau Decays, Rev. Mod. Phys.
+78 (2006) 1043–1109, [hep-ph/0507078].
+[19] D. Boito, M. Golterman, K. Maltman, S. Peris, M. V. Rodrigues, and W. Schaaf, Strong
+coupling at the 𝜏-mass scale from an improved vector isovector spectral function, in 16th
+International Workshop on Tau Lepton Physics , 12, 2021. arXiv:2112.05413.
+[20] F. Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE
+Transactions on Computers 66 (2017) 1281–1292.
+14
+
+The spectral reconstruction of inclusive rates
+John Bulava
+[21] ALEPH Collaboration, S. Schael et al., Branching ratios and spectral functions of tau
+decays: Final ALEPH measurements and physics implications, Phys. Rept. 421 (2005)
+191–284, [hep-ex/0506072].
+[22] M. T. Hansen and S. R. Sharpe, Lattice QCD and Three-particle Decays of Resonances, Ann.
+Rev. Nucl. Part. Sci. 69 (2019) 65–107, [arXiv:1901.00483].
+[23] M. Mai, U.-G. Meißner, and C. Urbach, Towards a theory of hadron resonances,
+arXiv:2206.01477.
+[24] XQCD Collaboration, J. Liang, T. Draper, K.-F. Liu, A. Rothkopf, and Y.-B. Yang, Towards
+the nucleon hadronic tensor from lattice QCD, Phys. Rev. D 101 (2020), no. 11 114503,
+[arXiv:1906.05312].
+[25] J. Karpie, K. Orginos, A. Rothkopf, and S. Zafeiropoulos, Reconstructing parton distribution
+functions from Ioffe time data: from Bayesian methods to Neural Networks, JHEP 04 (2019)
+057, [arXiv:1901.05408].
+[26] M. Cè, T. Harris, A. Krasniqi, H. B. Meyer, and C. Török, Photon emissivity of the
+quark-gluon plasma: A lattice QCD analysis of the transverse channel, Phys. Rev. D 106
+(2022), no. 5 054501, [arXiv:2205.02821].
+[27] P. Gambino, S. Hashimoto, S. Mächler, M. Panero, F. Sanfilippo, S. Simula, A. Smecca, and
+N. Tantalo, Lattice QCD study of inclusive semileptonic decays of heavy mesons, JHEP 07
+(2022) 083, [arXiv:2203.11762].
+[28] P. Gambino and S. Hashimoto, Inclusive Semileptonic Decays from Lattice QCD, Phys. Rev.
+Lett. 125 (2020), no. 3 032001, [arXiv:2005.13730].
+[29] R. Kellermann, A. Barone, S. Hashimoto, A. Jüttner, and T. Kaneko, Inclusive semi-leptonic
+decays of charmed mesons with Möbius domain wall fermions, in 39th International
+Symposium on Lattice Field Theory, 11, 2022. arXiv:2211.16830.
+[30] P. Gambino, S. Hashimoto, S. Mächler, M. Panero, F. Sanfilippo, S. Simula, A. Smecca, and
+N. Tantalo, Inclusive semileptonic 𝐵-decays from lattice QCD, in 39th International
+Symposium on Lattice Field Theory, 11, 2022. arXiv:2211.11833.
+[31] C. Alexandrou et al., Lattice calculation of the R-ratio smeared with Gaussian kernel, in 39th
+International Symposium on Lattice Field Theory, 12, 2022. arXiv:2212.12493.
+[32] A. Evangelista, R. Frezzotti, G. Gagliardi, V. Lubicz, F. Sanfilippo, S. Simula, and
+N. Tantalo, Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton, in
+39th International Symposium on Lattice Field Theory, 1, 2023. arXiv:2301.00796.
+[33] C. Alexandrou et al., Probing the 𝑅-ratio on the lattice, arXiv:2212.08467.
+[34] T. Ishikawa and S. Hashimoto, Spectral sum of current correlators from lattice QCD, Phys.
+Rev. D 104 (2021), no. 7 074521, [arXiv:2103.06539].
+15
+
+The spectral reconstruction of inclusive rates
+John Bulava
+[35] L. Del Debbio, A. Lupo, M. Panero, and N. Tantalo, Multi-Representation Dynamics of SU(4)
+Composite Higgs Models: Chiral Limit and Spectral Reconstructions, arXiv:2211.09581.
+[36] J. Bulava and M. T. Hansen, Scattering amplitudes from finite-volume spectral functions,
+Phys. Rev. D 100 (2019), no. 3 034521, [arXiv:1903.11735].
+[37] M. Bruno and M. T. Hansen, Variations on the Maiani-Testa approach and the inverse
+problem, JHEP 06 (2021) 043, [arXiv:2012.11488].
+[38] H. Fukaya, S. Hashimoto, T. Kaneko, and H. Ohki, Towards fully nonperturbative
+computations of inelastic ℓ𝑁 scattering cross sections from lattice QCD, Phys. Rev. D 102
+(2020), no. 11 114516, [arXiv:2010.01253].
+16
+

UtAyT4oBgHgl3EQf8voh/content/tmp_files/2301.00860v1.pdf.txt

@@ -0,0 +1,1105 @@
+arXiv:2301.00860v1  [hep-ph]  2 Jan 2023
+Dijet azimuthal decorrelation in e+e− annihilation
+Hana Benslamaa, Yazid Delendaa,∗, Kamel Khelifa-Kerfab
+aLaboratoire de Physique des Rayonnements et de leurs Interactions avec la Mati`ere,
+D´epartement de Physique, Facult´e des Sciences de la Mati`ere,
+Universit´e de Batna-1, Batna 05000, Algeria
+bLaboratoire de Math´ematique et Applications,
+D´epartement de Physique, Facult´e des Sciences et Technologies
+Universit´e Ahmed Zabana de Relizane, Relizane 48000, Algeria
+Abstract
+We examine non-global and clustering logarithms in the distribution of the azimuthal decorrelation between
+two jets in e+e− → dijet events, where the jets are defined with E-scheme recombination in the generalized kt
+algorithm. We calculate at one loop and to all orders the leading global single logarithms in the distribution
+of the said observable.
+We also compute at fixed order up to four loops the non-global and clustering
+logarithms, and numerically resum them to all orders in the large-Nc approximation.
+We compare our
+results at O(αs) and O(α2
+s) with those of the EVENT2 fixed-order Monte Carlo program and find agreement
+of the leading singular behavior of the azimuthal decorrelation distribution. We find that the impact of
+non-global logarithms on the resummed distribution in the anti-kt algorithm is substantial, while it is
+significantly smaller in the kt algorithm. Furthermore, the combined clustering and non-global logarithms
+in the kt algorithm have an even smaller effect on the distribution. Finally, we use the program Gnole
+to calculate the resummed distribution at NLL accuracy, thus achieving state-of-the-art accuracy for the
+resummation of this quantity.
+Keywords:
+QCD, Jets, Resummation
+1. Introduction
+The production of jets in e+e− collisions is a simple and clean environment, yet rich of physics, to test
+QCD and the Standard Model. It will be used in future colliders such as the ILC and FCC-ee in order to
+make precise measurements of QCD-related quantities, which together with detailed theoretical calculations
+will pave the way towards potential discovery of new-physics phenomena.
+At lowest order two correlated jets are produced back-to-back with a relative azimuthal angle equal to
+π. At higher orders the jets manifest a decorrelation of azimuthal angle δφ which is enhanced near the
+back-to-back limit. The quantity δφ, being sensitive to soft/collinear QCD effects, is of great interest in
+the phenomenology of perturbative and non-perturbative QCD dynamics. For instance it has been used
+to study unintegrated parton distribution functions in deep-inelastic e − p scattering (DIS) [1] and small-x
+BFKL effects [2], as well as measurements of the QCD coupling at various scales [3]. Many studies have
+been devoted to the distribution of δφ in various processes, such as dijet production in p − p [4, 5, 6] (and
+even p − pb [7]) collisions and DIS [2, 8]. Experimentally, boson-jet (in pp collisions) [9] and lepton-jet or
+photon-jet (in DIS) [10, 11] decorrelations have been measured.
+Near the back-to-back limit, the distribution of the azimuthal decorrelation is characterized by large
+logarithms preventing the convergence of the perturbative series, and thus need to be resummed to all
+∗Corresponding author.
+Email addresses: hana.benslama@univ-batna.dz (Hana Benslama), yazid.delenda@univ-batna.dz (Yazid Delenda,
+kamel.khelifakerfa@univ-relizane.dz (Kamel Khelifa-Kerfa)
+Preprint submitted to Physics Letters B
+January 4, 2023
+
+orders. Depending on the nature of the algorithm being used to define the jets, the leading logarithms in
+this distribution can be double or single logarithms. For instance, in pt-weighted recombination scheme of
+the kt [12, 13, 14], anti-kt [15] and Cambridge/Aachen algorithms [16, 17], the leading logarithms are double,
+αn
+s L2n, while in E-scheme recombination they are single, αn
+s Ln, with L = ln(1/δφ). In the former scheme,
+the jets recoil against emissions everywhere in the phase space, and in particular soft and collinear emissions
+to these jets, which leads to the double logarithms in δφ. On the other hand, in the latter (E-scheme), the
+jets recoil only against emissions that do not get clustered to them, and hence only away-from-jets emissions
+contribute to δφ, resulting in leading soft wide-angle single logarithmic contributions.
+In addition to this, the classification of the δφ observable, in E-scheme definition, falls in the “non-global”
+category, and as a consequence its distribution receives contributions from single non-global (NGLs) [18, 19]
+and/or clustering (CLs) [20, 21] logarithms. The resummation of these logarithms is not straightforward,
+and is usually performed numerically via Monte Carlo (MC) programs in the planar (large-Nc) limit.
+In this letter, we are interested in the calculation of NGLs and/or CLs for the δφ distribution both in the
+kt and anti-kt algorithms. We compute the coefficients of these logarithms as a function of the jet radius R
+up to O(α3
+s), and at O(α4
+s) in the anti-kt algorithm at small R. We use the fixed-order MC program EVENT2
+[22, 23] in order to compare the leading singular behavior of the δφ distribution with our results at O(αs)
+and O(α2
+s). We also compute the resummed NGLs and CLs at all orders in the large-Nc limit using the MC
+code of refs. [18, 19] as well as the recently-published program Gnole [24, 25] (in the anti-kt algorithm).
+The latter program is also used to compute the resummed differential δφ distribution at next-to-leading
+logarithmic (NLL) accuracy, in which we additionally control all the sub-leading logarithms αn+1
+s
+Ln in the
+exponent of the resummation, and quantify the corresponding scale uncertainties.
+This letter is organized as follows. In the next section we compute at O(αs) the leading-order distribution
+focusing on the logarithmic contribution, and compare with fixed-order MC programs at this order. In
+section 3, we present the calculation of NGLs and CLs at O(α2
+s) and show plots of the coefficients of these
+logarithms as a function of the jet radius and comment on the relative size of these coefficients. We also
+compare at this order the calculated δφ distribution with the output of the program EVENT2, thus confirming
+our results. In section 4 we extend the calculation to O(α3
+s) and (in the anti-kt algorithm and at small R)
+O(α4
+s), and point out the significantly different color structure of NGLs in kt clustering. In section 5 we
+present the all-orders resummation of the NGLs and/or CLs in the large-Nc limit up to LL accuracy for the
+kt algorithm, and NLL accuracy in the anti-kt clustering. Finally we draw our conclusions in section 6.
+2. One-loop calculation and the global form factor
+In this letter we consider the process of dijet production in e+e− annihilation at centre-of-mass energy
+√s. The jets are reconstructed with the kt [14] or anti-kt [15] algorithms, suited for e+e− annihilation, with
+merging and stopping distances dij and di defined by
+dij = min
+�
+E2p
+i , E2p
+j
+� 1 − cos θij
+1 − cos R ,
+di = E2p
+i
+,
+(1)
+where p = +1 for the kt algorithm and p = −1 for the anti-kt algorithm. Here R is the jet radius, Ei
+is the energy of the ith parton in the final state, and θij is the opening angle between partons i and j.
+The algorithm sequentially merges objects i and j whenever dij is the smallest of all merging and stopping
+distances, and if an object i has its stopping distance as the smallest then it gets admitted to the list of
+final inclusive jets. The algorithm keeps recursing until all partons are clustered into jets. In this letter,
+we assume the jet kinematics to be defined with E-scheme recombination, such that the 4-momentum of a
+merged object simply equals the vectorial sum of the momenta of its constituents.
+At the Born level, the two jets are produced back-to-back, and their relative azimuthal angle (with
+respect to the beam axis) is exactly π. The observable we are interested in is the deviation from π of this
+relative azimuthal angle, δφ, when soft gluons are emitted at higher orders. It is straightforward to obtain
+the following expression for δφ in terms of the transverse momenta of the emitted gluons κti and their
+2
+
+azimuthal angles ϕi, with respect to the beam axis
+δφ =
+������
+�
+i/∈jets
+κti
+pt
+sin ϕi
+������
+,
+(2)
+where pt is the jet transverse momentum. The (algebraic) sum is over all emitted gluons that are not clustered
+to any of the two measured (leading) jets. This definition is valid only at single leading logarithmic (LL)
+accuracy, and we shall give the proper definition, valid at NLL accuracy, in section 5. Furthermore, the
+expression of δφ in eq. (2) only applies in E-scheme recombination. Alternative jet recombination schemes
+exist for which the jet kinematics take a different form, e.g. the pt-weighted scheme, and the resummation
+takes an entirely different structure [8].
+At one loop the cumulative cross-section for events with azimuthal decorrelation δϕ less than some ∆,
+normalized to the Born cross-section, reads
+Σ1(∆) = −2 CF
+� αs(kt)
+π
+dkt
+kt
+d cos θ
+sin2 θ
+dφ
+2π ωk
+q¯q Θout(k) Θ (κt| sin ϕ|/pt − ∆) ,
+(3)
+where θ, φ and kt are the polar angle, azimuthal angle and transverse momentum of the emitted soft gluon
+k, with respect to the jet (thrust) axis (the back-to-back outgoing jets are aligned along the z axis), CF
+is the color factor associated with the emission of the gluon off the hard q¯q dipole, and αs is the strong
+coupling with argument kt. The invariant antenna function ωk
+q¯q is given by
+ωk
+q¯q = k2
+t
+2
+pq · p¯q
+(pq · k)(p¯q · k) = 1 ,
+(4)
+with pi denoting the momentum of particle i
+pq =
+√s
+2 (1, 0, 0, 1) ,
+(5a)
+p¯q =
+√s
+2 (1, 0, 0, −1) ,
+(5b)
+k = E (1, cos φ sin θ, sin φ sin θ, cos θ) ,
+(5c)
+where kt = E sin θ. The constraint Θout(k) restricts the emitted gluon to be outside the jets and forbids it
+from being clustered to any of them in order to induce a non-zero azimuthal decorrelation. It depends on
+the jet radius R and is given, in the generalized algorithm, by
+Θout(k) = Θ (cos R − | cos θ|) .
+(6)
+Since the soft emission is restricted to be outside both jets then there are no collinear logarithms as-
+sociated with this observable. This means that the leading logarithms are single, which allows us at LL
+accuracy to simply change Θ(κt| sin ϕ|/pt − ∆) → Θ(2 kt/√s − ∆), since any factor multiplying kt will only
+induce sub-leading logarithms. 1 We can then perform the integration over kt using the one-loop running
+of the coupling (which formally enters the distribution at higher orders), and write the result in terms of
+the evolution parameter t defined by
+t(∆) ≡
+� √s/2 αs(kt)
+π
+dkt
+kt
+Θ
+�
+2 kt/√s − ∆
+�
+= −
+1
+2πβ0
+ln(1 − 2λ) ,
+(7)
+where λ = αs(√s/2)β0 ln(1/∆) and β0 is the one-loop coefficient of the QCD beta function. The angular
+integration is straightforward and we obtain
+Σ1(∆) = −2 CF t(∆)
+� 1
+−1
+dc
+1 − c2 Θ (cos R − |c|) = −2 CF t(∆) ln 1 + cos R
+1 − cos R ,
+(8)
+1Notice that κt and ϕ are different from kt and φ.
+3
+
+with c standing for cos θ. Note that since only emissions in the inter-jet (gap) region are integrated over,
+this result may be cast in terms of the rapidity-gap width ∆η
+∆η ≡ ln 1 + cos R
+1 − cos R .
+(9)
+The all-orders resummed global form factor is simply the exponential of the one-loop distribution. That is
+Σglobal(∆) = exp [−2 CF t(∆) ∆η] .
+(10)
+An identical expression was also arrived at in ref. [26] for jet shapes in e+e− annihilation.
+The leading-order result can be verified by comparing it with the output of the MC program EVENT2 at
+O(αs) [22, 23]. Specifically we compare the differential distribution
+2π
+αs
+dΣ1
+dL = 4 CF ln 1 + cos R
+1 − cos R ,
+(11)
+where L = ln ∆, with the same MC distribution, for the chosen value of R = 0.5. We show in figure 1 a
+plot of the difference between the MC distribution and the expansion of the resummation at O(αs), where,
+as expected, this difference tends to zero in the logarithmically-enhanced region.
+Figure 1: The difference between the leading-order EVENT2 differential distribution 2π/αs dΣ1/dL and the resummed distribu-
+tion expanded at O(αs). The singular behavior of the MC distribution is exactly cancelled by the expanded result.
+3. Two-loops calculation: NGLs and CLs
+When employing the kt or anti-kt clustering algorithms with E-scheme recombination, the resummation
+of the azimuthal decorrelation distribution requires the treatment of NGLs and/or CLs. The corresponding
+cumulative distribution at O(α2
+s) can then be split into three contributions
+Σ2(∆) = 1
+2! [Σ1(∆)]2 + ΣNG
+2
+(∆) + ΣCL
+2 (∆) ,
+(12)
+with ΣCL
+2 (∆) = 0 for anti-kt clustering. Let us first discuss the NGLs contribution ΣNG
+2
+(∆) in both algo-
+rithms, and then compute the CLs contribution ΣCL
+2 (∆) for kt clustering.
+3.1. Calculation of NGLs
+The origin of NGLs at two loops is the emission of a soft gluon k1 inside any of the two outgoing jets
+which itself emits a softer gluon k2 outside the jets without being clustered back to them.
+While this
+configuration results in a non-zero δφ, its virtual correction (specifically when k2 is virtual) gives δφ = 0,
+4
+
+and thus we have a real-virtual mis-cancellation of the soft singularities. We express the contribution of the
+uncancelled virtual correction to the integrated azimuthal decorrelation distribution as follows
+ΣNG
+2
+(∆) = S2(R) t2
+2! ,
+(13a)
+S2(R) = −2 CF CA
+�
+dc1
+1 − c2
+1
+dφ1
+2π
+dc2
+1 − c2
+2
+dφ2
+2π A12
+q¯q ΞNG
+2
+(R) ,
+(13b)
+where CA is the color factor associated with the non-Abelian emission of gluon k2 off k1. The irreducible
+two-loops antenna function A12
+q¯q is given by [27]
+A12
+q¯q = ω1
+q¯q
+�
+ω2
+q1 + ω2
+1¯q − ω2
+q¯q
+�
+=
+1 − c1c2
+1 − c1c2 − s1s2 cos(φ1 − φ2) − 1 ,
+(14)
+with si ≡ sin θi. The function ΞNG
+2
+restricts the angular phase-space of integration and is given, in the
+anti-kt and kt algorithms respectively, by
+ΞNG, akt
+2
+= Θin(k1)Θout(k2) ,
+(15a)
+ΞNG, kt
+2
+= Θin(k1)Θout(k2)Θ(d12 − d2) .
+(15b)
+The step function Θ(d12 − d2) forbids gluon k2 from being clustered back to the jet in the kt algorithm. It
+is given by Θ(cos R − cos θ12), with cos θ12 = c1c2 + s1s2 cos(φ1 − φ2).
+In the anti-kt algorithm the integration is simple and its result can be expressed in the same form as
+that of the rapidity-gap NGLs coefficient found in ref. [19]
+Sakt
+2
+(R) = −CF CA
+�π2
+6 + 2∆η2 − 2∆η ln
+�
+e2∆η − 1
+�
+− Li2
+�
+e−2∆η�
+− Li2
+�
+1 − e2∆η��
+= −CF CA
+�π2
+3 − R4
+8 − R6
+24 − 29 R8
+2560 + O
+�
+R10��
+,
+(16)
+where ∆η(R) is defined in eq. (9). The above formula shows that in the limit R → 0 the two-loops NGLs
+coefficient does not vanish, but rather reaches its maximum value. This feature was observed in ref. [19]
+and was ascribed to the fact that NGLs originate from the edge of jets, since this is the phase-space region
+where gluons k1 and k2 are collinear and thus the amplitude squared (14) is most singular.
+In the kt algorithm, and at small values of R (using small angles), we can write the NGLs coefficient as
+Skt
+2 (R ∼ 0) = −8 CF CA
+� 1
+0
+dθ1
+� ∞
+1
+dθ2
+� 2π
+0
+dφ
+2π
+1
+(θ2
+1 + θ2
+2) sec φ − 2 θ1 θ2
+Θ
+�
+θ2
+1 + θ2
+2 − 2 θ1 θ2 cos φ − 1
+�
+= −2π2
+27 CF CA ,
+(17)
+where we made the following changes of variables: φ = φ1 − φ2 and θi → R θi. Away from the small-R
+limit one may perform the integration numerically to obtain the full-R result for the two-loops coefficient of
+NGLs. We present the results in the following subsection together with the CLs coefficient.
+3.2. CLs with kt clustering
+To compute the CLs we consider the Abelian primary emission of two strongly-ordered gluons directly
+off the hard q¯q dipole, whereby the harder gluon k1 is inside one of the two jets and the softer k2 is outside
+both of them, with the constraint d12 < d2, such that gluon k2 gets clustered into the jet by gluon k1, which
+leads to δφ = 0. However, when k1 is virtual then gluon k2 remains in the gap causing the hard jets to
+decorrelate. In this case we obtain a large single logarithmic contribution to the δφ distribution given by
+ΣCL
+2 (∆) = Ckt
+2 (R) t2
+2! ,
+(18a)
+Ckt
+2 (R) = 4 C2
+F
+�
+dc1
+1 − c2
+1
+dφ1
+2π
+dc2
+1 − c2
+2
+dφ2
+2π w1
+q¯q w2
+q¯q Θ(|c1| − cos R) Θ(cos R − |c2|) Θ(cos θ12 − cos R) .
+(18b)
+5
+
+Note again that Cakt
+2
+(R) = 0, as there are no CLs for anti-kt clustering.
+First, let us consider the small-R limit of this integral. In this case we may write
+Ckt
+2 (R ∼ 0) = 8 C2
+F
+� 1
+0
+dθ1
+θ1
+� ∞
+1
+dθ2
+θ2
+� 2π
+0
+dφ
+2π Θ
+�
+−θ2
+1 − θ2
+2 + 2 θ1 θ2 cos φ + 1
+�
+= 5π2
+27 C2
+F .
+(19)
+Away from this small-R limit we perform the integration numerically. We show in figure 2 a plot of the
+coefficients of NGLs and CLs at O(α2
+s) as a function of the jet radius R. Also shown is the combined
+coefficient of CLs and NGLs in the kt algorithm; Fkt
+2 = Skt
+2 + Ckt
+2 .
+Figure 2: CLs and NGLs coefficients at two loops with kt and anti-kt clustering.
+Notice that NGLs coefficient in the kt clustering algorithm is significantly smaller than that in the anti-
+kt. The CLs coefficient is positive and quite small, with the advantage that it cancels the NGLs coefficient,
+particularly at small values of R. Note that the overall coefficient F2 is less than 1 for values of R smaller
+than about 0.5. For small R values, the anti-kt NGLs coefficient computed here is identical to that found
+for the single hemisphere mass [28] and jet mass with a jet veto [26].
+3.3. Comparison to EVENT2
+We compare our two-loops results with the exact MC distribution at O(α2
+s) obtained with the EVENT2
+program. The latter splits the distribution at NLO into three color contributions; O(C2
+F), O(CF CA) and
+O(CF Tf nf), with Tf = 1/2 being the normalization constant of the generators of the SU(3) group in the
+fundamental representation and nf = 5 is the number of active quark flavors. The expansion of the resummed
+distribution at this order, including running coupling effects, is written, after differentiating with respect to
+L, as
+�2π
+αs
+�2 dΣ2
+dL =
+�
+C2
+F 4
+�
+4 ∆η2 + C2
+�
+− CF CA
+4
+3 (3 S2 + 11∆η) + CF Tf nf
+16
+3 ∆η
+�
+L + O(1) .
+(20)
+At this order this differential distribution is linear in L, but does not capture the O(1) constant, which, in
+the cumulative integrated distribution, is an NLL O(α2
+s L) term. We plot the difference between the MC
+distribution and the expansion (20) for each color contribution separately. All curves should tend towards a
+constant when L grows large and negative. The results are shown in figure 3. Here the O(C2
+F) part contains
+the CLs coefficient in kt clustering, while the O(CF CA) term contains the NGLs coefficient both in kt and
+anti-kt algorithms. The O(CF Tf nf) contribution is algorithm independent at LL accuracy, but the NLL
+O(1) constant does depend on the jet algorithm.
+6
+
+Figure 3: The difference between the NLO EVENT2 differential distribution (2π/αs)2 dΣ2/dL and the resummed distribution
+expanded at O(α2
+s). The leading logarithmic behavior of the MC distribution O(α2
+sL) is cancelled, leaving a constant behavior
+at large values of L.
+4. NGLs and CLs at three loops
+4.1. NGLs at three loops
+At O(α3
+s), the cumulative distribution can be written as follows
+Σ3(∆) = 1
+3! [Σ1(∆)]3 + Σ1(∆) × ΣNG
+2
+(∆) + Σ1(∆) × ΣCL
+2 (∆) + ΣNG
+3
+(∆) + ΣCL
+3 (∆) .
+(21)
+Focusing first on the pure NGLs contribution ΣNG
+3
+(i.e., excluding the cross-talk between the one-loop global
+and two-loops non-global logarithms) in the anti-kt algorithm, we may write it in the form
+ΣNG,akt
+3
+(∆) = Sakt
+3
+(R) t3
+3! ,
+(22a)
+Sakt
+3
+(R) = 2 CF C2
+A
+� � 3
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+�
+�
+A12
+q¯q ¯
+A13
+q¯q Θin(k1)Θout(k2)Θout(k3) − B123
+q¯q Θin(k1)Θin(k2)Θout(k3)
+�
+,
+(22b)
+where we define ¯
+A13
+q¯q = A13
+q¯q/ω1
+q¯q, and the 3-loops irreducible cascade antenna function is given by
+B123
+q¯q = ω1
+q¯q
+�
+A23
+q1 + A23
+1¯q − A23
+q¯q
+�
+.
+(23)
+It is worth mentioning that, for the cascade term, there is a non-negligible contribution from the configuration
+in which gluon k1 is in one jet emitting gluon k2 in the other jet which itself emits the softest gluon k3 in
+the gap between the two jets. At small values of R the integration is quite simple and yields the result
+Sakt
+3
+(R ∼ 0) = CF C2
+A 2 ζ3 .
+(24)
+Notice that this result is twice that found for the hemisphere mass observable [29]. Away from the small-R
+limit the integration can be performed numerically and we shall present the results in the next subsection.
+The pure NGLs contribution in the kt algorithm is given by an identical form to that of the anti-kt (22a)
+ΣNG,kt
+3
+(∆) = Skt
+3 (R) t3
+3! .
+(25)
+We perform for the first time in the literature a calculation of NGLs in the kt algorithm beyond two loops.
+Due to non-linearity of kt clustering we shall find a class of NGLs that have a non-standard color factor,
+namely C2
+F CA at O(α3
+s).
+Such terms usually (in anti-kt clustering) only arise in the cross-talk of the
+expansion of the primary-emission global form factor (10) at O(αs) together with NGLs at O(α2
+s), i.e., the
+term Σ1(∆) × ΣNG
+2
+(∆) in eq. (21). However, in the kt algorithm, we find them as “pure” irreducible NGLs.
+That is, they are part of the term ΣNG
+3
+(∆) in eq. (21).
+7
+
+To proceed, we consider three types of emissions at O(α3
+s), as shown in figure 4: (a) one primary + two
+correlated emissions, (b) ladder emissions, and (c) cascade emissions. In each case we consider all possible
+virtual-correction Feynman diagrams as well as angular configurations of the emitted gluons that affect the
+clustering procedure, and look for a mis-match between the soft divergences of these emissions.
+Figure 4: The three types of emissions to consider for NGLs calculation at O(α3
+s): (a) one primary + two correlated emissions,
+(b) ladder emissions, and (c) cascade emissions.
+Starting first with type (a) contributions, there are three possible permutations of the gluons.
+2
+For the permutation in which k3 is emitted in correlation with k2, and which has a squared amplitude
+4 C2
+F CA ω1
+q¯q A23
+q¯q, we find that the angular phase space of integration that yields a logarithmic contribution
+is given by
+ΞNG,kt
+31
+(R) = Θin(k1)Θin(k2)Θout(k3)Θ(d3 − d13)Θ(d23 − d3)+
++ Θout(k1)Θin(k2)Θout(k3)Θ(d23 − d3)+
+− Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d2 − d12) .
+(26)
+To see how one obtains this result let us give one example of angular configurations that result in a logarithmic
+contribution. There are four Feynman diagrams in this case, shown in figure 5. Consider the situation when
+particles k3 and k2 are outside the jet regions, d3j > d3 and d2j > d2, while particle k1 is inside, d1j < d1. In
+this scenario, both diagrams in which k1 is virtual (i.e., diagrams (3) and (4) in figure 5) yield δφ ̸= 0, since
+in both diagrams particle k2 is real and remains in the gap region after applying the clustering. However,
+these two diagrams contribute equally and with opposite signs, so they cancel each other. For the remaining
+two diagrams, (1) and (2), we have a mismatch when particle k2 gets pulled inside the jet by the real particle
+k1 while k3 remains in the gap. This happens when d12 is smaller than d2, and both d13 and d23 are greater
+than d3. While in diagram (1) we have a real unclustered gluon k3 in the gap region (i.e., it forms a jet
+on its own) giving δφ ̸= 0, in diagram (2) the gap is empty and the hard jets are exactly back-to-back
+with δφ = 0. The virtual-correction diagram (2) contributes fully to the cumulative distribution while the
+real-emission diagram (1) cancels this contribution only up to δφ = ∆, leaving uncancelled virtual-correction
+contributions with a negative sign. This is the last term in eq. (26).
+Figure 5: Feynman diagrams corresponding to the squared amplitude 4 C2
+F CA ω1
+q¯q A23
+q¯q.
+2Note that for all permutations the transverse momenta of the three gluons are strongly ordered as follows: kt1 ≫ kt2 ≫ kt3.
+8
+
+Similarly, we obtain for the second and third gluon permutations of type (a) diagrams, with squared
+amplitudes 4 C2
+F CA ω2
+q¯q A13
+q¯q and 4 C2
+F CA ω3
+q¯q A12
+q¯q, respectively, as well as type (b) (ladder-emission) contri-
+butions, with squared amplitude 2 CF C2
+A ¯
+A12
+q¯q A13
+q¯q, the same phase space function. It is given by
+ΞNG,kt
+32
+(R) = Θin(k1)Θin(k2)Θout(k3)Θ(d13 − d3)Θ(d3 − d23)+
++ Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d3 − d23)+
++ Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d12 − d2) .
+(27)
+Finally, for type (c) (cascade emission) contributions, corresponding to the squared amplitude 2 CF C2
+A B123
+q¯q ,
+the phase space function reads
+ΞNG,kt
+33
+(R) = − Θin(k1)Θin(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)−
+− Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d2 − d12) .
+(28)
+Before performing the integration, we subtract off the part of the phase space that produces the interfer-
+ence term between the one-loop global primary logarithm and the two-loops NGLs, which only comes from
+type (a) emissions. For the first permutation of gluons, this part of phase space is identified by the second
+term in eq. (26), Θout(k1)Θin(k2)Θout(k3)Θ(d23 − d3), where gluon k1 reproduces the one-loop global term
+(8) and the other correlated gluons, k2 and k3, give the two-loops NGLs (eq. (13b) with phase space (15b)).
+However, for the other two gluon permutations of type (a) emissions, the phase space (27) does not simply
+contain such interference terms. Strictly speaking, this means that NGLs do not cleanly factorize from the
+global form factor in the kt algorithm. Nevertheless, we can manually add and subtract the interference
+terms and write the total distribution in the factorizable form (21).
+We can then write the “pure” NGLs coefficient Skt
+3 , given in eq. (25), in the following form
+Skt
+3 (R) = S(a)
+3 (R) + S(b)+(c)
+3
+(R) ,
+(29)
+where we split the result according to the color factor, such that for type (a) emissions we have
+S(a)
+3 (R) = 4 C2
+F CA
+� � 3
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+� �
+ω1
+q¯q A23
+q¯q �Ξ31(R) + ω2
+q¯q A13
+q¯q �Ξ32(R) + ω3
+q¯q A12
+q¯q �Ξ33(R)
+�
+,
+(30)
+with modified phase space that subtracts away the interference terms
+�Ξ31(R) = ΞNG,kt
+31
+(R) − Θout(k1)Θin(k2)Θout(k3)Θ(d23 − d3) ,
+(31a)
+�Ξ32(R) = ΞNG,kt
+32
+(R) − Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3) ,
+(31b)
+�Ξ33(R) = ΞNG,kt
+32
+(R) − Θin(k1)Θout(k2)Θout(k3)Θ(d12 − d2) ,
+(31c)
+and for type (b) and (c) emissions
+S(b)+(c)
+3
+(R) = 2 CF C2
+A
+� � 3
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+� �
+A12
+q¯q ¯
+A13
+q¯q ΞNG,kt
+32
+(R) + B123
+q¯q ΞNG,kt
+33
+(R)
+�
+.
+(32)
+We are now in a position to perform the integrations numerically as a function of the jet radius R. We
+show the results in the next subsection, in which we also compute the clustering logarithmic contribution
+at O(α3
+s).
+4.2. CLs with kt clustering at three loops
+Following the same steps as for NGLs calculation, the phase space clustering function at O(α3
+s) for the
+CLs contribution, which results from the mismatch of soft singularities between real and virtual emissions
+9
+
+of three primary soft gluons, with a squared amplitude 8 C3
+F w1
+q¯q w2
+q¯q w3
+q¯q, is given by
+ΞCL,kt
+3
+(R) = − Θin(k1)Θin(k2)Θout(k3)Θ(d3 − d13)Θ(d3 − d23)−
+− Θout(k1)Θin(k2)Θout(k3)Θ(d3 − d23)−
+− Θin(k1)Θout(k2)Θout(k3)Θ(d3 − d13)−
+− Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d2 − d12) .
+(33)
+Note that the above phase-space clustering function is similar (but not exactly identical) to that found for
+the jet mass observable [30]. Extracting the interference terms between the CLs at two loops and the global
+logarithm at one loop, i.e., the term Σ1(∆) × ΣCL
+2 (∆) in eq. (21), we reduce the above phase space function
+to that of the “pure” CLs contribution as
+�ΞCL,kt
+3
+(R) = − Θin(k1)Θin(k2)Θout(k3)Θ(d3 − d13)Θ(d3 − d23)+
++ Θin(k1)Θout(k2)Θout(k3)Θ(d2 − d12)[1 − Θ(d13 − d3)Θ(d23 − d3)] .
+(34)
+Then, the clustering logarithmic contribution to the cumulative cross-section is given by
+ΣCL
+3 (∆) = Ckt
+3 (R) t3
+3! ,
+(35a)
+Ckt
+3 (R) = 8 C3
+F
+� � 3
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+�
+w1
+q¯q w2
+q¯q w3
+q¯q �ΞCL,kt
+3
+(R) .
+(35b)
+We show in figure 6 a plot of the coefficients of NGLs and CLs in the kt and anti-kt algorithms as a
+function of the jet radius R. Shown also is the combined coefficient Fkt
+3 = Skt
+3 + Ckt
+3 for the kt algorithm.
+Figure 6: CLs and NGLs coefficients at three loops with kt and anti-kt clustering.
+As in the previous section, we notice that the NGLs coefficient in the anti-kt algorithm at this order is
+also quite large. Clearly, the application of the kt clustering has reduced the significance of NGLs by almost
+a factor of 30 for values less than R ∼ 0.6. The CLs coefficient is also small such that the overall coefficient
+F3 is smaller in magnitude than the Sakt
+3
+by about a factor of 3 for most values of R.
+5. Four loops and beyond
+5.1. Four-loops NGLs with anti-kt at small R
+The calculation of NGLs with anti-kt clustering proceeds in a similar manner at fourth order, and can
+easily be deduced from previous calculations of NGLs in the literature. In fact, the phase space of integration
+10
+
+is similar to that of the hemisphere mass distribution reported in ref. [29], and thus the cumulative cross-
+section, at this order, may be cast in the following way
+Σakt
+4
+(∆) = 1
+4! [Σ1(∆)]4+ 1
+2! [Σ1(∆)]2 ΣNG,akt
+2
+(∆)+Σ1(∆) ΣNG,akt
+3
+(∆)+ 1
+2!
+�
+ΣNG,akt
+2
+(∆)
+�2
++ΣNG,akt
+4
+(∆) , (36)
+with pure NGLs contribution given by
+ΣNG,akt
+4
+(∆) = Sakt
+4
+(R) t4
+4! ,
+(37a)
+Sakt
+4
+(R) = 2 CF C3
+A
+� � 4
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+� �
+− A12
+q¯q ¯
+A13
+q¯q ¯
+A14
+q¯q Θin(k1)Θout(k2)Θout(k3)Θout(k4)+
++ 3 ¯
+A12
+q¯q B134
+q¯q Θin(k1)Θout(k2)Θin(k3)Θout(k4) + U1234
+q¯q
+Θin(k1)Θin(k2)Θout(k3)Θout(k4)−
+− C1234
+q¯q
+Θin(k1)Θin(k2)Θin(k3)Θout(k4)
+�
+−
+− 2 CF C3
+A
+�
+1 − 2 CF
+CA
+� � � 4
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+�
+A1234
+q¯q
+Θin(k1)Θin(k2)Θout(k3)Θout(k4) ,
+(37b)
+where the irreducible antenna functions read [27]
+C1234
+q¯q
+= w1
+q¯q
+�
+B234
+q1 + B234
+1¯q − B234
+q¯q
+�
+,
+(38a)
+U1234
+q¯q
+= w1
+q¯q
+�
+A23
+q1 ¯
+A24
+q1 + A23
+1¯q ¯
+A24
+1¯q − A23
+q¯q ¯
+A24
+q¯q
+�
+,
+(38b)
+A1234
+q¯q
+= ω1
+q¯q ω2
+q1
+� ¯
+A23
+q¯q − ¯
+A23
+q1
+� � ¯
+A24
+q1 − ¯
+A24
+1¯q
+�
++ ω1
+q¯q ω2
+1¯q
+� ¯
+A23
+q¯q − ¯
+A23
+1¯q
+� � ¯
+A24
+1¯q − ¯
+A24
+q1
+�
+−
+− ω1
+q¯q ω2
+q¯q
+� ¯
+A23
+q1 − ¯
+A23
+q¯q
+� � ¯
+A24
+1¯q − ¯
+A24
+q¯q
+�
++ k3 ↔ k4 .
+(38c)
+At small values of R the integration has been performed in ref. [29], and the corresponding result is
+given by
+Sakt
+4
+(R ∼ 0) = −CF C3
+A ζ4
+�29
+4 −
+�
+1 − 2 CF
+CA
+��
+.
+(39)
+5.2. LL resummation
+In this section we present numerical results for the resummation of NGLs and CLs in the large-Nc
+approximation. For this we use the MC code first developed in refs. [18, 19] with modification of kt clustering
+in terms of distances (1). The results are shown in figure 7 for the resummed cumulative distribution Σ as
+a function of the evolution parameter t (7), for the particular value of the jet radius R = 0.5. Shown in the
+figure are: results for the (primary) global distribution (black curve), obtained by running the MC program
+using anti-kt clustering and allowing only for primary emissions, and the full anti-kt distribution (solid pink
+curve) which additionally includes NGLs at large Nc. We observe the very large impact of NGLs on the
+distribution, reducing the global form factor by a factor of 10 for t = 0.3.
+We also show in the same figure the primary-emission distribution obtained by running the above-
+mentioned program with kt clustering (solid green curve), which includes the global form factor together
+with the resummed CLs, as well as the overall distribution in kt clustering which includes in addition the
+resummed NGLs (solid purple curve). We observe that the distribution in kt clustering is affected by both
+CLs and NGLs, but the combined impact of the two is noticeably small. This means that CLs tend to cancel
+NGLs in kt clustering, as noted with the fixed-order calculations performed above.
+Moreover, we show in figure 7 the analytical results for the resummed distribution in each case (dashed
+lines), which are estimated from the observed pattern of exponentiation
+Σ(∆) = exp [Σ1(∆)] exp
+� ∞
+�
+i=2
+ΣNG
+i
+(∆)
+�
+exp
+� ∞
+�
+i=2
+ΣCL
+i
+(∆)
+�
+.
+(40)
+11
+
+Figure 7: Numerically resummed NGLs and CLs at large Nc.
+It is clear that the truncation of the series at i = 3 in the exponent, though quite close to the numerical
+result, does not give an accurate fit of the MC distribution. This means that higher-order contributions
+cannot be ignored.
+5.3. NLL resummation with anti-kt
+We present here a resummed result for the azimuthal decorrelation distribution with anti-kt clustering
+at NLL accuracy in the large-Nc limit, obtained using the recently-published program Gnole [24, 25]. The
+distribution can be obtained from this program by defining our observable within the code, using the full
+definition rather than the LL approximation (2). Explicitly written the former reads
+δφ =
+������
+sin−1 �
+i/∈jets
+k⊥i
+ptr
+������
+,
+(41)
+where k⊥i is the component of the transverse momentum of the emission i perpendicular to the thrust (or
+leading jet) axis, and ptr is the recoiling jet’s transverse momentum. For simplicity we take the jets to be
+at threshold, i.e., transverse to the beam direction.
+Figure 8: NLL numerical resummation of the δφ distribution at large Nc.
+In figure 8 we present the results for the NLL resummed distribution together with the LL one, both
+obtained with Gnole. We note here that the LL distribution is obtained using the definition of the observable
+(41), while that obtained with the MC code of refs. [18, 19] (figure 7) is essentially equivalent to the transverse
+12
+
+energy distribution (in other words, definition 2 without the sin φ part). This does in fact numerically affect
+the distribution even at LL accuracy. 3
+We observe that the NLL corrections to the distribution are quite important, just like the transverse
+energy distribution shown in ref. [25]. Furthermore, the scale-uncertainty band, obtained by varying the
+renormalization and resummation scales by factors of 2 and 1/2 around their central values (µR = √s
+and Q0 = √s/2, respectively, with √s = MZ), gets significantly reduced in the NLL curve. It is worth
+mentioning that one still needs to account for the matching in order to fully control all sub-leading NLL
+logarithms at the tail of the distribution.
+We note that the actual distribution does not possess a Sudakov peak at low values of δφ, instead it tends
+towards a constant value. This is explained by the fact that the very low values of δφ are not suppressed
+by soft emissions, but are rather enhanced by vectorial cancellation of semi-hard emissions.
+6. Conclusions
+In this letter we have presented both fixed-order and all-orders results for the distribution of the azimuthal
+decorrelation observable for the specific QCD process e+e− annihilation into two jets. The said observable
+is of non-global nature and hence its distribution contains, in addition to the usual global logarithms, non-
+global and/or clustering logarithms. These logarithms are jet-algorithm dependent, and start to appear at
+two loops and are quite delicate to compute.
+For NGLs, we have calculated the full-R expression analytically at two loops and numerically at three
+loops for the anti-kt jet algorithm. At four loops, they have been determined only for small-R values for
+the same said jet algorithm. For the kt clustering algorithm, calculations have been performed up to three
+loops and only numerically.
+Moreover, CLs, which are absent in the anti-kt algorithm, have been computed numerically up to three
+loops. The usual reduction in the significance of NGLs due to kt clustering has been observed confirming
+previous findings. Furthermore, the combined impact of NGLs and CLs on the distribution is observed to
+be very small which has important phenomenological implications in terms of accuracy of the resummed
+distribution. Our results at two loops have been checked against the output of the MC program EVENT2.
+Numerical estimates of the all-orders distribution has been presented both at LL and NLL accuracy.
+The achievement of the latter accuracy has been made possible by the recently-published Gnole code. NLL
+resummation exhibits a better distribution both in terms of accuracy and scale uncertainty. It is thus worth
+investigating the impact of NLL effects with kt clustering. Of similar worthiness is computing NGLs and
+CLs at four loops with full-R dependence.
+Acknowledgements
+This work is supported by PRFU research project B00L02UN050120230003. The authors wish to thank
+the Algerian Ministry of Higher Education and Scientific Research and DGRSDT for financial support.
+The numerical calculations presented here have been performed in the HPC cluster at the University of
+Batna 2 (UB2-HPC).
+We thank Andrea Banfi for clarifications about using Gnole program.
+References
+[1] F. Hautmann, H. Jung, JHEP 10 (2008) 113. arXiv:0805.1049.
+[2] A. Aktas, et al. (H1), Eur. Phys. J. C 33 (2004) 477–493. arXiv:hep-ex/0310019.
+[3] M. Aaboud, et al. (ATLAS), Phys. Rev. D 98 (2018) 092004. arXiv:1805.04691.
+[4] G. Aad, et al. (ATLAS), Phys. Rev. Lett. 106 (2011) 172002. arXiv:1102.2696.
+[5] A. M. Sirunyan, et al. (CMS), Eur. Phys. J. C 78 (2018) 566. arXiv:1712.05471.
+[6] V. Abazov, et al. (D0), Phys. Rev. Lett. 94 (2005) 221801. arXiv:hep-ex/0409040.
+3Note that it is not possible to change the definition of the observable in the MC code of refs. [18, 19].
+13
+
+[7] M. Aaboud, et al. (ATLAS), Phys. Rev. C 100 (2019) 034903. arXiv:1901.10440.
+[8] A. Banfi, M. Dasgupta, Y. Delenda, Phys. Lett. B 665 (2008) 86–91. arXiv:0804.3786.
+[9] S. Chatrchyan, et al. (CMS), Phys. Lett. B 722 (2013) 238–261. arXiv:1301.1646.
+[10] X. Liu, F. Ringer, W. Vogelsang, F. Yuan, Phys. Rev. D 102 (2020) 094022. arXiv:2007.12866.
+[11] H. Abramowicz, et al. (ZEUS), JHEP 01 (2018) 032. arXiv:1712.04273.
+[12] S. Catani, Y. L. Dokshitzer, M. H. Seymour, B. R. Webber, Nucl. Phys. B 406 (1993) 187–224.
+[13] S. D. Ellis, D. E. Soper, Phys. Rev. D 48 (1993) 3160–3166. arXiv:hep-ph/9305266.
+[14] M. Cacciari, G. P. Salam, G. Soyez, Eur. Phys. J. C 72 (2012) 1896. arXiv:1111.6097.
+[15] M. Cacciari, G. P. Salam, G. Soyez, JHEP 04 (2008) 063. arXiv:0802.1189.
+[16] Y. L. Dokshitzer, G. D. Leder, S. Moretti, B. R. Webber, JHEP 08 (1997) 001. arXiv:hep-ph/9707323.
+[17] M. Wobisch, T. Wengler, in: Workshop on Monte Carlo Generators for HERA Physics (Plenary Starting Meeting), pp.
+270–279. arXiv:hep-ph/9907280.
+[18] M. Dasgupta, G. Salam, Phys. Lett. B 512 (2001) 323–330. arXiv:hep-ph/0104277.
+[19] M. Dasgupta, G. P. Salam, JHEP 03 (2002) 017. arXiv:hep-ph/0203009.
+[20] A. Banfi, M. Dasgupta, Phys. Lett. B 628 (2005) 49–56. arXiv:hep-ph/0508159.
+[21] Y. Delenda, R. Appleby, M. Dasgupta, A. Banfi, JHEP 12 (2006) 044. arXiv:hep-ph/0610242.
+[22] S. Catani, M. H. Seymour, Nucl. Phys. B 485 (1997) 291–419. arXiv:hep-ph/9605323, [Erratum: Nucl.Phys.B 510, 503–504
+(1998)].
+[23] S. Catani, M. H. Seymour, Phys. Lett. B 378 (1996) 287–301. arXiv:hep-ph/9602277.
+[24] A. Banfi, F. A. Dreyer, P. F. Monni, JHEP 10 (2021) 006. arXiv:2104.06416.
+[25] A. Banfi, F. A. Dreyer, P. F. Monni, JHEP 03 (2022) 135. arXiv:2111.02413.
+[26] K. Khelifa-Kerfa, JHEP 02 (2012) 072. arXiv:1111.2016.
+[27] Y. Delenda, K. Khelifa-Kerfa, Phys. Rev. D 93 (2016) 054027. arXiv:1512.05401.
+[28] A. Banfi, M. Dasgupta, K. Khelifa-Kerfa, S. Marzani, JHEP 08 (2010) 064. arXiv:1004.3483.
+[29] K. Khelifa-Kerfa, Y. Delenda, JHEP 03 (2015) 094. arXiv:1501.00475.
+[30] Y. Delenda, K. Khelifa-Kerfa, JHEP 09 (2012) 109. arXiv:1207.4528.
+14
+

V9AzT4oBgHgl3EQfJ_vP/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ecdb273477403d6ce012466ac7b13da23120e12d1cd4758bf601eb5fb1944b51
+size 2490413

WdE1T4oBgHgl3EQfJAOq/content/tmp_files/2301.02947v1.pdf.txt

@@ -0,0 +1,1384 @@
+arXiv:2301.02947v1  [cond-mat.mes-hall]  7 Jan 2023
+Chiral organic molecular structures supported by multilayer surfaces
+Alexander V. Savin1, 2, ∗ and Yuri S. Kivshar3, †
+1Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow 119991, Russia
+2 Plekhanov Russian University of Economics, Moscow 117997, Russia
+3Nonlinear Physics Center, Department of Fundamental and Theoretical Physics,
+Research School of Physics, Australian National University, Canberra ACT 2601, Australia
+We study numerically the dynamics of acetanilide (ACN) molecules placed on a flat surface of
+a multilayer hexagonal boron nitride structure. We demonstrate that the ACN molecules, being
+achiral in three dimensions, become chiral after being placed on the substrate. Homochirality of the
+ACN molecules leads to stable secondary structures stabilized by hydrogen bonds between peptide
+groups of the molecules. Numerical simulations of systems of such molecules reveal that the structure
+of the resulting hydrogen-bond chains depends on the isomeric composition of the molecules. If all
+molecules are homochiral (i.e. only one isomer is present), they form secondary structures (chains
+of hydrogen bonds in the shapes of arcs, circles, and spirals). If the molecules at the substrate
+form a racemic mixture, then no regular secondary structures appear, and only curvilinear chains of
+hydrogen bonds of random shapes can emerge. A hydrogen-bond chain can form a straight zigzag
+only if it has an alternation of isomers. Such chains can create two-dimensional (2D) regular lattices,
+or 2D crystals. The melting scenarios of such 2D crystals depend on density of its coverage of the
+substrate. At 25% coverage, melting occurs continuously in a certain temperature interval. For a
+complete coverage, melting occurs at 415 ÷ 470 K due to a shift of 11% of all molecules into the
+second layer of the substrate.
+I.
+INTRODUCTION
+Two-dimensional (2D) materials such as graphene (G)
+and hexagonal boron nitride (h-BN) have attracted a lot
+of attention due to their unique electronic [1–3] and me-
+chanical [4–7] properties. Currently, heterogeneous lay-
+ered materials of such 2D materials, which can exhibit
+various novel physical properties compared to their ho-
+mogeneous counterparts, became a special focus of such
+studies [8–10]. For example, the use of hybrid G/h-BN
+structures allows to achieve some desired electronic prop-
+erties [11, 12] and also reduce significantly friction be-
+tween the layers [13].
+In general, such multilayer het-
+erostructures are stabilized by van der Waals (vdW) in-
+teractions between atoms of the neighboring layers.
+The concept of vdW heterostructures can be extended
+to the integration of 2D materials with molecular struc-
+tures of different dimensions, such as nD/2D heterostruc-
+tures, where n stands for the dimension (n = 0, 1, or
+3) [14], describing flat molecules (n = 0), polymer chains
+(n = 1), or three-dimensional molecular objects (n = 3).
+For molecules and molecular chains with benzol rings,
+flat layers of G and h-BN are strong adsorbents [15–19].
+Theoretical studies reveal that molecules adsorbed on G
+and h-BN surfaces through non-covalent interactions can
+modify the properties of the surface as solid-liquid, solid-
+air, or solid-vacuum interfaces [20–22]. A strong stacking
+interaction with a flat substrate allows such molecules
+residing on a surface creating stable 2D supra-molecular
+systems, as shown in the characteristic example of Fig. 1
+∗asavin@chph.ras.ru
+†yuri.kivshar@anu.edu.au
+FIG. 1:
+Example of the molecular structures studied in
+this paper. A spiral structure of 52 R-isomers of the ACN
+molecules C6H5–NHCO–CH3 is stabilized on a h-BN multi-
+layer surface.
+By now,
+the behavior of multifunctional organic
+molecules placed on ideal metal surfaces has been stud-
+ied in detail [23]. Such organometallic systems may ex-
+hibit a variety of different structures induced by the sub-
+strate. In many cases, complex organic molecules (such
+as carboxylic acids, amino acids, anhydrides, and ring
+systems) become self-organized on metal surfaces creat-
+ing ordered super-structures stabilized by inter-molecular
+interactions. Chirality is of a particular interest that can
+appear for initially achiral metal surfaces by adsorbing
+organic molecules [24]. Similar behavior is expected for
+organic molecules adsorbed on flat surfaces of G and h-
+
+2
+BN molecular structures.
+In this paper, we study nu-
+merically the formation of supra-molecular complexes by
+acetanilide molecules placed on the surface of a multilayer
+h-BN sheet, see Fig. 1, serving as an introductory figure
+explaining our problem and results discussed below.
+For poly-cyclic aromatic hydrocarbons (for molecules
+of benzol C6H6, naphthalene C10H8, pyrene C16H10,...),
+graphene is a strong adsorbent [15, 17, 19, 25]. The in-
+teraction of graphene with such molecules often causes
+specific reactions that can be used in new types of sensors
+[26, 27]. Non-covalent functionalization of the graphene
+surface can significantly expand its potential range of
+applications [20, 21].
+It has been shown experimen-
+tally [25, 28] that benzol and pyrene molecules adsorbed
+on graphene form densely packed monolayers.
+For acetanilide (ACN, C6H5NHCOCH3) and parac-
+etamol (PCM, C6H4OHNHCOCH3) molecules, graphene
+and hexagonal boron nitride are also strong adsorbents.
+Due to possible medical applications, much attention has
+been paid to modeling the adsorption of PCM molecules,
+which is a strong analgesic, on h-BN sheets and nan-
+otubes [18, 29]. It has been shown in [30] that function-
+alized graphene can be used as a highly sensitive parac-
+etamol detection sensor.
+An example of a 1D/2D heterostructure is a graphene
+sheet with adsorbed Kevlar chains, kevlar-functionalized
+graphene [31]. The presence of planar C6H4 benzol rings
+and NHCO peptide groups in the polymer chain [–C6H4–
+NHCO–]∞ provides a strong non-covalent (vdW) inter-
+action of the chain with G and h-BN sheets. Such chains
+on the surface of sheets G and h-BN will lie parallel to
+the surface and form chains of hydrogen bonds between
+each other · · ·HNCO· · ·HNCO· · ·.
+Such 3D/2D heterostructures can form thin metal lay-
+ers on the G and h-BN surfaces. In particular, numerical
+modeling suggests that aluminum can form stable two-
+layer structures on the G surface [32].
+It has been shown in Refs. [33–42] that n-alkanes (lin-
+ear polymer chains CH3(CH2)lCH3 with internal units
+2 ≤ l ≤ 388) form a dense ordered monolayer of parallel
+linear chains on the graphite (graphene) surface. Inter-
+est in alkanes is due to the fact that they belong to the
+simplest families of polymer molecules, which members
+of which differ only in their length. Placing linear poly-
+mer chains on a flat graphite surface causes them to self-
+assemble into 2D crystals. The self-assembly mechanism
+depends on the chain length, temperature, and the level
+of coverage of the substrate with chains [36, 40].
+Adsorption by the surface of a long single-chain
+polyethylene molecule leads to its two-dimensional crys-
+tallization – it passes from the form of a three-
+dimensional globule into the form of a parallel folded
+linear chain lying in the plane parallel to the substrate
+surface [43–45].
+Thus, the flat surfaces of the G and h-BN substrates
+create a 2D platform for flat molecules adsorbed on them
+(for poly-cyclic aromatic hydrocarbons, for ACN and
+PCM molecules, Kevlar chains, ...)
+and linear poly-
+mer molecules. At low temperatures, the molecules move
+along the sheet, remaining parallel to its surface. They
+interact with each other and form two-dimensional supra-
+molecular structures. Such molecular adsorbents are con-
+venient systems for studying phase transitions caused by
+freedom restrictions.
+To date, only phase transitions in monolayers of n-
+alkanes have been well studied [33, 42, 46–48]. Modeling
+and experimental studies show that a monolayer always
+undergoes a transition from a solid-crystalline 2D phase
+to a liquid phase (the transition occurs at a temperature
+significantly lower than the desorption temperature of
+molecules). The melting scenario depends on the poly-
+mer chain length.
+The melting temperature increases
+monotonically with chain length, so for pentane, hep-
+tane and nonane (l = 3, 5, 7) the melting temperature
+is Tm = 92, 178 and 255K [46].
+A characteristic fea-
+ture associated with the adsorption of molecules is the
+continuity of melting of a 2D crystal – melting occurs in
+the temperature interval. Thus, for the longest synthe-
+sized monodisperse alkane C390H782 (l = 388), continu-
+ous melting occurs at 393 < T < 484 K [42].
+Despite the large number of theoretical and experimen-
+tal works on phase transitions in adsorbed monolayers of
+alkanes and their derivatives, as far as we know, there
+are no works on modeling phase transitions in adsorbed
+monolayers of ACN, PCM, and Kevlar (para-aramid)
+molecules.
+A detailed description of adsorption simu-
+lation methods is given in [49]. Unlike alkanes, the 2D
+structures of these molecules adsorbed on a flat surface
+are associated with the presence of chains of hydrogen
+bonds. Molecules including amide and hydroxyl groups
+can create 2D lattices and extended hydrogen chains.
+We notice that the ACN molecules are often consid-
+ered as a model system with chains of hydrogen bonds
+between HNCO peptide groups. Acetanilide crystallizes
+into an orthorhombic structure with ribbons of molecules
+linked by hydrogen bonds [50]. The chains of hydrogen
+bonds that stabilize the crystal structure are very similar
+to the chains that stabilize the alpha-helices and beta-
+sheets of proteins. Therefore, ACN was used as a model
+for modeling the energy transfer of vibrations of peptide
+groups along hydrogen bond chains in proteins [51–53].
+Living matter, unlike non-living matter, has chiral pu-
+rity: all proteins consist of left-handed amino acids, while
+DNA and RNA are built on right-handed ribose. In ex-
+periments on abiogenic synthesis, left and right isomers of
+sugars and proteins are formed in equal proportions. It is
+believed that if you try to build proteins from such a mix-
+ture, they will not be able to fold into a stable form and
+therefore will not work as enzymes. In three dimensions,
+the need for chiral purity to form stable protein struc-
+tures requires complex analysis.
+The situation is dra-
+matically simplified if we move from three-dimensional
+space to two-dimensional. Such a transition can be made
+if flat molecules are placed on a flat molecular sheet of
+graphene or hexagonal boron nitride (h-BN). Such a non-
+valent modification of the sheet surface actually creates
+
+3
+FIG. 2: When an ACN molecule is placed on a flat surface
+of a multilayer h-BN structure, it may create two isomers
+with the mirror symmetry (shown by a straight line). Vectors
+connecting the oxygen atom with the hydrogen atom of each
+peptide group show the dipole moments. For the L-isomer,
+the benzol ring is to the left of this vector, for the R-isomer
+it is to the right. Gray balls stand for carbon atoms, white
+balls – hydrogen, blue – nitrogen, red – oxygen, and green –
+bromine atoms.
+a 2D world for the flat molecules placed on it. At low
+temperatures, the molecules move along the sheet all the
+time remaining parallel to its surface. On the surface,
+they can form complex two-dimensional structures.
+An ACN molecule that is achiral in 3D becomes chiral
+after being placed on a flat substrate (the chirality de-
+pends on which side it lays on the surface of the sheet)
+– see Fig. 2. It will be shown that the homochirality of
+ACN molecules leads to the appearance on the surface
+of the sheet of stable secondary structures stabilized by
+hydrogen bonds: cyclic and spiral chains and complexes
+of them. Modeling the formation of such structures will
+make it possible to demonstrate the necessity of homochi-
+rality (chiral purity) of biomolecules for the formation of
+stable secondary molecular structures from them.
+As a flat substrate, we consider a surface of a multilayer
+h-BN structure, and for molecules we consider acetanilide
+(ACN) C6H5NHCOCH3, as shown in Figs. 1 and 2. The
+presence of a planar benzol ring C6H5 and a planar pep-
+tide group (PG) HNCO leads to large interaction energy
+of the molecule with the substrate, Esub = 0.762 eV.
+Molecules can create chains of hydrogen bonds between
+their peptide groups OCNH· · ·OCNH· · ·OCNH· · ·. Such
+chains of hydrogen bonds stabilize the secondary struc-
+tures of the protein molecules.
+The paper is organized as follows. In the next section,
+we describe our model. Section III is devoted to the study
+of secondary structures of the ACN molecules placed on
+a flat substrate. Self-assembly of such structures is simu-
+lated numerically in Sec. IV. Then, in Sec. V we analyze
+melting of 2D crystals. Section VI concludes the paper.
+FIG. 3: Construction of a coarse-grained model of the ACN
+molecule: (a) full-atomic view of the molecule, (b) coarse-
+grained model (the used numbering of the united atoms is
+shown).
+II.
+MODEL
+For modeling of the dynamics of a system of the ACN
+molecules, we will use the united-atoms approximation.
+Let us consider the molecular groups CH and CH3 as
+united atoms whose centers coincide with the centers of
+carbon atoms. In this approximation, the ACN molecule
+is described as a system of 11 united atoms – see Fig. 3.
+The values of the masses of the united atoms are shown
+in the table I.
+To model a ACN molecule, we use the force field in
+which distinct potentials describe the deformation of va-
+lence bonds and valence, torsion and dihedral angles, and
+non-valence atomic interacts [55]. In this model, the de-
+formation energy of the valence bonds C–CH, CH–CH,
+C–N, N–H, C=O and C–CH3 is described by the har-
+monic potential:
+V (ρ) = 1
+2K(ρ − ρ0)2,
+(1)
+where ρ and ρ0 are current and equilibrium bond lengths,
+K is the bond stiffness. The values of potential parame-
+ters for various valence bonds are presented in Table II.
+TABLE I:
+Masses and parameters of interaction potentials
+for united atoms X of the ACN molecule: i – atom number,
+Mi – atom mass (mp = 1.6603 × 10−27 kg – proton mass), εi
+and ri are the energy and radius of the LJ interaction, qi is
+the electric charge of the atom, ǫi and hi are the energy and
+equilibrium distance for the interaction of an atom with a flat
+substrate (with a crystal surface h- BN).
+X
+C
+CH
+N
+H
+C
+O
+CH3
+i
+1
+2,3,4,5,6
+7
+8
+9
+10
+11
+Mi (mp)
+12
+13
+14
+1
+12
+16
+15
+εi (meV)
+4.284
+4.284
+4.080 0.434 4.284 6.344 4.284
+ri (˚A)
+1.861
+1.861
+1.899 0.621 1.861 1.711 1.861
+qi (e)
+0.066
+0
+-0.463 0.286 0.580 -0.504 0.035
+ǫi (meV)
+61.5
+87.3
+47.7
+31.3
+61.5
+42.8
+87.3
+hi (˚A)
+3.52
+3.44
+3.43
+3.08
+3.52
+3.36
+3.44
+
+(a)
+H
+(b)
+CH
+6
+8
+5
+CH3
+4
+C
+N
+1
+9
+11
+3
+2
+10R
+T4
+FIG. 4: Dimer of (a) RR and (b)LR isomers of the ACN
+molecule on a flat substrate (shown in green). The vectors
+show the dipole moments of the peptide groups. Hydrogen
+bond energy of identical isomers Ehb = 0.330 eV, angle be-
+tween dipole moments φhb = 17.05◦, for different isomers
+Ehb = 0.322 eV, φhb = 33.98◦.
+Energies of the deformation of the valence angles X–
+Y–Z are described by the potential
+U(u1, u2, u3) = U(φ) = ǫa(cos φ − cos φ0)2,
+(2)
+where the cosine of the valence angle φ is defined as
+cos φ = −(v1, v2)/ρ1ρ2, with the vectors v1 = u2 − u1,
+v2 = u3 − u2 and bond lengths ρ1 = |v1|, ρ2 = |v2|, the
+vectors u1, u2, u3 specify the coordinates of the atoms
+forming the valence angle φ, φ0 is the value of equilib-
+rium valence angle. The values of potential parameters
+used for various valence angles are presented in Table III.
+Deformation of dihedral angles are described by the
+potential
+W(u1, u2, u3, u4) = ǫd(1 + zd cos kϕ),
+(3)
+where cos ϕ = (w1w2)/|w1||w2|, with the vectors w1 =
+(u2 − u1) × (u3 − u2), w2 = (u3 − u2) × (u4 − u3). The
+values of potential parameters used for various dihedral
+angles are presented in Table IV.
+For pairs of atoms Xi,Xj (i, j are the numbers of atoms
+in the molecule) participating in the formation of the
+dihedral angle Xi–Y–Z–Xj, their non-valence interaction
+is also taken into account described by the Lennard-Jones
+(LJ) potential
+W0(r) = ε0[(r0/r)12 − 2(r0/r)6],
+(4)
+with halved interaction energy ε0 = √εiεj/2, where r is
+current distance between interacting atoms, equilibrium
+distance r0 = ri + rj. The LJ interaction of an oxygen
+atom (i = 10) with two combined atoms CH (i = 2, 6)
+was also taken into account with interaction energy ǫ0 =
+TABLE II: Values of the harmonic potential parameters (1)
+for different valence bonds X—Y.
+X—Y
+C–CH, CH–CH
+C–N
+N–H
+C=O C–CH3
+K (N/m)
+469
+427
+434
+570
+317
+ρ0 (˚A)
+1.39
+1.405
+1.007
+1.222
+1.505
+√ε2ε10 and equilibrium distance r0 = r2+r10. Parameter
+values εi and ri are shown in Table I.
+The interaction of two ACN molecules is described by
+the potential
+U(X1, X2) =
+11
+�
+i=1
+11
+�
+j=1
+{εij[(¯rij/rij)12 − 2(¯rij/rij)6]
++κqiqj/rij},
+(5)
+where the 33-dimensional vector Xk = {uk,i}11
+i=1 (k =
+1, 2) defines the coordinates of atoms of the k-th ACN
+(vector uk,i specifies the coordinates of the i-th atom of
+the k-th molecule), distance between atoms rij = |u1,i −
+u2,j|.
+Here energy εij = √εiεj, equilibrium distances
+¯rij = ri + rj, qi is the electric charge of i-th atom (i, j =
+1, ..., 11), coefficient κ = 14.400611 eV˚A/e2. The values
+of the parameters εi, ri and qi are shown in Table I. All
+values of parameters of interaction potentials (1), (2), (3)
+and (5) are obtained from force field AMBER [55].
+The van der Waals interactions of the atoms of the
+ACN molecule with flat substrate are described by the
+LJ potential (m, l)
+W(X) =
+11
+�
+i=1
+Wi(zi) =
+11
+�
+i=1
+ǫi
+l − m[m(hi/zi)l − n(hi/zi)m],
+(6)
+where zi is the distance from i-th atom to the outer sur-
+face of the substrate, which is plane z = 0. Potential
+Wi(zi) in Eq. (6) is the interaction energy of i-th atom as
+a function of the distance to the substrate. This energy
+was found numerically for different substrates [56, 57].
+The calculations showed that interaction energy with
+substrate Wi(z) can be described with a high accuracy
+by LJ potential (6) with power l > k. Potential Wi(z)
+has the minimum Wi(hi) = −ǫi (ǫi is the binding en-
+ergy of the i-th atom with substrate). For the surface of
+the h-BN crystal l = 10, m = 4.25. The values of the
+parameters ǫi, hi, i = 1, ..., 11, are given in the table I.
+The 10-layer fragment of h-BN crystal was used to find
+values of this parameters. The interaction energy of an
+atom with a substrate was found as the sum of all LJ
+potentials (4) with parameters from the force field UFF
+[58].
+Thus,
+the Hamiltonian of a system of N
+ACN
+molecules located on the flat surface of h-BN crystal has
+the form
+H =
+N
+�
+n=1
+1
+2(M ˙Xn, ˙Xn) + P,
+(7)
+where the first term specifies the kinetic and the second
+– potential energy of the system
+P =
+N
+�
+n=1
+[V (Xn) + W(Xn)] +
+N−1
+�
+n=1
+N
+�
+k=n+1
+U(Xn, Xk). (8)
+Here the vector Xn = {un,i}11
+i=1 specifies the coordinates
+of the atoms of n-th ACN molecule, M is the diago-
+nal matrix of atom masses of the molecule, V (Xn) and
+
+a
+b5
+TABLE III: Values of the parameters of the potential of the valence angle X–Y–Z (2) for different atoms.
+X–Y–Z
+C–C–C
+C–C–N
+C–N–H
+C–N–C
+N–C–O
+N–C–C
+O–C–C
+ǫa (eV)
+3.643
+3.823
+2.781
+4.888
+4.932
+3.758
+4.625
+ϕ0 (◦)
+120
+117
+118
+128
+123
+116
+120
+TABLE IV: Values of the parameters of the potential of the dihedral angle X–Y–Z–W (3) for different atoms.
+X–Y–Z–W
+C–C–C–C
+C–C–C–N
+C–C–N–H
+C–C–N–C
+C–N–C–O
+C–N–C–CH3
+ǫd (eV)
+0.63
+0.63
+0.21
+0.21
+0.42
+0.42
+zd
+-1
+1
+-1
+-1
+-1
+1
+k
+1
+1
+2
+2
+1
+1
+W(Xn) are deformation energy and energy of interac-
+tion with the substrate of n-th molecule, U(Xn, Xk) is
+the interaction energy of n and k molecules.
+III.
+SECONDARY STRUCTURES OF ACN
+MOLECULES ON A FLAT SUBSTRATE
+The ACN molecule is achiral, but it becomes chiral
+when placed on a flat substrate.
+Depending on which
+side it lies on the substrate, it can be either right (when
+the benzol ring is located to the right of the dipole mo-
+ment vector of the peptide group
+⃗
+OH) or left (the ben-
+zol ring is located to the left). Two mirror-symmetrical
+isomers of the molecule are shown in Fig. 2. To trans-
+fer a molecule from one isomer to another, it must be
+partially torn off the substrate and be placed on the sub-
+strate with its other side. All this requires overcoming
+the energy barrier ∆E = 0.466 eV. The total energy of
+interaction with the h-BN substrate (desorption energy)
+Esub = 0.762 eV. Therefore, the spontaneous transition
+of the ACN molecule lying on the substrate from the L
+to the R form and vice versa is possible only at temper-
+atures T > 300 K. At lower temperatures, the molecule
+will always stay on the substrate, adjoining it with the
+same side, i.e. without changing the isomer type.
+To find the stationary state of the system of N ACN
+molecules lying on a flat h-BN substrate, it is necessary
+to find the state of the system with a minimum potential
+energy
+P → min : {Xn}N
+n=1.
+(9)
+The minimization problem (9) is solved numerically by
+the conjugate gradient method. Choosing the starting
+point of the minimization procedure, one can obtain all
+the main stationary states of the molecular system.
+Peptide groups of neighboring molecules can form hy-
+drogen bonds, creating dimers – see Fig. 4. The numeri-
+cal solution of the problem (9) shows that when molecules
+are located on a flat substrate, two types of dimers are
+FIG. 5:
+Typical secondary structures of homochiral ACN
+molecules on a flat substrate:
+(a) arc (number of atoms
+N = 15), (b) circle (N = 21), (c) single-beam spiral (N = 34),
+(d) two-beam spiral (N = 23 + 23), (e) three-beam spiral
+(N = 15 + 15 + 15), (f) nested circles (N = 12 + 24).
+possible: dimers of molecules of the same and different
+chirality. If a dimer is formed by identical isomers, then
+its binding energy is slightly higher: the hydrogen bond
+energy for RR and LL isomers is Ehb = 0.330 eV, and
+for RL and LR isomers Ehb = 0.322 eV. This is due to
+the fact that in this case the benzol rings C6H5 of the
+molecules are on the same side and they make a larger
+contribution to the interaction energy.
+The hydrogen
+bond angle also depends on the chirality of the dimer
+molecules. For dimer molecules of the same chirality, the
+angle between the dipole moments of the peptide groups
+forming a hydrogen bond is φhb = 17◦, and for molecules
+of different chirality φhb = 34◦.
+Chains of hydrogen bonds of molecules of the same chi-
+rality will always have benzol rings on one (outer) side,
+so they will twist in the opposite (inner) direction and
+have approximately the same curvature. On a plane, cir-
+cular arcs, circles, and spirals have such properties. The
+solution of the problem (9) has shown that on a flat sub-
+
+b
+a
+C6
+10
+20
+30
+40
+50
+−0.3
+−0.2
+−0.1
+1
+2
+3
+4
+5
+6
+Es (eV)
+N
+FIG. 6: Dependence of the specific energy of the secondary
+structure of the homochiral ACN molecules Es on the num-
+ber of molecules N for an arc, a circle, one-beam, two-beam,
+three-beam spiral and nested two circles (curves 1, 2, 3, 4,
+5 and 6). For the structure of two nested circles (curve 6),
+the dependence on the number of atoms of the outer circle is
+shown.
+strate molecules of the same chirality form stable shape
+structures with little changing curvature: arcs, spirals,
+circles – see Fig. 1 and 5. Left isomers form structures
+with a twist to the right, right – to the left.
+Hydrogen bond chains of N ≤ 16 ACN molecules of
+the same chirality form circular arcs of the same radius.
+The step of such a chain (the distance between the oxy-
+gen atoms of neighboring peptide groups) is a = 4.76 ˚A,
+the angle between neighboring links is ϕ = 162◦, the ra-
+dius of curvature by oxygen atoms is R = 15.2 ˚A – see
+Fig. 5 (a). The specific energy of the chain Es = E/N
+decreases monotonically with the growth of the number
+of molecules N – see Fig. 6. When the number of links
+is N > 16, the arcs cease to be stable; they either close
+and form circular chains, or touch their ends and form
+flat spirals – see Fig. 5 (b), (c).
+Stable cyclic chains can be formed from N
+≥ 5
+molecules of the same chirality. The dependence of the
+specific energy of the cyclic chain Es on the number of
+its links N is shown in Fig. 6. As can be seen from the
+figure, the most energetically favorable are cyclic chains
+of N = 20, 21, 22 links. Such chains form ring struc-
+tures with inner R1 = 27.3, 28.8, 30.5 and outer radii
+R2 = 40.4, 41.9, 43.6 ˚A.
+The dependence Es(N) for a one-beam spiral actually
+continues the dependence for an arc (see Fig. 6, curves
+1 and 3). Two-beam and three-beam spirals are bound
+states of arc structures.
+The specific energy of helical
+structures decreases monotonically with an increase in
+the number of molecules. For N > 27, helical structures
+are more energy efficient than ring structures (this is due
+to their denser structure).
+The most energy-efficient are nested structures of two
+circles with the number of atoms N = 14 + 26, 15+27,
+17+29 (the first number in the sum corresponds the num-
+ber of atoms in the inner circle, the second – in the outer
+FIG. 7: 2D crystals of ACN molecules on a flat substrate:
+(a) with parallel packing of linear chains of hydrogen bonds
+(periods ax = 9.81, ay = 10.24), (b) with antiparallel packing
+(ax = 9.79, ay = 20.43 ˚A). On the surface of a flat substrate,
+the chain of hydrogen bonds can be linear only if the L and
+R isomers alternate. A crystal is formed by 6 linear chains of
+24 molecules (the total number of molecules is N = 6 × 24).
+The flat substrate is shown in green.
+circle) – see Fig. 6, curve 6.
+If the chain of hydrogen bonds consists of a random
+sequence of isomers, then it will look like an irregular
+broken line. The chain has the shape of a straight zigzag
+only if there is a strict alternation of L and R isomers.
+In this case, the zigzag step (the distance between the
+oxygen atoms of neighboring molecules) is a = 4.85 ˚A,
+the zigzag angle is ϕ = 170◦. Such chains on a flat sub-
+strate surface can form two-dimensional regular lattices
+(2D crystals) with parallel and antiparallel packing of
+neighboring chains – see Fig. 7. With parallel packing,
+the crystal periods are ax = 9.81˚A, ay = 10.24˚A, the
+specific energy is Es = −0.325 eV. With antiparallel
+packing, periods are ax = 9.79˚A, ay = 20.43˚A, energy
+is Es = −0.322 eV.
+IV.
+SELF-ASSEMBLY OF MOLECULAR
+STRUCTURES
+Let us simulate the self-organization of the molecular
+structures of ACN molecules on the flat surface of the h-
+
+a
+b7
+100
+200
+300
+400
+500
+0.4
+0.6
+0.8
+1
+nhb
+(a)
+100
+200
+300
+400
+500
+1
+1.2
+1.4
+T (K)
+c
+(b)
+FIG. 8: Dependence of (a) the normalized number of hydro-
+gen bonds nhb and (b) the dimensionless heat capacity c on
+temperature T for a system of N = 1024 ACN molecules
+located on a flat substrate with the periodic square compu-
+tational cell of size 33 × 33 nm2. Solid (blue) lines show de-
+pendencies for a system of homochiral molecules, dotted (red)
+lines – dependencies for a racemic mixture of molecules. Verti-
+cal dotted straight lines correspond to temperatures T = 250,
+330, and 480 K.
+BN crystal. To do this, we take a square periodic cell of
+size 33 × 33 nm2 on the surface of the substrate and ran-
+domly place N = 1024 ACN molecules into it. Then we
+immerse this molecular system in a Langevin thermostat
+of temperature T and numerically simulate the dynamics
+of the system during the time t = 10 ns. To do this, we
+numerically integrate the system of Langevin equations
+M ¨Xn = −
+∂
+∂Xn
+H − ΓM ˙Xn − Ξn,
+n = 1, ..., N,
+(10)
+where Γ
+=
+1/tr is the friction coefficient,
+Ξn
+=
+{ξn,i,k}11, 3
+i=1,k=1 is 33-dimensional vector of normally dis-
+tributed random Langevin forces with the following cor-
+relations:
+⟨ξn1,i,k(t1)ξn2,j,l(t2)⟩ = 2MikBT Γδn1n2δijδklδ(t1 − t2).
+Here Mi is mass of i-th atom of ACN molecule, kB is
+Boltzmann constant, T is temperature of the Langevin
+thermostat (temperature of the substrate), numbers
+n1, n2 = 1, ..., N, i, j = 1, ..., 11, k, l = 1, 2, 3.
+The parameter tr characterizes the intensity of energy
+exchange between the molecular system and the ther-
+mostat. Simulation of the dynamics of ACN molecules
+on an h-BN sheet, taking into account the mobility of
+the sheet atoms, makes it possible to estimate the relax-
+ation time tr ∼ 100 ps. For the convenience of numer-
+ical integration, we will use a smaller value tr = 10 ps.
+FIG. 9: Structure of N = 1024 homochiral ACN molecules
+appearing on the flat surface of the substrate at T = 240K.
+The straight lines show the boundaries of the periodic calcu-
+lation cell of size 33 × 33 nm2. The substrate surface is not
+shown.
+This makes it possible to significantly reduce the time of
+numerical integration, which is sufficient to obtain reli-
+able average values. After the dynamics of the molec-
+ular system reaches the steady state, we will find the
+time averages of the system energy ¯E(T ) and the num-
+ber of hydrogen bonds ¯Nhb(T ). We assume that two ACN
+molecules form a hydrogen bond if their interaction en-
+ergy is greater than half of the hydrogen bond energy:
+U(X1, X2) < −Ehb/2 = −0.16 eV.
+The state of the system can be conveniently character-
+ized by its dimensionless heat capacity
+c =
+1
+33NkB
+d ¯E(T )
+dT
+,
+(11)
+and the normalized number of hydrogen bonds nhb =
+¯Nhb(T )/N. The dependence of these quantities on tem-
+perature is shown in Fig. 8.
+Numerical simulation shows the existence of three
+characteristic temperature values T1 < T2 < T3.
+At
+T < T1 = 250K the molecules on a flat substrate form a
+stable system of chains of hydrogen bonds. Here, almost
+every molecule participates in the formation of one hy-
+drogen bond (number nhb ≈ 1). The dimensionless heat
+capacity of the system is c = 1. At T1 < T < T2 = 330K,
+a slight decrease in the number of nhb bonds and a
+monotonous increase in heat capacity begin to occur –
+the process of melting of hydrogen bond chains begins.
+At a temperature T > T2, the number of hydrogen bonds
+
+8
+FIG. 10: Structure of N = 1024 homochiral ACN molecules
+appearing on the flat surface of the substrate at T = 300K.
+The straight lines show the boundaries of the periodic calcu-
+lation cell of size 33 × 33 nm2.
+decreases in proportion to the increase in temperature,
+and the heat capacity reaches a constant value c ≈ 1.23.
+Here we have a melt of short chains of hydrogen bonds
+(the average length of the chains decreases proportion-
+ally to the increase in temperature). At T > T3 = 480K,
+individual molecules can already be detached from the
+substrate – desorption of molecules begins.
+The structure of the resulting system of hydrogen
+bond chains depends on the isomeric composition of the
+molecules. If all molecules are homochiral (only one iso-
+mer is present), then secondary structures form on the
+surface of the substrate.
+These structures are circular
+and spiral hydrogen bond chains – see Figs. 9 and 10. The
+substrate surface become optically active, the left isomers
+form chains with a right twist, and the right ones with a
+left twist. As can be seen from Fig. 9 for T = 240K on
+a flat substrate all possible circular secondary structures
+(arcs, circles, spirals) are formed (Fig. 5). An increase in
+temperature leads, first of all, to the destruction of spiral
+structures. As a result, the number of cyclic chains of
+average radius increases since they are the most stable –
+see Fig. 10.
+If the isomers of molecules are taken randomly, we get
+their racemic mixture. In this case, no secondary struc-
+tures are formed. There are only curvilinear chains of
+random shape – see Fig. 11.
+FIG. 11: Structure formed on the flat surface of the sub-
+strate at T = 240K from a racemic mixture of N = 1024
+ACN molecules. The straight lines show the boundaries of
+the periodic calculation cell of size 33 × 33 nm2.
+V.
+MELTING OF 2D CRYSTALS
+To simulate the dynamics of 2D crystals of ACN
+molecules on a flat h-BN substrate, consider a crystal
+formed by 22 linear chains of hydrogen bonds of 48
+molecules (total number of molecules N = 22 × 48 =
+1056). A crystal with parallel chain packing has dimen-
+sions of 23.3 × 22.5 nm2.
+We place the crystal in the
+center of the calculated periodic cell size 23.5×22.6 nm2.
+In this case, the chains of hydrogen bonds located par-
+allel to the x axis are closed, and the first chain begins
+to come into contact with the last one – the 2D crystal
+form a dense packing on the substrate that has no edges
+and has normalized number of hydrogen bonds nhb = 1
+(the number of hydrogen bonds is equal to the number of
+molecules). To simulate a rectangular crystal with edges
+(square crystallite), we take a periodic computational cell
+of size 47 × 45.2 nm2. In this case, the 2D crystal (crys-
+tallite) covers only 25% substrate surface, the chains of
+hydrogen bonds are not closed, the normalized number
+of hydrogen bonds is nhb = (N − 22)/N = 0.979.
+Then we numerically integrate the system of equations
+of motion (10) with the initial condition corresponding to
+the stationary state of the crystal. At different thermo-
+stat temperatures, we find the average values of the sys-
+tem energy ¯E(T ), the normalized number of hydrogen
+bonds nhb, and the fraction of molecules that left the
+substrate from the first layer p (the fraction of molecules
+located on the substrate at distance z > 5 ˚A). Next, using
+
+9
+100
+200
+300
+400
+500
+0.4
+0.6
+0.8
+1
+1
+2
+(a)
+nhb
+100
+200
+300
+400
+500
+1
+1.5
+2
+2.5
+3
+4
+(b)
+c
+100
+200
+300
+400
+500
+0
+5
+10
+6
+5
+T (K)
+p (%)
+(c)
+FIG. 12: Dependence of (a) the normalized number of hy-
+drogen bonds nhb, (b) the dimensionless heat capacity c and
+(c) the fraction of molecules that left the first layer p on the
+temperature T for a 2D crystal of N = 1056 ACN molecules
+located on a flat substrate with periodic computational cell
+of size 46.6 × 45 nm2 (curves 1, 3, 5; 25% substrate coverage)
+and 23.3 × 22.5 nm2 (curves 2, 4 , 6; 100% substrate cover-
+age). Vertical dotted straight lines show temperature values
+T = 295, 335, 365, 415, 445 and 470 K.
+the formula (11), we find the temperature dependence of
+the heat capacity of the molecular system c.
+The dependence of nhb, c, and p on the thermostat
+temperature (on the substrate temperature) T is shown
+in Fig. 12. As can be seen from the figure, at 25% cover-
+age of the substrate, the square crystallite begins to melt
+at a temperature of T1 = 295K. At T < T1, the crystal
+structure is preserved, the number of bonds, heat capac-
+ity, and density of the first layer of the substrate coating
+practically do not change with increasing temperature:
+nhb(T ) ≡ nhb(0), c(T ) ≡ 1, p(T ) ≡ 0. Crystallite melt-
+ing occurs in the temperature range T1 < T < T2 where
+the upper temperature is T2 = 365 K. Here, a slight de-
+crease in the number of bonds and an increase in heat
+capacity begin to occur. The heat capacity reaches its
+maximum value at the temperature Tm = 335 K. In the
+temperature interval [T1, T2], an increasing destruction
+of the edges of the initial crystallite occurs. The ends of
+the chains peel off from the central part of the crystallite,
+100
+200
+300
+400
+500
+1
+1.5
+2
+2.5
+c
+1
+2
+3
+4
+T (K)
+FIG. 13: Dependence of the dimensionless heat capacity c on
+temperature T for a 2D square crystallite of N = 480, 1056,
+2376 of the ACN molecules located on a flat substrate with
+periodic computational cell of the size 31.2× 30.68, 46.6× 45,
+70.2 × 67.5 nm2 (curves 1, 3, 5; 25% substrate coverage) and
+for infinite 2D crystal (curve 4). Vertical dotted straight lines
+mark the temperatures T = 315, 335, 365, and 445 K.
+then break off and go to the free part of the substrate.
+As a result of this ”continuous” melting, the crystallite
+transforms into a melt consisting of short chains of hy-
+drogen bonds, uniformly covering the entire substrate.
+For T > T2, the number of hydrogen bonds decreases
+in proportion to the increase in temperature, while the
+heat capacity remains almost constant: c ≈ 1.4. It can
+be concluded that a complete transition of the molecu-
+lar system from a low-energy and low-entropy crystalline
+state to a high-energy and high-entropy liquid state has
+taken place. As a result of this transition, the substrate
+is uniformly covered with a ”solution” of short chains of
+hydrogen bonds. The chain lengths decreases monotoni-
+cally with increasing temperature (on the Fig. 12 (a) this
+manifests itself in a monotonous decrease in the number
+of bonds). All molecules remain directly adjacent to the
+substrate (p = 0), insignificant desorption is observed
+only at T > 480 K – see Fig. 12 (c).
+With 100% coverage of the substrate, due to the ab-
+sence of edges in the crystal, its melting occurs at higher
+temperatures and happens according to a different sce-
+nario.
+Here melting also occurs ”continuously” in the
+temperature interval [T1, T2], where T1 = 415, T2 =
+470 K. The peak of the heat capacity at Tm = 445 K
+becomes more pronounced. Melting occurs due to the
+expulsion of some molecules from the first layer to the
+second, which manifests itself in a monotonous increase
+in the fraction of displaced molecules p at T1 < T < T2
+– see Fig. 12 (c).
+As a result, the density of the first
+layer during melting decreases by 11%, and after melting
+a dense melt of molecules is formed on the substrate, in
+which 11% of the molecules are located on the second
+layer from the substrate.
+Conventionally, the temperature value Tm, at which
+
+10
+the heat capacity has reached its maximum value, can
+be considered as the melting temperature of a 2D crystal.
+However melting occurs not discretely, but continuously
+in the temperature interval [T1, T2]. The value Tm is in
+the center of this interval.
+To study the dependence of the melting temperature
+on the crystallite size, we analyze numerically the melt-
+ing of 2D square crystallite of N = 480 and 2376 ACN
+molecules placed on a flat substrate with periodic com-
+putational cell of size 31.2 × 30.68 and 70.2 × 67.5 nm2.
+Our results show that melting occurs continuously for all
+crystallite sizes. When the crystallite size is increased,
+the melting interval [T1, T2] shifts to the right and, in
+the limit, coincides with the melting temperature inter-
+val of a 2D crystal with 100% coverage of the substrate,
+as shown in Fig. 13.
+Let us note that the continuum melting scenario also
+takes place for 2D n-alkane crystals lying on a flat sur-
+face of graphite [42].
+This allows us to conclude that
+the quasi-continuous melting scenario is a characteristic
+feature of 2D systems of molecules adsorbed by a flat
+surface.
+VI.
+CONCLUSION
+Our numerical simulations of the dynamics of the sys-
+tem of acetanilide molecules have revealed that the struc-
+tures achiral in three-dimensional space become chiral
+when being placed on a flat substrate: Depending on the
+side it touches the substrate, the molecule has two iso-
+mers L and R. The homochirality of the molecules leads
+to the appearance of stable secondary structures stabi-
+lized by hydrogen bonds on a flat substrate in the form
+of arc, cyclic, and helical chains of hydrogen bonds and
+their complexes.
+Hydrogen-bond chains of N ≤ 16 molecules of the same
+chirality form circular arcs of the same radius.
+When
+the number of molecules in the chain is N > 16, the
+arcs becomes unstable, they either collapse into circles,
+or touch their ends and form flat spirals.
+In addition
+to single-beam spirals, two- and three-beam spirals may
+exist being bound states of arc chains.
+Stable cyclic chains can be formed from N
+≥ 5
+molecules of the same chirality.
+The most energeti-
+cally favorable are cyclic chains of 20, 21 and 22 links.
+The structures of two circles with the number of atoms
+N = 14 + 26, 15 + 27, 17 + 29 (where the first number of
+the sum is the number of atoms in the inner circle, and
+the second number is the number of atoms in the outer
+circle) are more energy efficient.
+If the chain of hydrogen bonds consists of a random
+sequence of isomers, it look like an irregular broken line.
+A chain can take the form of a rectilinear zigzag only if
+there is a strict alternation of the L and R isomers. Such
+chains can form regular two-dimensional crystals with
+parallel and antiparallel packing of the adjacent chains.
+Simulation of the dynamics of a system of molecules
+shows that the homochirality of molecules(the presence
+of only one isomer) leads to the appearance of stable sec-
+ondary structures on the surface of the substrate, i.e. to
+the appearance of chains of hydrogen bonds in the form
+of arcs, circles and spirals.
+As a result, the substrate
+surface becomes optically active, the left isomers form
+chains with a right twist, and the right ones form chain
+with a left twist. At temperature T ≤ 240 K, all possible
+secondary structures are formed on the substrate. An
+increase in temperature leads, first of all, to the disinte-
+gration of spiral structures. As a result, the number of
+more stable circular chains increases.
+If the molecules on the substrate form a racemic mix-
+ture, no regular secondary structures are formed, so only
+curvilinear chains of hydrogen bonds of random shape
+can appear. Thus, our results demonstrate the impor-
+tance of homochirality (chiral purity) of biomolecules for
+the formation of stable secondary molecular structures.
+Numerical simulations of the dynamics of a 2D crystal
+of the ACN molecules shows that the scenario of crystal
+melting depends on the density of its coverage of the
+substrate.
+At 25% coverage of the substrate the melting occurs
+”continuously” in the temperature interval [T1, T2] from
+the edges of the initial crystallite (for crystallite of 1056
+ACN molecules temperatures T1 = 295, T2 = 365 K).
+The ends of the chains peel off from the central part of
+the crystallite, then break off and go to the free part
+of the substrate. As a result, the crystallite transforms
+into a melt consisting of short chains of hydrogen bonds,
+uniformly covering the entire substrate. When the size of
+the crystallite is increased, the melting interval [T1, T2]
+shifts to the right and, in the limit, will coincides with
+the melting temperature interval of infinite crystal with
+100% coverage of the substrate.
+For 100% coverage of the substrate with no crystal
+edges, the melting occurs at higher temperatures 415 ÷
+470 K by a shift of some molecules from the first to the
+second layer in the substrate. As a result, the density of
+the first layer during melting decreases by 11%, and after
+melting, 11% of the molecules move to the second layer
+formed on the substrate.
+A continual melting scenario (the presence of a melting
+temperature interval)has also been found in 2D n-alkane
+crystals lying on a flat surface of graphite [42]. All this
+leads us to conclusion that the quasi-continuous melt-
+ing scenario is a characteristic feature of 2D systems of
+molecules adsorbed by a flat surface.
+ACKNOWLEDGMENTS
+AVS acknowledges the use of the computational facilities
+provided by the Interdepartmental Supercomputer Cen-
+ter of the Russian Academy of Science. YSK acknowl-
+edges a support from the Australian Research Council
+(grant DP210101292) and the Strategic Fund of the Aus-
+tralian National University.
+
+11
+[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.
+Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov.
+Electric Field Effect in Atomically Thin Carbon Films.
+Science 306 (5696), 666-669 (2004).
+[2] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.
+Novoselov and A. K. Geim. The electronic properties of
+graphene. Rev. Mod. Phys. 81, 109-162 (2009).
+[3] E. Koren, I. Leven, E. Lortscher, A. Knoll, O. Hod and
+U. Duerig. Coherent commensurate electronic states at
+the interface between misoriented graphene layers. Na-
+ture Nanotech. 11, 752-757 (2016).
+[4] J. C. Meyer, A. K. Geim, M. Katsnelson, K. Novoselov,
+T. Booth and S. Roth. The structure of suspended
+graphene sheets. Nature 446, 60-63 (2007).
+[5] C. Lee, X. Wei, J. W. Kysar and J. Hone. Measurement
+of the Elastic Properties and Intrinsic Strength of Mono-
+layer Graphene. Science 321, 385-388 (2008).
+[6] A. Falin, Q. Cai, E. J. G. Santos, D. Scullion, D. Qian, R.
+Zhang, Z. Yang, S. Huang, K. Watanabe, T. Taniguchi,
+M. R. Barnett, Y. Chen, R. S. Ruoff and L. H. Li.
+Mechanical properties of atomically thin boron nitride
+and the role of interlayer interactions. Nat. Commun. 8,
+15815 (2017).
+[7] E. Han, J. Yu, E. Annevelink, J. Son, D. A. Kang, K.
+Watanabe, T. Taniguchi, E. Ertekin, P. Y. Huang and
+A. M. van der Zande. Ultrasoft slip-mediated bending in
+few-layer graphene. Nature Materials 19, 305-309 (2020).
+[8] I. Leven, D. Krepel, O. Shemesh, and O. Hod. Robust Su-
+perlubricity in Graphene/h-BN Heterojunctions. J. Phys.
+Chem. Lett. 4, 115-120 (2013).
+[9] A. Geim and I. Grigorieva. Van der Waals heterostruc-
+tures. Nature 499, 419-425 (2013).
+[10] K. S. Novoselov, A. Mishchenko, A. Carvalho and A.
+H. Castro Neto. 2D materials and van der Waals het-
+erostructures. Science 353 (6298), 461 (2016).
+[11] C. R. Woods, L. Britnell, A. Eckmann, R. S. Ma, J.
+C. Lu, H. M. Guo, X. Lin, G. L. Yu, Y. Cao, R. V.
+Gorbachev, A. V. Kretinin, J. Park, L. A. Ponomarenko,
+M. I. Katsnelson, Y. N. Gornostyrev, K. Watanabe, T.
+Taniguchi, C. Casiraghi, H. J. Gao, A. K. Geim and K. S.
+Novoselov. Commensurate-incommensurate transition in
+graphene on hexagonal boron nitride. Nature Phys. 10,
+451-456 (2014).
+[12] G. J. Slotman, M. M. van Wijk, P. L. Zhao, A. Fasolino,
+M. I. Katsnelson and S. Yuan. Effect of Structural Relax-
+ation on the Electronic Structure of Graphene on Hexag-
+onal Boron Nitride. Phys. Rev. Lett. 115, 186801 (2015).
+[13] D. Mandelli, I. Leven, O. Hod and M. Urbakh. Sliding
+friction of graphene/hexagonal-boron nitride heterojunc-
+tions: a route to robust superlubricity. Sci. Rep. 7 (1),
+10851 (2017).
+[14] D. Jariwala, T. J. Marks and M. C. Hersam. Mixed-
+dimensional van der Waals heterostructures. Nature
+Mater 16, 170-181 (2017).
+[15] O. V. Ershova, T. C. Lillestolen and E. Bichoutskaia.
+Study of polycyclic aromatic hydrocarbons adsorbed on
+graphene using density functional theory with empiri-
+cal dispersion correction. Phys. Chem. Chem. Phys., 12,
+6483-6491 (2010).
+[16] E. G. Gordeev, M. V. Polynski and V. P. Ananikov.
+Fast and accurate computational modeling of adsorption
+on graphene: a dispersion interaction challenge. Phys.
+Chem. Chem. Phys., 15, 18815-18821 (2013).
+[17] J. Wang, Z. Chen and B. Chen. Adsorption of polycyclic
+aromatic hydrocarbons by graphene and graphene oxide
+nanosheets. Environ. Sci. Technol., 48, 4817-4825 (2014).
+[18] M. Castro and E. Chigo-Anota. BN and BN oxide
+nanosheets based nanosensor for Paracetamol adsorp-
+tion: A Density Functional investigation. Mex. J. Mat.
+Sci. Eng., 1, 21-29 (2014).
+[19] P. P. Zhou and R. Q. Zhang. Physisorption of benzene
+derivatives on graphene: critical roles of steric and stere-
+oelectronic effects of the substituent. Phys. Chem. Chem.
+Phys., 17, 12185-12193 (2015).
+[20] V. Georgakilas, M. Otyepka, A. B. Bourlinos, V. Chan-
+dra, N. Kim, K. C. Kemp, P. Hobza, R. Zboril and K.
+S. Kim. Functionalization of Graphene: Covalent and
+Non-Covalent Approaches, Derivatives and Applications.
+Chem. Rev., 112, 6156-6214 (2012).
+[21] V. Georgakilas, J. N. Tiwari, K. C. Kemp, J. A. Perman,
+A. B. Bourlinos, K. S. Kim, and R. Zboril. Noncovalent
+Functionalization of Graphene and Graphene Oxide for
+Energy Materials, Biosensing, Catalytic, and Biomedical
+Applications. Chem. Rev., 116, 5464-5519 (2016).
+[22] R. Thakkar, S. Gajaweera and J. Comer. Organic con-
+taminants and atmospheric nitrogen at the graphene-
+water interface: a simulation study. Nanoscale Adv., 4,
+1741 (2022).
+[23] S. M. Barlow, R. Raval. Complex organic molecules at
+metal surfaces: bonding, organisation and chirality. Sur-
+face Science Reports, 50, 201-341 (2003).
+[24] H. Cao, M.-P. Van Den Eede, G. Koeckelberghs, K. S.
+Mali and S. De Feyter. Direct observation of the influence
+of chirality on the microstructure of regioregular poly-
+(3-alkylthiophene)s at the liquid/solid interface. Chem.
+Commun., 53, 153-156 (2017).
+[25] E. Bahn, H. Hedgeland, A. P. Jardine, P. F. Henry, T. C.
+Hansen and P. Fouquet. The structure of deuterated ben-
+zene films adsorbed on the graphite (0001) basal plane:
+what happens below and above the monolayer coverage?
+Phys. Chem. Chem. Phys., 16, 22116 (2014).
+[26] F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill,
+P. Blake, M. I. Katsnelson, K. S. Novoselov. Detection
+of individual gas molecules adsorbed on graphene. Nat.
+Mater., 6, 652-655 (2007).
+[27] Y. Shao, J. Wang, H. Wu, J. Liu, I. a. Aksay and Y. Lin.
+Graphene Based Electrochemical Sensors and Biosensors:
+A Review. Electroanalysis, 22, 1027-1036 (2010).
+[28] X. V. Zhen, E. G. Swanson, J. T. Nelson, Y. Zhang, Q.
+Su, S. J. Koester, and P. Buhlmann. Noncovalent Mono-
+layer Modification of Graphene Using Pyrene and Cy-
+clodextrin Receptors for Chemical Sensing. ACS Appl.
+Nano Mater. 1, 2718-2726 (2018).
+[29] Z. Iranmanesh-Zarandy and M. Dehestani. Molecular
+Dynamics Simulation of Paracetamol Drug Adsorption
+on Boron Nitride Nanotube:
+Effects of Temperature,
+Nanotube Length, Diameter, and Chirality. ChemistryS-
+elect, 4, 7866 -7873 (2019).
+[30] X. Kang, J. Wang, H. Wu, J. Liu, I. A. Aksay, Y. Lin.
+A graphene-based electrochemical sensor for sensitive de-
+tection of paracetamol. Talanta, 81, 754-759 (2010).
+[31] M. Lian,
+J. Fan,
+Z. Shi,
+H. Li,
+J. Yin. Kevlar-
+
+12
+functionalized graphene nanoribbon for polymer rein-
+forcement. Polymer, 55(10), 2578-2587 (2014).
+[32] S. Kumar. Graphene Engendered 2-D Structural Mor-
+phology of Aluminium Atoms: Molecular Dynamics Sim-
+ulation study. Materials Chemistry and Physics, 202,
+329-339 (2017).
+[33] M. W. Roth, C. L. Pint and C. Wexler. Phase transitions
+in hexane monolayers physisorbed onto graphite. Phys.
+Rev. B, 71, 155427 (2005).
+[34] Cary L. Pint, M. W. Roth and C. Wexler. Behavior of
+hexane on graphite at near-monolayer densities: Molec-
+ular dynamics study. Phys. Rev. B, 73, 085422 (2006).
+[35] K. E. Becker. Accelerated molecular dynamics simulation
+of the thermal desorption of n-alkanes from the basal
+plane of graphite. J. Chem. Phys. 125, 184706 (2006).
+[36] Q. Chen, H.-J. Yan, C.-J. Yan, G.-B. Pan, L.-J. Wan, G.-
+Y. Wen, D.-Q. Zhang. STM investigation of the depen-
+dence of alkane and alkane (C18H38, C19H40) derivatives
+self-assembly on molecular chemical structure on HOPG
+surface. Surface Science, 602 1256-1266 (2008).
+[37] M. J. Connolly, M. W. Roth, P. A. Gray, and C. Wexler.
+Explicit Hydrogen Molecular Dynamics Simulations of
+Hexane Deposited onto Graphite at Various Coverages.
+Langmuir, 24(7), 3228-3234 (2008).
+[38] T. Yang, S. Berber, J.-F. Liu, G. P. Miller and D.
+Tomanek. Self-assembly of long chain alkanes and their
+derivatives on graphite. J. Chem. Phes., 128, 124709
+(2008).
+[39] M. W. Roth, L. Firlej, B. Kuchta, M. J. Connolly, E.
+Maldonado, and C. Wexler. Simulation and Characteri-
+zation of Tetracosane on Graphite: Molecular Dynamics
+Beyond the Monolayer. J. Phys. Chem. C, 120(2), 984-
+994 (2016).
+[40] T. K. Piskorz, C. Gobbo, S. J. Marrink, S. De Feyter, A.
+H. de Vries, and J. H. van Esch. Nucleation Mechanisms
+of Self-Assembled Physisorbed Monolayers on Graphite.
+J. Phys. Chem. C 123(28), 17510-17520 (2019).
+[41] L. Y. Fang, Y. Hua, Z. Z. Meng, Z. Hui. Molecular
+dynamics simulations on the orientation of n-alkanes
+with different lengths on graphene. Surface Science, 690,
+121468 (2019).
+[42] R. Zhang, W. S. Fall, K. W. Hall, G. A. Gehring, X. Zeng,
+G. Ungar. Quasi-continuous melting of model polymer
+monolayers prompts reinterpretation of polymer melting.
+Nat. Commun., 12, 1710 (2021).
+[43] H. Yang, X. J. Zhao, and M. Sun. Induced crystalliza-
+tion of single-chain polyethylene on a graphite surface:
+Molecular dynamics simulation. Phys. Rev. E, 84, 011803
+(2011).
+[44] M. Gulde, A. N. Rissanou, V. Harmandaris, M. M¨uller,
+S. Sch¨afer, and C. Ropers. Dynamics and Structure of
+Monolayer Polymer Crystallites on Graphene. Nano Lett.
+16(11), 6994-7000 (2016).
+[45] Y. F. Liu, H. Yang, H. Zhang. Molecular dynamics simu-
+lation of the folding of single alkane chains with different
+lengths on single-walled carbon nanotubes and graphene.
+Journal of Molecular Modeling, 24, 140 (2018).
+[46] C. L. Pint. Different melting behavior in pentane and
+heptane monolayers on graphite:
+Molecular dynamics
+simulations. Phys. Rev. B, 73, 045415 (2006).
+[47] H.Yang, Z.-S. Li, Z.-Y. Lu, C.-C. Sun. Molecular dynam-
+ics simulations study on the melting process of n-heptane
+layer(s) adsorbed on the graphite (0 0 1) surface. Surface
+Science, 600(6), 1213-1220 (2006).
+[48] C. Wexler, L. Firlej, B. Kuchta, and M. W. Roth. Melting
+of Hexane Monolayers Adsorbed on Graphite: The Role
+of Domains and Defect Formation. Langmuir, 25(12),
+6596-6598 (2009).
+[49] M. Pykal, P. Jurecka, F. Karlicky, and M. Otyepka.
+Modelling of Graphene Functionalization. Phys. Chem.
+Chem. Phys., 18, 6351-6372 (2016).
+[50] S. W. Johnson, J. Eckert, M. Barthes, R. K. McMullan,
+and M. Muller. Crystal Structure of Acetanilide at 15
+and 295 K by Neutron Diffraction. Lack of Evidence for
+Proton Transfer along the N-H...O Hydrogen Bond. J.
+Phys. Chem. 99(44), 16253-16260 (1995).
+[51] G. Careri, U. Buontempo, F. Galluzzi, A. C. Scott, E.
+Gratton, and E. Shyamsunder. Spectroscopic evidence
+for Davydov-like solitons in acetanilide. Phys. Rev. B 30,
+4689 (1984).
+[52] A. Scott. Davydov’s soliton. Physics Reports 217(1), 1-
+67 (1992).
+[53] L. Cruzeiro, H. Freedman. The temperature dependent
+amide I band of crystalline acetanilide. Physics Letters
+A 377, 1593-1596 (2013).
+[54] L. Cruzeiro. The Amide I Band of Crystalline Ac-
+etanilide:
+Old Data Under New Light. In:
+Archilla,
+J.,
+Jimenez,
+N., Sanchez-Morcillo,
+V., Garcia-Raffi,
+L.
+(eds)
+Quodons
+in
+Mica.
+Springer
+Series
+in
+Materials
+Science,
+221,
+Springer,
+Cham.
+(2015)
+https://doi.org/10.1007/978-3-319-21045-2 17
+[55] W. D. Cornell, W. P. Cieplak, C. I. Bayly, I. R. Gould, K.
+M. Merz, D. M. Ferguson, D. C. Spellmeyer, T. Fox, J.
+W. Caldwell, P. A. Kollman. A second generation force
+field for the simulation of proteins, nucleic acids, and
+organic molecules, J. Am. Chem. Soc. 117(19) 5179-5197
+(1995).
+[56] A. V. Savin, E. A. Korznikova, and S. V. Dmitriev.
+Dynamics of surface graphene ripplocations on a flat
+graphite substrate. Phys. Rev. B 99, 235411 (2019).
+[57] A. V. Savin. Eigenmodes and resonance vibrations of
+graphene nanomembranes. Phys. Rev. B, 103, 195435
+(2021).
+[58] A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. God-
+dard III, and W. M. Skiff. UFF, a Full Periodic Table
+Force Field for Molecular Dynamics Simulations., J. Am.
+Chem. Soc. 114(25), 10024-10035 (1992).
+

XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ddfdeaf75fd96d5755f9110944c3fc8c7af9949e260adc01f3a80165c13f3cef
+size 1047073

XNAyT4oBgHgl3EQfWPfd/content/tmp_files/2301.00161v1.pdf.txt

@@ -0,0 +1,1135 @@
+Active RISs: Signal Modeling, Asymptotic
+Analysis, and Beamforming Design
+Zijian Zhang∗, Linglong Dai∗, Fellow, IEEE, Xibi Chen∗, Changhao Liu∗, Fan Yang∗, Fellow, IEEE,
+Robert Schober†, Fellow, IEEE, and H. Vincent Poor§, Life Fellow, IEEE
+∗Beijing National Research Center for Information Science and Technology (BNRist)
+Department of Electronic Engineering, Tsinghua University, China
+†Institute for Digital Communications, Friedrich-Alexander University Erlangen-N¨urnberg, Germany
+§Department of Electrical and Computer Engineering, Princeton University, USA
+E-mails: zhangzj20@mails.tsinghua.edu.cn; daill@tsinghua.edu.cn; cxb17@tsinghua.org.cn; liuch17@tsinghua.org.cn;
+fan yang@tsinghua.edu.cn; robert.schober@fau.de; poor@princeton.edu
+Abstract—Reconfigurable
+intelligent
+surfaces
+(RISs)
+have
+emerged as a candidate technology for future 6G networks.
+However, due to the “multiplicative fading” effect, the existing
+passive RISs only achieve a negligible capacity gain in environ-
+ments with strong direct links. In this paper, the concept of
+active RISs is studied to overcome this fundamental limitation.
+Unlike the existing passive RISs that reflect signals without
+amplification, active RISs can amplify the reflected signals via
+amplifiers integrated into their elements. To characterize the
+signal amplification and incorporate the noise introduced by the
+active components, we verify the signal model of active RISs
+through the experimental measurements on a fabricated active
+RIS element. Based on the verified signal model, we formulate
+the sum-rate maximization problem for an active RIS aided
+multi-user multiple-input single-output (MU-MISO) system and
+a joint transmit precoding and reflect beamforming algorithm is
+proposed to solve this problem. Simulation results show that, in
+a typical wireless system, the existing passive RISs can realize
+only a negligible sum-rate gain of 3%, while the active RISs can
+achieve a significant sum-rate gain of 62%, thus overcoming the
+“multiplicative fading” effect. Finally, we develop a 64-element
+active RIS aided wireless communication prototype, and the
+significant gain of active RISs is validated by field test.
+I. INTRODUCTION
+From the first generation (1G) to 5G wireless communica-
+tions, the wireless channel has been considered to be uncon-
+trollable. Recently, due to the advances in meta-materials, re-
+configurable intelligent surfaces (RISs) have been proposed [1]
+for the purpose of intelligently controlling wireless channels
+to achieve improved communication performance. Specifically,
+an RIS is an array composed of a very large number of passive
+elements that reflects electromagnetic signals in a desired
+manner so as to reconfigure the propagation properties of
+wireless environment. As an important advantage of RIS, the
+negligible noise introduced by passive RISs enables a high
+array gain. Benefiting from this advantage, RISs are expected
+to introduce significant capacity gains in wireless systems [2].
+However, in practice, the expected capacity gains are typ-
+ically only observed in communication environments where
+the direct link between transmitter and receiver is completely
+blocked or very weak. By contrast, in many scenarios where
+the direct link is not weak, conventional RISs can only
+achieve negligible capacity gains [3]. The reason behind this
+phenomenon is the “multiplicative fading” effect introduced
+by RISs, i.e., the equivalent path loss of the transmitter-RIS-
+receiver link is the product (instead of the sum) of the path
+losses of the transmitter-RIS and RIS-receiver links, which is
+usually thousands of times larger than that of the direct link
+[1]–[3]. As a result, the “multiplicative fading” effect makes
+it almost impossible for passive RISs to achieve noticeable
+capacity gains in many wireless environments. Therefore,
+to advance the practicability of RISs in future 6G wireless
+networks, a critical issue for RISs to be addressed is: How to
+break the fundamental performance bottleneck caused by the
+“multiplicative fading” effect?
+To overcome the fundamental physical limitation of con-
+ventional passive RISs imposed by the “multiplicative fading”
+effect, in this paper, we investigate the concept of active RISs
+to overcome the “multiplicative fading” effect. Different from
+the existing passive RISs that passively reflect signals without
+amplification, the key feature of active RISs is their ability
+to actively reflect signals with amplification at the expense
+of additional power consumption. Firstly, through the experi-
+mental measurements on a fabricated active RIS element, we
+verify the signal model of active RISs, which characterizes the
+amplification of the incident signal and accounts for the non-
+negligible thermal noise introduced by the active elements.
+Based on the verified signal model, we further analyze the
+asymptotic performance of active RISs and formulate a sum-
+rate maximization problem for an active RIS aided multi-
+user multiple-input single-output (MU-MISO) system. Then,
+a joint transmit precoding and reflect beamforming algorithm
+is proposed to solve this problem. Simulation results show
+that, in a typical wireless system, the existing passive RISs
+achieve only a negligible sum-rate gain of 3%, while the active
+RISs are able to achieve a substantial sum-rate gain of 62%.
+Finally, we develop a 64-element active RIS aided wireless
+communication prototype, and field tests are conducted to
+validate the significant gain of active RISs.
+The rest of this paper is organized as follows. In Section II,
+the concept of RISs and their signal models are introduced.
+In Section III, the asymptotic performance of active RISs is
+analyzed. In Section IV, a sum-rate maximization problem is
+978-1-6654-3540-6/22 © 2022 IEEE
+arXiv:2301.00161v1  [cs.IT]  31 Dec 2022
+
+formulated, and a joint precoding and beamforming design is
+proposed to solve the problem. In Section V, simulation results
+and experimental measurements are presented to validate the
+signal model and evaluate the performance of active RISs.
+Finally, conclusions are drawn in Section VI.
+II. PASSIVE RISS AND ACTIVE RISS
+A. Conventional Passive RISs
+The RISs widely studied in most existing works are passive
+[1]–[3]. In general, each passive RIS element consists of a re-
+flective patch terminated with an impedance-adjustable circuit
+for phase shifting. Thanks to the passive mode of operation,
+the thermal noise at passive RISs is usually negligible [2].
+Thereby, the signal model of an N-element passive RIS widely
+used in the literature is given as follows:
+y = Θx,
+(1)
+where x
+∈
+CN denotes the incident signal, y
+∈
+CN
+denotes
+the
+signal
+reflected
+by
+the
+RIS,
+and
+Θ
+:=
+diag
+�
+ejθ1, · · · , ejθN �
+∈ CN×N denotes the phase shift matrix
+of the RIS with diag(·) being the diagonalization operation.
+By properly adjusting Θ to manipulate the N signals reflected
+by the N RIS elements to coherently add with the same phase
+at the receiver, a high array gain proportional to N 2 can be
+achieved, which is expected to significantly increase the signal-
+to-noise ratio (SNR) [1] at the receiver.
+Unfortunately, this expected high capacity gain often cannot
+be realized in practice, especially in communication scenarios
+where the direct link between the transmitter and the receiver is
+strong. The reason for this negative result is the “multiplicative
+fading” effect introduced by passive RISs. Specifically, the
+equivalent path loss of the transmitter-RIS-receiver reflected
+link is the product (instead of the sum) of the path losses of
+the transmitter-RIS and RIS-receiver links, and therefore, it is
+thousands of times larger than that of the unobstructed direct
+link. Thereby, for an RIS to realize a noticeable capacity gain,
+thousands (or even millions) of RIS elements are required to
+compensate for this extremely large path loss [3]. The resulting
+high signaling overhead for channel estimation and the high
+complexity of real-time beamforming make the application
+of such a large number of passive RIS elements in practical
+wireless networks very challenging.
+B. Concept of Active RISs
+To overcome the fundamental performance bottleneck
+caused by the “multiplicative fading” effect of RISs, we study
+the concept of active RISs as a promising solution1. As shown
+in Fig. 1, similar to the existing passive RISs, active RISs
+can also reflect the incident signals with reconfigurable phase
+shifts. Different from passive RISs that just reflect the incident
+signals without amplification, active RISs can further amplify
+the reflected signals. To achieve this goal, the key component
+1Note that active RISs are fundamentally different from relay-type RISs
+equipped with RF components and relays. Due to space constraints, we refer
+to the journal version of this paper for a more detailed discussion [4, Remark
+1].
+input
+BS
+RIS
+Active RIS
+user 1
+user k
+incident 
+signal
+reflected signal
+with amplification 
+phase-
+shift
+circuit
+patch
+reflection-type 
+amplifier
+output
+active element
+power
+supply
+Fig. 1. The downlink transmission in an active RIS aided MU-MISO system.
+of an active RIS element is the additionally integrated active
+reflection-type amplifier, which can be realized by different
+existing active components, such current-inverting converters
+or some integrated circuits [5].
+With reflection-type amplifiers supported by a power supply,
+the reflected and amplified signal of an N-element active RIS
+can be modeled as follows:
+y =
+PΘx
+� �� �
+Desired signal
++
+PΘv
+� �� �
+Dynamic noise
++
+ns
+����
+Static noise
+,
+(2)
+where P := diag (p1, · · · , pN) ∈ RN×N denotes the amplifi-
+cation factor matrix of the active RIS, wherein each element pn
+can be larger than one thanks to the integrated reflection-type
+amplifier. Due to the use of active components, active RISs
+consume additional power for amplifying the reflected signals,
+and the thermal noise introduced by active RIS elements
+cannot be neglected as is done for passive RISs. Particularly,
+as shown in (2), the introduced noise can be classified into
+dynamic noise and static noise [5]. Specifically, v is related to
+the input noise and the inherent device noise of the active RIS
+elements [5], while the static noise ns is unrelated to P and
+is usually negligible compared to the dynamic noise PΘv, as
+will be verified by experimental results in Section V-A. Thus,
+here we neglect ns and model v as v ∼ CN
+�
+0N, σ2
+vIN
+�
+,
+where CN(µ, Σ) denotes the complex multivariate Gaussian
+distribution with mean µ and variance Σ, IL is an L × L
+identity matrix, and 0L is an L × 1 zero vector.
+C. Active RIS Aided MU-MISO System
+Consider an active RIS aided downlink MU-MISO system
+as shown in Fig. 1, where an M-antenna base station (BS)
+serves K single-antenna users simultaneously with the aid
+of an N-element active RIS. Let s := [s1, · · · , sK]T ∈ CK
+denote the transmitted symbol vector for the K users and let
+wk ∈ CM×1 denote the BS precoding vector for symbol sk.
+According to (2), signal rk ∈ C received at user k can be
+modeled as follows:
+rk = (
+hH
+k
+����
+Direct link
++ f H
+k PΘG
+�
+��
+�
+Reflected link
+)
+�K
+j=1wjsj +
+f H
+k PΘv
+�
+��
+�
+Noise introduced by active RIS
+
++
+zk
+����
+Noise introduced at user k
+,
+(3)
+where [·]H denotes the conjugate-transpose operation; G ∈
+CN×M, hH
+k
+∈ C1×M, and f H
+k
+∈ C1×N characterize the
+channels between the BS and the RIS, between the BS and
+user k, and between the RIS and user k, respectively; and zk
+denotes the additive white Gaussian noise (AWGN) at user k
+with zero mean and variance σ2.
+III. PERFORMANCE ANALYSIS
+In this section, we analyze the performance of active RISs to
+reveal their notable capacity gains compared to passive RISs.
+To this end, in order to make the problem analytically tractable
+and get insightful results, in this section, we consider a single-
+user single-input single-output (SU-SISO) system with M = 1
+BS antenna and K = 1 user, while the general MU-MISO case
+is studied in Section IV.
+A. Asymptotic SNR for Passive RISs and Active RISs
+To illustrate the capacity gain provided by passive/active
+RIS aided reflected links, for the moment, we ignore the
+direct link by setting hk to zero, as was done in, e.g., [6].
+Furthermore, for simplicity, we assume that each active RIS
+element has the same amplification factor (i.e., pn := p). For
+a fair comparison with the asymptotic performance of passive
+RISs, similar to [6], we assume Rayleigh-fading channels.
+We first redefine the BS-RIS channel matrix and the RIS-
+user channel vector as G := g ∈ CN×1 and fk := f ∈ CN×1,
+respectively. Then, we recall the following lemma from [6] for
+the asymptotic SNR achieved by passive RISs.
+Lemma 1 (Asymptotic SNR for passive RISs): Assuming
+f ∼ CN
+�
+0N, ϱ2
+fIN
+�
+, g ∼ CN
+�
+0N, ϱ2
+gIN
+�
+and letting N →
+∞, the asymptotic SNR γpassive of a passive RIS aided SU-
+SISO system is given by
+γpassive → N 2 P max
+BS
+π2ϱ2
+fϱ2
+g
+16σ2
+,
+(4)
+where P max
+BS
+denotes the maximum transmit power at the BS.
+Proof: The proof can be found in [6, Proposition 2].
+For comparison, under the same transmission conditions, we
+provide the asymptotic SNR of an active RIS aided SU-SISO
+system in the following lemma.
+Lemma 2 (Asymptotic SNR for active RISs): Assuming
+f ∼ CN
+�
+0N, ϱ2
+fIN
+�
+, g ∼ CN
+�
+0N, ϱ2
+gIN
+�
+and letting N →
+∞, the asymptotic SNR γactive of an active RIS aided SU-SISO
+system is given by
+γactive → N
+P max
+BS
+P max
+A
+π2ϱ2
+fϱ2
+g
+16
+�
+P max
+A
+σ2vϱ2
+f + P max
+BS
+σ2ϱ2g + σ2σ2v
+�,
+(5)
+where P max
+A
+denotes the maximum reflect power of the active
+RIS.
+Proof: Please see the journal version [4, Appendix A].
+Remark 1: From (5) we observe that, the asymptotic SNR
+of an active RIS aided SU-SISO system depends on both the
+BS transmit power P max
+BS
+and the reflect power of the active
+RIS P max
+A
+. When P max
+BS
+→ ∞, the asymptotic SNR will be
+upper-bounded by γactive → NP max
+A
+π2ϱ2
+f/
+�
+16σ2�
+, which is
+independent of the BS-RIS channel g and the noise power at
+the active RIS σ2
+v. Similarly, if P max
+A
+→ ∞, the asymptotic
+SNR will be upper-bounded by γactive → NP max
+BS π2ϱ2
+g/16σ2
+v,
+which is independent of the RIS-user channel f and the noise
+power at the user σ2. These results reveal that, to increase the
+sum-rate of active RIS aided systems, the negative impact of
+small g and large σ2
+v on system performance can be alleviated
+by increasing the BS transmit power P max
+BS
+, and the negative
+impact of small f and large σ2 can be reduced by increasing
+the reflect power of the active RIS P max
+A
+.
+B. Comparisons between Passive RISs and Active RISs
+We can observe from Lemma 1 and Lemma 2 that, compared
+to the asymptotic SNR for passive RISs γpassive in (4) which is
+proportional to N 2, the asymptotic SNR for active RISs γactive
+in (5) is proportional to N due to the noises introduced by
+the use of active components. At first glance, it seems that the
+SNR achieved by passive RISs γpassive always exceeds the SNR
+achieved by active RISs γactive. However, this is actually not the
+case. The reason behind this counterintuitive behavior is that,
+due to the large path loss caused by the “multiplicative fading”
+effect and thanks to the use of the reflection-type amplifiers
+in active RISs, only when N is unaffordably large can passive
+RISs outperform active RISs.
+To illustrate this claim, let us consider two different SU-
+SISO systems, which are aided by an active RIS and a passive
+RIS, respectively. Then, the following lemma specifies the
+condition that has to be met for passive RISs to outperform
+active RISs.
+Lemma 3 (Case when passive RISs outperform active
+RISs): Assuming the number of RIS elements N is large,
+the required number of elements N for a passive RIS to
+outperform an active RIS has to satisfy
+N ≥ P max
+BS-A
+P max
+BS-P
+P max
+A
+σ2
+�
+P max
+A
+σ2vϱ2
+f + P max
+BS-Aσ2ϱ2g + σ2σ2v
+�,
+(6)
+where P max
+BS-A denotes the maximum BS transmit power for the
+active RIS aided system and P max
+BS-P denotes that for the passive
+RIS aided system.
+Proof: Please see the journal version [4, Appendix B].
+Next, we consider a specific setup to compare the user’s
+achievable SNRs in the above two systems. For a fair com-
+parison, we constrain the total power consumption P max of
+the two systems to 2 W by setting P max
+BS-P = 2 W for the
+passive RIS aided system and P max
+BS-A = P max
+A
+= 1 W for
+the active RIS aided system, respectively. Therefore, when
+σ2 = σ2
+v = −70 dBm and ϱ2
+f = ϱ2
+g = −70 dB, the required
+number of elements N for the passive RIS to outperform the
+active RIS is 2.5 × 106 according to (6), which is impractical
+to realize with current technology. Conversely, for a more
+practical number of elements of N = 256, according to (5)
+and (4), the SNR achieved by the passive RIS is γpassive ≈ 9.0
+dB, while the SNR achieved by the active RIS is γactive ≈ 49.0
+dB, which is about 10, 000 times higher than γpassive.
+
+IV. JOINT TRANSMIT PRECODING AND REFLECT
+BEAMFORMING DESIGN
+To investigate the capacity gain enabled by the use of
+active RISs in typical wireless communication scenarios, in
+this section, we consider more general MU-MISO systems.
+According to the model in (3), the signal-to-interference-plus-
+noise ratio (SINR) at user k can be obtained as
+γk =
+��¯hH
+k wk
+��2
+�K
+j=1,j̸=k
+��¯hH
+k wj
+��2 +
+��f H
+k PΘ
+��2σ2v + σ2 ,
+(7)
+wherein ¯hH
+k = hH
+k + fk
+HPΘG ∈ C1×M is the equivalent
+channel from the BS to user k, which includes both the direct
+link and the reflected link. Therefore, the original problem of
+sum-rate maximization, subject to the power constraints at the
+BS and the active RIS, can be formulated as follows:
+Po : max
+w,P,Θ Rsum(w, P, Θ) =
+�K
+k=1 log2 (1 + γk),
+(8a)
+s.t. C1 :
+�K
+k=1 ∥wk∥2 ≤ P max
+BS
+,
+(8b)
+C2 :
+�K
+k=1∥PΘGwk∥2+∥PΘ∥2 σ2
+v ≤P max
+A
+, (8c)
+where w :=
+�
+wT
+1 , · · · , wT
+K
+�T is the overall transmit precoding
+vector for the K users; C1 and C2 are the power constraints at
+the BS and active RIS, respectively. Due to the non-convexity
+and the highly coupled variables in problem Po in (8), the
+joint design of w, P, and Θ is challenging.
+To efficiently solve the above problem, we develop a joint
+precoding and beamforming algorithm based on alternating
+optimization and fractional programming (FP). Note that P
+and Θ always appear in product form in problem Po in
+(8). Therefore, P and Θ can be merged as Ψ = PΘ =
+diag
+�
+p1ejθ1, · · · , pNejθN �
+∈ CN×N. We refer to Ψ as the
+RIS beamforming matrix. Next, to deal with the non-convex
+sum-of-logarithms and fractions in (8), we exploit the FP
+methods proposed in [7] to decouple the variables in problem
+Po in (8). This leads to the following lemma.
+Lemma 4 (Equivalent problem for sum-rate maximiza-
+tion): By introducing auxiliary variables ρ := [ρ1, · · · , ρK] ∈
+RK and ϖ := [ϖ1, · · · , ϖK] ∈ CK, the original problem Po
+in (8) can be equivalently reformulated as follows
+P1 :
+max
+w,Ψ,ρ,ϖ R′
+sum(w, Ψ, ρ, ϖ) =
+�K
+k=1 ln (1 + ρk)−
+�K
+k=1 ρk +
+�K
+k=1 g(w, Ψ, ρk, ϖk),
+s.t. C1, C2,
+(9)
+where function g(w, Ψ, ρk, ϖk) is defined as
+g(w, Ψ, ρk, ϖk) = 2
+�
+(1 + ρk)R
+�
+ϖ∗
+k¯hH
+k wk
+�
+−
+|ϖk|2
+��K
+j=1
+��¯hH
+k wj
+��2 +
+��f H
+k Ψ
+��2σ2
+v + σ2
+�
+.
+(10)
+Proof: Constructive proof can be found in [7, Subsection
+III-C].
+Strong convergence of the FP methods was proved in [7].
+Thus, a locally optimal solution to (9) can be obtained by
+alternately optimizing the variables. For clarity, we summarize
+the proposed joint precoding and beamforming algorithm in
+Algorithm 1, and the specific solutions for variables w, Ψ,
+ρ, and ϖ are given in the following four steps, respectively.
+Algorithm 1 Proposed joint transmit precoding and reflect
+beamforming algorithm
+Input:
+Channels G, hk, and fk, ∀k ∈ {1, · · · , K}.
+Output:
+Optimized BS precoding vector w, amplification
+factor matrix of active RIS P, phase shift matrix of active
+RIS Θ, and sum-rate Rsum.
+1: Randomly initialize w, P and Θ;
+2: while no convergence of Rsum do
+3:
+Update ρ by (11);
+4:
+Update ϖ by (12);
+5:
+Update w by solving problem P2 in (14);
+6:
+Update Ψ by solving problem P3 in (15);
+7: end while
+8: Obtain P and Θ from Ψ;
+9: return Optimized w, P, Θ, and Rsum.
+1) Fix (w, Ψ, ϖ) and optimize ρ: After fixing precoding
+vector w, beamforming matrix Ψ, and auxiliary variable ϖ,
+the optimal ρ can be obtained by solving ∂R′
+sum
+∂ρk
+= 0 as
+ρopt
+k
+= ξ2
+k + ξk
+�
+ξ2
+k + 4
+2
+,
+∀k ∈ {1, · · · , K},
+(11)
+where ξk = ℜ
+�
+ϖ∗
+k¯hH
+k wk
+�
+.
+2) Fix (w, Ψ, ρ) and optimize ϖ: After fixing the precod-
+ing vector w, beamforming matrix Ψ, and auxiliary variable
+ρ, the optimal ϖ can be derived by solving ∂R′
+sum
+∂ϖk
+= 0 as
+ϖopt
+k
+=
+�
+(1 + ρk)¯hH
+k wk
+�K
+j=1
+��¯hH
+k wj
+��2+
+��f H
+k Ψ
+��2σ2v + σ2 .
+(12)
+3) Fix (Ψ, ρ, ϖ) and optimize w: To simplify the nota-
+tions, we first introduce the following definitions:
+bH
+k =
+�
+(1 + ρk)ϖ∗
+k¯hH
+k , b =
+�
+bT
+1 , bT
+2 , · · · , bT
+N
+�T ,
+(13a)
+A=IK ⊗
+�K
+k=1|ϖk|2¯hk¯hH
+k , Ξ=IK ⊗
+�
+GHΨHΨG
+�
+, (13b)
+P max
+m
+= P max
+A
+− ∥Ψ∥2σ2
+v,
+(13c)
+where ⊗ denotes the Kronecker product. Then, problem P1 in
+(9) can be reformulated as follows:
+P2 :
+max
+w
+R
+�
+2bHw
+�
+− wHAw,
+s.t.
+C1 : ∥w∥2 ≤ P max
+BS
+,
+C2 : wHΞw ≤ P max
+m
+.
+(14)
+Note that P2 in (14) is a standard quadratic constraint
+quadratic programming (QCQP) problem, which can be solved
+by alternating direction method of multipliers (ADMM).
+4) Fix
+(w, ρ, ϖ)
+and
+optimize
+Ψ:
+Define
+ψ
+=
+�
+p1ejθ1, · · · , pNejθN �H as the vectorized RIS beamforming
+matrix Ψ, i.e., diag
+�
+ψH�
+:= Ψ. While fixing w and ρ and
+ϖ, problem P1 in (9) can be reformulated as follows:
+P3 :
+max
+ψ
+R
+�
+2ψHυ
+�
+− ψHΩψ,
+s.t. C2 : ψHΠψ ≤ P max
+A
+,
+(15)
+
+spectrum 
+analyzer
+LNA
+vector network 
+analyzer
+DC source
+active RIS
+element
+circulator
+pump
+source
+noise source
+Fig. 2. The experimental devices and environment used for validating the signal model (2) of active RISs.
+2.359
+2.3595
+2.36
+2.3605
+2.361
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+30
+Fig. 3. Experimental measurement result for reflection gain G versus signal
+frequency f.
+wherein
+υ =
+�K
+k=1
+�
+(1 + ρk)diag
+�
+ϖ∗
+kf H
+k
+�
+Gwk−
+�K
+k=1 |ϖk|2diag
+�
+f H
+k
+�
+G
+�K
+j=1 wjwH
+j hk,
+(16a)
+Ω =
+�K
+k=1 |ϖk|2diag
+�
+f H
+k
+�
+diag (fk) σ2
+v+
+�K
+k=1 |ϖk|2 �K
+j=1diag
+�
+f H
+k
+�
+GwjwH
+j GHdiag (fk), (16b)
+Π =
+�K
+k=1 diag (Gwk) (diag (Gwk))H + σ2
+vIN.
+(16c)
+Note that problem P3 in (15) is also a standard QCQP
+problem. Thus, the optimal solution ψopt can be obtained by
+adopting ADMM.
+V. VALIDATION RESULTS
+A. Validation Results for Signal Model
+To validate the signal model (2), we designed and fabri-
+cated an active RIS element with an integrated reflection-
+type amplifier for experimental measurements in [8]. Note
+that this design can be directly extended to the large-array
+case. Particularly, since the phase-shifting ability of RISs has
+been widely verified, we focus on studying the reflection gain
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+-170
+-165
+-160
+-155
+-150
+-145
+-140
+-135
+-130
+Fig. 4.
+Experimental measurement result for the density of noise power
+Gσ2
+v + σ2
+s versus reflection gain G.
+and the noise introduced by an active RIS element. Thus, the
+validation of signal model (2) is equivalent to validating
+Py =
+GPx
+����
+Desired-signal power
++ Gσ2
+v + σ2
+s
+�
+��
+�
+noise power
+,
+(17)
+where Py is the power of the signals reflected by the active
+RIS element; Px is the power of the incident signal; G := p2 is
+the reflection gain of the active RIS element; Gσ2
+v and σ2
+s are
+the powers of the dynamic noise and static noise introduced
+by the active RIS element, respectively.
+1) Hardware platform: To validate the model in (17), we
+first establish the hardware platform used for our experimental
+measurements in Fig. 2. Due to space constraints, we refer
+the reader to the journal version of this paper [4, Fig. 4] for
+detailed information about the hardware platform.
+2) Reflection gain measurement: Using the measurement
+system for the reflection gain depicted in [4, Fig. 4 (b)], we
+first investigate the reflection gain G of the active RIS element.
+Note that the reflection gain G can be reconfigured by the
+input power of the pump source Pp. By setting the input
+power of the vector network analyzer as Px = −50 dBm, the
+reflection gain G as a function of the signal frequency can be
+directly measured via a vector network analyzer. Then, in Fig.
+
+-10
+-5
+0
+5
+10
+15
+20
+25
+30
+0
+10
+20
+30
+40
+50
+60
+507%
+Fig. 5. Simulation results for the sum-rate versus total power consumption
+P max in scenario 1 with a weak direct link..
+3, we show the measurement results for reflection gain G as
+a function of signal frequency f for different input powers of
+the pump source Pp. We observe that the active RIS element
+can achieve a reflection gain G of more than 25 dB, when
+Pp = 18.24 dBm, which confirms the significant reflection
+gains enabled by active RISs.
+3) Noise power measurement: We further study the noise
+power introduced by the active RIS element, i.e., Gσ2
+v + σ2
+s
+in (17), where Gσ2
+v and σ2
+s are the powers of the dynamic
+noise and the static noise introduced at the active RIS element,
+respectively. Using the noise measurement system in [4, Fig. 4
+(c)], we show the measurement results for the spectral density
+of noise power Gσ2
+v + σ2
+s as a function of G for different
+operating frequencies in Fig. 4. We can observe that the noise
+power increases nearly linearly with G, which verifies the
+noise model Gσ2
+v + σ2
+s in (17). Particularly, for f = 2.3601
+GHz, the spectral density of σ2
+s is about −174 dBm/Hz, while
+that of σ2
+v is about −160 dBm/Hz, which is about 15 dB
+higher. The reason for this is that the input noise is amplified
+by the noise factor, and additional noises are also introduced
+by the other active components such as the DC source used
+to power the active RIS.
+B. Simulation Results for Sum-Rate
+1) Simulation setup: We consider an active RIS aided MU-
+MISO system. Particularly, we consider two scenarios with dif-
+ferent channel conditions. In scenario 1, the direct link is weak
+due to severe obstruction, while the direct link is strong in
+scenario 2. To be specific, two different path loss models from
+the 3GPP TS 36.814 standard are utilized to characterize the
+large-scale fading of the channels: i) PLs = 37.3+22.0 log d;
+ii) PLw = 41.2 + 28.7 log d, where d is the distance between
+two devices. Path loss model PLw is used to generate the weak
+BS-user link in scenario 1, while PLs is used to generate the
+strong BS-user link in scenario 2. For both scenarios, PLs is
+used to generate the BS-RIS and the RIS-user channels. To
+account for small-scale fading, we adopt the Ricean fading
+channel model for all channels involved and we assume the
+Ricean factor as κ = 1.
+-10
+-5
+0
+5
+10
+15
+20
+25
+30
+0
+10
+20
+30
+40
+50
+60
+62%
+Fig. 6. Simulation results for the sum-rate versus total power consumption
+P max in scenario 2 with a strong direct link.
+The BS and the active/passive RIS are located at (0, -60
+m) and (200 m, 30 m), respectively. Four users are randomly
+located in a circle with a radius of 5 m from the center (200
+m, 0). The numbers of BS antennas and RIS elements are set
+as M = 4 and N = 256, respectively. The noise power is set
+as σ2 = σ2
+v = −70 dBm. For fair comparison, we constrain
+the total power consumption P max := P max
+BS
++ P max
+A
+. For the
+active RIS, Algorithm 1 is employed for joint precoding and
+beamforming design, while for the passive RIS, the algorithm
+from [2] is adopted.
+2) Simulation results: In Fig. 5 and Fig. 6, we plot the
+sum-rate versus the total consumed power P max for the two
+considered scenarios, where the direct link is weak and strong,
+respectively. Firstly, in scenario 1 with a weak direct link, the
+passive RIS can indeed achieve a performance improvement,
+while the active RIS achieves a much higher sum-rate gain.
+Secondly, in scenario 2 with a strong direct link, the passive
+RIS achieves only a negligible sum-rate gain, while the active
+RIS still realizes a noticeable sum-rate gain. For example,
+when P max = 10 dBW, the capacities without RIS, with
+passive RIS, and with active RIS in scenario 1 are 5.34 bps/Hz,
+7.00 bps/Hz, and 32.41 bps/Hz respectively, while in scenario
+2, these values are 19.87 bps/Hz, 20.51 bps/Hz, and 32.18
+bps/Hz, respectively. In this case, the passive RIS provides a
+31% gain in scenario 1 and a negligible 3% gain in scenario 2.
+By contrast, the active RIS achieves noticeable sum-rate gains
+of 507% in scenario 1 and 62% in scenario 2, which are much
+higher than those achieved by the passive RIS.
+C. Field Test for a 64-Element Active RIS Aided Wireless
+Communication Prototype
+1) 64-element active RIS aided communication prototype:
+To validate the significant gain of active RISs, we develop a
+64-element active RIS aided wireless communication proto-
+type, as shown in Fig. 7. Specifically, the hardware structure
+of this prototype consists of three parts including a BS, a 64-
+element active RIS, and a user. For the BS and the user, two
+horn antennas with 13 dBi antenna gain are used to transmit
+
+Fig. 7. A photograph of the developed 64-element active RIS aided wireless
+communication system.
+and receive the signals, and the universal software radio
+peripherals (USRPs) are deployed to generate and process the
+baseband and RF signals (hardware version: USRP-2953R).
+By periodically expanding the active RIS elements designed
+in [8], the 64-element active RIS is an 8×8 plane array, of
+which each element has a reflection gain of G = 10 dB.
+2) Experimental environment: Based on the developed pro-
+totype, we establish the experimental environment for further
+validation. To match the transceivers, we configure the oper-
+ating frequency of the active RIS to f = 3.5 GHz and the
+bandwidth to 40 MHz by adjusting the circuit impedance of
+active elements. The polarization of the antenna at the BS
+and that at the user are selected as vertical and horizontal,
+respectively. The transmit power is set to −10 mW. We fix the
+heights of the BS, the RIS, and the user as 1 m. The horizontal
+distance of the BS-RIS link and that of the RIS-user link are
+set to 2 m and 3.5 m, respectively. The angle of arrival (AoA)
+at the active RIS is fixed as 0◦, and the angle of departure
+(AoD) will be specified to evaluate the performance gain of
+active RISs at different orientations. To observe the reflection
+gain of the active RIS, we use a metal plate with the same
+aperture size as the active RIS for performance comparison.
+3) Experimental results: By moving the user at different
+AoDs and configuring the phase shift of the active RIS with
+discrete Fourier transform (DFT) codebook, we obtain the
+experimental results shown in Table I. One can observe that,
+compared with the received power for the metal plate, the
+active RIS can always achieve a gain of about 10 dB. The data
+rate for the active RIS can hold at about 30 Mbps, while that
+for the metal plate only ranges from 1 Mbps to 2Mbps. The
+reason is that, the beamforming at the active RIS can make the
+reflected beam with high array gain and reflection gain, while
+the metal plate can only reflect the incident signals randomly
+without in-phase combination or amplification, which validates
+the significant gain of active RISs.
+VI. CONCLUSIONS
+In this paper, we have studied the concept of active RISs
+to overcome the fundamental limitation of the “multiplicative
+fading” effect. Firstly, we have verified the signal model of
+TABLE I
+EXPERIMENTAL RESULTS FOR THE DEVELOPED PROTOTYPE
+AoD
+Device
+Received Power
+Data Rate
+15◦
+Metal plate
+-110 dBm
+1.2 Mbps
+Active RIS
+-100 dBm
+28.5 Mbps
+30◦
+Metal plate
+-105 dBm
+1.5 Mbps
+Active RIS
+-98 dBm
+30.5 Mbps
+45◦
+Metal plate
+-105 dBm
+1.5 Mbps
+Active RIS
+-95 dBm
+30 Mbps
+60◦
+Metal plate
+-108 dBm
+2 Mbps
+Active RIS
+-90 dBm
+32 Mbps
+active RISs through the experimental measurements on a fab-
+ricated active RIS element. Based on the verified signal model,
+we have formulated the sum-rate maximization problem for
+an active RIS aided MU-MISO system and a joint precoding
+and beamforming algorithm has been proposed to solve this
+problem. Simulation results have shown that, in a typical
+application scenario, the existing passive RIS can realize only
+a negligible sum-rate gain of about 3%, while the active RIS
+can achieve a substantial sum-rate gain of about 62%, thus
+indeed overcoming the “multiplicative fading” effect. Finally,
+we have developed a communication wireless communication
+prototype aided by a 64-element active RIS, and the significant
+gain of active RISs is validated by field test. In the future,
+many research directions for active RISs are worth pursuing,
+such as hardware design, prototype development, channel
+estimation, and energy efficiency analysis.
+ACKNOWLEDGMENT
+This work was supported in part by the National Key
+Research and Development Program of China (Grant No.
+2020YFB1805005), in part by the National Natural Science
+Foundation of China (Grant No. 62031019), and in part by the
+European Commission through the H2020-MSCA-ITN META
+WIRELESS Research Project under Grant 956256.
+REFERENCES
+[1] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen,
+“Reconfigurable intelligent surfaces for energy efficiency in wireless
+communication,” IEEE Trans. Wireless Commun., vol. 18, no. 8, pp.
+4157–4170, Aug. 2019.
+[2] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan,
+and L. Hanzo, “Multicell MIMO communications relying on intelligent
+reflecting surfaces,” IEEE Trans. Wireless Commun., vol. 19, no. 8, pp.
+5218–5233, Aug. 2020.
+[3] M. Najafi, V. Jamali, R. Schober, and H. V. Poor, “Physics-based mod-
+eling and scalable optimization of large intelligent reflecting surfaces,”
+IEEE Trans. Commun., vol. 69, no. 4, pp. 2673–2691, Apr. 2021.
+[4] Z. Zhang, L. Dai, X. Chen, C. Liu, F. Yang, R. Schober, and H. V. Poor,
+“Active RIS vs. passive RIS: Which will prevail in 6G?” IEEE Trans.
+Commun., Dec. 2022.
+[5] J. Bousquet, S. Magierowski, and G. G. Messier, “A 4-GHz active
+scatterer in 130-nm CMOS for phase sweep amplify-and-forward,” IEEE
+Trans. Circuits Syst. I, vol. 59, no. 3, pp. 529–540, Mar. 2012.
+[6] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless
+network via joint active and passive beamforming,” IEEE Trans. Wireless
+Commun., vol. 18, no. 11, pp. 5394–5409, Nov. 2019.
+[7] K. Shen and W. Yu, “Fractional programming for communication sys-
+tems—part I: Power control and beamforming,” IEEE Trans. Signal
+Process., vol. 66, no. 10, pp. 2616–2630, May 2018.
+[8] X. Chen and F. Yang, “Nonlinear electromagnetic surfaces: Theory,
+design and application,” Master Thesis in Tsinghua University, May 2020,
+[Online] Available: http://etds.lib.tsinghua.edu.cn/Thesis.
+
+3.5GHz
+有源RIS
+8 X 8 Active RIS
+BS
+User

XNAyT4oBgHgl3EQfWPfd/content/tmp_files/load_file.txt

@@ -0,0 +1,389 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf,len=388
+page_content='Active RISs: Signal Modeling, Asymptotic  Analysis, and Beamforming Design  Zijian Zhang∗, Linglong Dai∗, Fellow, IEEE, Xibi Chen∗, Changhao Liu∗, Fan Yang∗, Fellow, IEEE,  Robert Schober†, Fellow, IEEE, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Vincent Poor§, Life Fellow, IEEE  ∗Beijing National Research Center for Information Science and Technology (BNRist)  Department of Electronic Engineering, Tsinghua University, China  †Institute for Digital Communications, Friedrich-Alexander University Erlangen-N¨urnberg, Germany  §Department of Electrical and Computer Engineering, Princeton University, USA  E-mails: zhangzj20@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' daill@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' cxb17@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' liuch17@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  fan yang@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' robert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='schober@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' poor@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='edu  Abstract—Reconfigurable  intelligent  surfaces  (RISs)  have  emerged as a candidate technology for future 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  However, due to the “multiplicative fading” effect, the existing  passive RISs only achieve a negligible capacity gain in environ-  ments with strong direct links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In this paper, the concept of  active RISs is studied to overcome this fundamental limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Unlike the existing passive RISs that reflect signals without  amplification, active RISs can amplify the reflected signals via  amplifiers integrated into their elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To characterize the  signal amplification and incorporate the noise introduced by the  active components, we verify the signal model of active RISs  through the experimental measurements on a fabricated active  RIS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Based on the verified signal model, we formulate  the sum-rate maximization problem for an active RIS aided  multi-user multiple-input single-output (MU-MISO) system and  a joint transmit precoding and reflect beamforming algorithm is  proposed to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation results show that, in  a typical wireless system, the existing passive RISs can realize  only a negligible sum-rate gain of 3%, while the active RISs can  achieve a significant sum-rate gain of 62%, thus overcoming the  “multiplicative fading” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Finally, we develop a 64-element  active RIS aided wireless communication prototype, and the  significant gain of active RISs is validated by field test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' INTRODUCTION  From the first generation (1G) to 5G wireless communica-  tions, the wireless channel has been considered to be uncon-  trollable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Recently, due to the advances in meta-materials, re-  configurable intelligent surfaces (RISs) have been proposed [1]  for the purpose of intelligently controlling wireless channels  to achieve improved communication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Specifically,  an RIS is an array composed of a very large number of passive  elements that reflects electromagnetic signals in a desired  manner so as to reconfigure the propagation properties of  wireless environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' As an important advantage of RIS, the  negligible noise introduced by passive RISs enables a high  array gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Benefiting from this advantage, RISs are expected  to introduce significant capacity gains in wireless systems [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  However, in practice, the expected capacity gains are typ-  ically only observed in communication environments where  the direct link between transmitter and receiver is completely  blocked or very weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' By contrast, in many scenarios where  the direct link is not weak, conventional RISs can only  achieve negligible capacity gains [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The reason behind this  phenomenon is the “multiplicative fading” effect introduced  by RISs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', the equivalent path loss of the transmitter-RIS-  receiver link is the product (instead of the sum) of the path  losses of the transmitter-RIS and RIS-receiver links, which is  usually thousands of times larger than that of the direct link  [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' As a result, the “multiplicative fading” effect makes  it almost impossible for passive RISs to achieve noticeable  capacity gains in many wireless environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Therefore,  to advance the practicability of RISs in future 6G wireless  networks, a critical issue for RISs to be addressed is: How to  break the fundamental performance bottleneck caused by the  “multiplicative fading” effect?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  To overcome the fundamental physical limitation of con-  ventional passive RISs imposed by the “multiplicative fading”  effect, in this paper, we investigate the concept of active RISs  to overcome the “multiplicative fading” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Different from  the existing passive RISs that passively reflect signals without  amplification, the key feature of active RISs is their ability  to actively reflect signals with amplification at the expense  of additional power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Firstly, through the experi-  mental measurements on a fabricated active RIS element, we  verify the signal model of active RISs, which characterizes the  amplification of the incident signal and accounts for the non-  negligible thermal noise introduced by the active elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Based on the verified signal model, we further analyze the  asymptotic performance of active RISs and formulate a sum-  rate maximization problem for an active RIS aided multi-  user multiple-input single-output (MU-MISO) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Then,  a joint transmit precoding and reflect beamforming algorithm  is proposed to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation results show  that, in a typical wireless system, the existing passive RISs  achieve only a negligible sum-rate gain of 3%, while the active  RISs are able to achieve a substantial sum-rate gain of 62%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Finally, we develop a 64-element active RIS aided wireless  communication prototype, and field tests are conducted to  validate the significant gain of active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In Section II,  the concept of RISs and their signal models are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  In Section III, the asymptotic performance of active RISs is  analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In Section IV, a sum-rate maximization problem is  978-1-6654-3540-6/22 © 2022 IEEE  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='00161v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='IT]  31 Dec 2022  formulated, and a joint precoding and beamforming design is  proposed to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In Section V, simulation results  and experimental measurements are presented to validate the  signal model and evaluate the performance of active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Finally, conclusions are drawn in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' PASSIVE RISS AND ACTIVE RISS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Conventional Passive RISs  The RISs widely studied in most existing works are passive  [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In general, each passive RIS element consists of a re-  flective patch terminated with an impedance-adjustable circuit  for phase shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Thanks to the passive mode of operation,  the thermal noise at passive RISs is usually negligible [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Thereby, the signal model of an N-element passive RIS widely  used in the literature is given as follows:  y = Θx,  (1)  where x  ∈  CN denotes the incident signal, y  ∈  CN  denotes  the  signal  reflected  by  the  RIS,  and  Θ  :=  diag  �  ejθ1, · · · , ejθN �  ∈ CN×N denotes the phase shift matrix  of the RIS with diag(·) being the diagonalization operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  By properly adjusting Θ to manipulate the N signals reflected  by the N RIS elements to coherently add with the same phase  at the receiver, a high array gain proportional to N 2 can be  achieved, which is expected to significantly increase the signal-  to-noise ratio (SNR) [1] at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Unfortunately, this expected high capacity gain often cannot  be realized in practice, especially in communication scenarios  where the direct link between the transmitter and the receiver is  strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The reason for this negative result is the “multiplicative  fading” effect introduced by passive RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Specifically, the  equivalent path loss of the transmitter-RIS-receiver reflected  link is the product (instead of the sum) of the path losses of  the transmitter-RIS and RIS-receiver links, and therefore, it is  thousands of times larger than that of the unobstructed direct  link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Thereby, for an RIS to realize a noticeable capacity gain,  thousands (or even millions) of RIS elements are required to  compensate for this extremely large path loss [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The resulting  high signaling overhead for channel estimation and the high  complexity of real-time beamforming make the application  of such a large number of passive RIS elements in practical  wireless networks very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Concept of Active RISs  To overcome the fundamental performance bottleneck  caused by the “multiplicative fading” effect of RISs, we study  the concept of active RISs as a promising solution1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' As shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 1, similar to the existing passive RISs, active RISs  can also reflect the incident signals with reconfigurable phase  shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Different from passive RISs that just reflect the incident  signals without amplification, active RISs can further amplify  the reflected signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To achieve this goal, the key component  1Note that active RISs are fundamentally different from relay-type RISs  equipped with RF components and relays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Due to space constraints, we refer  to the journal version of this paper for a more detailed discussion [4, Remark  1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  input  BS  RIS  Active RIS  user 1  user k  incident  signal  reflected signal  with amplification  phase-  shift  circuit  patch  reflection-type  amplifier  output  active element  power  supply  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The downlink transmission in an active RIS aided MU-MISO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  of an active RIS element is the additionally integrated active  reflection-type amplifier, which can be realized by different  existing active components, such current-inverting converters  or some integrated circuits [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  With reflection-type amplifiers supported by a power supply,  the reflected and amplified signal of an N-element active RIS  can be modeled as follows:  y =  PΘx  � �� �  Desired signal  +  PΘv  � �� �  Dynamic noise  +  ns  ����  Static noise  ,  (2)  where P := diag (p1, · · · , pN) ∈ RN×N denotes the amplifi-  cation factor matrix of the active RIS, wherein each element pn  can be larger than one thanks to the integrated reflection-type  amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Due to the use of active components, active RISs  consume additional power for amplifying the reflected signals,  and the thermal noise introduced by active RIS elements  cannot be neglected as is done for passive RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Particularly,  as shown in (2), the introduced noise can be classified into  dynamic noise and static noise [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Specifically, v is related to  the input noise and the inherent device noise of the active RIS  elements [5], while the static noise ns is unrelated to P and  is usually negligible compared to the dynamic noise PΘv, as  will be verified by experimental results in Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Thus,  here we neglect ns and model v as v ∼ CN  �  0N, σ2  vIN  �  ,  where CN(µ, Σ) denotes the complex multivariate Gaussian  distribution with mean µ and variance Σ, IL is an L × L  identity matrix, and 0L is an L × 1 zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Active RIS Aided MU-MISO System  Consider an active RIS aided downlink MU-MISO system  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 1, where an M-antenna base station (BS)  serves K single-antenna users simultaneously with the aid  of an N-element active RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Let s := [s1, · · · , sK]T ∈ CK  denote the transmitted symbol vector for the K users and let  wk ∈ CM×1 denote the BS precoding vector for symbol sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  According to (2), signal rk ∈ C received at user k can be  modeled as follows:  rk = (  hH  k  ����  Direct link  + f H  k PΘG  �  ��  �  Reflected link  )  �K  j=1wjsj +  f H  k PΘv  �  ��  �  Noise introduced by active RIS  +  zk  ����  Noise introduced at user k  ,  (3)  where [·]H denotes the conjugate-transpose operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' G ∈  CN×M, hH  k  ∈ C1×M, and f H  k  ∈ C1×N characterize the  channels between the BS and the RIS, between the BS and  user k, and between the RIS and user k, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' and zk  denotes the additive white Gaussian noise (AWGN) at user k  with zero mean and variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' PERFORMANCE ANALYSIS  In this section, we analyze the performance of active RISs to  reveal their notable capacity gains compared to passive RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  To this end, in order to make the problem analytically tractable  and get insightful results, in this section, we consider a single-  user single-input single-output (SU-SISO) system with M = 1  BS antenna and K = 1 user, while the general MU-MISO case  is studied in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Asymptotic SNR for Passive RISs and Active RISs  To illustrate the capacity gain provided by passive/active  RIS aided reflected links, for the moment, we ignore the  direct link by setting hk to zero, as was done in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Furthermore, for simplicity, we assume that each active RIS  element has the same amplification factor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', pn := p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For  a fair comparison with the asymptotic performance of passive  RISs, similar to [6], we assume Rayleigh-fading channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  We first redefine the BS-RIS channel matrix and the RIS-  user channel vector as G := g ∈ CN×1 and fk := f ∈ CN×1,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Then, we recall the following lemma from [6] for  the asymptotic SNR achieved by passive RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Lemma 1 (Asymptotic SNR for passive RISs): Assuming  f ∼ CN  �  0N, ϱ2  fIN  �  , g ∼ CN  �  0N, ϱ2  gIN  �  and letting N →  ∞, the asymptotic SNR γpassive of a passive RIS aided SU-  SISO system is given by  γpassive → N 2 P max  BS  π2ϱ2  fϱ2  g  16σ2  ,  (4)  where P max  BS  denotes the maximum transmit power at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Proof: The proof can be found in [6, Proposition 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  For comparison, under the same transmission conditions, we  provide the asymptotic SNR of an active RIS aided SU-SISO  system in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Lemma 2 (Asymptotic SNR for active RISs): Assuming  f ∼ CN  �  0N, ϱ2  fIN  �  , g ∼ CN  �  0N, ϱ2  gIN  �  and letting N →  ∞, the asymptotic SNR γactive of an active RIS aided SU-SISO  system is given by  γactive → N  P max  BS  P max  A  π2ϱ2  fϱ2  g  16  �  P max  A  σ2vϱ2  f + P max  BS  σ2ϱ2g + σ2σ2v  �,  (5)  where P max  A  denotes the maximum reflect power of the active  RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Proof: Please see the journal version [4, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Remark 1: From (5) we observe that, the asymptotic SNR  of an active RIS aided SU-SISO system depends on both the  BS transmit power P max  BS  and the reflect power of the active  RIS P max  A  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' When P max  BS  → ∞, the asymptotic SNR will be  upper-bounded by γactive → NP max  A  π2ϱ2  f/  �  16σ2�  , which is  independent of the BS-RIS channel g and the noise power at  the active RIS σ2  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Similarly, if P max  A  → ∞, the asymptotic  SNR will be upper-bounded by γactive → NP max  BS π2ϱ2  g/16σ2  v,  which is independent of the RIS-user channel f and the noise  power at the user σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' These results reveal that, to increase the  sum-rate of active RIS aided systems, the negative impact of  small g and large σ2  v on system performance can be alleviated  by increasing the BS transmit power P max  BS  , and the negative  impact of small f and large σ2 can be reduced by increasing  the reflect power of the active RIS P max  A  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Comparisons between Passive RISs and Active RISs  We can observe from Lemma 1 and Lemma 2 that, compared  to the asymptotic SNR for passive RISs γpassive in (4) which is  proportional to N 2, the asymptotic SNR for active RISs γactive  in (5) is proportional to N due to the noises introduced by  the use of active components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' At first glance, it seems that the  SNR achieved by passive RISs γpassive always exceeds the SNR  achieved by active RISs γactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' However, this is actually not the  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The reason behind this counterintuitive behavior is that,  due to the large path loss caused by the “multiplicative fading”  effect and thanks to the use of the reflection-type amplifiers  in active RISs, only when N is unaffordably large can passive  RISs outperform active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  To illustrate this claim, let us consider two different SU-  SISO systems, which are aided by an active RIS and a passive  RIS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Then, the following lemma specifies the  condition that has to be met for passive RISs to outperform  active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Lemma 3 (Case when passive RISs outperform active  RISs): Assuming the number of RIS elements N is large,  the required number of elements N for a passive RIS to  outperform an active RIS has to satisfy  N ≥ P max  BS-A  P max  BS-P  P max  A  σ2  �  P max  A  σ2vϱ2  f + P max  BS-Aσ2ϱ2g + σ2σ2v  �,  (6)  where P max  BS-A denotes the maximum BS transmit power for the  active RIS aided system and P max  BS-P denotes that for the passive  RIS aided system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Proof: Please see the journal version [4, Appendix B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Next, we consider a specific setup to compare the user’s  achievable SNRs in the above two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For a fair com-  parison, we constrain the total power consumption P max of  the two systems to 2 W by setting P max  BS-P = 2 W for the  passive RIS aided system and P max  BS-A = P max  A  = 1 W for  the active RIS aided system, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Therefore, when  σ2 = σ2  v = −70 dBm and ϱ2  f = ϱ2  g = −70 dB, the required  number of elements N for the passive RIS to outperform the  active RIS is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 × 106 according to (6), which is impractical  to realize with current technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Conversely, for a more  practical number of elements of N = 256, according to (5)  and (4), the SNR achieved by the passive RIS is γpassive ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='0  dB, while the SNR achieved by the active RIS is γactive ≈ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='0  dB, which is about 10, 000 times higher than γpassive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' JOINT TRANSMIT PRECODING AND REFLECT  BEAMFORMING DESIGN  To investigate the capacity gain enabled by the use of  active RISs in typical wireless communication scenarios, in  this section, we consider more general MU-MISO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  According to the model in (3), the signal-to-interference-plus-  noise ratio (SINR) at user k can be obtained as  γk =  ��¯hH  k wk  ��2  �K  j=1,j̸=k  ��¯hH  k wj  ��2 +  ��f H  k PΘ  ��2σ2v + σ2 ,  (7)  wherein ¯hH  k = hH  k + fk  HPΘG ∈ C1×M is the equivalent  channel from the BS to user k, which includes both the direct  link and the reflected link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Therefore, the original problem of  sum-rate maximization, subject to the power constraints at the  BS and the active RIS, can be formulated as follows:  Po : max  w,P,Θ Rsum(w, P, Θ) =  �K  k=1 log2 (1 + γk),  (8a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' C1 :  �K  k=1 ∥wk∥2 ≤ P max  BS  ,  (8b)  C2 :  �K  k=1∥PΘGwk∥2+∥PΘ∥2 σ2  v ≤P max  A  , (8c)  where w :=  �  wT  1 , · · · , wT  K  �T is the overall transmit precoding  vector for the K users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' C1 and C2 are the power constraints at  the BS and active RIS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Due to the non-convexity  and the highly coupled variables in problem Po in (8), the  joint design of w, P, and Θ is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  To efficiently solve the above problem, we develop a joint  precoding and beamforming algorithm based on alternating  optimization and fractional programming (FP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Note that P  and Θ always appear in product form in problem Po in  (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Therefore, P and Θ can be merged as Ψ = PΘ =  diag  �  p1ejθ1, · · · , pNejθN �  ∈ CN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' We refer to Ψ as the  RIS beamforming matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Next, to deal with the non-convex  sum-of-logarithms and fractions in (8), we exploit the FP  methods proposed in [7] to decouple the variables in problem  Po in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' This leads to the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Lemma 4 (Equivalent problem for sum-rate maximiza-  tion): By introducing auxiliary variables ρ := [ρ1, · · · , ρK] ∈  RK and ϖ := [ϖ1, · · · , ϖK] ∈ CK, the original problem Po  in (8) can be equivalently reformulated as follows  P1 :  max  w,Ψ,ρ,ϖ R′  sum(w, Ψ, ρ, ϖ) =  �K  k=1 ln (1 + ρk)−  �K  k=1 ρk +  �K  k=1 g(w, Ψ, ρk, ϖk),  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' C1, C2,  (9)  where function g(w, Ψ, ρk, ϖk) is defined as  g(w, Ψ, ρk, ϖk) = 2  �  (1 + ρk)R  �  ϖ∗  k¯hH  k wk  �  −  |ϖk|2  ��K  j=1  ��¯hH  k wj  ��2 +  ��f H  k Ψ  ��2σ2  v + σ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  (10)  Proof: Constructive proof can be found in [7, Subsection  III-C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Strong convergence of the FP methods was proved in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Thus, a locally optimal solution to (9) can be obtained by  alternately optimizing the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For clarity, we summarize  the proposed joint precoding and beamforming algorithm in  Algorithm 1, and the specific solutions for variables w, Ψ,  ρ, and ϖ are given in the following four steps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Algorithm 1 Proposed joint transmit precoding and reflect  beamforming algorithm  Input:  Channels G, hk, and fk, ∀k ∈ {1, · · · , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Output:  Optimized BS precoding vector w, amplification  factor matrix of active RIS P, phase shift matrix of active  RIS Θ, and sum-rate Rsum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  1: Randomly initialize w, P and Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2: while no convergence of Rsum do  3:  Update ρ by (11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  4:  Update ϖ by (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  5:  Update w by solving problem P2 in (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  6:  Update Ψ by solving problem P3 in (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  7: end while  8: Obtain P and Θ from Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  9: return Optimized w, P, Θ, and Rsum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  1) Fix (w, Ψ, ϖ) and optimize ρ: After fixing precoding  vector w, beamforming matrix Ψ, and auxiliary variable ϖ,  the optimal ρ can be obtained by solving ∂R′  sum  ∂ρk  = 0 as  ρopt  k  = ξ2  k + ξk  �  ξ2  k + 4  2  ,  ∀k ∈ {1, · · · , K},  (11)  where ξk = ℜ  �  ϖ∗  k¯hH  k wk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2) Fix (w, Ψ, ρ) and optimize ϖ: After fixing the precod-  ing vector w, beamforming matrix Ψ, and auxiliary variable  ρ, the optimal ϖ can be derived by solving ∂R′  sum  ∂ϖk  = 0 as  ϖopt  k  =  �  (1 + ρk)¯hH  k wk  �K  j=1  ��¯hH  k wj  ��2+  ��f H  k Ψ  ��2σ2v + σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  (12)  3) Fix (Ψ, ρ, ϖ) and optimize w: To simplify the nota-  tions, we first introduce the following definitions:  bH  k =  �  (1 + ρk)ϖ∗  k¯hH  k , b =  �  bT  1 , bT  2 , · · · , bT  N  �T ,  (13a)  A=IK ⊗  �K  k=1|ϖk|2¯hk¯hH  k , Ξ=IK ⊗  �  GHΨHΨG  �  , (13b)  P max  m  = P max  A  − ∥Ψ∥2σ2  v,  (13c)  where ⊗ denotes the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Then, problem P1 in  (9) can be reformulated as follows:  P2 :  max  w  R  �  2bHw  �  − wHAw,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  C1 : ∥w∥2 ≤ P max  BS  ,  C2 : wHΞw ≤ P max  m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  (14)  Note that P2 in (14) is a standard quadratic constraint  quadratic programming (QCQP) problem, which can be solved  by alternating direction method of multipliers (ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  4) Fix  (w, ρ, ϖ)  and  optimize  Ψ:  Define  ψ  =  �  p1ejθ1, · · · , pNejθN �H as the vectorized RIS beamforming  matrix Ψ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', diag  �  ψH�  := Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' While fixing w and ρ and  ϖ, problem P1 in (9) can be reformulated as follows:  P3 :  max  ψ  R  �  2ψHυ  �  − ψHΩψ,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' C2 : ψHΠψ �� P max  A  ,  (15)  spectrum  analyzer  LNA  vector network  analyzer  DC source  active RIS  element  circulator  pump  source  noise source  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The experimental devices and environment used for validating the signal model (2) of active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='359  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='3595  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='36  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='3605  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='361  15  10  5  0  5  10  15  20  25  30  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Experimental measurement result for reflection gain G versus signal  frequency f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  wherein  υ =  �K  k=1  �  (1 + ρk)diag  �  ϖ∗  kf H  k  �  Gwk−  �K  k=1 |ϖk|2diag  �  f H  k  �  G  �K  j=1 wjwH  j hk,  (16a)  Ω =  �K  k=1 |ϖk|2diag  �  f H  k  �  diag (fk) σ2  v+  �K  k=1 |ϖk|2 �K  j=1diag  �  f H  k  �  GwjwH  j GHdiag (fk), (16b)  Π =  �K  k=1 diag (Gwk) (diag (Gwk))H + σ2  vIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  (16c)  Note that problem P3 in (15) is also a standard QCQP  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Thus, the optimal solution ψopt can be obtained by  adopting ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' VALIDATION RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Validation Results for Signal Model  To validate the signal model (2), we designed and fabri-  cated an active RIS element with an integrated reflection-  type amplifier for experimental measurements in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Note  that this design can be directly extended to the large-array  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Particularly, since the phase-shifting ability of RISs has  been widely verified, we focus on studying the reflection gain  15  10  5  0  5  10  15  20  25  170  165  160  155  150  145  140  135  130  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Experimental measurement result for the density of noise power  Gσ2  v + σ2  s versus reflection gain G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  and the noise introduced by an active RIS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Thus, the  validation of signal model (2) is equivalent to validating  Py =  GPx  ����  Desired-signal power  + Gσ2  v + σ2  s  �  ��  �  noise power  ,  (17)  where Py is the power of the signals reflected by the active  RIS element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Px is the power of the incident signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' G := p2 is  the re��ection gain of the active RIS element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Gσ2  v and σ2  s are  the powers of the dynamic noise and static noise introduced  by the active RIS element, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  1) Hardware platform: To validate the model in (17), we  first establish the hardware platform used for our experimental  measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Due to space constraints, we refer  the reader to the journal version of this paper [4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 4] for  detailed information about the hardware platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2) Reflection gain measurement: Using the measurement  system for the reflection gain depicted in [4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 4 (b)], we  first investigate the reflection gain G of the active RIS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Note that the reflection gain G can be reconfigured by the  input power of the pump source Pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' By setting the input  power of the vector network analyzer as Px = −50 dBm, the  reflection gain G as a function of the signal frequency can be  directly measured via a vector network analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Then, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  10  5  0  5  10  15  20  25  30  0  10  20  30  40  50  60  507%  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation results for the sum-rate versus total power consumption  P max in scenario 1 with a weak direct link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='.  3, we show the measurement results for reflection gain G as  a function of signal frequency f for different input powers of  the pump source Pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' We observe that the active RIS element  can achieve a reflection gain G of more than 25 dB, when  Pp = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='24 dBm, which confirms the significant reflection  gains enabled by active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  3) Noise power measurement: We further study the noise  power introduced by the active RIS element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', Gσ2  v + σ2  s  in (17), where Gσ2  v and σ2  s are the powers of the dynamic  noise and the static noise introduced at the active RIS element,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Using the noise measurement system in [4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 4  (c)], we show the measurement results for the spectral density  of noise power Gσ2  v + σ2  s as a function of G for different  operating frequencies in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' We can observe that the noise  power increases nearly linearly with G, which verifies the  noise model Gσ2  v + σ2  s in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Particularly, for f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='3601  GHz, the spectral density of σ2  s is about −174 dBm/Hz, while  that of σ2  v is about −160 dBm/Hz, which is about 15 dB  higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The reason for this is that the input noise is amplified  by the noise factor, and additional noises are also introduced  by the other active components such as the DC source used  to power the active RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation Results for Sum-Rate  1) Simulation setup: We consider an active RIS aided MU-  MISO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Particularly, we consider two scenarios with dif-  ferent channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In scenario 1, the direct link is weak  due to severe obstruction, while the direct link is strong in  scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To be specific, two different path loss models from  the 3GPP TS 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='814 standard are utilized to characterize the  large-scale fading of the channels: i) PLs = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='3+22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='0 log d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  ii) PLw = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='2 + 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='7 log d, where d is the distance between  two devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Path loss model PLw is used to generate the weak  BS-user link in scenario 1, while PLs is used to generate the  strong BS-user link in scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For both scenarios, PLs is  used to generate the BS-RIS and the RIS-user channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To  account for small-scale fading, we adopt the Ricean fading  channel model for all channels involved and we assume the  Ricean factor as κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  10  5  0  5  10  15  20  25  30  0  10  20  30  40  50  60  62%  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation results for the sum-rate versus total power consumption  P max in scenario 2 with a strong direct link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  The BS and the active/passive RIS are located at (0, -60  m) and (200 m, 30 m), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Four users are randomly  located in a circle with a radius of 5 m from the center (200  m, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The numbers of BS antennas and RIS elements are set  as M = 4 and N = 256, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The noise power is set  as σ2 = σ2  v = −70 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For fair comparison, we constrain  the total power consumption P max := P max  BS  + P max  A  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For the  active RIS, Algorithm 1 is employed for joint precoding and  beamforming design, while for the passive RIS, the algorithm  from [2] is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2) Simulation results: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 6, we plot the  sum-rate versus the total consumed power P max for the two  considered scenarios, where the direct link is weak and strong,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Firstly, in scenario 1 with a weak direct link, the  passive RIS can indeed achieve a performance improvement,  while the active RIS achieves a much higher sum-rate gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Secondly, in scenario 2 with a strong direct link, the passive  RIS achieves only a negligible sum-rate gain, while the active  RIS still realizes a noticeable sum-rate gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For example,  when P max = 10 dBW, the capacities without RIS, with  passive RIS, and with active RIS in scenario 1 are 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='34 bps/Hz,  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='00 bps/Hz, and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='41 bps/Hz respectively, while in scenario  2, these values are 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='87 bps/Hz, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='51 bps/Hz, and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='18  bps/Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In this case, the passive RIS provides a  31% gain in scenario 1 and a negligible 3% gain in scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  By contrast, the active RIS achieves noticeable sum-rate gains  of 507% in scenario 1 and 62% in scenario 2, which are much  higher than those achieved by the passive RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Field Test for a 64-Element Active RIS Aided Wireless  Communication Prototype  1) 64-element active RIS aided communication prototype:  To validate the significant gain of active RISs, we develop a  64-element active RIS aided wireless communication proto-  type, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Specifically, the hardware structure  of this prototype consists of three parts including a BS, a 64-  element active RIS, and a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For the BS and the user, two  horn antennas with 13 dBi antenna gain are used to transmit  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' A photograph of the developed 64-element active RIS aided wireless  communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  and receive the signals, and the universal software radio  peripherals (USRPs) are deployed to generate and process the  baseband and RF signals (hardware version: USRP-2953R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  By periodically expanding the active RIS elements designed  in [8], the 64-element active RIS is an 8×8 plane array, of  which each element has a reflection gain of G = 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2) Experimental environment: Based on the developed pro-  totype, we establish the experimental environment for further  validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To match the transceivers, we configure the oper-  ating frequency of the active RIS to f = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 GHz and the  bandwidth to 40 MHz by adjusting the circuit impedance of  active elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The polarization of the antenna at the BS  and that at the user are selected as vertical and horizontal,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The transmit power is set to −10 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' We fix the  heights of the BS, the RIS, and the user as 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The horizontal  distance of the BS-RIS link and that of the RIS-user link are  set to 2 m and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The angle of arrival (AoA)  at the active RIS is fixed as 0◦, and the angle of departure  (AoD) will be specified to evaluate the performance gain of  active RISs at different orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To observe the reflection  gain of the active RIS, we use a metal plate with the same  aperture size as the active RIS for performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  3) Experimental results: By moving the user at different  AoDs and configuring the phase shift of the active RIS with  discrete Fourier transform (DFT) codebook, we obtain the  experimental results shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' One can observe that,  compared with the received power for the metal plate, the  active RIS can always achieve a gain of about 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The data  rate for the active RIS can hold at about 30 Mbps, while that  for the metal plate only ranges from 1 Mbps to 2Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The  reason is that, the beamforming at the active RIS can make the  reflected beam with high array gain and reflection gain, while  the metal plate can only reflect the incident signals randomly  without in-phase combination or amplification, which validates  the significant gain of active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' CONCLUSIONS  In this paper, we have studied the concept of active RISs  to overcome the fundamental limitation of the “multiplicative  fading” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Firstly, we have verified the signal model of  TABLE I  EXPERIMENTAL RESULTS FOR THE DEVELOPED PROTOTYPE  AoD  Device  Received Power  Data Rate  15◦  Metal plate  110 dBm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='2 Mbps  Active RIS  100 dBm  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 Mbps  30◦  Metal plate  105 dBm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 Mbps  Active RIS  98 dBm  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 Mbps  45◦  Metal plate  105 dBm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 Mbps  Active RIS  95 dBm  30 Mbps  60◦  Metal plate  108 dBm  2 Mbps  Active RIS  90 dBm  32 Mbps  active RISs through the experimental measurements on a fab-  ricated active RIS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Based on the verified signal model,  we have formulated the sum-rate maximization problem for  an active RIS aided MU-MISO system and a joint precoding  and beamforming algorithm has been proposed to solve this  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation results have shown that, in a typical  application scenario, the existing passive RIS can realize only  a negligible sum-rate gain of about 3%, while the active RIS  can achieve a substantial sum-rate gain of about 62%, thus  indeed overcoming the “multiplicative fading” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Finally,  we have developed a communication wireless communication  prototype aided by a 64-element active RIS, and the significant  gain of active RISs is validated by field test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In the future,  many research directions for active RISs are worth pursuing,  such as hardware design, prototype development, channel  estimation, and energy efficiency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  ACKNOWLEDGMENT  This work was supported in part by the National Key  Research and Development Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2020YFB1805005), in part by the National Natural Science  Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 62031019), and in part by the  European Commission through the H2020-MSCA-ITN META  WIRELESS Research Project under Grant 956256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  REFERENCES  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Huang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Zappone, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Alexandropoulos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Debbah, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Yuen,  “Reconfigurable intelligent surfaces for energy efficiency in wireless  communication,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  4157–4170, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Pan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Ren, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Elkashlan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Nallanathan,  and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Hanzo, “Multicell MIMO communications relying on intelligent  reflecting surfaces,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  5218–5233, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Najafi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Jamali, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Schober, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Poor, “Physics-based mod-  eling and scalable optimization of large intelligent reflecting surfaces,”  IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2673–2691, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  [4] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Dai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Liu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Yang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Schober, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Poor,  “Active RIS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' passive RIS: Which will prevail in 6G?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Bousquet, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Magierowski, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Messier, “A 4-GHz active  scatterer in 130-nm CMOS for phase sweep amplify-and-forward,” IEEE  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Circuits Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' I, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 529–540, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  [6] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Wu and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Zhang, “Intelligent reflecting surface enhanced wireless  network via joint active and passive beamforming,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Wireless  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 5394–5409, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  [7] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Shen and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Yu, “Fractional programming for communication sys-  tems—part I: Power control and beamforming,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Signal  Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2616–2630, May 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  [8] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Chen and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Yang, “Nonlinear electromagnetic surfaces: Theory,  design and application,” Master Thesis in Tsinghua University, May 2020,  [Online] Available: http://etds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='lib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='cn/Thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5GHz  有源RIS  8 X 8 Active RIS  BS  User' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}

ZdE5T4oBgHgl3EQfeA8r/content/tmp_files/2301.05615v1.pdf.txt

@@ -0,0 +1,475 @@
+Linear Computation Coding:
+Exponential Search and Reduced-State Algorithms
+Hans Rosenberger, Johanna S. Fr¨ohlich, Ali Bereyhi and Ralf R. M¨uller
+Institute for Digital Communications (IDC)
+Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg
+Erlangen, Germany
+{hans.rosenberger, johanna.froehlich, ali.bereyhi, ralf.r.mueller}@fau.de
+Abstract
+Linear computation coding is concerned with the compression of multidimensional linear
+functions, i.e. with reducing the computational effort of multiplying an arbitrary vector to
+an arbitrary, but known, constant matrix. This paper advances over the state-of-the art,
+that is based on a discrete matching pursuit (DMP) algorithm, by a step-wise optimal search.
+Offering significant performance gains over DMP, it is however computationally infeasible
+for large matrices and high accuracy. Therefore, a reduced-state algorithm is introduced
+that offers performance superior to DMP, while still being computationally feasible even for
+large matrices. Depending on the matrix size, the performance gain over DMP is on the
+order of at least 10 %.
+Introduction
+Multiplying a vector by a constant matrix is an ubiquitous task performed in various
+technical and scientific applications. The main body of earlier work is focused on
+speeding up the calculation of matrix-vector multiplications in a structure-oriented
+fashion. A well-known example is the fast implementation of the discrete fourier
+transform (DFT). Here, the structure of the DFT matrix is exploited to eliminate
+redundant computations and reduce the number of required operations as compared
+to a naive implementation. For arbitrary constant matrices, redundancies within the
+finite-precision representation of the matrix entries can be exploited as well, a method
+that is typically known as common subexpression sharing/elimination. Earlier work
+in this respect has either targeted special cases of constant multiplication [1, 2] or has
+proposed schemes with high computational complexity, such that their implementation
+in practice is difficult for medium to large size matrices [3, 4].
+Recently, linear computation coding (LCC) has been proposed in [5, 6, 7]. This
+framework develops an information-theoretic scheme for the efficient calculation
+of matrix-vector products that is especially well-suited for the implementation on
+reconfigurable hardware, such as field programmable gate arrays (FPGAs) [8]. Similar
+to rate-distortion theory, LCC is concerned with the tradeoff between distortion
+and compression. However, instead of compressing data, LCC deals with the lossy
+compression of multidimensional linear functions under a given fidelity constraint. An
+This work was supported by Deutsche Forschungsgemeinschaft (DFG) under the project Compu-
+tation Coding (MU-3735/8-1).
+arXiv:2301.05615v1  [cs.IT]  13 Jan 2023
+
+instance can be found in [7], where an optimal decomposition scheme is first defined
+in terms of classical metrics for computation and distortion. A greedy approach is
+then developed to approximate the proposed scheme sub-optimally with tractable
+complexity.
+Contributions
+In this paper, we develop a new LCC scheme. Similar to earlier approaches discussed
+in [7], the optimal decomposition deals with an exponentially complex problem. We first
+address this problem via an exhaustive search procedure with a careful optimization.
+This enables us to evaluate the performance of the optimal scheme for reasonable
+matrix sizes. We then present a computationally tractable scheme by proposing a
+reduced-state algorithm for the underlying search problem. Our investigations show
+that the proposed algorithm can achieve a computation-distortion tradeoff close to the
+exponentially-complex optimal scheme while drastically reducing the decomposition
+complexity.
+Notation
+Vectors are denoted as lower-case boldface letters x and matrices as upper-case boldface
+letters X. The Euclidean and the Frobenius norm are denoted by ∥ · ∥2 and ∥ · ∥F,
+respectively. The symbol 0N×K denotes an N × K matrix with all zero elements,
+IN×K denotes the augmented identity matrix of dimension N × K and 1j,K denotes
+the j-th row unit vector in K dimensions.
+Problem Formulation
+We consider the problem of matrix-vector multiplication, i.e the calculation
+y = Ax
+(1)
+for an arbitrary input vector x ∈ RK×1 and a constant matrix A ∈ RN×K. Commonly,
+matrices are approximated by quantizing their entries independently. By using the
+canonically signed digit (CSD) binary representation the quantization error can be
+decreased on average by a factor of
+√
+28 per CSD [9]. This still leaves room for
+improvement. LCC instead suggests to approximate A by a product of matrices, i.e.
+finding W and C such that
+A ≈ W C.
+(2)
+The matrix C ∈ AN×K is termed the codebook matrix and W ∈ AN×N is termed the
+wiring matrix in the sequel1. The entries of the wiring matrix are restricted to the set
+of zero and signed powers of two (A ⊆ {0, ±2Z}).
+Obtaining the wiring and codebook matrix jointly is typically NP-hard and infea-
+sible. To overcome this computational intractability, [7] proposes a scheme where the
+1In [6] the multiplication order of the decomposed matrices is reversed. Please note that this
+change makes no difference to the general idea of the decomposition and to the following algorithms.
+It is equal to the transposed version of the algorithm presented in [7].
+
+n-th row of the wiring matrix is determined by solving the following sparse recovery
+problem for some design parameter S < N controlling the cost between distortion and
+computation effort [10]
+wn =
+argmin
+ω∈{ω=�S
+s=1 is1js,N: is∈A, js∈{1,...,N} ∀s}
+∥an − ωC∥2
+∀n.
+(3)
+The new scheme is still NP-hard, but not in N, anymore, but in S. Thus, small values
+of S are required, in practice.
+In order to have a high accuracy despite small values of S, the factorization
+procedure can be applied multiple times. Then the product Ci = W iCi−1 of the
+previous wiring step acts as the new codebook for obtaining the following matrix
+factor W i of the current wiring step. Hence, by setting2 C0 = IN×K, we obtain the
+approximated matrix P after I wiring steps:
+A ≈ P =
+� I�
+i=1
+W i
+�
+C0.
+(4)
+To quantify the accuracy of a given approximation P we use the signal-to-
+quantization-noise-ratio (SQNR)
+SQNR(A, P ) =
+∥A∥2
+F
+∥A − P ∥2
+F
+.
+(5)
+Computational Cost
+In a binary number representation the multiplication by a signed power of two
+corresponds only to a bitshift. On reconfigurable hardware, this shift can be realized
+simply by appropriate wiring without the need for dedicated processing elements
+such as adders [8]. The parameter S in (3) determines the number of vectors from
+the codebook to be used in forming the linear combination to approximate a row
+an of A. It therefore directly controls the computational cost, as in computing the
+linear combination, exactly S − 1 additions are required. No multiplications, except
+by signed powers of two, are necessary due to the specific structure of the wiring
+matrix. Therefore, the separate product of the decomposed matrices with the input
+vector y ≈ W (Cx) is much simpler to compute than calculating the product in (1)
+straightforwardly.
+The total computational cost Cadd of a decomposition in (4) is given by the number
+of additions (or subtractions) required to form the linear combinations
+Cadd = IN(S − 1).
+(6)
+2In [6] this choice is termed the self-designing codebook. It was found to work very well for a
+wide range of matrices.
+
+Algorithms
+In this section we will briefly look at the state of the art for solving the optimization
+problem in (3) to obtain the wiring matrices and then introduce two improved novel
+algorithms.
+State-of-the-Art: Discrete Matching Pursuit
+The discrete matching pursuit (DMP) follows the matching pursuit approach to
+successively determine the wiring coefficients. The algorithm can be summarized in
+the following key steps; for details see [6].
+1. Start with iteration s ← 0. Initialize ω ← 01×N
+2. Update ω in at most a single component, such that ∥an − ωC∥2 is minimized.
+3. Increment s.
+4. If s ≤ S, go to step 2, otherwise the procedure terminates.
+It is straightforward to show that the time complexity of the DMP algorithm for
+computing a single matrix factor scales with O(N 3S).
+Exponential Search Algorithm
+The row-wise optimization problem in (3) is NP-hard. However, for small S, reasonable
+matrix sizes and some careful optimization it can be solved in a tractable timeframe. We
+limit the set of scaling factors to a finite set of signed powers of two (Aexp ⊂ {0, ±2Z}),
+as an exhaustive search over the whole set is infeasible. As the search procedure has
+to be performed for each row of the target matrix individually, the time complexity
+for the computation of each wiring step is given by O(N S|Aexp|S).
+Generally, which and how many coefficients are included in the subset Aexp is a
+design parameter and needs to be adapted to each specific decomposition. It depends
+primarily on two factors. First, the current wiring step plays a crucial role. For
+each additional wiring layer, the error between each row of the target matrix and
+the approximated matrix decreases. Hence, for any subsequent wiring step, smaller
+coefficients are needed to scale the rows of the newly found codebook matrix to
+appropriately approximate the residual error. This also means that for high desired
+accuracy the coefficient set needs to be chosen large, i.e. to include also many small
+coefficients, to accurately approximate the error. Furthermore, relative variations
+in the length of the row vectors of the target matrix require a larger coefficient
+set to compensate for differences. Still, to keep the decomposition computationally
+feasible, the number of elements in Aexp needs to be chosen as small as possible, as
+the computational complexity scales exponentially in S with the product of the size
+of the coefficient set |Aexp| and N as base.
+A promising approach for further research is to adapt the coefficient set for each
+wiring step dynamically based on the current fidelity of the approximation. The
+coefficient set may then be determined from the probability distribution of the likely
+entries of the wiring matrix.
+
+Reduced-State Algorithm
+The runtime of the exponential algorithm for a matrix with N = 64 rows is on the
+order of an hour3, for larger matrices the runtime scales up accordingly and can
+become infeasible. Hence, for some applications the computational burden of the
+exponential algorithm might consequently not be feasible. A reasonable compromise
+between complexity and performance is desirable.
+Unlike in DMP, we do not immediately update the components of the wiring
+vector for the reduced-state algorithm. Instead, we keep updating in each iteration a
+list of the M best vectors, which minimize (3) and select at the termination of the
+algorithm the vector with minimum error from the list. Specifically, we apply the
+following successive, greedy procedure for the optimization problem in (3):
+1. Start with s ← 0. Initialize the set Ω with M all-zero vectors.
+2. For each ω ∈ Ω find the set Ωm of M mutually distinct vectors ω ˜m with
+˜m ∈ {1, ..., M} that minimize ∥an − ω ˜mC∥2 and differ from ω in at most a
+single component.
+3. Update the set Ω by selecting from �M
+m=1 Ωm the M distinct vectors ω with
+minimum ∥an − ωC∥2
+4. Increment s.
+5. If s ≤ S, go to step 2, else, continue
+6. Return argmin
+ω∈Ω
+∥an − ωC∥2
+By tuning M, we are able to adjust the space of possible combinations that the
+algorithm explores. For each wiring step, we have to evaluate O(SN 3M 2) combinations.
+Compared to DMP the complexity is increased by a factor of M 2. Note that for
+M = 1 the algorithm reduces to the DMP.
+Numerical Evaluation
+In this section, we compare the performance of the proposed algorithms to the baseline
+DMP [7]. We decompose matrices whose entries are drawn i.i.d. from a Gaussian
+distribution with zero mean and unit variance. Similar to DMP, the performance of the
+improved versions of the algorithm hardly depends on the distribution of the matrix
+elements. The algorithms are invariant to scaling of the variance, however it is crucial
+that for the exponential search algorithm the coefficient set Aexp is scaled appropriately
+as well, as to not compromise performance. Throughout the simulations, we select the
+coefficient set for the exponential search algorithm to Aexp = {±2−40, .., ±23}.
+As a first experiment, we compare all three algorithms for fixed matrix sizes in
+Figure 1. We choose matrices with N = 64 rows and vary the number of columns K
+3For a multithreaded implementation in Python with Numba acceleration executed on an Intel
+i9-12900@2.40 GHz and parameters S = 3, Aexp =
+�
+±2−40, . . . , ±23�
+.
+
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Cumulative number of additions
+0
+20
+40
+60
+80
+100
+120
+SQNR [dB]
+64x4
+64x6
+64x8
+Figure 1: Performance comparison for matrices with 64 rows and different aspect ratios.
+Matrices of dimension 64×4 are indicated by crosses, 64×6 is indicated by squares and 64×8
+is indicated by triangles. The solid lines refer to the exponential search algorithm (S = 3),
+the dashed lines to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced state
+algorithm with memory size M = 10 and S = 3. Results are averaged over 104 matrix entries
+for the exponential search algorithm and over 105 matrix entries for the other algorithms.
+from four to eight. Hence, we can compare the performance for different aspect ratios
+of the matrices4. To quantify performance, we plot the tradeoff between distortion
+and computational cost. The latter being measured by the number of cumulative
+additions Cadd required for a given wiring step.
+As Figure 1 shows, for all three matrix sizes there is a performance gain by both
+proposed algorithms against DMP. Further, for memory size M = 10, the reduced
+state algorithm performs only slightly worse than exhaustive search.
+For the first two wiring steps, the performance of all three algorithms is equal
+for a given matrix size. This is due to the fact that for the first two steps we use
+DMP with S = 2 for an initial refinement of the codebook5. Applying any of the
+4LCC works best for matrices with an exponential aspect ratio, i.e. for K ≈ log2 N. For square
+matrices it is beneficial to cut these into multiple tall matrices and decompose each slice independently,
+see [8] for details.
+5Using any of the novel algorithms with S = 2 is possible as well with very similar performance, i.e.
+a slight increase in SQNR by 0.2 dB to 0.5 dB for the novel algorithms and matrix sizes considered.
+
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Cumulative number of additions
+0
+10
+20
+30
+40
+50
+60
+70
+80
+SQNR [dB]
+Increasing M
+(2, 5, 10, 50)
+DMP
+Exponential Search
+Reduced State (Variable M)
+Figure 2: Performance comparison for different memory sizes M of the reduced state
+algorithm of a 64 × 6 matrix. The solid line refers to the exponential search algorithm
+(S = 3), the dashed line to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced
+state algorithm with 4 different memory sizes and S = 3. Results are averaged over 104
+matrix entries for the exponential search algorithm and over 105 matrix entries for the other
+algorithms.
+algorithms with S ≥ 3 directly to the initial codebook C0 = IN×K would lead to
+degraded performance for the first few wiring steps. Instead it is beneficial to set S = 2
+to allow for more frequent updates of the codebook in the beginning. From empirical
+investigations it seems that two wiring iterations with S = 2 are most beneficial for
+the overall performance.
+As the next experiment, we compare the performance of the reduced state algorithm
+for different choices of the memory parameter M for given matrix size of 64 × 6 in
+Figure 2.
+From the figure we observe that even for small M the reduced state
+algorithm offers a noticeable performance gain over DMP. As M grows, the reduced
+state algorithm approaches the performance of the exponential search. The large
+advantage of the reduced state algorithm is that the computation time is reduced
+drastically6. For M = 1 the reduced state algorithm reduces to DMP and both lines
+6For the considered matrix size the execution time differs from an hour for the exponential
+algorithm to a few seconds for the reduced state algorithm in practice.
+
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Cumulative number of additions
+0
+20
+40
+60
+80
+100
+120
+SQNR [dB]
+Increasing S
+(3, 4, 8)
+DMP
+Exponential search (S = 3)
+Reduced State (S = 3, 4, 8)
+Figure 3: Performance comparison for different choices of the parameter S of a matrix
+with dimension 64 × 4. The solid line refers to the exponential search algorithm (S = 3),
+the dashed line to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced state
+algorithm with different choices of the parameter S and M = 10. Results are averaged over
+104 matrix entries for the exponential search algorithm and over 105 matrix entries for the
+other algorithms.
+coincide for any given matrix size.
+Table 1 lists the relative performance gains over DMP for various matrix sizes and
+configurations of the algorithms.
+Practical Considerations for the Choice of S
+Figure 3 shows the performance of the reduced state algorithm for varying S. Due to
+the exponentially growing complexity in S, the exponential algorithm is not feasible
+for S > 3, except for very small matrices. We can observe, that by choosing S = 4
+for the reduced state algorithm, we approximately achieve the same distortion-cost
+tradeoff as for the exponential algorithm with S = 3. For choosing S even larger the
+gains increase likewise.
+However for a practicable implementation S should not be chosen arbitrarily. In [8],
+the performance of DMP is validated in an implementation on reconfigurable hardware
+with S = 2. This means that on an FPGA exactly N adders are required per wiring
+matrix. With the inputs depending only on the outputs of the previous wiring matrix,
+
+Table 1: Relative average gain in terms of SQNR of the novel algorithms over the baseline
+DMP algorithm with S = 2 (For at least 8 bit signed integer accuracy (10 log(SQNR) ≥
+47 dB)). Results are averaged over 104 matrix entries for the exponential search algorithm
+and over 105 matrix entries for the other algorithms.
+Exponential
+Reduced State
+search
+S = 3
+S = 4
+S = 8
+Matrix size
+S = 3
+M = 5
+M = 10
+M = 5
+M = 10
+M = 5
+M = 10
+16 × 2
+17.8 %
+10.4 %
+13.4 %
+14.1 %
+17.8 %
+16.7 %
+21.9 %
+16 × 4
+34.5 %
+16.0 %
+24.5 %
+25.8 %
+32.7 %
+25.4 %
+34.4 %
+32 × 4
+15.7 %
+10.5 %
+12.9 %
+14.0 %
+17.2 %
+18.4 %
+22.3 %
+32 × 6
+25.3 %
+15.5 %
+19.0 %
+19.7 %
+24.5 %
+19.4 %
+26.0 %
+64 × 4
+12.9 %
+7.5 %
+9.8 %
+11.0 %
+13.8 %
+13.5 %
+16.8 %
+64 × 6
+14.9 %
+9.6 %
+11.4 %
+13.6 %
+16.5 %
+15.6 %
+19.0 %
+the decomposition is well suited for parallel execution and pipelining [8]. Due to the
+greedy, step-wise nature of DMP S > 2 does not offer significant performance gains
+over S = 2. Both novel algorithms behave differently.
+From an implementation point of view, S = 3 is even more suitable than S = 2
+for an effective implementation in hardware due to the availability of efficient adders
+with three inputs [11]. Interestingly, on modern FPGAs, these adders do not require
+more hardware resources, in terms of Lookup-Tables (LUTs), than an adder with two
+inputs. Any powers of two and three (S = 4, 8, 9, . . . ) can be realized efficiently as well
+by the use of adder trees. However, it is questionable if choosing S > 4 is beneficial,
+as performance gains over S = 3 or S = 4 are small and the desired fidelity of the
+approximation cannot be chosen in a fine granularity anymore7.
+Conclusion
+In this paper, we have proposed two new algorithms for LCC, a framework for
+the lossy compression of multidimensional linear functions. While the exponential
+search algorithm shows the best performance, it is generally infeasible especially for
+large matrices. The proposed reduced-state algorithm, performs close to exponential
+search at a fraction of the computational complexity. The time complexity of the
+decomposition compared to the baseline algorithm from earlier works is only mildly
+increased, while the performance gains over the baseline DMP algorithm are on the
+order of at least 10 %.
+References
+[1] Jason Thong and Nicola Nicolici, “An optimal and practical approach to single constant
+multiplication,” IEEE Transactions on Computer-Aided Design of Integrated Circuits
+and Systems, vol. 30, no. 9, pp. 1373–1386, Sep 2011.
+[2] Yevgen Voronenko and Markus P¨uschel, “Multiplierless multiple constant multiplication,”
+ACM Transactions on Algorithms, vol. 3, no. 2, May 2007.
+7For a matrix of dimension 64 × 4 the reduced state algorithm with S = 8 improves the SQNR
+approximately by 70 dB per matrix factor.
+
+[3] N. Boullis and A. Tisserand, “Some optimizations of hardware multiplication by constant
+matrices,” in Proceedings 2003 16th IEEE Symposium on Computer Arithmetic, 2003,
+pp. 20–27.
+[4] Levent Aksoy, Paulo Flores, and Jose Monteiro, “A novel method for the approximation
+of multiplierless constant matrix vector multiplication,” EURASIP Journal on Embedded
+Systems, , no. 12, May 2016.
+[5] Ralf R. M¨uller, Bernhard G¨ade, and Ali Bereyhi, “Efficient matrix multiplication:
+The sparse power-of-2 factorization,” in 2020 Information Theory and Applications
+Workshop (ITA), 2020, pp. 1–6.
+[6] Ralf R. M¨uller, Bernhard G¨ade, and Ali Bereyhi, “Linear computation coding,” 2021,
+arXiv:2102.00398.
+[7] Ralf R. M¨uller, Bernhard M. W. G¨ade, and Ali Bereyhi, “Linear computation coding:
+A framework for joint quantization and computing,” Algorithms, vol. 15, no. 7, 2022.
+[8] Alexander Lehnert, Philipp Holzinger, Simon Pfenning, Ralf R. M¨uller, and Marc
+Reichenbach, “Most ressource efficient matrix vector multiplication on FPGA,” IEEE
+Access, 2022, Early Access.
+[9] A. D. Booth, “A signed binary mutliplication technique,” The Quarterly Journal of
+Mechanics and Applied Mathematics, vol. 4, no. 2, pp. 236 – 240, Jan 1951.
+[10] Simon Foucart and Holger Rauhut,
+A Mathematical Introduction to Compressive
+Sensing, Springer New York, 2013.
+[11] James M. Simkins and Brian D. Philofsky, “Structures and methods for implementing
+ternary adders/subtractors in programmable logic devices,” Sep 2007,
+US Patent
+7,274,211.
+

ZdE5T4oBgHgl3EQfeA8r/content/tmp_files/load_file.txt

@@ -0,0 +1,277 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf,len=276
+page_content='Linear Computation Coding:  Exponential Search and Reduced-State Algorithms  Hans Rosenberger, Johanna S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Fr¨ohlich, Ali Bereyhi and Ralf R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' M¨uller  Institute for Digital Communications (IDC)  Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg  Erlangen, Germany  {hans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='rosenberger, johanna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='froehlich, ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='bereyhi, ralf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='mueller}@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='de  Abstract  Linear computation coding is concerned with the compression of multidimensional linear  functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' with reducing the computational effort of multiplying an arbitrary vector to  an arbitrary, but known, constant matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This paper advances over the state-of-the art,  that is based on a discrete matching pursuit (DMP) algorithm, by a step-wise optimal search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Offering significant performance gains over DMP, it is however computationally infeasible  for large matrices and high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Therefore, a reduced-state algorithm is introduced  that offers performance superior to DMP, while still being computationally feasible even for  large matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Depending on the matrix size, the performance gain over DMP is on the  order of at least 10 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Introduction  Multiplying a vector by a constant matrix is an ubiquitous task performed in various  technical and scientific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The main body of earlier work is focused on  speeding up the calculation of matrix-vector multiplications in a structure-oriented  fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' A well-known example is the fast implementation of the discrete fourier  transform (DFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Here, the structure of the DFT matrix is exploited to eliminate  redundant computations and reduce the number of required operations as compared  to a naive implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For arbitrary constant matrices, redundancies within the  finite-precision representation of the matrix entries can be exploited as well, a method  that is typically known as common subexpression sharing/elimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Earlier work  in this respect has either targeted special cases of constant multiplication [1, 2] or has  proposed schemes with high computational complexity, such that their implementation  in practice is difficult for medium to large size matrices [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Recently, linear computation coding (LCC) has been proposed in [5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This  framework develops an information-theoretic scheme for the efficient calculation  of matrix-vector products that is especially well-suited for the implementation on  reconfigurable hardware, such as field programmable gate arrays (FPGAs) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Similar  to rate-distortion theory, LCC is concerned with the tradeoff between distortion  and compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' However, instead of compressing data, LCC deals with the lossy  compression of multidimensional linear functions under a given fidelity constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' An  This work was supported by Deutsche Forschungsgemeinschaft (DFG) under the project Compu-  tation Coding (MU-3735/8-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='05615v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='IT]  13 Jan 2023  instance can be found in [7], where an optimal decomposition scheme is first defined  in terms of classical metrics for computation and distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' A greedy approach is  then developed to approximate the proposed scheme sub-optimally with tractable  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Contributions  In this paper, we develop a new LCC scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Similar to earlier approaches discussed  in [7], the optimal decomposition deals with an exponentially complex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We first  address this problem via an exhaustive search procedure with a careful optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  This enables us to evaluate the performance of the optimal scheme for reasonable  matrix sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We then present a computationally tractable scheme by proposing a  reduced-state algorithm for the underlying search problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Our investigations show  that the proposed algorithm can achieve a computation-distortion tradeoff close to the  exponentially-complex optimal scheme while drastically reducing the decomposition  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Notation  Vectors are denoted as lower-case boldface letters x and matrices as upper-case boldface  letters X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The Euclidean and the Frobenius norm are denoted by ∥ · ∥2 and ∥ · ∥F,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The symbol 0N×K denotes an N × K matrix with all zero elements,  IN×K denotes the augmented identity matrix of dimension N × K and 1j,K denotes  the j-th row unit vector in K dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Problem Formulation  We consider the problem of matrix-vector multiplication, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e the calculation  y = Ax  (1)  for an arbitrary input vector x ∈ RK×1 and a constant matrix A ∈ RN×K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Commonly,  matrices are approximated by quantizing their entries independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' By using the  canonically signed digit (CSD) binary representation the quantization error can be  decreased on average by a factor of  √  28 per CSD [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This still leaves room for  improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' LCC instead suggests to approximate A by a product of matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  finding W and C such that  A ≈ W C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  (2)  The matrix C ∈ AN×K is termed the codebook matrix and W ∈ AN×N is termed the  wiring matrix in the sequel1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The entries of the wiring matrix are restricted to the set  of zero and signed powers of two (A ⊆ {0, ±2Z}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Obtaining the wiring and codebook matrix jointly is typically NP-hard and infea-  sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' To overcome this computational intractability, [7] proposes a scheme where the  1In [6] the multiplication order of the decomposed matrices is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Please note that this  change makes no difference to the general idea of the decomposition and to the following algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  It is equal to the transposed version of the algorithm presented in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  n-th row of the wiring matrix is determined by solving the following sparse recovery  problem for some design parameter S < N controlling the cost between distortion and  computation effort [10]  wn =  argmin  ω∈{ω=�S  s=1 is1js,N: is∈A, js∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=',N} ∀s}  ∥an − ωC∥2  ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  (3)  The new scheme is still NP-hard, but not in N, anymore, but in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Thus, small values  of S are required, in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  In order to have a high accuracy despite small values of S, the factorization  procedure can be applied multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Then the product Ci = W iCi−1 of the  previous wiring step acts as the new codebook for obtaining the following matrix  factor W i of the current wiring step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Hence, by setting2 C0 = IN×K, we obtain the  approximated matrix P after I wiring steps:  A ≈ P =  � I�  i=1  W i  �  C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  (4)  To quantify the accuracy of a given approximation P we use the signal-to-  quantization-noise-ratio (SQNR)  SQNR(A, P ) =  ∥A∥2  F  ∥A − P ∥2  F  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  (5)  Computational Cost  In a binary number representation the multiplication by a signed power of two  corresponds only to a bitshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' On reconfigurable hardware, this shift can be realized  simply by appropriate wiring without the need for dedicated processing elements  such as adders [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The parameter S in (3) determines the number of vectors from  the codebook to be used in forming the linear combination to approximate a row  an of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' It therefore directly controls the computational cost, as in computing the  linear combination, exactly S − 1 additions are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' No multiplications, except  by signed powers of two, are necessary due to the specific structure of the wiring  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Therefore, the separate product of the decomposed matrices with the input  vector y ≈ W (Cx) is much simpler to compute than calculating the product in (1)  straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  The total computational cost Cadd of a decomposition in (4) is given by the number  of additions (or subtractions) required to form the linear combinations  Cadd = IN(S − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  (6)  2In [6] this choice is termed the self-designing codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' It was found to work very well for a  wide range of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Algorithms  In this section we will briefly look at the state of the art for solving the optimization  problem in (3) to obtain the wiring matrices and then introduce two improved novel  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  State-of-the-Art: Discrete Matching Pursuit  The discrete matching pursuit (DMP) follows the matching pursuit approach to  successively determine the wiring coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The algorithm can be summarized in  the following key steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' for details see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Start with iteration s ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Initialize ω ← 01×N  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Update ω in at most a single component, such that ∥an − ωC∥2 is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Increment s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' If s ≤ S, go to step 2, otherwise the procedure terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  It is straightforward to show that the time complexity of the DMP algorithm for  computing a single matrix factor scales with O(N 3S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Exponential Search Algorithm  The row-wise optimization problem in (3) is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' However, for small S, reasonable  matrix sizes and some careful optimization it can be solved in a tractable timeframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We  limit the set of scaling factors to a finite set of signed powers of two (Aexp ⊂ {0, ±2Z}),  as an exhaustive search over the whole set is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' As the search procedure has  to be performed for each row of the target matrix individually, the time complexity  for the computation of each wiring step is given by O(N S|Aexp|S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Generally, which and how many coefficients are included in the subset Aexp is a  design parameter and needs to be adapted to each specific decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' It depends  primarily on two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' First, the current wiring step plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For  each additional wiring layer, the error between each row of the target matrix and  the approximated matrix decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Hence, for any subsequent wiring step, smaller  coefficients are needed to scale the rows of the newly found codebook matrix to  appropriately approximate the residual error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This also means that for high desired  accuracy the coefficient set needs to be chosen large, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' to include also many small  coefficients, to accurately approximate the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Furthermore, relative variations  in the length of the row vectors of the target matrix require a larger coefficient  set to compensate for differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Still, to keep the decomposition computationally  feasible, the number of elements in Aexp needs to be chosen as small as possible, as  the computational complexity scales exponentially in S with the product of the size  of the coefficient set |Aexp| and N as base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  A promising approach for further research is to adapt the coefficient set for each  wiring step dynamically based on the current fidelity of the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The  coefficient set may then be determined from the probability distribution of the likely  entries of the wiring matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Reduced-State Algorithm  The runtime of the exponential algorithm for a matrix with N = 64 rows is on the  order of an hour3, for larger matrices the runtime scales up accordingly and can  become infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Hence, for some applications the computational burden of the  exponential algorithm might consequently not be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' A reasonable compromise  between complexity and performance is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Unlike in DMP, we do not immediately update the components of the wiring  vector for the reduced-state algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Instead, we keep updating in each iteration a  list of the M best vectors, which minimize (3) and select at the termination of the  algorithm the vector with minimum error from the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Specifically, we apply the  following successive, greedy procedure for the optimization problem in (3):  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Start with s ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Initialize the set Ω with M all-zero vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For each ω ∈ Ω find the set Ωm of M mutually distinct vectors ω ˜m with  ˜m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=', M} that minimize ∥an − ω ˜mC∥2 and differ from ω in at most a  single component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Update the set Ω by selecting from �M  m=1 Ωm the M distinct vectors ω with  minimum ∥an − ωC∥2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Increment s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' If s ≤ S, go to step 2, else, continue  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Return argmin  ω∈Ω  ∥an − ωC∥2  By tuning M, we are able to adjust the space of possible combinations that the  algorithm explores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For each wiring step, we have to evaluate O(SN 3M 2) combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Compared to DMP the complexity is increased by a factor of M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Note that for  M = 1 the algorithm reduces to the DMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Numerical Evaluation  In this section, we compare the performance of the proposed algorithms to the baseline  DMP [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We decompose matrices whose entries are drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' from a Gaussian  distribution with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Similar to DMP, the performance of the  improved versions of the algorithm hardly depends on the distribution of the matrix  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The algorithms are invariant to scaling of the variance, however it is crucial  that for the exponential search algorithm the coefficient set Aexp is scaled appropriately  as well, as to not compromise performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Throughout the simulations, we select the  coefficient set for the exponential search algorithm to Aexp = {±2−40, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='., ±23}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  As a first experiment, we compare all three algorithms for fixed matrix sizes in  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We choose matrices with N = 64 rows and vary the number of columns K  3For a multithreaded implementation in Python with Numba acceleration executed on an Intel  i9-12900@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='40 GHz and parameters S = 3, Aexp =  �  ±2−40, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' , ±23�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  0  100  200  300  400  500  600  700  800  Cumulative number of additions  0  20  40  60  80  100  120  SQNR [dB]  64x4  64x6  64x8  Figure 1: Performance comparison for matrices with 64 rows and different aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Matrices of dimension 64×4 are indicated by crosses, 64×6 is indicated by squares and 64×8  is indicated by triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The solid lines refer to the exponential search algorithm (S = 3),  the dashed lines to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced state  algorithm with memory size M = 10 and S = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Results are averaged over 104 matrix entries  for the exponential search algorithm and over 105 matrix entries for the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  from four to eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Hence, we can compare the performance for different aspect ratios  of the matrices4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' To quantify performance, we plot the tradeoff between distortion  and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The latter being measured by the number of cumulative  additions Cadd required for a given wiring step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  As Figure 1 shows, for all three matrix sizes there is a performance gain by both  proposed algorithms against DMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Further, for memory size M = 10, the reduced  state algorithm performs only slightly worse than exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  For the first two wiring steps, the performance of all three algorithms is equal  for a given matrix size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This is due to the fact that for the first two steps we use  DMP with S = 2 for an initial refinement of the codebook5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Applying any of the  4LCC works best for matrices with an exponential aspect ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' for K ≈ log2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For square  matrices it is beneficial to cut these into multiple tall matrices and decompose each slice independently,  see [8] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  5Using any of the novel algorithms with S = 2 is possible as well with very similar performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  a slight increase in SQNR by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='2 dB to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 dB for the novel algorithms and matrix sizes considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  0  100  200  300  400  500  600  700  800  Cumulative number of additions  0  10  20  30  40  50  60  70  80  SQNR [dB]  Increasing M  (2, 5, 10, 50)  DMP  Exponential Search  Reduced State (Variable M)  Figure 2: Performance comparison for different memory sizes M of the reduced state  algorithm of a 64 × 6 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The solid line refers to the exponential search algorithm  (S = 3), the dashed line to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced  state algorithm with 4 different memory sizes and S = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Results are averaged over 104  matrix entries for the exponential search algorithm and over 105 matrix entries for the other  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  algorithms with S ≥ 3 directly to the initial codebook C0 = IN×K would lead to  degraded performance for the first few wiring steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Instead it is beneficial to set S = 2  to allow for more frequent updates of the codebook in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' From empirical  investigations it seems that two wiring iterations with S = 2 are most beneficial for  the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  As the next experiment, we compare the performance of the reduced state algorithm  for different choices of the memory parameter M for given matrix size of 64 × 6 in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  From the figure we observe that even for small M the reduced state  algorithm offers a noticeable performance gain over DMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' As M grows, the reduced  state algorithm approaches the performance of the exponential search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The large  advantage of the reduced state algorithm is that the computation time is reduced  drastically6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For M = 1 the reduced state algorithm reduces to DMP and both lines  6For the considered matrix size the execution time differs from an hour for the exponential  algorithm to a few seconds for the reduced state algorithm in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  0  100  200  300  400  500  600  700  800  Cumulative number of additions  0  20  40  60  80  100  120  SQNR [dB]  Increasing S  (3, 4, 8)  DMP  Exponential search (S = 3)  Reduced State (S = 3, 4, 8)  Figure 3: Performance comparison for different choices of the parameter S of a matrix  with dimension 64 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The solid line refers to the exponential search algorithm (S = 3),  the dashed line to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced state  algorithm with different choices of the parameter S and M = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Results are averaged over  104 matrix entries for the exponential search algorithm and over 105 matrix entries for the  other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  coincide for any given matrix size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Table 1 lists the relative performance gains over DMP for various matrix sizes and  configurations of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Practical Considerations for the Choice of S  Figure 3 shows the performance of the reduced state algorithm for varying S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Due to  the exponentially growing complexity in S, the exponential algorithm is not feasible  for S > 3, except for very small matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We can observe, that by choosing S = 4  for the reduced state algorithm, we approximately achieve the same distortion-cost  tradeoff as for the exponential algorithm with S = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For choosing S even larger the  gains increase likewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  However for a practicable implementation S should not be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' In [8],  the performance of DMP is validated in an implementation on reconfigurable hardware  with S = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This means that on an FPGA exactly N adders are required per wiring  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' With the inputs depending only on the outputs of the previous wiring matrix,  Table 1: Relative average gain in terms of SQNR of the novel algorithms over the baseline  DMP algorithm with S = 2 (For at least 8 bit signed integer accuracy (10 log(SQNR) ≥  47 dB)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Results are averaged over 104 matrix entries for the exponential search algorithm  and over 105 matrix entries for the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Exponential  Reduced State  search  S = 3  S = 4  S = 8  Matrix size  S = 3  M = 5  M = 10  M = 5  M = 10  M = 5  M = 10  16 × 2  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='1 %  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='7 %  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='9 %  16 × 4  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='7 %  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  32 × 4  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='7 %  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='9 %  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='2 %  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='3 %  32 × 6  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='3 %  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='7 %  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  64 × 4  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='9 %  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  64 × 6  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='9 %  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='6 %  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='6 %  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='6 %  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  the decomposition is well suited for parallel execution and pipelining [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Due to the  greedy, step-wise nature of DMP S > 2 does not offer significant performance gains  over S = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Both novel algorithms behave differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  From an implementation point of view, S = 3 is even more suitable than S = 2  for an effective implementation in hardware due to the availability of efficient adders  with three inputs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Interestingly, on modern FPGAs, these adders do not require  more hardware resources, in terms of Lookup-Tables (LUTs), than an adder with two  inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Any powers of two and three (S = 4, 8, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' ) can be realized efficiently as well  by the use of adder trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' However, it is questionable if choosing S > 4 is beneficial,  as performance gains over S = 3 or S = 4 are small and the desired fidelity of the  approximation cannot be chosen in a fine granularity anymore7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Conclusion  In this paper, we have proposed two new algorithms for LCC, a framework for  the lossy compression of multidimensional linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' While the exponential  search algorithm shows the best performance, it is generally infeasible especially for  large matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The proposed reduced-state algorithm, performs close to exponential  search at a fraction of the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The time complexity of the  decomposition compared to the baseline algorithm from earlier works is only mildly  increased, while the performance gains over the baseline DMP algorithm are on the  order of at least 10 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  References  [1] Jason Thong and Nicola Nicolici, “An optimal and practical approach to single constant  multiplication,” IEEE Transactions on Computer-Aided Design of Integrated Circuits  and Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 1373–1386, Sep 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  [2] Yevgen Voronenko and Markus P¨uschel, “Multiplierless multiple constant multiplication,”  ACM Transactions on Algorithms, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 2, May 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  7For a matrix of dimension 64 × 4 the reduced state algorithm with S = 8 improves the SQNR  approximately by 70 dB per matrix factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Boullis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Tisserand, “Some optimizations of hardware multiplication by constant  matrices,” in Proceedings 2003 16th IEEE Symposium on Computer Arithmetic, 2003,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 20–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  [4] Levent Aksoy, Paulo Flores, and Jose Monteiro, “A novel method for the approximation  of multiplierless constant matrix vector multiplication,” EURASIP Journal on Embedded  Systems, , no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 12, May 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  [5] Ralf R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' M¨uller, Bernhard G¨ade, and Ali Bereyhi, “Efficient matrix multiplication:  The sparse power-of-2 factorization,” in 2020 Information Theory and Applications  Workshop (ITA), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  [6] Ralf R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' M¨uller, Bernhard G¨ade, and Ali Bereyhi, “Linear computation coding,” 2021,  arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='00398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  [7] Ralf R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' M¨uller, Bernhard M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' G¨ade, and Ali Bereyhi, “Linear computation coding:  A framework for joint quantization and computing,” Algorithms, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 7, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  [8] Alexander Lehnert, Philipp Holzinger, Simon Pfenning, Ralf R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' M¨uller, and Marc  Reichenbach, “Most ressource efficient matrix vector multiplication on FPGA,” IEEE  Access, 2022, Early Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Booth, “A signed binary mutliplication technique,” The Quarterly Journal of  Mechanics and Applied Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' 236 – 240, Jan 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  [10] Simon Foucart and Holger Rauhut,  A Mathematical Introduction to Compressive  Sensing, Springer New York, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  [11] James M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Simkins and Brian D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Philofsky, “Structures and methods for implementing  ternary adders/subtractors in programmable logic devices,” Sep 2007,  US Patent  7,274,211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}

_9E5T4oBgHgl3EQfSQ7q/content/tmp_files/2301.05528v1.pdf.txt

@@ -0,0 +1,725 @@
+Harold Costales  et al.,  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 
+7076 
+ 
+ 
+ 
+ABSTRACT 
+ 
+Rice is the number one staple food in the country, as this 
+serves as the primary livelihood for thousands of Filipino 
+households. However, as the tradition continues, farmers are 
+not familiar with the different types of rice leaf diseases that 
+might compromise the entire rice crop. The need to address 
+the common bacterial leaf blight in rice is a serious disease 
+that can lead to reduced yields and even crop loss of up to 
+75%. This paper is a design and development of a rice leaf 
+disease detection mobile application prototype using an 
+algorithm used for image analysis. The researchers also used 
+the Rice Disease Image Dataset by Huy Minh Do available at 
+https://www.kaggle.com/ 
+to 
+train 
+state-of-the-art 
+convolutional neural networks using transfer learning. 
+Moreover, we used image augmentation to increase the 
+number of image samples and the accuracy of the neural 
+networks as well. 
+ 
+Key words : deep neural networks, convolutional neural 
+networks, agriculture, transfer learning, rice diseases 
+ 
+1 INTRODUCTION 
+ 
+Across the 20th century, our prospects of engineering have 
+rapidly changed. It was once thought of as impossible for 
+everyone, as all we had seen on earth showed us a manual 
+process that is uneasy for us; we would recognize as 
+technology. However, in the 21st century, our machines 
+continue to improve in what we conceive "Artificial 
+Intelligence," which makes it more even feasible for devices 
+to learn from experience, adjust to a high new level input and 
+perform human-like tasks. Methods such as the use of 
+convolutional neural networks may hold the key to 
+developing software applications, particularly in the field of 
+agriculture. 
+ 
+Portability, efficiency, and affordability of agricultural 
+technology and information continue to be a major 
+interference for improving agricultural productivity among 
+small enterprises in the country. The Department of 
+Agriculture, in partnership with the Department of 
+Information and Communications Technology (DICT), has 
+furnished a possible solution to enhance this kind of situation. 
+ 
+ 
+Recently, they have launched the HACKATON to address 
+further innovations that integrate software applications and 
+systems into the development of agricultural technology, 
+especially to farmers, which have been conducted nationwide. 
+They have currently established outstanding software 
+applications and systems to aid the farmers for better 
+understanding when it comes to e-agriculture. 
+ 
+This paper emphasizes the role of ICT and the functional 
+benefaction of software development to agriculture in the 
+Philippines. Data from Rice Disease Image Dataset by Huy 
+Minh Do available at https://www.kaggle.com are used to 
+train data using a convolutional neural network algorithm 
+using transfer learning. Moreover, we used image 
+augmentation to increase the number of image samples and 
+the neural network accuracy as well. It turns out that the 
+modified neural networks achieved a state-of-the-art result 
+with an accuracy closed to human-level performance. The 
+researchers have developed a prototype application for Rice 
+Leaf Disease Detection using Convolutional Neural Networks 
+(CNNs) for farmers in the local community. This application 
+will primarily install and use the said application on their 
+smartphone, simply took a picture of the infected area of the 
+rice leaf. Then the app gives a percentage of accuracy of rice 
+disease infected in rice leaf. 
+ 
+Farmers were also able to grasp the knowledge of the different 
+rice leaf diseases because of the information that the 
+application is providing. This paper recommends the adoption 
+of such software applications by institutions such as the 
+Department of Agriculture to improve provision for 
+appropriate decision making by agricultural farmers in the 
+country.  
+ 
+Related studies on rice leaf disease detection using neural 
+networks have been on-trend. According to [1] , "the use of 
+computational intelligence-based techniques has proven 
+successful in recent times for automated rice-disease detection 
+(p.21)". Another related study stated that [3] "an automated 
+system could have a feature on detection of diseases present in 
+a rice leaf using color image analysis". Furthermore, [4] cited 
+that "the management of perennial fruit crops requires close 
+monitoring especially for the management of diseases that can 
+affect production significantly and subsequently the 
+post-harvest life (p. 1)". Corollary to the contexts presented, 
+the researchers came up with a rice disease detection that 
+ 
+Development of a Prototype Application for Rice Disease 
+Detection Using Convolutional Neural Networks 
+Harold Costales1, Arpee Callejo-Arruejo2, Noel Rafanan3 
+1 University of Northern Philippines, Philippines, hlcostales@unp.edu.ph 
+2 University of Northern Philippines, Philippines, arpee.callejo@unp.edu.ph 
+3 University of Northern Philippines, Philippines, noel.rafanan@unp.edu.ph 
+        ISSN  2347 - 3983 
+Volume 8. No. 10, October 2020 
+International Journal of Emerging Trends in Engineering Research 
+Available Online at http://www.warse.org/IJETER/static/pdf/file/ijeter708102020.pdf 
+https://doi.org/10.30534/ijeter/2020/708102020 
+ 
+ 
+ 
+
+WARSEHarold Costales  et al.,  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 
+7077 
+ 
+ 
+integrates the use of an algorithm, specifically CNNs and an 
+image analysis, for immediate monitoring of the rice crops. 
+ 
+Thus, this mobile application is needed to bring virtual IT 
+experts into the field to determine, examine, and to give 
+accurate results (on rice leaf diseases) that would inform the 
+farmers what to do next without any further expenses. 
+1.1 Objectives of the Study 
+This study aimed to design and develop a mobile application 
+for rice disease detection. Specifically, it sought to answer the 
+following objectives: 
+1. Propose an application to address the problem, issues 
+and challenges encountered; 
+2. Identify an appropriate algorithm for the application; and 
+3. Features for the Mobile application. 
+ 
+1.2 Conceptual Framework 
+ 
+ 
+ 
+Figure 1: Paradigm of the Study 
+ 
+Figure 1 shows researcher used the input-process-output (IPO) 
+model as a guide in conducting the study. The Application is 
+proposed for the problems and issues encountered, 
+appropriate algorithm to be incorporated, and features for the 
+mobile application are the input of the study (input), which 
+served as the guide in designing the application. The software 
+development methodology, which is the Feature-Driven 
+Development Methodology, is the process for the 
+"Development of a Prototype Application for Rice Disease 
+Detection Using Convolutional Neural Networks"(output). 
+ 
+2 METHODS 
+ 
+This part presents the algorithm and the software development 
+methodology used in the development of the mobile 
+application.  
+ 
+For the algorithm, the researchers chose Convolutional Neural 
+Networks (CNNs) for the mobile application. As stated by 
+[12], technological advancements in Computer Vision and 
+Deep Learning a subset of the Artificial Intelligence gains 
+more importance in the last decade especially in the field of 
+object detection using Convolutional Neural Networks (CNN) 
+became popular in many fields to address the current societal 
+issues. 
+2.1 Convolutional Neural Networks Deep Learning 
+Algorithm for the Prototype Mobile App 
+ 
+A Convolutional Neural Network (CNN) is a deep learning 
+algorithm in which images serve as input to a learnable 
+process to analyze various aspects of the image and be able to 
+differentiate one from the other. A rice leaf is an input, and 
+images of diseases of rice leaves are stored in the database to 
+match the input. As cited by [11], many researchers use deep 
+learning method to classify an image automatically. The 
+purpose of classification is to arrange objects that will be 
+observed into categories that have been defined. Furthermore, 
+[14] cited that the detection technique assisted by simple 
+image processing in the evidence is very interesting to be 
+further researched. The researcher also utilized the method of 
+[16] for data processing for the realization of the study.  
+ 
+Convolutional Neural Networks (CNNs) is the most 
+appropriate model of Rice Disease Image Dataset since it uses 
+image analysis. It was proven by [6] that convolutional neural 
+networks (CNNs) have used in the field of computer vision for 
+decades. Moreover, [5] proposes a convolutional autoencoder 
+deep learning framework to support unsupervised image 
+features learning for lung nodule through unlabeled data, 
+which only needs a small amount of labeled data for efficient 
+feature learning. Below are the steps on how the researchers 
+came up with the integration of the CNNs for the developed 
+application following the process of the CNNs in the study of 
+[6]. 
+ 
+ 
+ 
+Figure 2. A Typical Architecture of CNNs by Tajbakhsh 
+(2016) 
+ 
+ 
+Figure 2 presents a typical architecture of a CNN. It consists 
+of an input layer (images), convolutional layer, pooling layer, 
+a fully connected layer, a classifier and an output layer.  
+ 
+The convolutional layer is the most important layer in the 
+CNN hence the name Convolutional Neural Networks. It acts 
+as an automatic feature that extracts meaningful features of 
+the object in the image. In mathematics, convolution is the 
+operation of two functions to produce a third modified 
+function. 
+ 
+               
+                      (1) 
+
+WNPOI
+PROCESS
+OUIPUTAplicationcanbeproposedtoaddress
+Development
+icationttheproblem,issues
+Methodology
+eafDiseasepuoue
+·DevelopanOverallencounterea
+Model
+CIgorithmforthe
+·BuildaFeatureListdevelopedapplication;
+PlanbyFeatureand
+·DesignbyFeatureeFeaturesofthe
+·BuildbyFeatureproposedsystembird
+Prird
+sunset
+Psunset
+dog
+cat
+Pol
+o
+convolution+
+maxpooling
+vec
+nonlinearity
+convolution+pooling layers
+fully connected layers
+Nxbinaryclassification(f*g)(t) ≤
+f(T)g(t - T) dT
+00Harold Costales  et al.,  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 
+7078 
+ 
+ 
+ 
+In the context of CNNs, convolutional operation is simply a 
+matrix operation, specifically matrix multiplication and 
+addition. The byproduct of this operation called the feature 
+map. 
+ 
+The pooling layer is used to down sample the feature map to 
+reduce computational complexity. It is common in the 
+literature to add the pooling layer after the convolution layer. 
+There are two main types of pooling the max pooling and the 
+average pooling. In the literature max pooling is preferred. It 
+calculates the maximum value for each patch of the feature 
+map. 
+ 
+ 
+                  (2) 
+ 
+The fully connected layer, abbreviated FC is responsible for 
+vectorizing the features from the convolution and pooling 
+layers. This transforms the multi-dimensional feature map 
+into a row vector to be fed into a classifier.  
+ 
+The classifier is responsible in outputing the probibilities of 
+the output layer. There are two commonly used activation 
+functions used in classifiers, the sigmoid activation function 
+and the softmax activation function. Since the problem of 
+detecting disease on rice leaf images is a multi-class 
+classification problem, the appropriate function to be used is 
+the softmax activation function.  
+ 
+ 
+                             
+                                     (3) 
+ 
+ 
+Another is the use of a software development methodology 
+which helped the researchers identify the problems, created an 
+overview of the concept for the mobile application and built 
+the application. 
+ 
+2.2 Software Development Methodology 
+ 
+The researcher used the Feature-driven Development (FDD) 
+under the family of Agile methodology for the software 
+development because it is the most suitable and it has a 
+customer-centric process. Its iterative feature allowed the 
+researchers to develop the application while it is being tested. 
+In the study of [7] stated that, “the iterative feature of the 
+methodology allowed the proponents to capitalize on the 
+learning that was accumulated during the development of 
+earlier parts or versions of the solution”. The figure below 
+presents the phases of FDD methodology. 
+ 
+Figure 3: Feature-Driven Development Methodology 
+ 
+Figure 
+3 
+shows 
+the 
+Feature-Driven 
+Development 
+Methodology which has five phases, which include the (a) 
+Development of an Overall Model, (b) Build a Feature List, 
+(c) Plan by Feature, (d) Design by Feature, and (e) Build by 
+Feature. The phases of development are presented as follows. 
+ 
+2.2.1 Develop Overall Model 
+ 
+A stage where the researchers created a fundamental 
+foundation of the application and the variables it required. 
+This stage also identified the initial development of the 
+application. The researchers designed the application's overall 
+model and served it as a guideline in developing the 
+application. As the primary objective, the researchers develop 
+the mobile app for the farmers for the early detection of rice 
+diseases of rice-crops. 
+ 
+2.2.2 Build Feature List 
+ 
+The researchers created concrete plans for each feature of the 
+mobile application. The researchers created a prototype of the 
+user interfaces for the conceptualization of the mobile 
+application. 
+ 
+2.2.3 Plan by Feature 
+ 
+The researchers created concrete plans for each feature of the 
+mobile application. The researchers created a prototype of the 
+user interfaces for the conceptualization of the mobile 
+application. 
+ 
+2.2.4 Design by Feature 
+ 
+The researchers designed the features individually using the 
+needed preferences. To fulfill this stage, the researchers 
+designed the identified features with the help of the needed 
+tools and processes. 
+ 
+2.2.5 Build by Feature 
+The last stage of the methodology wherein the researchers 
+created the planned and designed features for the application. 
+With the previous step as a guide, the researchers turned the 
+details of the features into a working application, with the help 
+of the tools and processes. After an inspection via a test run by 
+the researchers, the researchers created the planned and 
+designed features for the application. 
+ 
+ 
+
+Mf(p) =max(f(q)-dmax(p-q))
+52berj
+1xa7Modeling
+Initial
+Model
+Storming
+DevelopanOverall
+Bulld a Features
+Planby
+Designby
+Buildby
+Model
+List
+Feature
+Feature
+Feature
+(moreshape
+Alistoffeatures
+A developmentplan
+Adesign package
+Completed
+thancontant)
+grouped into sets
+Class owners
+client-valued
+and subjectareas
+Features setowners
+function
+Anobjectmodel
+(addmorecontent
++notes
+to the object model)Harold Costales  et al.,  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 
+7079 
+ 
+ 
+3 RESULTS 
+3.1 Prototype Mobile Application for Rice Disease 
+Detection 
+ 
+ 
+Figure 4: The Prototype of the Mobile Application 
+ 
+Figure 4 shows the rice disease detection of the mobile 
+application. The researchers designed and develop a mobile 
+application to assist the farmers in disease detection. It aims to 
+help the farmers who are the backbone of the country to 
+increase rice crop productivity. The lack of knowledge of 
+farmers in the detection of rice diseases made the researchers 
+conceptualize the integration of the field of agriculture and 
+medicine in detecting diseases of rice crops. Also, Duterte's 
+Administration supports Research and Development along 
+with agriculture. It is stated in the Philippine Development 
+Plan (PDP) 2017-2022 that there is a need to conduct 
+researches in the field because of rice-crop yield losses. Thus, 
+the PDP 2017-2022 established a plan to support the 
+initiatives of the agriculture sector along with R&D. 
+ 
+Moreover, the application is different from other similar apps 
+available in the market because it was developed specifically 
+for the Philippine agriculture setting. The application will be 
+stored with rice disease data exclusively for agriculture in the 
+northern Philippines and further validated by pathologists as 
+the researchers continue to gather data that will be for the 
+database of the system. 
+3.2 Convolutional Neural Networks Deep Learning 
+Algorithm for the Prototype Application 
+ 
+The researchers utilized an Image Processing typical machine 
+learning workflow in the development of the model which 
+consists of the following Data Acquisition, Data Preparation, 
+and Training and Validation.  As cited by [10] Image 
+processing is the method of using different manipulation 
+techniques and algorithms so that a desired features can be 
+extracted such as morphology, color and texture from an 
+image. Another study stated that [13] CNN proved to be best 
+among the results in terms of accuracy when compared to 
+other classifiers 
+ 
+3.2.1. Image Acquisition 
+ 
+The 
+dataset 
+by 
+Huy 
+Minh 
+Do 
+available 
+at 
+https://www.kaggle.com/ containing 1,260 labeled rice leaf 
+images was used for creating the model. The images were 
+labeled as Leaf Blast, Brown Spot, and Hispa, which are the 
+(3) three rice leaf diseases that the model will try to classify. 
+ 
+3.2.2. Image Preprocessing 
+ 
+The images were downscaled to 500 x 500 pixels to reduce 
+computational complexity. Image dataset were split into train 
+dataset (80% of the entire samples) and validation dataset 
+(20% of the entire samples). Image augmentation such as 
+flipping, shearing, and rescaling in order to increase dataset 
+samples as neural networks are data crunchers. 
+ 
+3.2.3. Training and Validation 
+ 
+To fast track the training process the researchers employed 
+transfer learning. Transfer learning, in general, is the process 
+of taking a previously trained model used in a problem and 
+apply in to another related problem. It is referring to the 
+knowledge transfer from pretrained network in one domain to 
+your own problem in a different domain. .[8] 
+ 
+The researcher used MobileNet 2 [9] as the base model. It is a 
+state-of-the-art CNN model trained from ImageNet and one of 
+the winners of the ImageNet Large Scale Visual Recognition 
+Challenge (ILSVRC), a prestigious computer vision 
+competition. Since the target dataset is small and somewhat 
+different from the dataset where MobileNet 2 was trained, the 
+top layer of the network needs to be frozen. 
+ 
+First Iteration 
+Base model mobile net Version 2.0 
+Hyper-parameters: 
+Top layer = false 
+Initialweights 
+ 
+ 
+
+.a58%12:57.PM
+agDetect
+blast
+50.13%
+brownspol
+bacterial_leaf_blight
+0.14%Harold Costales  et al.,  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 
+7080 
+ 
+ 
+Second Iteration 
+Base model mobile net Version 2.0 
+Hyper-parameters: 
+Top layer = false 
+Initialweights = imagenet 
+Loss = categorical cross entropy 
+Metrics = accuracy 
+Optimization = ADAM  
+EPOCHS= 10 
+TRAINABLE = FALSE 
+Result= 97% trained data set, 94% validation 
+Suffered from overfitting 
+ 
+Third Iteration 
+Base model mobile net Version 2.0,  
+image augmentation 
+Hyper-parameters: 
+Top layer = false 
+Initialweights = imagenet 
+Loss = categorical cross entropy 
+Metrics = accuracy 
+Optimization = ADAM  
+EPOCHS= 20 
+TRAINABLE = true 
+Result= 98.9% trained data set, 98% validation 
+Overfitting was solved 
+ 
+The third iteration proved that the process for the image 
+analysis on the training and testing for the data sets was 
+validated with an accuracy rate of 98% from 94% from the 
+second iteration. This proved that the CNNs algorithm of the 
+mobile application was proven for use. 
+3.3 Features of the Prototype Mobile Application 
+ 
+The features of the application include (1) real-time detection, 
+(2) artificial intelligence, and (3) image analysis. 
+ 
+3.3.1 Real-time Detection 
+ 
+One of the features of the mobile application is a real-time 
+detection. The application helps in the identification of the 
+rice leaf diseases using convolutional neural networks that 
+match the input data and the data stored in the database of the 
+application.  
+ 
+The embed camera of the mobile application can detect the 
+diseases in real-time. This feature helps farmers to detect the 
+diseases even without the instruction of plant pathologists. 
+ 
+3.3.2 Artificial Intelligence 
+ 
+Experts in the field of Agriculture are marginal, especially in 
+rice leaf disease detection. This feature of the mobile 
+application replicates the knowledge about rice leaf disease 
+detection in the field, which is one of the features of the 
+mobile application. As data grows on the database of the 
+mobile application, the more accurate results it can give.   
+ 
+3.3.3 Image Analysis 
+ 
+Another feature is Image Analysis. This feature uses the 
+CNNs algorithm aspects and features of images were stored, 
+enabling the mobile application to analyze the rice leaf for 
+disease detection through the following steps of the algorithm. 
+4 CONCLUSION 
+In summary, the researchers have developed a mobile 
+application, which could help the agricultural sector. 
+Specifically, the researchers conclude that the mobile 
+application will help in the realization of the high-yield of rice 
+crops against rice diseases for preventive measures. Thus, this 
+study is helpful to the country, particularly the agricultural 
+sector. 
+5 RECOMMENDATION 
+ 
+The 
+researchers 
+came 
+up 
+with 
+the 
+following 
+recommendations: 
+(1) The output of the Mobile Application will have its 
+validation of experts in rice disease for accuracy of the 
+information from the mobile application, (2) the mobile 
+application will have a series of tests for the implementation 
+for future use. Lastly, (3) the researchers highly recommend 
+that the application shall be introduced to the Department of 
+Agriculture for dissemination. 
+ 
+ACKNOWLEDGEMENT 
+ 
+The researchers would like to thank the following people: the 
+active UNP Research Office- Science and Technology 
+Coordinator, Prof. Redentor S. Rojas, the ever-supportive 
+UNP Research Director, Dr. Edelyn Cadorna and the dynamic 
+UNP President, Dr. Erwin F. Cadorna for pushing us to 
+enhance our research knowledge and capabilities. Also, the 
+researchers would like to thank the IMPACT HACKATON 
+2050 for organizing the IMPACT HACKATON 2019, where 
+they conceptualized this research study. Above all, thanks to 
+Almighty God for all the blessings for our families and the 
+society. 
+REFERENCES 
+1. E. G. Emberda, D. L. Dumas, and T. M. Rentillo, 
+Forecasting Coconut Yield: A Comparative Study 
+between the Use of Traditional Forecasting and Feed 
+Forward 
+Back 
+Propagation 
+Artificial 
+Neural 
+Network, UIC Research Journal, 18(2), 2012. 
+2. K. B. Francisco, J. D. Concepcion, A. R. Mojica, and , S. 
+B Montes. Philippines First: An Edutainment 3d 
+Game 
+For 
+Android 
+Mobile 
+Platform 
+Using 
+Separating Axis Theorem (Sat) Algorithm, Innovatus, 
+2(1). 2019. 
+3. Pugoy, Reinald Adrian D. L. and Mariano Vladimir Y. 
+Automated rice leaf disease detection using color image 
+
+Harold Costales  et al.,  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 
+7081 
+ 
+ 
+analysis, 
+Proc. 
+SPIE 
+8009, 
+Third 
+International 
+Conference on Digital Image Processing. 2011. 
+4. P. Revathi and M. Hemalatha. Advance computing 
+enrichment evaluation of cotton leaf spot disease 
+detection using Image Edge detection, 2012 Third 
+International Conference on Computing, Communication 
+and Networking Technologies. 2012. 
+5. Talingdan, J. A., & Llanda, C. R., Assessment of the 
+effectiveness of learning theories using gamified 
+Android app in teaching C programming. Int. Proc. 
+IOP Conf. Ser. Mater. Sci. Eng, 482-1,1-6, 2019. 
+6. S. Chen, B. Mulgrew, and P. M. Grant. A clustering 
+technique 
+for 
+digital 
+communications 
+channel 
+equalization using radial basis function networks,, 
+IEEE Trans. on Neural Networks, 4, 570-578, 1993. 
+7. N. Tajbakhsh. Convolutional Neural Networks for 
+Medical Image Analysis: Full Training or Fine 
+Tuning?, IEEE Transactions on Medical Imaging, 35-5, 
+1299-1312, 2016. 
+8. A. Aitken and V. Ilango A Comparative Analysis of 
+Traditional Software Engineering and Agile Software 
+Development, 46th Hawaii International Conference on 
+System Sciences, Wailea, Maui, HI, 4751-4760, 2013. 
+9. M. Elgendy, Deep Learning for Vision Systems. 
+Manning Publications, 2019. 
+10. Olga Russakovsky, et al. ImageNet Large Scale Visual 
+Recognition 
+Challenge, 
+International 
+Journal 
+of 
+Comput. Vis. 115. 1-42, 2015. 
+11. W. Eustaquio and J. Dioses Jr.,  Artificial Neural 
+Network For Classification Of Immature And 
+Mature 
+Coffee 
+Beans 
+Using 
+RGB 
+Values, 
+International Journal of Emerging Technologies in 
+Engineering 
+Research, 
+9-4, 
+2020. 
+https://doi.org/10.30534/ijeter/2020/41882020 
+12. Devi Devani, Alethea Suryadibrata, Julio Christian, 
+Young, Implementation of Siamese Convolutional 
+Neural Network from Cell Images for Malaria 
+Disease 
+Identification. 
+International 
+Journal 
+of 
+Advanced Trends in Computer Science and Engineering, 
+9-4, 2020. https://doi.org/10.30534/ijatcse/2020/0494 
+13. Handayani, Indri, Supriyanti, Dedeh, Maulani, Giandari, 
+Liutfani, Ninda, Real Time Multi-Scale Facial Mask 
+Detection and Classification Using Deep Transfer 
+Learning 
+Techniques. 
+International 
+Journal 
+of 
+Advanced Trends in Computer Science and Engineering, 
+9-4, 
+2020. 
+https://doi.org/10.30534/ijatcse/2020/33942020 
+14. Simhadri Chinna Gopi et al., Fuzzy Based Classification 
+of X-Ray Images with Convolution Neural Network, 
+International Journal of Emerging Trends in Engineering 
+Research, 
+8(8), 
+4433 
+- 
+4436 
+, 
+2020. 
+https://doi.org/10.30534/ijeter/2020/63882020 
+15. Fitria Hidayanti et al., Upper Stomach Disorder 
+Detection System using Backpropagation Artificial 
+Neural Network, International Journal of Emerging 
+Trends in Engineering Research, 8(8), 4426 - 4432 , 
+2020. https://doi.org/10.30534/ijeter/2020/62882020 
+16. Lamarca, Bryan Irvin J, et al, The Development of a 
+Performance Appraisal System Using Decision Tree 
+Analysis and Fuzzy Logic, International Journal of 
+Intelligent Engineering and Systems, 11-4, 2018. DOI: 
+10.22266/ijies2018.0831.02 
+

_9E5T4oBgHgl3EQfSQ7q/content/tmp_files/load_file.txt

@@ -0,0 +1,315 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf,len=314
+page_content='Harold Costales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 7076  \uf020  ABSTRACT  Rice is the number one staple food in the country, as this serves as the primary livelihood for thousands of Filipino households.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' However, as the tradition continues, farmers are not familiar with the different types of rice leaf diseases that might compromise the entire rice crop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The need to address the common bacterial leaf blight in rice is a serious disease that can lead to reduced yields and even crop loss of up to 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This paper is a design and development of a rice leaf disease detection mobile application prototype using an algorithm used for image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researchers also used the Rice Disease Image Dataset by Huy Minh Do available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='com/ to train state-of-the-art convolutional neural networks using transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Moreover, we used image augmentation to increase the number of image samples and the accuracy of the neural networks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Key words : deep neural networks, convolutional neural networks, agriculture, transfer learning, rice diseases  1 INTRODUCTION  Across the 20th century, our prospects of engineering have rapidly changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It was once thought of as impossible for everyone, as all we had seen on earth showed us a manual process that is uneasy for us;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' we would recognize as technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' However, in the 21st century, our machines continue to improve in what we conceive "Artificial Intelligence," which makes it more even feasible for devices to learn from experience, adjust to a high new level input and perform human-like tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Methods such as the use of convolutional neural networks may hold the key to developing software applications, particularly in the field of agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Portability, efficiency, and affordability of agricultural technology and information continue to be a major interference for improving agricultural productivity among small enterprises in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The Department of Agriculture, in partnership with the Department of Information and Communications Technology (DICT), has furnished a possible solution to enhance this kind of situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Recently, they have launched the HACKATON to address further innovations that integrate software applications and systems into the development of agricultural technology, especially to farmers, which have been conducted nationwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' They have currently established outstanding software applications and systems to aid the farmers for better understanding when it comes to e-agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  This paper emphasizes the role of ICT and the functional benefaction of software development to agriculture in the Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Data from Rice Disease Image Dataset by Huy Minh Do available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='com are used to train data using a convolutional neural network algorithm using transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Moreover, we used image augmentation to increase the number of image samples and the neural network accuracy as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It turns out that the modified neural networks achieved a state-of-the-art result with an accuracy closed to human-level performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researchers have developed a prototype application for Rice Leaf Disease Detection using Convolutional Neural Networks (CNNs) for farmers in the local community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This application will primarily install and use the said application on their smartphone, simply took a picture of the infected area of the rice leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Then the app gives a percentage of accuracy of rice disease infected in rice leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Farmers were also able to grasp the knowledge of the different rice leaf diseases because of the information that the application is providing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This paper recommends the adoption of such software applications by institutions such as the Department of Agriculture to improve provision for appropriate decision making by agricultural farmers in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Related studies on rice leaf disease detection using neural networks have been on-trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' According to [1] , "the use of computational intelligence-based techniques has proven successful in recent times for automated rice-disease detection (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='21)".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Another related study stated that [3] "an automated system could have a feature on detection of diseases present in a rice leaf using color image analysis".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Furthermore, [4] cited that "the management of perennial fruit crops requires close monitoring especially for the management of diseases that can affect production significantly and subsequently the post-harvest life (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 1)".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Corollary to the contexts presented, the researchers came up with a rice disease detection that  Development of a Prototype Application for Rice Disease Detection Using Convolutional Neural Networks Harold Costales1, Arpee Callejo-Arruejo2, Noel Rafanan3 1 University of Northern Philippines, Philippines, hlcostales@unp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='ph 2 University of Northern Philippines, Philippines, arpee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='callejo@unp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='ph 3 University of Northern Philippines, Philippines, noel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='rafanan@unp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='ph ISSN  2347 - 3983 Volume 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 10, October 2020 International Journal of Emerging Trends in Engineering Research Available Online at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='warse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='org/IJETER/static/pdf/file/ijeter708102020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='pdf https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='30534/ijeter/2020/708102020  WARSEHarold Costales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 7077  integrates the use of an algorithm, specifically CNNs and an image analysis, for immediate monitoring of the rice crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Thus, this mobile application is needed to bring virtual IT experts into the field to determine, examine, and to give accurate results (on rice leaf diseases) that would inform the farmers what to do next without any further expenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1 Objectives of the Study This study aimed to design and develop a mobile application for rice disease detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Specifically, it sought to answer the following objectives: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Propose an application to address the problem, issues and challenges encountered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Identify an appropriate algorithm for the application;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Features for the Mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2 Conceptual Framework  Figure 1: Paradigm of the Study  Figure 1 shows researcher used the input-process-output (IPO) model as a guide in conducting the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The Application is proposed for the problems and issues encountered, appropriate algorithm to be incorporated, and features for the mobile application are the input of the study (input), which served as the guide in designing the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The software development methodology, which is the Feature-Driven Development Methodology, is the process for the "Development of a Prototype Application for Rice Disease Detection Using Convolutional Neural Networks"(output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2 METHODS  This part presents the algorithm and the software development methodology used in the development of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  For the algorithm, the researchers chose Convolutional Neural Networks (CNNs) for the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' As stated by [12], technological advancements in Computer Vision and Deep Learning a subset of the Artificial Intelligence gains more importance in the last decade especially in the field of object detection using Convolutional Neural Networks (CNN) became popular in many fields to address the current societal issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1 Convolutional Neural Networks Deep Learning Algorithm for the Prototype Mobile App  A Convolutional Neural Network (CNN) is a deep learning algorithm in which images serve as input to a learnable process to analyze various aspects of the image and be able to differentiate one from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' A rice leaf is an input, and images of diseases of rice leaves are stored in the database to match the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' As cited by [11], many researchers use deep learning method to classify an image automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The purpose of classification is to arrange objects that will be observed into categories that have been defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Furthermore, [14] cited that the detection technique assisted by simple image processing in the evidence is very interesting to be further researched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researcher also utilized the method of [16] for data processing for the realization of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Convolutional Neural Networks (CNNs) is the most appropriate model of Rice Disease Image Dataset since it uses image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It was proven by [6] that convolutional neural networks (CNNs) have used in the field of computer vision for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Moreover, [5] proposes a convolutional autoencoder deep learning framework to support unsupervised image features learning for lung nodule through unlabeled data, which only needs a small amount of labeled data for efficient feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Below are the steps on how the researchers came up with the integration of the CNNs for the developed application following the process of the CNNs in the study of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' A Typical Architecture of CNNs by Tajbakhsh (2016)  Figure 2 presents a typical architecture of a CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It consists of an input layer (images), convolutional layer, pooling layer, a fully connected layer, a classifier and an output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  The convolutional layer is the most important layer in the CNN hence the name Convolutional Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It acts as an automatic feature that extracts meaningful features of the object in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' In mathematics, convolution is the operation of two functions to produce a third modified function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  (1)  WNPOI PROCESS OUIPUTAplicationcanbeproposedtoaddress Development icationttheproblem,issues Methodology eafDiseasepuoue ·DevelopanOverallencounterea Model CIgorithmforthe ·BuildaFeatureListdevelopedapplication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' PlanbyFeatureand ·DesignbyFeatureeFeaturesofthe ·BuildbyFeatureproposedsystembird Prird sunset Psunset dog cat Pol o convolution+ maxpooling vec nonlinearity convolution+pooling layers fully connected layers Nxbinaryclassification(f*g)(t) ≤ f(T)g(t - T) dT 00Harold Costales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 7078  In the context of CNNs, convolutional operation is simply a matrix operation, specifically matrix multiplication and addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The byproduct of this operation called the feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  The pooling layer is used to down sample the feature map to reduce computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It is common in the literature to add the pooling layer after the convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' There are two main types of pooling the max pooling and the average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' In the literature max pooling is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It calculates the maximum value for each patch of the feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  (2)  The fully connected layer, abbreviated FC is responsible for vectorizing the features from the convolution and pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This transforms the multi-dimensional feature map into a row vector to be fed into a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  The classifier is responsible in outputing the probibilities of the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' There are two commonly used activation functions used in classifiers, the sigmoid activation function and the softmax activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Since the problem of detecting disease on rice leaf images is a multi-class classification problem, the appropriate function to be used is the softmax activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  (3)  Another is the use of a software development methodology which helped the researchers identify the problems, created an overview of the concept for the mobile application and built the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2 Software Development Methodology  The researcher used the Feature-driven Development (FDD) under the family of Agile methodology for the software development because it is the most suitable and it has a customer-centric process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Its iterative feature allowed the researchers to develop the application while it is being tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' In the study of [7] stated that, “the iterative feature of the methodology allowed the proponents to capitalize on the learning that was accumulated during the development of earlier parts or versions of the solution”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The figure below presents the phases of FDD methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Figure 3: Feature Driven Development Methodology  Figure 3 shows the Feature-Driven Development Methodology which has five phases, which include the (a) Development of an Overall Model, (b) Build a Feature List, (c) Plan by Feature, (d) Design by Feature, and (e) Build by Feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The phases of development are presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1 Develop Overall Model  A stage where the researchers created a fundamental foundation of the application and the variables it required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This stage also identified the initial development of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=" The researchers designed the application's overall model and served it as a guideline in developing the application." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' As the primary objective, the researchers develop the mobile app for the farmers for the early detection of rice diseases of rice-crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2 Build Feature List  The researchers created concrete plans for each feature of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researchers created a prototype of the user interfaces for the conceptualization of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3 Plan by Feature  The researchers created concrete plans for each feature of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researchers created a prototype of the user interfaces for the conceptualization of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='4 Design by Feature  The researchers designed the features individually using the needed preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' To fulfill this stage, the researchers designed the identified features with the help of the needed tools and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='5 Build by Feature The last stage of the methodology wherein the researchers created the planned and designed features for the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' With the previous step as a guide, the researchers turned the details of the features into a working application, with the help of the tools and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' After an inspection via a test run by the researchers, the researchers created the planned and designed features for the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Mf(p) =max(f(q)-dmax(p-q)) 52berj 1xa7Modeling Initial Model Storming DevelopanOverall Bulld a Features Planby Designby Buildby Model List Feature Feature Feature (moreshape Alistoffeatures A developmentplan Adesign package Completed thancontant) grouped into sets Class owners client-valued and subjectareas Features setowners function Anobjectmodel (addmorecontent +notes to the object model)Harold Costales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 7079  3 RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1 Prototype Mobile Application for Rice Disease Detection  Figure 4: The Prototype of the Mobile Application  Figure 4 shows the rice disease detection of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researchers designed and develop a mobile application to assist the farmers in disease detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It aims to help the farmers who are the backbone of the country to increase rice crop productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The lack of knowledge of farmers in the detection of rice diseases made the researchers conceptualize the integration of the field of agriculture and medicine in detecting diseases of rice crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=" Also, Duterte's Administration supports Research and Development along with agriculture." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It is stated in the Philippine Development Plan (PDP) 2017-2022 that there is a need to conduct researches in the field because of rice-crop yield losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Thus, the PDP 2017-2022 established a plan to support the initiatives of the agriculture sector along with R&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Moreover, the application is different from other similar apps available in the market because it was developed specifically for the Philippine agriculture setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The application will be stored with rice disease data exclusively for agriculture in the northern Philippines and further validated by pathologists as the researchers continue to gather data that will be for the database of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2 Convolutional Neural Networks Deep Learning Algorithm for the Prototype Application  The researchers utilized an Image Processing typical machine learning workflow in the development of the model which consists of the following Data Acquisition, Data Preparation, and Training and Validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' As cited by [10] Image processing is the method of using different manipulation techniques and algorithms so that a desired features can be extracted such as morphology, color and texture from an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Another study stated that [13] CNN proved to be best among the results in terms of accuracy when compared to other classifiers  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Image Acquisition  The dataset by Huy Minh Do available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='com/ containing 1,260 labeled rice leaf images was used for creating the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The images were labeled as Leaf Blast, Brown Spot, and Hispa, which are the (3) three rice leaf diseases that the model will try to classify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Image Preprocessing  The images were downscaled to 500 x 500 pixels to reduce computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Image dataset were split into train dataset (80% of the entire samples) and validation dataset (20% of the entire samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Image augmentation such as flipping, shearing, and rescaling in order to increase dataset samples as neural networks are data crunchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Training and Validation  To fast track the training process the researchers employed transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Transfer learning, in general, is the process of taking a previously trained model used in a problem and apply in to another related problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It is referring to the knowledge transfer from pretrained network in one domain to your own problem in a different domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' [8]  The researcher used MobileNet 2 [9] as the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It is a state-of-the-art CNN model trained from ImageNet and one of the winners of the ImageNet Large Scale Visual Recognition Challenge (ILSVRC), a prestigious computer vision competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Since the target dataset is small and somewhat different from the dataset where MobileNet 2 was trained, the top layer of the network needs to be frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  First Iteration Base model mobile net Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='0 Hyper-parameters: Top layer = false Initialweights  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='a58%12:57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='PM agDetect blast 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='13% brownspol bacterial_leaf_blight 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='14%Harold Costales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 7080  Second Iteration Base model mobile net Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='0 Hyper-parameters: Top layer = false Initialweights = imagenet Loss = categorical cross entropy Metrics = accuracy Optimization = ADAM EPOCHS= 10 TRAINABLE = FALSE Result= 97% trained data set, 94% validation Suffered from overfitting  Third Iteration Base model mobile net Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='0, image augmentation Hyper-parameters: Top layer = false Initialweights = imagenet Loss = categorical cross entropy Metrics = accuracy Optimization = ADAM EPOCHS= 20 TRAINABLE = true Result= 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='9% trained data set, 98% validation Overfitting was solved  The third iteration proved that the process for the image analysis on the training and testing for the data sets was validated with an accuracy rate of 98% from 94% from the second iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This proved that the CNNs algorithm of the mobile application was proven for use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3 Features of the Prototype Mobile Application  The features of the application include (1) real-time detection, (2) artificial intelligence, and (3) image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1 Real time Detection  One of the features of the mobile application is a real-time detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The application helps in the identification of the rice leaf diseases using convolutional neural networks that match the input data and the data stored in the database of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  The embed camera of the mobile application can detect the diseases in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This feature helps farmers to detect the diseases even without the instruction of plant pathologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2 Artificial Intelligence  Experts in the field of Agriculture are marginal, especially in rice leaf disease detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This feature of the mobile application replicates the knowledge about rice leaf disease detection in the field, which is one of the features of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' As data grows on the database of the mobile application, the more accurate results it can give.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3 Image Analysis  Another feature is Image Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This feature uses the CNNs algorithm aspects and features of images were stored, enabling the mobile application to analyze the rice leaf for disease detection through the following steps of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 4 CONCLUSION In summary, the researchers have developed a mobile application, which could help the agricultural sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Specifically, the researchers conclude that the mobile application will help in the realization of the high-yield of rice crops against rice diseases for preventive measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Thus, this study is helpful to the country, particularly the agricultural sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 5 RECOMMENDATION  The researchers came up with the following recommendations: (1) The output of the Mobile Application will have its validation of experts in rice disease for accuracy of the information from the mobile application, (2) the mobile application will have a series of tests for the implementation for future use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Lastly, (3) the researchers highly recommend that the application shall be introduced to the Department of Agriculture for dissemination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  The researchers would like to thank the following people: the active UNP Research Office- Science and Technology Coordinator, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Redentor S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Rojas, the ever-supportive UNP Research Director, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Edelyn Cadorna and the dynamic UNP President, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Erwin F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Cadorna for pushing us to enhance our research knowledge and capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Also, the researchers would like to thank the IMPACT HACKATON 2050 for organizing the IMPACT HACKATON 2019, where they conceptualized this research study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Above all, thanks to Almighty God for all the blessings for our families and the society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Emberda, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Dumas, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Rentillo, Forecasting Coconut Yield: A Comparative Study between the Use of Traditional Forecasting and Feed Forward Back Propagation Artificial Neural Network, UIC Research Journal, 18(2), 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Francisco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Concepcion, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Mojica, and , S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' B Montes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Philippines First: An Edutainment 3d Game For Android Mobile Platform Using Separating Axis Theorem (Sat) Algorithm, Innovatus, 2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Pugoy, Reinald Adrian D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' and Mariano Vladimir Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Automated rice leaf disease detection using color image  Harold Costales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 7081  analysis, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' SPIE 8009, Third International Conference on Digital Image Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Revathi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Hemalatha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Advance computing enrichment evaluation of cotton leaf spot disease detection using Image Edge detection, 2012 Third International Conference on Computing, Communication and Networking Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Talingdan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=', & Llanda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=', Assessment of the effectiveness of learning theories using gamified Android app in teaching C programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' IOP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Eng, 482-1,1-6, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Mulgrew, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' A clustering technique for digital communications channel equalization using radial basis function networks,, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' on Neural Networks, 4, 570-578, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Tajbakhsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Convolutional Neural Networks for Medical Image Analysis: Full Training or Fine Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=', IEEE Transactions on Medical Imaging, 35-5, 1299-1312, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Aitken and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Ilango A Comparative Analysis of Traditional Software Engineering and Agile Software Development, 46th Hawaii International Conference on System Sciences, Wailea, Maui, HI, 4751-4760, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Elgendy, Deep Learning for Vision Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Manning Publications, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Olga Russakovsky, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' ImageNet Large Scale Visual Recognition Challenge, International Journal of Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 1-42, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Eustaquio and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Dioses Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  Artificial Neural Network For Classification Of Immature And Mature Coffee Beans Using RGB Values, International Journal of Emerging Technologies in Engineering Research, 9-4, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='30534/ijeter/2020/41882020 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Devi Devani, Alethea Suryadibrata, Julio Christian, Young, Implementation of Siamese Convolutional Neural Network from Cell Images for Malaria Disease Identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' International Journal of Advanced Trends in Computer Science and Engineering, 9-4, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='30534/ijatcse/2020/0494 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Handayani, Indri, Supriyanti, Dedeh, Maulani, Giandari, Liutfani, Ninda, Real Time Multi-Scale Facial Mask Detection and Classification Using Deep Transfer Learning Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' International Journal of Advanced Trends in Computer Science and Engineering, 9-4, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='30534/ijatcse/2020/33942020 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Simhadri Chinna Gopi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=', Fuzzy Based Classification of X-Ray Images with Convolution Neural Network, International Journal of Emerging Trends in Engineering Research, 8(8), 4433 - 4436 , 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='30534/ijeter/2020/63882020 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Fitria Hidayanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=', Upper Stomach Disorder Detection System using Backpropagation Artificial Neural Network, International Journal of Emerging Trends in Engineering Research, 8(8), 4426 - 4432 , 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='30534/ijeter/2020/62882020 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Lamarca, Bryan Irvin J, et al, The Development of a Performance Appraisal System Using Decision Tree Analysis and Fuzzy Logic, International Journal of Intelligent Engineering and Systems, 11-4, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='22266/ijies2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='0831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='02' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}

adE3T4oBgHgl3EQfdAqs/content/tmp_files/2301.04531v1.pdf.txt

@@ -0,0 +1,925 @@
+arXiv:2301.04531v1  [physics.optics]  11 Jan 2023
+Unified Model for Nonlinear Pulse Propagation in Composites and Optimization of
+THz Generation
+A. Husakou∗
+Max Born Institute, Max Born Str.
+2a, D-12489 Berlin, Germany
+O. Fedotova, R. Rusetsky, and O. Khasanov
+Scientific-Practical Materials Research Centre of National Academy
+of Sciences of Belarus, P.Brovki str.
+19, 220072 Minsk, Belarus
+T. Smirnova and A. Fedotov
+Belarus State University, Niezalieˇznasci ave 4, 220030 Minsk, Belarus
+T. Apostolova
+Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,
+Tsarigradsko Chausse 72, 1784 Sofia, Bulgaria and
+Institute for Advanced Physical Studies, New Bulgarian University, 1618 Sofia, Bulgaria
+I. Babushkin
+Institute of Quantum Optics, Leibniz University Hannover, Welfengarten 1, 30167, Hannover, Germany
+Cluster of Excellence PhoenixD (Photonics, Optics,
+and Engineering-Innovation Across Disciplines), 30167, Hannover, Germany and
+Max Born Institute, Max-Born-Str. 2a, 12489, Berlin, Germany
+U. Sapaev
+Tashkent State Technical University, University street 2, 100097 Tashkent, Uzbekistan
+We describe a unified numerical model which allows fast and accurate simulation of nonlinear light
+propagation in nanoparticle composites, including various effects such as group velocity dispersion,
+second- and third-order nonlinearity, quasi-free-carrier formation and plasma contribution, exciton
+dynamics, scattering and so on. The developed software package SOLPIC is made available for
+the community. Using this model, we analyze and optimize efficient generation of THz radiation
+by two-color pulses in ZnO/fused silica composite, predicting an efficiency of 3%. We compare the
+role of various nonlinear effects contributing to the frequency conversion, and show that optimum
+conditions of THz generation differ from those expected intuitively.
+I.
+INTRODUCTION
+THz technology has attracted a lot of attention in
+the recent years, since it provides unique experimental
+tools and techniques in nonlinear and time-domain spec-
+troscopy, biology and medicine, remote sensing, security
+screening, as well as information and communication sys-
+tems (see e.g. [1–4]). For generation of THz radiation,
+different techniques were proposed, such as two-color ion-
+izing femtosecond pulses in gases [5–10] and plasmas,
+produced at the surface of solids (e.g. metal foil) with
+high intensities [11] that are free from optical damage
+threshold and phase-matching drawbacks; optical rectifi-
+cation of the transmitted intense ultrashort laser pulses
+in non-linear crystals [12–16], which provide a basis for
+compact low-intensity devices.
+The needs of the THz
+technology require, however, extension of the range of
+the available techniques and materials, in order to pro-
+vide flexible designs required in multifarious applications.
+∗ gusakov@mbi-berlin.de
+Following this line, investigations of THz generation in
+various media such as water [17] and strongly magnetized
+plasmas [18] were performed. Emission of terahertz radi-
+ation with broad bandwidth by femtosecond photoexcita-
+tion of spintronic materials (ferromagnetic and synthetic
+multiferroic heterostructures) was also reported recently
+[19, 20].
+Nanoparticle composites were actively investigated in
+the past as a nonlinear material, e.g. [21–23], and their
+particular strength lies in the flexibility of their design
+leading to unusual properties such as e.g. negative re-
+fractive index [24].
+However, surprisingly, up to our
+knowledge they have not attracted attention as a medium
+for THz generation. In this paper, we close this gap by
+developing a numerical model suitable for simulation of
+THz generation in nanocomposites.
+A range of linear
+and nonlinear effects such as group velocity dispersion,
+second- and third-order nonlinearity, quasi-free-carrier
+formation, exciton dynamics and so on are encompassed
+by the developed model. We use it to explore THz gen-
+eration by two-color pulses in nanoparticle composites,
+to elucidate the contributions of different frequency con-
+version mechanisms, and to predict efficiencies in few-
+
+2
+percent range.
+The applicability of the above model is, in fact, much
+broader than mere simulation of THz generation; a wide
+range of nonlinear effects such as soliton dynamics and
+supercontinuum generation, frequency conversion, multi-
+level dynamics and electromagnetically-induced trans-
+parency and so on can be studied using this unified ap-
+proach. With this is mind, we have created the extensive
+documentation of the code and made the code publicly
+available [25], in a hope that it will be useful to the optical
+community for investigations of the nonlinear processes
+in nanocomposites and other materials.
+The paper is organized as follows. In Section 2, we
+present the numerical model, including detailed formal-
+ism for all the relevant mechanisms.
+In Section 3, we
+optimize the THz generation by two-color pulses, and
+analyze the role of different parameters. A summary of
+the paper is given in the conclusion.
+II.
+THEORETICAL MODEL
+We consider a composite consisting of two components,
+a homogeneous host material and spherical nanoparticles
+(inclusions) randomly distributed in space. We assume
+a sufficiently low (typically few percent or below) filling
+fraction of the inclusions so that neither percolation nor
+interaction between the inclusions play a role. We con-
+sider homogeneous inclusions to be sufficiently small with
+diameter below the light wavelength so that effective-
+medium theory can be applied.
+Note that we do not
+place any limitations on the nature of host and inclu-
+sion materials, i.e., either of them could be a dielectric,
+a metal, or a semiconductor. We do not require point
+symmetry in host material or in the inclusions, so that
+the second-order susceptibility can be non-zero in either
+material.
+The model is designed to simulate propaga-
+tion over relatively short distances of few millimeters, be-
+low the damage threshold, and without back-reflection,
+therefore (1+1)D treatment using unidirectional propa-
+gation equation [26] is the most suitable.
+Under these conditions, the following effects have to
+be taken into account: linear dispersion including intrin-
+sic and scattering losses, second- and third-order optical
+nonlinearities and photoionization accompanied by ion-
+ization losses and plasma dynamics. In addition, transi-
+tions between excitonic states can play a significant role
+in the inclusion response, in particular for the genera-
+tion of new frequencies in the THz range. Among the
+effects which were neglected in this treatment are ther-
+mal effects (due to slow ns-scale response), coupling to
+phonons (because of relatively slow ps-scale response),
+Raman scattering (which is typically weaker than in-
+stantaneous nonlinearities), anisotropy of the host ma-
+terial (due to manufacturing limitations for composites),
+deviations of the inclusion form from a sphere (because
+of typical manufacturing conditions), and generation of
+high-order harmonics (because of the considered inten-
+sity ranges).
+The following unidirectional propagation equation is
+used to model the light propagation in a homogeneous
+medium [26, 27]:
+∂E(z, ω)
+∂z
+= −i
+�
+[
+�
+ǫ(ω) − ng]ω
+c
+− β(ω0)
+�
+E(z, ω)
+−
+iω
+2c
+�
+ǫ(ω)
+PNL(z, ω),
+(1)
+where E(z, ω) = ˆFE(z, t) =
+� ∞
+−∞ E(z, t) exp(−iωt)dt
+is the Fourier transform ˆF of the electric field E(z, t), z
+is the propagation coordinate, ǫ(ω) is the linear dielec-
+tric permittivity (generally speaking, complex-valued to
+include loss mechanisms), ng is the group refractive in-
+dex, ω0 is a characteristic frequency of the pulse spec-
+trum, β(ω) =
+�
+ǫ(ω)ω/c, and PNL(z, ω) is the Fourier
+transform of the nonlinear part of the polarization. We
+would like to empathise that no slowly-varying envelope
+approximation is used, and E(z, t) represents the real-
+valued field including the carrier oscillations. This ap-
+proach yields a unified treatment for a pulse with arbi-
+trary spectral content, which is particularly important
+for extremely broad spectra.
+A.
+Linear dispersion
+The effective-medium theory allows to substitute the
+composite material by a homogenised medium with ap-
+propriately defined effective material parameters.
+The
+effective refractive index of a composite can be expressed
+as [27]
+neff =
+�
+(1 − f)ǫh + fǫi
+3ǫh
+2ǫh + ǫi
++ 2i
+� ǫh − ǫi
+2ǫh + ǫi
+�2 �rNPω√ǫh
+c
+�3�1/2
+,
+(2)
+where f is the volume filling factor of the inclusions,
+rNP is their radius, and ǫh,i are the frequency-dependent
+dielectric functions of the host and of the inclusions, cor-
+respondingly. The last term in the square brackets de-
+scribes scattering losses.
+B.
+Second- and third-order nonlinearities
+The second- and third-order nonlinear processes can
+also be described in the framework of the effective-
+medium theory. The expressions for the effective second-
+order susceptibility looks like [28]
+
+3
+χ(2)
+eff (ω1 = ω2 + ω3; ω2, ω3) = (1 − f)χ(2)
+h
++
++ fx(ω1)x(ω2)x(ω3)χ(2)
+i ,
+(3)
+where χ(2)
+h
+and χ(2)
+i
+are the susceptibilities of host and
+inclusion materials, correspondingly. Note that we ne-
+glected the frequency dependence of the susceptibilities
+of host and inclusions, which is a good assumption far
+from resonances. Quantity x(ω) is the ratio of local field
+inside the inclusion and the incident field:
+x(ω) =
+3ǫh(ω)
+2ǫh(ω) + ǫi(ω).
+(4)
+Here we note that, due to photoionization as described
+below, the ǫi(ω) and therefore x, strictly speaking, de-
+pend on time due to buildup of plasma during the pulse.
+However, in the current simulation we neglect this de-
+pendence, assuming that corresponding change of ǫi(ω)
+is small and that we are far from the plasmonic resonance
+given by 2ǫh(ω) = −ǫi(ω).
+Similarly, for the effective third-order susceptibility we
+write [28]
+χ(3)
+eff (ω1 = ω2 + ω3 + ω4; ω2, ω3, ω4) = (1 − f)χ(3)
+h
++
++ fx(ω1)x(ω2)x(ω3)x(ω4)χ(3)
+i ,
+(5)
+where χ(3)
+h
+and χ(3)
+i
+are the susceptibilities of host and
+inclusion materials, correspondingly. The final expres-
+sions which were used to calculate the corresponding po-
+larizations look like
+Pχ(2)(z, ω) = (1 − f)ǫ0χ(2)
+h
+ˆFE(z, t)2 +
++ fǫ0χ(2)
+i
+ˆF[ ˆF −1{E(z, ω)x(ω)}2],
+(6)
+Pχ(3)(z, ω) = (1 − f)ǫ0χ(3)
+h
+ˆFE(z, t)3 +
++ fǫ0χ(3)
+i
+ˆF[ ˆF −1{E(z, ω)x(ω)}3].
+(7)
+C.
+Plasma dynamics
+Let us turn to the description of plasma formation and
+dynamics. In the framework of SOLPIC, we consider a
+case when the ionization potential Ip of the inclusions
+is lower than that of the host material, so that due to
+the sensitive dependence of the polarization rate on the
+ionization potential we can neglect plasma formation in
+host material.
+The contribution from the plasma is determined by the
+average displacement ⟨d⟩(z, t) of the electron from the
+equilibrium position in the parent ”molecule”, whereby
+by a ”molecule” we denote an atom or a group of atoms of
+the solid-state material which can provide a single ioniza-
+tion event. Furthermore, it is determined by the relative
+ionization of the solid state ρ(z, t), which is the ratio of
+the conduction-band electron density to the density of
+”molecules”:
+Pplasma(z, ω) = −Nmole ˆF[⟨d⟩(z, t)ρ(z, t)]
+(8)
+Here Nmol is the concentration of the molecules and
+e = 1.6×10−19 is the electron charge. The above expres-
+sion would be valid in a homogeneous medium; however,
+as it refers to a polarization which occurs inside of inclu-
+sions, in contrast to averaged macroscopic polarization,
+in the case of effective-medium theory it has to be addi-
+tionally multiplied by x(ω). For the origin of this factor
+and further details see Ref. [28].
+The dynamics of the quantity ⟨d⟩(z, t)ρ(z, t) is given
+by [33]
+∂(⟨d⟩(z, t)ρ(z, t))
+∂t
+= ⟨v⟩(z, t) + x0Γ(t),
+(9)
+where ⟨v⟩ is the average velocity of electrons and
+x0 ≃ −Ip/eE(t) is the initial displacement of the elec-
+tron immediately after the ionization event, Ip being the
+bandgap. It can be shown that the second term describes
+the energy loss of the pulse due to photoionization. The
+dynamics of the ⟨v⟩(z, t)ρ(z, t) is given by second New-
+ton’s law as
+∂(⟨d⟩(z, t)ρ(z, t))
+∂t
+= −eE(z, t)
+me
+ρ,
+(10)
+where me is the effective electron mass near the bot-
+tom of the conduction band. Here we neglect the initial
+displacement and velocity of electron just after the ion-
+ization.
+The dynamics of the relative plasma density ρ is given
+by
+∂ρ
+∂t = Γ( ˆF −1[x(ω)E(z, ω)]),
+(11)
+where x(ω)E(z, ω) is the local field inside of inclusions
+which determines the photoionization rate Γ.
+D.
+Ionization rate
+Depending on the relation between the frequency of
+pump light and the ionization potential of inclusions, we
+consider two models for the ionization rate. For the case
+when the energy of pump photons is much smaller than
+the ionization potential, the photoionization occurs ei-
+ther by multiphoton regime or by tunneling regime, as
+determined by intensity and Keldysh parameter. Here we
+
+4
+10-4
+10-3
+10-2
+0.1
+1
+ 1
+ 2
+ 3
+ 4
+ 5
+(a)
+I(ω) (arb. units)
+ω (fs-1)
+-6
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+-40
+-20
+ 0
+ 20
+ 40
+ 0
+ 0.005
+ 0.01
+(b)
+E (GV/m)
+ρ
+t (fs)
+FIG. 1. Dependence of the spectra (a) and the electric field
+(b) on the propagation length. 15-fs pulses at 2.26 fs−1 and
+4.58 fs−1 are considered, with intensity of 1 TW/cm2 and
+propagation length of 0.75 µm (magenta curves), 2.25 µm
+(green curves), and 6.75 µm (bluecurves). In (b), additionally
+the relative plasma density is shown for propagation length
+of 6.75 µm. A composite of ZnO inclusions with f = 0.03 in
+fused silica is considered.
+utilize so-called Yudin-Ivanov model [29], which provides
+a formalism for both of these regimes in a unified way.
+This model was initially developed for isolated atoms; its
+use for solid state is justified in a case a negligible an-
+harmonicity of the bands in the center of the Brillouin
+zone.
+The cycle-resolved ionization rate Γ is given (in atomic
+units, that is, with frequency ω, time t and field E mea-
+sured in the corresponding Hartree units ωa = 0.26
+rad/as, ta = 24.2 as, xa = 0.0529 nm, and Ea = 514.2
+V/nm) by
+Γ(z, t) = π
+τT
+exp
+�
+−σ0
+⟨2E(z, t)2⟩
+ω3
+� �
+2κ3
+�
+⟨2E(z, t)2⟩
+�2Z/κ
+× exp
+�
+−E(z, t)2
+2ω3
+σ1
+�
+.
+(12)
+Here τT = κ/E(z, t), κ =
+�
+Ip/(ℏωa), σ0 =
+1
+2(γ2 +
+1
+2) ln C − 1
+2γ
+�
+1 + γ2, γ = ωτT , Z is the effective atomic
+charge, C = 1 + 2γ
+�
+1 + γ2 + 2γ2, and σ1 = ln C-
+2γ/
+�
+1 + γ2.
+The quantity ⟨E(z, t)2⟩ is the averaged
+value of the squared electric field over few past periods
+(5 fs is assumed in this work).
+The Yudin-Ivanov model was initially derived for gases;
+its applicability for solid state, while generally justi-
+fied for materials with tight binding, is not strictly es-
+tablished.
+We have benchmarked Yudin-Ivanov model
+by comparing it to the numerical solution of the time-
+dependent Schrodinger equation in single active electron
+approximation [30]. In this approach the empirical pseu-
+dopotential method was used for describing the electron
+band structure of ZnO [31]. We have found that the dif-
+ference of the ionization rate does not typically exceed
+one order of magnitude. This difference is, in fact, not
+very significant: because of the threshold-like behavior of
+the ionization rate, it leads to only a slight shift of the
+intensity at which a strong plasma generation is reached.
+For the specal case when the energy of pump photons
+is around two ionization potentials, it is preferable to
+use the two-photon formalism [32] and write the cycle-
+resolved ionization rate Γ (in SI units) as
+Γ(z, t) =
+2e4x4
+aν
+ℏ4ω2
+0[(2ω0 − Ip/ℏ)2 + ν2 ⟨E(z, t)2⟩E(z, t)2,
+(13)
+where ν is the relaxation constant of the two-photon
+transition.
+E.
+Contribution by excitons
+Finally, we include the nonlinear polarization due to
+excitons into treatment. We consider multiple excitonic
+levels and utilize the standard Bloch equations for the de-
+scription of the ionization. The dynamics of the density
+matrix ρe is given by (see e.g. [32])
+iℏ∂ρe
+∂t = [H, ρe],
+(14)
+where H = H0 + Hint, H0 is the Hamiltonian of the
+system in the absence of excitation, Hint is the interaction
+Hamiltonian, which components Hij are related with the
+corresponding dipole transition moments eMij:
+Hij = eMij ˆF −1[E(z, ω)x(ω)].
+(15)
+In addition, polarization decay (decay of the off-
+diagonal elements of ρe) with decay time T2 and decay
+of the population to the ground state with decay time T1
+are included. In order to avoid numerical instabilities,
+the normalization of the density matrix ρ is performed
+each few steps in time, by a) enforcing 0 ≤ ρe,ii ≤ 1, b)
+enforcing T r(ρe) = 1, and c) adjusting the non-diagonal
+elements which exceed the maximum possible value de-
+termined by the corresponding level populations.
+The excitonic polarization is then defined in a standard
+way as
+Pexc(z, ω) = x(ω) ˆF[fTr(ρeM)].
+(16)
+We solve the propagation equation by an extended
+split-step method, whereby each of the contributions to
+
+5
+the polarization is treated subsequently, which allows to
+reduce the accumulation of numerical error. Nonlinear
+steps are performed using the Runge-Kutta approach,
+the order of which can be selected between 1,2, and 4.
+Fixed step of the grid both in time and in the propa-
+gation coordinate is used. The appearance of numerical
+artifacts during the propagation is monitored by tracing
+the total pulse energy as well as the total energy absorbed
+at the boundaries of the numerical time window.
+III.
+NUMERICAL RESULTS AND DISCUSSION
+In order to exemplify the above model and function-
+ing of SOLPIC, we present in this section a simulation of
+THz generation. We consider a composite of ZnO inclu-
+sions in SiO2 matrix. Phenomenological Sellmeyer-type
+expressions were used to describe dispersion on ZnO [35]
+and SiO2 [36]. Similarly, experimental data on second-
+order [37] and third-order [38] susceptibility of bulk ZnO
+and third-order susceptibility of SiO2 [40] were used. We
+estimated the value of T2 as 50 fs from Ref. [41] and used
+T1 = 2T2. For ZnO, the typical exciton size is larger than
+interatomic distance, meaning that we are dealing with
+Wannier-Mott type of excitons. In a case of sufficiently
+small inclusions, the exciton is bounded by the inclu-
+sion boundaries, therefore its wavefunctions (as well as
+energy levels and dipole momenta) are better described,
+instead of hydrogen-like potential, by a constant poten-
+tial inside a sphere [42] with a step on its boundary. We
+have taken into account 5 lowest excitonic levels, and
+typical values of the off-diagonal dipole momenta, as cal-
+culated by this approach, are around 3×10−28 C·m, for
+same-size nanoparticles with a radius of 2.5 nm which
+are considered here and hereafter.
+For the permanent
+dipole momenta of ZnO, we have adopted a typical value
+of 6.66×10−30 C·m per ZnO molecule, which was used
+to define the on-diagonal elements of the dipole matrix.
+We used the ionization potential of 3.37 eV equal to the
+bandgap of ZnO to characterize the transition from va-
+lence band to conduction band, and all the presented
+numerical results correspond to the conditions below the
+damage threshold of ZnO [43].
+The evolution of the field profile and spectra with prop-
+agation is illustrated in Fig. 1, for two-color pulsed exci-
+tation with pump pulses around 800 nm and 400 nm, for
+conditions given in the caption. In Fig. 1(a) one can see
+that initial stages of the propagation are characterized by
+self-phase modulation with typical spectral side lobes. At
+later stages, spectrum becomes irregular and transform
+into a supercontinuum extending up to the absorption
+edge given by the bandgap. The evolution of the tempo-
+ral profile, shown in Fig. 1(b), shows gradual reduction
+of the enegry of electric field, as well as significantly ir-
+regular envelope for longer propagation. This reduction
+of the maximum field determines the saturation of the
+THz generation efficiency and is caused both by strong
+group-velocity dispersion for broad spectrum and energy
+absorption due to transition to conduction band. One
+can see from the red curve in Fig.
+1(b) that relative
+plasma density reaches values of roughly 0.01 after the
+pulse, which is sufficient to induce significant energy ab-
+sorption.
+10-4
+10-3
+10-2
+0.1
+1
+ 10
+ 20
+ 30
+I(ν) (arb. units)
+ ν (THz)
+FIG. 2. Dependence of the spectra in the THz range on the
+propagation distance. We consider 1-TW/cm2, 15-fs pump
+pulses at 2.26 fs−1 and 4.58 fs−1, in a composite of ZnO
+nanoparticles with filling fraction of f = 0.03 in a fused-silica
+matrix, after propagation length of 5 µm (magenta curve),
+15 µm (green curve), 45 µm (blue curve), and 50 µm (yellow
+curve).
+In Fig. 2 the evolution of the spectrum in the THz
+range is shown. One can see that while the spectrum is
+flat at early stages of propagation, for larger propagation
+lengths the spectrum is localized around 28 THz, proba-
+bly due to phase-matching effects with a phase-mismatch
+length of 10.5 µm for the four-wave mixing between the
+two photons at 2.26 fs−1, one photon at 4.58 fs−1, and a
+THz photon. Losses around 15 THz and below can also
+contribute to saturation of generation. After 45 µm prop-
+agation length, the efficiency of the generation reaches
+3.05%, which is sufficiently high for practical applica-
+tions.
+In order to determine the optimum conditions of the
+THz generation, in Fig. 3 we plot the dependence of the
+generation efficiency on the distribution of energy be-
+tween the 830-nm pulse and 412-nm pulse (a), intensity
+of pulses (b), and wavelength of the short-wavelength
+pulse (c). One can see that the efficiency of THz gen-
+eration is non-zero but very small for the cases when
+only one of the pulses is present (energy fraction of 0
+or 1). This indicates that the optical rectification based
+the second-order susceptibility of ZnO cannot efficiently
+generate THz for the considered conditions, and that
+the dominant contribution comes from the third-order
+susceptibility of ZnO nanoparticles, third-order suscep-
+tibility of SiO2 being comparatively weak. In an ideal
+case without pump pulses modification, the efficiency of
+the THz generation is proportional to E2
+830(Etot − E830),
+where E830 is the energy of the pulse at 830 nm and
+Etot = E830 + E412 is the total energy of the pulses. The
+maximum efficiency is then reached at E830/Etot = 1/3,
+
+6
+0%
+0.5%
+1%
+1.5%
+2%
+2.5%
+3%
+ 0
+ 0.25
+ 0.5
+ 0.75
+ 1
+(a)
+THz efficiency
+Energy fraction of 800-nm pulse
+0%
+0.5%
+1%
+1.5%
+2%
+2.5%
+3%
+ 0
+ 0.25
+ 0.5
+ 0.75
+ 1
+ 1.25
+ 1.5
+(b)
+THz efficiency
+I, TW/cm2
+0%
+0.5%
+1%
+1.5%
+2%
+2.5%
+3%
+ 405
+ 410
+ 415
+ 420
+ 425
+ 430
+(c)
+THz efficiency
+λ (nm)
+FIG. 3. Dependence of the THz generation efficiency on en-
+ergy fraction of the 800-nm pulse (a), intensity of each of the
+pump pulses (b), and the wavelength of the second-harmonic
+pulse (c). A composite of ZnO nanoparticles with f = 0.03 in
+fused silica is considered. In (a), 15-fs pulses at 2.26 fs−1 and
+4.58 fs−1 are considered, with total intensity of 2 TW/cm2
+and propagation length of 50 µm. In (b) we consider 15-fs
+(red curve) and 150-fs (green curve) pulses at 2.26 fs−1 and
+4.58 fs−1. In (c), 1 TW/cm2, 15-fs pulses are considered, with
+IR pulse frequency of 2.26 fs−1 and propagation length of 50
+µm.
+however, as shown in Fig. 3(a), maximum numerical ef-
+ficiency is achieved for E830/Etot = 0.5. This could be
+due to strong SPM-induced spectral spreading of high-
+frequency pulse during the propagation, which needs to
+be compensated by relatively higher value of E412. In
+Fig. 3(b), the dependence of the efficiency on the pulse
+intensity is shown, exhibiting saturation and decrease
+after a certain intensity as well as lower efficiencies for
+longer pulses. We attribute these features to detrimen-
+tal contribution of the accumulated plasma, which grows
+with intensity and pulse duration [cf. Fig. 4(a)]. In Fig.
+3(c), the dependence of the efficiency on the wavelength
+of the short-wavelength pulse exhibits several maxima.
+Note that while one might expect an optimum THz gen-
+eration for 415 nm, which would correspond to generation
+of frequencies near zero, our simulation in fact predict a
+minimum around this value, determined most probably
+by phase mismatch and losses below 15 THz.
+Finally, in order to access the role of plasma and ex-
+citons in the THz generation in composites, in Fig. 4
+we compare the spectra for plasma contribution (a) and
+exciton contribution (b) switched on/off. One can see
+that the plasma contribution is significant, both due to
+contribution to refractive index and due to losses, and
+absence of plasma contribution leads to a notable (more
+than twofold) increase of the efficiency.
+On the other
+hand, from Fig.
+4(b) one can see that exciton polar-
+ization do not provide a strong contribution to the effi-
+ciency for the considered parameters. Also, additionally
+including the permanent dipole momenta, described in
+10-3
+10-2
+0.1
+1
+ 10
+ 20
+ 30
+(a)
+I(ν) (arb. units)
+ ν (THz)
+10-3
+10-2
+0.1
+1
+ 10
+ 20
+ 30
+(b)
+I(ν) (arb. units)
+ ν (THz)
+FIG. 4. Spectra in the THz range with (a) plasma contri-
+bution on (red) and off (green) and (b) exciton contribution
+on (red) and off (green), as well including both exciton con-
+tribution and permanent dipole moment (blue). We consider
+1-TW/cm2, 15-fs pump pulses at 2.26 fs−1 and 4.58 fs−1, in a
+composite of ZnO inclusions with filling fraction of f = 0.03
+in a fused-silica matrix, after propagation length of 10 µm (a)
+and 0.75 µm (b).
+the model above, does not significantly increase the effi-
+ciency of THz generation, as indicated by the blue curve
+in Fig. 4(b) which is close to the red and green curves.
+We note, however, that this conclusion is of limited gen-
+erality; for other parameters of the medium excitons can
+provide the key mechanism of THz generation (see e.g.
+[44, 45] and references therein).
+IV.
+CONCLUSION
+In this paper we have established a comprehensive nu-
+merical model for the simulation of light propagation in
+composites, including all the relevant physical effects for
+a broad range of parameters, such as linear dispersion
+of the composite, second- and third-order nonlinear ef-
+fects, plasma contribution, excitons contribution and so
+on. The model was applied to simulate the generation of
+THz radiation in a ZnO-SiO2 composite. We have per-
+formed optimization of the frequency conversion process,
+predicting an efficiency of 3.05%.
+We show that sim-
+ulations provide insights into the optimization, such as
+the power distribution between the pump pulses, which
+would not be accessible intuitively. We hope that the nu-
+merical model and the corresponding software solution,
+which we make available for the community, will con-
+tribute to the capacity of the simulations in the area of
+nonlinear optics.
+ACKNOWLEDGMENTS
+Authors acknowledge financial support
+from Eu-
+ropean Union project H2020-MSCA-RISE-2018-823897
+”Atlantic”. I.B. thanks Cluster of Excellence PhoenixD
+(EXC 2122, project ID 390833453) for financial support.
+
+7
+[1] M. Tonouchi, ”Cutting-edge terahertz technology” Nat.
+Photon. 1, 97–105 (2007).
+[2] K. Sakai (ed.),
+Terahertz Optoelectronics,
+Springer,
+Berlin, 2005.
+[3] D.
+Mittleman,
+Sensing
+with
+Terahertz
+Radiation
+Springer, Berlin, 2002.
+[4] J. Liu, J. Dai, S. L. Chin, and X. C. Zhang, ”Broadband
+terahertz wave remote sensing using coherent manipula-
+tion of fluorescence from asymmetrically ionized gases”,
+Nat Photon 4, 627–631 (2010).
+[5] D. J. Cook and R. M. Hochstrasser, ”Intense terahertz
+pulses by four-wave rectification in air”, Opt. Lett. 25,
+1210–1212 (2000).
+[6] M. Kress, T. Loffler, S. Eden, M. Thomson, and H. G.
+Roskos, ”Terahertz-pulse generation by photoionization
+of airwith laser pulses composed of both fundamental
+and second-harmonic waves”, Opt. Lett. 29, 1120–1122
+(2004).
+[7] T. Bartel, P. Gaal, K. Reimann, M. Woerner, and T. El-
+saesser, ”Generation of single-cycle THz transients with
+high electric-field amplitudes”, Opt. Lett. 30, 2805–2807
+(2005).
+[8] X. Xie, J. Dai, and X.-C. Zhang, ”Coherent control of
+THz wave generation in ambient air”, Phys. Rev. Lett.
+96, 075005 (2006).
+[9] M. D. Thomson, M. Kress, T. Loffler, and H. G. Roskos,
+”Broadband THz emission from gas plasmas induced by
+femtosecond optical pulses: From fundamentals to appli-
+cations”, Laser Photon. Rev. 1, 349–368 (2007).
+[10] I. Babushkin, S. Skupin, A. Husakou, C. Kohler, E.
+Cabrera-Granado, L. Berge and J. Herrmann, ”Tailoring
+terahertz radiation by controlling tunnel photoionization
+events in gases”, New J. Phys. 13, 123029 (2011).
+[11] S. Herzer, A. Woldegeorgis, J. Polz, A. Reinhard, M. Al-
+massarani, B. Beleites, F. Ronneberger, R. Grosse, G.G.
+Paulus, U. H¨ubner, T. May, A. Gopal, “An investigation
+on THz yield from laser-produced solid density plasmas
+at relativistic laser intensities”, New J. Phys. 20 , 063019,
+((2018)
+[12] F. Blanchard, B. E. Schmidt, X. Ropagnol, N. Thir´e,
+T. Ozaki, R. Morandotti, D. G. Cooke, and F. Legare,
+“Terahertz pulse generation from bulk GaAs by a tilted-
+pulse-front excitation at 1.8 µm,” Appl. Phys. Lett. 105,
+241106 (2014).
+[13] K. Aoki, J. Savolainen, and M. Havenith, “Broadband
+terahertz pulse generation by optical rectification in GaP
+crystals,” Appl. Phys. Lett. 110, 201103 (2017).
+[14] G. L. Dakovski, B. Kubera, and J. Shan, “Localized ter-
+ahertz generation via optical rectification in ZnTe,” J.
+Opt. Soc. Am. B 22, 1667–1670 (2005).
+[15] S.-C. Zhong, J. Li, Z.-H. Zhai, L.-G. Zhu, J. Li, P. W.
+Zhou, J. H. Zhao, and Z. R. Li, “Generation of 0.19-mJ
+THz pulses in LiNbO3 driven by 800-nm femtosecond
+laser,” Opt. Express 24, 14828–14835 (2016).
+[16] T. Apostolova, B. Obreshkov “High harmonic generation
+in crystalline silicon irradiated by an intense ultrashort
+laser pulse,” The European Physical Journal D. 75, 1
+(2021).
+[17] Q. Jin, E. Yiwen, K. Williams, J. Dai, and X.-C. Zhang,
+”Observation of broadband terahertz wave generation
+from liquid water”, Appl. Phys. Lett. 111, 071103 (2017).
+[18] C. Tailliez, X. Davoine, A. Debayle, L. Gremillet, and
+L. Berge, ”Terahertz Pulse Generation by Strongly Mag-
+netized, Laser-Created Plasmas”, Phys. Rev. Lett. 128,
+174802.
+[19] C. Zhou, Y. P. Liu, Z. Wang, S. J. Ma, M. W. Jia, R.
+Q. Wu, L. Zhou, W. Zhang, M. K. Liu, Y. Z. Wu, J. Qi,
+Phys. Rev. Lett. , 121, 086801, 2018
+[20] P. Agarwal, L. Huang, S. Ter Lim, R. Singh, Nat. Com-
+mun. , 13, 4072, 2022
+[21] D.Ricard, ”Nonlinear optics in composites and in hetero-
+geneous media”, Physica A 157, 301-308 (1989).
+[22] V. M. Shalaev, Nonlinear Optics of Random Media,
+Springer, Berlin, 2000.
+[23] R. Driben, A. Husakou, and J. Herrmann, ”Supercon-
+tinuum generation in aqueous colloids containing silver
+nanoparticles”, Opt. Lett. 34, 2132-2134 (2009).
+[24] J. B. Pendry, ”Negative Refraction Makes a Perfect Lens”
+Phys. Rev. Lett. 85, 3966-3969 (2000).
+[25] https://github.com/F1Ulia/SOLPIC/tree/main
+[26] A. Husakou and J. Herrmann, ”Supercontinuum Gen-
+eration of Higher-Order Solitons by Fission in Photonic
+Crystal Fibers”, Physical Review Letters 87 (20), 203901
+(2001).
+[27] J. C. Maxwell Garnett, ”Colours in Metal Glasses and in
+Metallic Films”, Philos. Trans. R. Soc. London 203, 385
+(1904).
+[28] J. E. Sipe and R. W. Boyd, ”Nonlinear susceptibility
+of composite optical materials in the Maxwell Garnett
+model”, Phys. Rev. A 46, 1614 (1992).
+[29] G. L. Yudin and M. Y. Ivanov, ”Nonadiabatic tunnel
+ionization: Looking inside a laser cycle”, Phys. Rev. A
+64, 013409 (2001).
+[30] T. Apostolova and B. Obreshkov, ”High harmonic gener-
+ation from bulk diamond driven by intense femtosecond
+laser pulse”, Diam. Relat. Mater. 82, 165 (2018).
+[31] Fan, W. J.; Abiyasa, A. P.; Tan, S. T.; Yu, S. F.; Sun,
+X. W.; Xia, J. B.; Yeo, Y. C.; Li, M. F.; Chong, T. C,
+”Electronic structures of wurtzite ZnO and ZnO/MgZnO
+quantum well” Journal of Crystal Growth 287, 28 (2006)
+[32] L. S. Meng, ”Continuous-wave Raman laser in H2 : semi-
+classical theory and diode-pumping experiments”, PhD
+thesis, Montana State University, 2002.
+[33] A. Husakou and J. Herrmann,
+”High-power,
+high-
+coherence supercontinuum generation in dielectric-coated
+metallic hollow waveguides”, Opt. Express 17, 12481
+(2009).
+[34] P. J¨urgens, B. Liewehr, B. Kruse, C. Peltz, D. Engel,
+A. Husakou, T. Witting, M. Ivanov, M. J. J. Vrakking,
+T. Fennel and A. Mermillod-Blondin, ”Origin of strong-
+field-induced low-order harmonic generation in amor-
+phous quartz”, Nature Physics 16, 1035–1039 (2020).
+[35] I. Bodurov, I. Vlaeva, A. Viraneva, T. Yovcheva, S.
+Sainov,
+”Modified design of a laser refractometer”,
+Nanoscience Nanotechnology 16, 31-33 (2016).
+[36] C. Z. Tan, ”Determination of refractive index of silica
+glass for infrared wavelengths by IR spectroscopy”, J.
+Non-Cryst. Solids 223, 158-163 (1998).
+[37] G. Wang, G. K. L. Wong, and J. B. Ketterson, ”Rede-
+termination of second-order susceptibility of zinc oxide
+single crystals,” Appl. Opt. 40, 5436-5438 (2001).
+
+8
+[38] M. C. Larciprete, D. Haertle, A. Belardini, M. Bertolotti,
+F. Sarto and P. G¨unter, ”Characterization of second and
+third order optical nonlinearities of ZnO sputtered films”,
+Appl. Phys. B 82, 431–437 (2006).
+[39] V. V. Multian, J. Riporto, M. Urbain, Y. Mugnier, G.
+Djanta, S. Beauquis, C. Galez, V. Ya Gayvoronsky, and
+R. Le Dantec, ”Averaged third-order susceptibility of
+ZnO nanocrystals from Third Harmonic Generation and
+Third Harmonic Scattering”, Optical Materials 84, 579-
+585 (2018).
+[40] D. Milam, ”Review and assessment of measured values of
+the nonlinear refractive-index coefficient of fused silica,”
+Appl. Opt. 37, 546-550 (1998).
+[41] D. Golde, T. Meier, and S. W. Koch, ”High harmon-
+ics generated in semiconductor nanostructures by the
+coupled dynamics of optical inter- and intraband exci-
+tations”, Phys. Rev. B 77, 075330 (2008).
+[42] I. Babushkin, L. Shi, A. Demircan, U. Morgner, J. Her-
+rmann, and A. Husakou, ”Metallic nanostructures as
+electronic billiards”, https://arxiv.org/abs/2104.14637
+[43] C. Li, D. Feng, T. Jia, H. Sun, X. Li, S. Xu, X. Wang,
+and Z. Xu, ”Ultrafast dynamics in ZnO thin films irradi-
+ated by femtosecond lasers”, Solid State Communications
+136, 389-394 (2005).
+[44] O. Khasanov, O. Fedotova, G. Rusetsky and V. Niki-
+forov, ”Echo-Spectroscopy of Nanocomposites”, Laser
+Physics 29, 124011 (2019).
+[45] O. Fedotova, A. Husakou, G. Rusetsky, O. Khasanov,
+A. Fedotov, T. Smirnova, U. Sapaev, and I. Babushkin,
+”Mechanisms of Terahertz Generation under Femtosec-
+ond Pulses propagation in Nanocomposites”, jsii 1 4,
+CLEO-Europe - European Quantum Electronics Confer-
+ence, Munich, Germany (2021).
+

b9E_T4oBgHgl3EQfzxyC/content/tmp_files/2301.08325v1.pdf.txt

@@ -0,0 +1,1411 @@
+Advanced Scaling Methods for VNF deployment with
+Reinforcement Learning
+Namjin Seoa, DongNyeong Heoa, Heeyoul Choia
+aHandong Global University, Pohang, 37554, Gyeongbuk, South Korea
+Abstract
+Network function virtualization (NFV) and software-defined network (SDN)
+have become emerging network paradigms, allowing virtualized network func-
+tion (VNF) deployment at a low cost. Even though VNF deployment can be
+flexible, it is still challenging to optimize VNF deployment due to its high
+complexity. Several studies have approached the task as dynamic program-
+ming, e.g., integer linear programming (ILP). However, optimizing VNF de-
+ployment for highly complex networks remains a challenge. Alternatively, re-
+inforcement learning (RL) based approaches have been proposed to optimize
+this task, especially to employ a scaling action-based method which can de-
+ploy VNFs within less computational time. However, the model architecture
+can be improved further to generalize to the different networking settings.
+In this paper, we propose an enhanced model which can be adapted to more
+general network settings. We adopt the improved GNN architecture and a
+few techniques to obtain a better node representation for the VNF deploy-
+ment task. Furthermore, we apply a recently proposed RL method, phasic
+policy gradient (PPG), to leverage the shared representation of the service
+function chain (SFC) generation model from the value function. We evaluate
+the proposed method in various scenarios, achieving a better QoS with min-
+imum resource utilization compared to the previous methods. Finally, as a
+qualitative evaluation, we analyze our proposed encoder’s representation for
+the nodes, which shows a more disentangled representation.
+Keywords:
+Network Function Virtualization, Software Defined Networking,
+Reinforcement Learning, Graph Neural Network
+Preprint submitted to
+January 23, 2023
+arXiv:2301.08325v1  [cs.NI]  19 Jan 2023
+
+1. Introduction
+Softwarization of the Internet network, such as software-defined network-
+ing (SDN) and network function virtualization (NFV), has emerged as a
+new network paradigm. In this paradigm, the network functions provided
+by hardware-based middleboxes (e.g., Network Address Translation (NAT),
+Firewall, Proxy) are replaced with virtualized network functions (VNFs),
+running on virtual machines as VNF instances. By decoupling the network
+functions from the hardware, NFV allows the network service providers to
+deploy VNFs with low capital expenses (CAPEX) and operating expenses
+(OPEX). In addition, the traffics and network devices are managed and
+monitored by NFV orchestration (NFVO) in the centralized NFV frame-
+work. Hence, the VNF instances can be deployed by NFVO adaptable to the
+traffic requirements, which is the VNF deployment task.
+Even though VNF deployment can be accomplished dynamically with
+flexibility, requirements for the VNF deployment task are also getting more
+complex.
+The first requirement is that the service function chain (SFC)
+should be generated efficiently while maintaining an acceptable quality of
+service (QoS). SFC requires the traffics of a request to be routed through
+multiple stages of VNFs in NFV. Fig. 1 shows an example of SFC where the
+SFC request (Req1) demands an SFC: “NAT (‘N’) → Firewall(‘F’)”. This
+SFC request should be sequentially processed at these VNF instances as it
+travels from its ingress to its egress.
+These types of VNF should be de-
+ployed, taking into account the path of the requests, to meet the pre-defined
+service-level agreement (SLA) for QoS. Another requirement is to optimize
+resource utilization while satisfying the QoS. Redundantly deployed VNF in-
+stances could preserve high QoS, but it incurs unnecessary operating costs
+and energy consumption. Therefore, it is essential to improve the QoS while
+maintaining minimized resource consumption for the efficient management
+of VNF deployment.
+To achieve such efficient management of VNF deployment, the existing
+works have exploited dynamic programming algorithms, like integer linear
+programming (ILP) [1, 2, 3]. However, even though the ILP-based approach
+exhibits acceptable performance in networks with a low level of complex-
+ity, its computational cost becomes too expensive as the network scales up.
+Therefore, the ILP-based approach is not suitable for large-scale networks
+and the dynamical adjustment for traffic lifespans.
+As an alternative solution, ML-based approaches have been proposed
+2
+
+Figure 1: Overview of VNF Deployment task: SFC requests are required to pass through
+the VNFs sequentially as well as satisfy the SLA. Optimal deployment (Right) takes
+into account the paths of SFC requests, while sub-optimal deployment (Left) is deployed
+regardless of the paths of SFC request. Red boxes indicate inefficiently deployed VNF
+instances, which are on the wrong path and redundant.
+employing deep learning models, and reinforcement learning (RL) for VNF
+deployment [4, 5, 6, 7]. Especially, an RL-based method for VNF deployment
+was proposed to scale in and out VNFs on the network nodes [8].
+They
+adapted the graph neural network (GNN) model and RL algorithms on top
+of previously deployed VNF instances. Their action controls the number of
+VNFs on each node by scaling in, scaling out, or doing nothing. It showed
+that the agent was able to make appropriate scaling decisions for all the nodes
+and VNF types with a single forwarding process, involving fewer computation
+times for VNF deployment compared to other ML-based VNF deployment
+approaches. However, their model could not obtain the node representations
+in a general deployment setting where the networks include some nodes which
+cannot have VNF instances (e.g., switches) deployed on them due to its model
+setting for the node features.
+In this paper, we propose to enhance the neural network model and adapt
+new learning techniques so that the model can be adapted to the general
+deployment setting.
+First, we redesign the GNN architecture, and adapt
+a few techniques motivated from other domains, such as natural language
+processing and image processing [9, 10, 11, 12, 13], to effectively obtain a
+node representation of the networking information for the VNF deployment
+task. As the new GNN-based architecture, we employ the graph attention
+network (GAT) [14] and separately process different types of nodes: VNF
+deployable or non-deployable nodes.
+This architecture allows an effective
+propagation of node information from the network with different types of
+nodes. In addition, we adapt the positional encoding [11] to preserve the
+3
+
+SFC
+Qos
+SFC
+Qos
+Delay > SLA
+SRC→NF→DST
+Delay < SLA
+F
+N
+F
+Npositional information in the node representations.
+Furthermore, we apply a recently proposed RL algorithm, phasic policy
+gradient (PPG) [15], which is a variation of the policy gradient algorithm to
+optimize the policy network efficiently. By jointly optimizing the objectives
+of the policy network and the value network, the policy network can share
+representation with the value network, which helps the policy network obtain
+a more effective representation for the VNF deployment task.
+In the experiment, our approach optimizes the policy network to obtain a
+higher reward with rapid convergence compared to the previous approaches.
+Also, it proves that our approach can work in various scenarios. In addition,
+we analyze the result of our approach to show the robustness in the various
+topology and compare the processing time with ILP to show the practicality
+for the dynamic adaptation of the traffic lifespans. Finally, as a qualitative
+evaluation, we analyze the representation from the encoder of the policy
+network, and it demonstrates that our method obtains more disentangled
+representations of nodes.
+2. Background
+In this section, we briefly review graph neural networks and reinforcement
+learning for VNF deployment.
+2.1. Graph Neural Network
+Graph neural networks (GNN) [16] were proposed to effectively handle
+the graph-structured data consisting of a set of nodes V = {v1, · · · , v|V|} and
+edges E = {ˆeij|vi, vj ∈ V} [17, 18, 19, 14, 20, 21]. GNNs aim to address
+graph-related tasks (e.g., node/edge/graph classification) in an end-to-end
+manner with neural network model [17]. In the training process, GNNs are
+trained to extract nodes’ representations which reflect the graph information
+for the target tasks.
+In GNNs, a graph is represented as the following two matrix forms: the
+adjacency matrix A and the node feature matrix X. The adjacency matrix
+A ∈ R|V|×|V| represents the connections between pairs of nodes. The node
+feature matrix X contains node features xi ∈ R|V|×d. With A, X is trans-
+formed into node representation H = [h1, ..., h|V|]. Finally, these node repre-
+sentations are adapted to the target task in an end-to-end manner, which is
+implemented with a multi-layer perceptron (MLP) and a softmax layer.
+4
+
+GNN models are designed under the assumption that a node represen-
+tation for the target task is trained by propagating node information to its
+neighbors [17]. There are many different types of GNN models depending on
+the way how the node information propagates, like recurrent connection, con-
+volution, or attention. Among many GNN models, we review and compare
+gated graph neural network (GGNN) and graph attention network (GAT)
+which are used in the baseline and our approach, respectively.
+2.1.1. Gated Graph Neural Network
+In gated graph neural network (GGNN) [19], propagation between neigh-
+bors is implemented as recurrent connection as in recurrent neural networks
+(RNNs), and the node representation is calculated iteratively by RNN-like
+updates. For the recurrent connection, they use gated recurrent unit (GRU)
+[9], and the node representations are updated as follows:
+h′
+i = GRU(hi,
+�
+j∈Ni
+Whj),
+(1)
+where GRU is the GRU-like update function, Ni are the neighbors of node
+vi, W is a weight matrix, and the initial node representation is set to X.
+The recurrent updates are repeated a fixed number of times. Then, node
+representations are fed into the output layers.
+2.1.2. Graph Attention Network
+Graph attention network (GAT) [14] was proposed to update node repre-
+sentations with attention mechanism [10], while node information is propa-
+gated through recurrent steps in GGNN. GAT computes the attention scores
+of its neighbors so that the node representations can be updated according
+to the different importance of its neighbors as follows:
+h′
+i =
+�
+j∈Ni∪{i}
+ˆαijWhi,
+(2)
+ˆαij =
+exp(σ(a⊤ [Whi||Whj||Weˆeij]))
+�
+j′∈Ni∪{i} exp(σ(a⊤ [Whi||Whj′||Weˆeij′])),
+(3)
+where αij is the attention score of node vj for node vi, and ˆeij is edge at-
+tributes between nodes vi and vj. W, We, and a are weight matrices, σ is a
+non-linear function, and [·||·] indicates concatenation.
+5
+
+2.2. Reinforcement Learning
+Reinforcement learning (RL) trains an agent to maximize the expected
+reward by interacting with the environment. It formulates the task as Markov
+decision process (MDP) described as a sequence of state st, action at, and
+reward rt over discrete time steps. At discrete time-step t, the agent observes
+the state st of the environment and takes an action at. Subsequently, the
+current state st transitions into the next state st+1 along with the transition
+probability P(st+1|st, at), and finally, the agent receives a reward rt+1. This
+process is repeated until the agent reaches the terminal state which ends one
+episode. As the agent experiences many episodes, the agent could maximize
+the return, Gt = rt+1 + γrt+2 + γ2rt+3 + · · · , where γ is the discount factor.
+To maximize Gt over an episode, the RL agent needs to update policy
+π(at|st), a probability distribution of possible actions given state st.
+To
+estimate how good the action is at discrete time t, the agent should learn the
+state value V (st) = E[Gt|st] (or action-state value Q(st, at) at action-state
+pair (at, st)). In other words, the goal of an RL agent is to optimize its π(a|s)
+based on the trained state value V (s) (or action value Q(s, a)) to maximize
+the expected return in any episode.
+2.2.1. Policy Gradient
+Policy gradient (PG) is an RL method to model policy π(a|s) as a param-
+eterized function with neural networks called the policy network πθ(a|s). It
+trains the parameter of the policy network optimizing the objective function
+defined as follows:
+LPG(θ) = Et
+�
+log πθ(at|st) ˆAt
+�
+,
+(4)
+where ˆAt is the advantage estimator. For example, the REINFORCE algo-
+rithm [22] trains the parameters of the policy network defining advantage
+estimator ˆAt as return Gt.
+2.2.2. Proximal Policy Optimization
+Proximal policy optimization (PPO) [23] is one of the actor-critic (AC)
+based RL algorithms [24] with the trust-region method [25].
+In addition
+to the policy network, the AC algorithm employs the value network Vθ(s)
+estimating the state value V (st) in order to mitigate the variance in the
+training process. Furthermore, PPO has an additional constraint on the step
+size of the policy update to prevent inappropriate update of parameters,
+6
+
+called the trust-region method [25, 26]. Let ˆrt(θ) be the probability ratio
+ˆrt(θ) =
+πθ(at|st)
+πθold(at|st), then PPO maximizes
+LPPO(θ) = Et
+�
+min(ˆrt(θ) ˆAt, clip(ˆrt(θ), 1 − ϵ, 1 + ϵ) ˆAt)
+�
+,
+(5)
+where clip(·, x, y) is an operation clamping in range over x and y, and ϵ
+is a hyper-parameter of the clipping. The clipping operation constrains the
+probability ratio ˆrt to be close to 1 so that the parameters cannot be updated
+radically.
+2.2.3. Phasic Policy Gradient
+The phasic policy gradient (PPG) [15] algorithm was proposed to share
+parameters between the policy and value networks in the AC algorithms.
+Even though parameter sharing allows both networks to learn features jointly
+with a higher level of sample reuses, it is unclear whether it optimizes effi-
+ciently both networks jointly or not. Thus, they employ two value networks
+to avoid conflicts between competing objectives of the policy and value net-
+works. The one is value network VθV (s) not sharing parameters, and the
+other shares the parameter, called the auxiliary value head Vθπ(s).
+The PPG algorithm divides the policy and value network training phases
+into policy and auxiliary phases. In the policy phase, the policy is optimized
+in the same manner as PPO. Then, in every NPPG
+th iterations, an auxiliary
+loss is minimized in the auxiliary phase, and the auxiliary loss is defined by
+Laux = Et
+�1
+2(Vθπ(st) − VθV (st))2
+�
++ βcloneEt [KL [πθold(·|st), πθ(·|st)]] ,
+(6)
+where βclone is the hyper-parameter controlling how much the original policy
+is preserved.
+2.3. Scaling Agent for VNF Deployment
+Scaling of VNF deployment monitors VNF instances deployed in a net-
+work and adjusts the number of instances to maintain reliable QoS at min-
+imized cost by scaling in (removing instances), keeping, and scaling out
+(adding new instances). An RL-based method was proposed for VNF scaling
+[8] to optimize QoS and resource utilization, simultaneously. They adapted
+the graph neural network (GNN) models and RL algorithms on top of pre-
+viously deployed VNF instances. The GNN-based model inputs the network
+7
+
+information, and then produces the probabilities of scaling actions: scale
+in/out and keep. The parameters of the model were updated by PG-based
+RL algorithms, such as REINFORCE [22] and PPO [23]. In this section, we
+briefly review the model of the policy network and the RL formulation in [8].
+2.3.1. GNN based Policy Network
+The policy network has an encoder-decoder architecture. The encoder re-
+ceives the network information (e.g., network topology, deployed VNFs, and
+SFC requests.) and outputs the node representation reflecting the network
+information.
+The decoder reads the representations and produces action
+probabilities for ‘scaling in/out and keep’ of nodes and VNF types. To com-
+pute node representations that reflect diverse network topologies, the encoder
+is designed based on GNN [27], which is implemented with Gated Graph
+Neural Network (GGNN) [19]. Additionally, the network topology-related
+knowledge is transferred from the trained model for the SFC task, and the
+parameters are frozen so that the model could obtain proper representations
+of the network information.
+2.3.2. RL formulation for VNF scaling
+In the RL formulation, the RL agent observes the VNF deployment and
+a set of SFC requests as the initial state and produces the scaling actions for
+all the nodes and VNF types from the policy network. Then, after scaling
+from the initial VNF deployment, the reward is computed 1, which is defined
+as follows:
+R = − 1
+Nreq
+�
+k
+δk
+δk
+SLA
+− α
+�
+vnf
+Nvnf,
+(7)
+where Nreq and Nvnf are the number of SFC requests and the instances of
+VNF type vnf ∈ {Firewall, NAT, · · · }. δk and δk
+SLA are the traversing delay
+and SLA of request k, and α is the coefficient of penalty for restriction of
+resource utilization. The RL scenario is illustrated in Fig. 2.
+3. Proposed Method
+In this paper, we propose to enhance the neural network model and adapt
+new learning techniques. First, we enhance the previous VNF scaling model
+1The traversing path of the requests is generated by GNN-based SFC model [27].
+8
+
+Figure 2: Overview of the RL pipeline for VNF scaling: 1) The initial deployment is ob-
+served as the state from the environment. 2) Then, the RL agent updates the deployment
+by scaling action. 3) Finally, the reward is given, and one episode ends.
+[8] so that the model can be adaptable to more general deployment settings
+which includes nodes which cannot have any VNF instances deployed on
+them. Then, we apply new RL algorithms like PPG to train the model more
+efficiently.
+3.1. Model Architecture
+As in [8], the encoder-decoder based model reads the network information
+and produces the action probabilities for all the nodes and VNF types. The
+pipeline of the model architecture is presented in Fig. 3.
+The encoder computes the node representations H(k) by forwarding the
+pair of the adjacency matrix A and node feature matrix X(k) for each SFC
+request, where X(k) = [x(k)
+1 , · · · , x(k)
+|V|] is the node feature matrix for SFC
+request k. Specifically, node vi’s feature is defined by
+x(k)
+i
+= [src(k)(i), N (k)
+i,Firewall, N (k)
+i,NAT, · · · , dst(k)(i)],
+(8)
+where src(k)(i) and dst(k)(i) indicate whether node vi is the ingress or egress
+of request k. N (k)
+i,Firewall, N (k)
+i,NAT, · · · are the instance numbers of VNF type
+vnf ∈ {Firewall, NAT, · · · } deployed on node vi if needed for request k,
+otherwise set to 0. After encoding all the SFC requests, the encoder outputs
+the sets of node representations {H(k)}. Then, {H(k)} are averaged into a
+single representation ¯H, which is broadcasted and concatenated with each
+VNF embeddings evnf ∈ {eFirewall, eNAT, · · · } to make target VNF represen-
+tations ¯hi,vnf at the target nodes.
+The RNN-based decoder consisting of GRU and MLP layers reads the
+target representation ¯hi,vnf at each decoding step i, then outputs the action
+probabilities of the target VNF type vnf on target node vi. The actions
+9
+
+GNN-based
+RNN-based
+Encoder
+Decoder
+Initial
+Scaled
+Compute
+Deployment
+Deployment
+QoS&Resource(a) Encoding
+(b) Decoding (Example of Firewall as target VNF type)
+Figure 3: The proposed model architecture and the pipeline. (a) Node representations
+{H(k)} are obtained from all the pairs of an SFC request and the VNF deployment,
+and the target representation ¯hi,vnf for the target VNF type vnf on target node vi is
+obtained by averaging the node representations {H(k)} and concatenating with target
+VNF embeddings evnf as well as a positional encoding li (“P.E.”). (b) The decoder reads
+the target representation ¯hi,vnf to make a decision for scaling action of the target at the
+decoding step i. Over every decoding step, the decoder generates actions for the target
+VNF type, and the iteration stops after the number of target nodes. Scaling for all the
+VNF types can be performed in parallel.
+include three types of scaling actions for the target: ‘scale-in’, ‘scale-keep’,
+and ‘scale-out’.
+Over every decoding step, the decoder generates actions
+for the target VNF type, and the iteration stops after the number of target
+nodes. Scaling for all the VNF types can be performed in parallel, with this
+single process.
+3.1.1. GAT-based Encoder
+We apply the graph attention network (GAT) for the encoder instead of
+GGNN [27]. GAT has several training advantages compared to GGNN for
+VNF deployment. First, GAT could effectively propagate the node informa-
+tion from the previous layers through the attention mechanism, while GGNN
+10
+
+Ex.)
+Reqk5→NEP→1
+X
+N
+X
+GAT-based
+Encoder
+d
+Auxiliary
+ModuleDecoder
+Decoder
+Decoder
+N
+N
+Erelies on RNN-like updates. Next, GAT updates the node representation with
+different importance of the neighbor nodes as the attention score, which is
+helpful to train the model for requests with multiple hops between ingress
+and egress. Due to the fact that the VNF deployment is highly related to
+the paths of the requests, it is essential to train the importance of neighbors
+on the paths.
+The input of the GAT encoder is the adjacency matrix A and the in-
+put node feature matrix X given the input pair of the VNF deployment
+and an SFC request.
+The encoder produces node representation matrix
+H =
+�
+h1, ..., h|V|
+�
+.
+The input node features and edge weights defined as
+link latency are first forwarded through a shared linear transformation, pa-
+rameterized by weight matrices W, We, where W, We are weight matrices
+for node features and edge latency.
+Then, self-attention is performed on
+the nodes, where attention scores αij between node vi and vj are computed
+only for neighbor nodes, which is defined by Eq. 3. In addition, we employ
+multi-head attention to stabilize the learning process [14]. Finally, the node
+representation hi is updated as a weighted sum with the attention scores over
+nodes.
+3.1.2. Auxiliary Encoding Module with Node Embedding
+In the general deployment setting, the nodes in the network can be classi-
+fied into 2 different types: non-resource-allocating nodes (“non-deployable”)
+and resource-allocating nodes (“deployable”). As defined in Eq. 8, node fea-
+ture xi contains the number of instances for each VNF type, which are always
+zero-valued for the non-deployable nodes (N (k)
+i,vnf = 0). Fig. 4a represents an
+example of node features.
+Given the different node types, the model cannot update enough the
+representation of non-deployable nodes since the zero-valued features cannot
+have enough attention, which could cause a bottleneck when propagating
+node information through these nodes. To overcome the issue, we designed
+an auxiliary encoding module that provides additional neighbor information
+to the GNN-based encoder, as illustrated in Fig. 4b. This information is then
+utilized by the next module of the encoder to complement for information
+propagation when updating the representation of deployable nodes.
+The auxiliary encoding module consists of vectorized embeddings and a
+GAT layer. We first initialize vectorized embeddings {n1, n2, · · · } randomly
+and assign them to each non-deployable node, and then update them along
+with other parameters. As these embeddings are optimized during the train-
+11
+
+(a)
+Illustrative
+example
+of
+how
+non-
+deployable nodes could be bottlenecks,
+which hinder the information propagation
+from node#5 to node#1.
+(b) The auxiliary module provides the neighboring infor-
+mation of non-deployable nodes so that the GNN-based
+encoder could update node representation efficiently.
+Figure 4: Example of the general deployment setting and architecture of the encoder.
+ing process, the model could provide the non-deployable nodes’ neighboring
+information to mitigate the bottleneck effect. We refer to these embeddings
+as node embeddings (‘N.E.’). In addition, we define and assign the same node
+embedding n0 to all deployable nodes, allowing the module to focus on non-
+deployable cases. Subsequently, we process these node embeddings through
+a GAT layer to further reflect relations between multi-hops neighbors. We
+implement the GAT layer of this auxiliary module with less parameters than
+the encoder’s GAT layers to avoid over-fitting.
+Finally, we concatenate the output of the auxiliary module and the output
+of the GNN-based encoder’s first layer (refer to Fig. 4b) and forward to the
+second layer of the encoder to get a final nodes representations Hk.
+3.1.3. Positional Encoding
+To pass spatial information of nodes, we employ positional encoding, as in
+the Transformer [11]. Positional encoding could alleviate the decoder’s bur-
+den of transferring knowledge from previous nodes to decode hidden states.
+Before forwarding the aggregated representation ¯H, the location vector li is
+computed with sine and cosine functions of different frequencies as in the
+Transformer, then concatenated with ¯hi.
+Finally, the decoder inputs the
+representation of node vi partitioned into three parts: aggregated node vi’s
+representation ¯hi, target VNF embeddings evnf, and location vector li.
+12
+
+Ex.)
+Reqk
+5 
+N
+1→1
+Bottleneck
+N
+P
+XE.
+#1
+0
+0
+1
+0
+0
+X#2
+0
+0
+0
+0
+0
+0
+GAT
+GAT
+H
+...
+1#5
+0
+0
+0
+0
+0
+Additional Info
+Node Feature Mat. Xk
+forefficientpropagation
+#1
+n 
+X#2 -
+ni
+X#3 >
+n2 
+GAT
+#5一
+no
+NodeEmb3.2. RL for VNF Deployment
+In this section, we discuss how we optimize the policy network with RL
+algorithms. We follow the setting for RL formation and the perturbation
+scenario as in [8]. In the perturbation scenario where the VNF deployment
+is perturbed from the optimal deployment, the RL agent needs to remove
+the redundant VNFs or add necessary VNF instances to reconstruct optimal
+deployment by making scaling decisions for all the nodes and VNF types in
+the current deployment.
+In the perturbation scenario, we optimize the proposed model as the
+policy network. Moreover, we propose an architecture of the value network
+to apply the PPG algorithm, with which the policy network can leverage the
+shared representation of the SFC generation from the value network. The
+new value network architecture is described in the next section.
+3.2.1. Auxiliary Value Head for PPG
+The PPG algorithm jointly optimizes the objectives of the policy network
+and value function, where the policy network and value function share the
+parameters. In the optimization process, the objective of the value function
+is to minimize the error of the estimation for the state value, and the value
+function extracts the useful representation for the estimation of the SFC
+generation. The policy network can incorporate the representation from the
+value function to achieve a better policy.
+However, to avoid a conflict between both objectives of the policy and
+value network, PPG utilizes the auxiliary value head Vaux(s) as well as the
+value network V (s) to estimate the state value. The value head shares the
+parameters with the policy network, while the value network is implemented
+with the separated parameters. Thus, we design these two value functions
+and then apply the PPG to update the policy network.
+Fig. 5 presents the architectures of the value network and the auxiliary
+value head. The value network is implemented as a 2-layer MLP forwarding
+the mean of the aggregated representations ¯H to produce the state value
+V (st). For the auxiliary value head, we design the architecture to reflect the
+action’s state value for the node at each decoding step. The value head is
+implemented as a 2-layer MLP which is connected to the GRU layer of the
+decoder of the policy network. The value head estimates Vi when the policy
+network produces the node vi’s action probabilities. Finally, the state value
+13
+
+Vaux(st) is computed by averaging ¯Vi over nodes. This can be formulated by
+Vaux(s) = 1
+|V|
+�
+vi∈V
+Vi,
+(9)
+where Vi = f aux(�
+vnf zi,vnf), zi,vnf is the output of the decoder’s GRU layer
+from target VNF representation ¯hi,vnf, and f aux is the MLP layer of the
+auxiliary value head.
+Figure 5: Architectures of the value network and the auxiliary value head: PPO optimizes
+the policy and the value networks (orange box). In addition, PPG optimizes the auxiliary
+value head (green box) sharing parameters with the policy network.
+We apply the PPG algorithms with the value network and the auxiliary
+value head. After one episode, the agent gets reward R from the updated
+deployment, and then it stores the tuple (R, st, πθ(at|st)) in the buffer. Then,
+for NPPO episodes, the agent updates the parameter of the policy network
+to optimize the surrogate objective defined in Eq. 5. At the same time,
+the RL agent updates the parameter of the value network to minimize the
+mean squared error (MSE) between the return and the estimated state value.
+Lastly, for NPPG episodes, the parameter of the auxiliary value head is up-
+dated to minimize the auxiliary loss, defined in Eq. 6, which jointly updates
+the parameter of the policy network.
+14
+
+GRU4. Experiments
+4.1. Dataset and Configuration
+In this section, we describe the experiment settings and configurations
+to train the proposed model. The dataset was created from two networks:
+Internet2 [28] and Mobile Edge Computing (MEC), represented in Fig. 6.
+In the Internet2 network, the topology consists of 12 nodes with 15 links.
+SFC requests are created by normalizing Abilene traffic matrices proposed
+by [4]. In the MEC network, the topology consists of 14 nodes with 13 links.
+We followed [4] for other configurations, and specifically for the generation
+of SFC requests, we set the SLA as 95% of the latency computed from the
+SFC path generated by ILP [1]. The SFC requests contain the ingress and
+egress nodes, and the SFC path includes 3 or 4 services out of 5 VNF types.
+(a) Internet2
+(b) MEC
+Figure 6: Internet2 and MEC networks in the experiment settings. The nodes with ‘X’
+are non-deployable nodes like switches.
+First, we generated the ILP-based deployment to determine the optimal
+number and location of VNF instances for the set of active requests at each
+interval. Then, we created a VNF deployment dataset of which each entry
+contains an ILP-based deployment and a set of SFC requests. The dataset
+was then divided into training, validation, and testing sets with the ratio
+of 8:1:1.
+Lastly, we implemented a simulation environment for the same
+topology with each network and calculated the latency of the requests in the
+same manner as [27].
+Then, we trained the models on perturbed deployments by perturbing the
+ILP-based deployment. To perturb the deployment, an integer noise -1, 0 or
++1 is added on nodes and VNF types, and these deployments are set as the
+15
+
+X
+70
+23
+86
+209
+26
+129
+54
+63
+58
+84
+36
+90
+X
+117
+6812000
+1000
+1000
+5000
+5000
+1000
+X
+100
+100
+100
+100
+100
+100
+100initial states. Furthermore, we made random and zero deployments to evalu-
+ate the generalizability of the models. We set the number of VNF instances
+as 0 or 1 for the random deployment and only 0 for the zero deployment.
+For the details of hyper-parameters, we mainly followed the setting of [27]
+to pre-train the encoder and the configurations shown in Table 1 to train the
+decoder. The decoder architecture is implemented with layer normalization,
+1-layer GRU with 64 hidden units.
+We use three linear layers with the
+same number of hidden units in the decoder for the value function. As the
+activation function, ReLU (Rectified Linear Unit) is used. On each epoch,
+we measured the average reward in the validation set, and after the training,
+we selected the final model with the best reward in the validation set.
+Table 1: Hyper-Parameters for Training.
+Parameter
+Value
+Learning rate (LR)2
+3e-4
+Decoder dim. (GRU)
+32
+Decoder dim. (decoder-layer1)
+32
+Decoder dim. (decoder-layer2)
+32
+Decoder dim. (VNF embedding)
+5
+Decoder dim. (Positional Encoding)
+4
+Discount factor (γ)
+0.995
+PPO value network dim. (layer1)
+128
+PPO value network dim. (layer2)
+64
+PPO aux. value head dim. (layer1)
+32
+PPO aux. value head dim. (layer2)
+32
+PPO epsilon (ϵ)
+0.2
+PPO epoch (K)
+4
+PPO minibatch size (M)
+4
+PPO interval & PPG interval (policy phase) (NP P O)
+16
+PPG interval (auxiliary phase) (NP P G)
+64
+PPG hyper-parameter (βclone)
+1
+4.2. Quantitative Results
+We evaluate the performance of the method in both networks: Internet2
+and MEC.
+Internet2 Network:
+Table 2 shows an ablation study on Internet2 where we trained the models
+with PPO and tested them. For the metric, we measured the average number
+16
+
+of VNF instances (‘Avg. #VNF’), the average of delay time, and the average
+SLA violation ratio (‘Avg. SLAV’), as well as a reward (Eq. 7). Furthermore,
+we train the agent with different coefficient value (α) of instance penalty in
+Eq. 7 to show the effect of restriction for the level of the resource utilization,
+which controls the trade-off between the resource utilization and the QoS. We
+trained the agent with different seeds three times and reported the averages.
+Models (SFC)
+α
+Reward
+Avg. #VNF
+Avg. Delay
+Avg. SLAV
+ILP (GGNN)
+-
+-0.922
+12.86
+575.57
+0.197
+ILP (GAT)
+-
+-0.881
+12.86
+545.42
+0.167
+ILP (GAT+N.E.)
+-
+-0.876
+12.86
+542.11
+0.162
+GGNN (GGNN)
+0.15
+-1.796
+19.89
+1158.01
+0.279
+GGNN (GGNN)
+0.20
+-1.877
+18.44
+1226.94
+0.260
+GGNN (GGNN)
+0.25
+-2.345
+16.57
+1580.09
+0.354
+GAT(GAT)
+0.15
+-1.376
+17.81
+868.16
+0.141
+GAT(GAT)
+0.20
+-1.580
+17.58
+1018.22
+0.179
+GAT(GAT)
+0.25
+-2.018
+16.40
+1343.65
+0.255
+GAT+N.E. (GAT+N.E.)
+0.15
+-1.062
+17.71
+641.21
+0.086
+GAT+N.E. (GAT+N.E.)
+0.20
+-1.314
+16.04
+836.31
+0.117
+GAT+N.E. (GAT+N.E.)
+0.25
+-1.345
+15.75
+860.86
+0.122
+GAT+P.E. (GAT)
+0.15
+-1.084
+16.59
+665.61
+0.093
+GAT+P.E. (GAT)
+0.20
+-1.283
+15.89
+815.00
+0.122
+GAT+P.E. (GAT)
+0.25
+-1.409
+15.17
+911.44
+0.135
+GAT+P.E.+N.E. (GAT+N.E.)
+0.15
+-1.029
+17.39
+619.91
+0.081
+GAT+P.E.+N.E. (GAT+N.E.)
+0.20
+-1.240
+15.85
+784.09
+0.108
+GAT+P.E.+N.E. (GAT+N.E.)
+0.25
+-1.499
+14.87
+978.89
+0.139
+Table 2: Comparison of different models in the Internet2 network. The models are used
+to adjust VNF deployment based on scaling in/out/keep, while the methods in the paren-
+theses indicate how the SFC path was created for evaluation.
+Each row indicates the method to make VNF deployments and the method
+to generate SFC paths. For example, the row “ILP (GAT)” shows the re-
+sults when VNFs were deployed by the ILP method and the deployments
+were evaluated on the path generated by GAT for SFC. For the RL models,
+the same encoders were used for the policy networks and for the SFC path
+generation, though positional encoding was not used for SFC.
+From the table, we can see that the GAT-based models (‘GAT’) outper-
+form the baseline models (i.e., GGNN-based models). For example, ‘GAT
+(GAT)’ decreases the SLA violation rates from 0.279 (by ‘GGNN(GGNN)’)
+to 0.141 when the coefficient α for the instance penalty is 0.15. Furthermore,
+it improved the level of QoS and decreased the delay time with less number of
+instances. We believe that GAT models can be trained more efficiently, while
+GGNN-based models are hard to be optimized on our deployment settings,
+including many non-deployable nodes.
+17
+
+Moreover, ‘GAT+N.E.’ models outperform GGNN as well as GAT mod-
+els.
+They decrease the SLA violation rate, the number of instances and
+the delay time. This means that the node embedding-based approach, de-
+scribed in Section 3.1.2 can get better representations separating the VNF
+deployable and non-deployable nodes. In addition, the positional encoding
+(‘GAT+P.E.’) shows an even further improvement on SLAV compared to
+the corresponding methods. It implies that the positional encoding provides
+more distinguishable information for the policy network to find a better pol-
+icy for deployment. Lastly, we trained GAT models with both positional en-
+coding and node embedding (GAT+P.E.+N.E.). This method increases the
+reward and SLAV, while keeping the same level of the other metrics (‘Avg.
+#VNF’ and ‘Avg. Delay’), compared to the GAT-based models using only
+the positional encoding.
+MEC network:
+We experimented in the MEC network, where 5 nodes out of a total of
+12 nodes are non-deployable.
+We train the models with PPG, and com-
+pare GAT-based models to ILP approaches, which is summarized in Table
+3. The reward is computed using the SFC model, which employs the same
+architecture as the policy network. In the table, we can see that Positional
+Encoding (‘GAT+P.E.’) significantly decreases the SLA violation rate, the
+number of instances as well as the delay time. For the node embedding-
+based approach, ‘GAT+P.E.+N.E.’ has similar performance as ‘GAT+P.E.’.
+It means node embedding does not have additional information compared
+to positional embedding. Actually, as shown in Fig. 6, MEC has a regu-
+lar structure separating deployable and non-deployable nodes, which might
+be a possible explanation of why ‘N.E.’ does not additionally increase the
+performance on top of ‘GAT+P.E.’
+For the GGNN-based approach, it does not optimize the reward effec-
+tively, because links of the MEC network contain high latency3 and high
+variance, which hinders updating node representation in the model. Espe-
+cially, GGNN uses a weighted sum of its node representation with the edge
+attribute when updating the node representation, which causes it to under-
+estimate of the information from nodes connected with high latency. On the
+other hand, the GAT-based approach employs the attention mechanism to
+3The edge attribute is set as the reciprocal of latency.
+18
+
+Models (SFC)
+α
+Reward
+Avg. #VNF
+Avg. Delay
+Avg. SLAV
+ILP (GAT)
+0
+-0.702
+13.00
+29097.86
+0.046
+ILP (GAT+N.E.)
+-
+-0.707
+13.00
+29282.68
+0.045
+GAT (GAT)
+0.15
+-1.658
+20.33
+74682.66
+0.121
+GAT (GAT)
+0.20
+-1.867
+18.75
+87157.00
+0.146
+GAT (GAT)
+0.25
+-2.136
+18.75
+97987.34
+0.174
+GAT+P.E. (GAT)
+0.15
+-0.887
+16.84
+37044.12
+0.052
+GAT+P.E. (GAT)
+0.20
+-0.956
+16.25
+41100.82
+0.060
+GAT+P.E. (GAT)
+0.25
+-1.316
+15.41
+58091.53
+0.096
+GAT+P.E.+N.E. (GAT+N.E.)
+0.15
+-0.915
+16.55
+38930.56
+0.055
+GAT+P.E.+N.E. (GAT+N.E.)
+0.20
+-1.119
+16.29
+48965.09
+0.075
+GAT+P.E.+N.E. (GAT+N.E.)
+0.25
+-1.350
+15.95
+60878.52
+0.099
+Table 3: Comparison of different models in the MEC network. The positional encoding
+helps the training process more effective, while the node embedding does not additionally
+increase the performance on top of ‘GAT+P.E.’.
+compute the different importance of its neighbors and the edges so that it
+could mitigate the underestimation problem.
+4.2.1. Comparison of Various RL algorithms
+Since we propose to use the PPG algorithms, we compare them to other
+RL algorithms in the Internet2 network. In the experiment, we trained the
+GAT-based models with positional encoding with the various RL algorithms.
+As a result, the PPG algorithms can obtain better rewards, compared to
+DQN and PPO. PPG decreases the SLA violation rate and the delay time
+with a similar number of VNFs as presented in Table 4.
+Method
+α
+Reward
+Avg. #VNF
+Avg. Delay
+Avg. SLAV
+DQN
+0.15
+-1.317
+32.00
+718.56
+0.112
+DQN
+0.20
+-1.311
+25.00
+765.22
+0.174
+DQN
+0.25
+-2.701
+21.90
+1793.06
+0.285
+PPO
+0.15
+-1.084
+16.59
+665.62
+0.093
+PPO
+0.20
+-1.283
+15.89
+815.00
+0.122
+PPO
+0.25
+-1.409
+15.17
+911.44
+0.135
+PPG
+0.15
+-1.053
+16.30
+644.77
+0.091
+PPG
+0.20
+-1.164
+16.19
+726.24
+0.101
+PPG
+0.25
+-1.353
+15.51
+868.61
+0.130
+Table 4: Comparison of the various RL algorithms. PPG can optimize the reward more
+effectively compared to DQN and PPO.
+Moreover, as shown in Fig.
+7, PPG can optimize the average loss of
+the policy network more effectively than PPO. We believe that the auxiliary
+19
+
+Figure 7: Comparison of policy loss (α = 0.2) in training process.
+The loss of PPG
+decreases faster than PPO after around 40k iterations.
+Initial Deployment
+Reward
+Avg. #VNF
+Avg. Delay
+Avg. SLAV
+ILP-perturbed
+-1.029
+17.03
+622.59
+0.087
+Random
+-1.060
+16.95
+646.12
+0.090
+Zero
+-1.060
+16.94
+646.00
+0.090
+Table 5: Evaluations from different initial deployments in testing for the trained GAT
+model with α = 0.2.
+value head helps the policy network converge more efficiently, even though
+parameter sharing might make the training slow at the beginning of the
+training process.
+4.2.2. Analysis with Random and Zero Deployments
+To analyze further the performance of the trained models, we trained the
+GAT-based model (α = 0.2) optimized with PPG as before, and tested the
+trained model on different settings, where initial deployment is random or no
+VNFs are deployed (random deployment or zero deployment). The experi-
+ment results are presented in Table 5, where “Random” and “Zero” initial
+deployments are compared to the ILP-perturbed initial deployment. As the
+result, the performance on both initial deployments shows the same level of
+QoS and resource utilization as the ILP-perturbed case. It demonstrates that
+our approach can work on any sub-optimal initial deployment. Furthermore,
+our method can deploy VNFs without the ILP-based initial deployment.
+20
+
+Avg. Policy Loss
+PPG
+ agent: PPO
+0
+-0.01
+-0.02
+-0.03
+-0.04
+-0.05
+Valid/iter
+20k
+40k
+60k
+80kMethod
+Avg. Time (sec)
+ILP
+14.06
+GAT
+0.431
+Table 6: Average of execution times of 10 VNF-deployments.
+Figure 8: The VNF deployment generated from the model (Left), target SFC requests
+(Middle), and generated paths (Right). Given the target SFC request, VNF deployment
+and path are generated by the proposed method and the SFC model.
+4.2.3. Execution Time
+To show how fast our method works, we measured the average execution
+times to make decisions for VNF deployments and compared the proposed
+GAT-based model to ILP. The execution time includes the VNF deployment
+actions by the agent as well as SFC path generation by the SFC model [27].
+Each approach makes 10 deployments on a machine with 24 x 12-Core AMD
+Ryzen 9 3900X CPUs. Table 6 presents the average results of each approach.
+Our method can process VNF deployment about 30 times faster than ILP.
+4.3. Qualitative Evaluation
+In this section, we discuss the quality of the results by our approach. We
+analyze the VNF deployment generated from the agent and plot t-SNE [29]
+of the representation computed from the encoder of the policy networks.
+4.3.1. Generated Deployment
+We plot the VNF deployment generated from the GAT-based model
+trained with PPG as presented in Fig. 8, which shows VNF deployment
+generated from the SFC requests, and paths generated for the requests by
+the SFC model [27]. The VNF deployment is generated with more than 20
+requests, so we omitted the other requests in the figure for simplicity.
+21
+
+Req1
+6 →
+个N
+个
+6→1→1→11→8 (275.0 / 735)
+NW
+Req2
+←m
+N
+个
+个山
+7
+3→10→4→4→7 (765.0 / 722)
+23
+86
+209
+26
+N
+Req3
+←
+个N
+H个山
+个
+7
+5→1→4→4→7 (420.0 /723)
+129
+63
+Req4
+2 →
+个N
+8个
+2→ 11→ 11→ 11→ 8 (172.0 / 729)
+主E
+W
+36
+90
+WNPN
+117
+189
+Req23
+S个d个个
+4→1→1→5 (225.0 /700)
+Req24
+个个M个F个N个
+0→1→1→1→4→4(179.0 /705)The most VNF instances are deployed in the middle of the shortest paths
+for the traffics. For example, the first traffic (‘Req1’) has node 6 and node 8
+as the ingress and egress, and is required to pass through NAT (‘N’), Firewall
+(‘F’), and IDS (‘I’). In the network, the traffic goes through NAT at node 1,
+Firewall at node 11, and IDS at node 8. This traffic takes 275 ms while SLA
+for the SFC path is 735 ms. That is, all the VNFs of ‘Req1’ are deployed on
+the shortest path of its ingress and its egress.
+In addition, the generated deployment needs to meet the QoS with the
+optimized amount of resources. As shown in Fig. 8, VNF instances are de-
+ployed on the intersection of the shortest paths for the traffics. For example,
+the VNF instances at node 1 process more than four requests, like NAT for
+‘Req1’, ‘Req2’, ‘Req23’ and ‘Req24’. That is, our approach can deploy the
+instances at the shared nodes of the paths for many requests, which can
+reduce the number of VNF instances.
+4.3.2. Node Representations by t-SNE
+In the section, to analyze how the network information is represented by
+the encoder, we plot the 2D representation of node embeddings by t-SNE [29]
+as in Fig. 9, which shows the representations from the GGNN-based encoder
+and our proposed GAT-based encoder.
+Each dot represents one node of
+network, and the nodes are located in the Internet2 network as shown in Fig.
+6a.
+(a) GGNN
+(b) GAT
+Figure 9: Node representations by t-SNE. The GAT-based model gets more disentangled
+representations, compared to the GGNN-based model.
+Even though the GGNN-based encoder has multiple clusters, the nodes
+within each cluster are not distinguishable. However, the GAT-based en-
+coder’s representations are disentangled so that each cluster contains only
+22
+
+0
+60
+1
+2
+40
+3
+4
+5
+20
+6
+0
+8
+9
+-20
+10
+-40
+-60
+-80
+-60
+-40
+-20
+0
+20
+40
+600
+60
+1
+2
+40
+3
+4
+20
+5
+6
+7
+0
+8
+9
+-20
+10
+11
+-40
+-60
+-80
+-60
+-40
+-20
+0
+20
+40
+60
+80one or two nodes. Furthermore, the representations of neighbors in the net-
+work are close to each other in the plot.
+For example, clusters for node
+8 (‘brown’) and node 2 (‘green’) are close to each other, which reflects its
+neighborhood with the same node type as presented in Fig. 6a. This demon-
+strates the representation by GAT-based model reflects the network topology
+effectively.
+5. Conclusion
+In this paper, we proposed enhanced models which can be adapted to
+more general network settings. We proposed an improved GNN architecture
+and a few techniques to obtain a better node representation for the VNF
+deployment task. Furthermore, we optimized the model with PPG, a variant
+policy gradient-based algorithm. In the experiment, we evaluated the pro-
+posed method in various scenarios, achieving a better QoS with minimum
+resource utilization compared to the previous methods. Finally, we analyzed
+the generated VNF deployment and compare the node representations of our
+model to the baseline.
+Acknowledgement
+This research was supported by Basic Science Research Program through
+the National Research Foundation of Korea funded by the Ministry of Edu-
+cation (NRF-2022R1A2C1012633), and by Institute for Information & com-
+munications Technology Promotion (IITP) grant funded by the Korea gov-
+ernment(MSIT) (No. 2018-0-00749, Development of virtual network man-
+agement technology based on artificial intelligence).
+References
+[1] M. F. Bari, S. R. Chowdhury, R. Ahmed, R. Boutaba, On orchestrating
+virtual network functions, in: 2015 11th International Conference on
+Network and Service Management (CNSM), 2015, pp. 50–56. doi:10.
+1109/CNSM.2015.7367338.
+[2] H. Moens, F. D. Turck, VNF-P: A model for efficient placement of virtu-
+alized network functions, in: 10th International Conference on Network
+and Service Management (CNSM) and Workshop, 2014, pp. 418–423.
+doi:10.1109/CNSM.2014.7014205.
+23
+
+[3] M. A. Abdelaal, G. A. Ebrahim, W. R. Anis, Efficient placement of
+service function chains in cloud computing environments, Electronics
+10 (3) (2021). doi:10.3390/electronics10030323.
+URL https://www.mdpi.com/2079-9292/10/3/323
+[4] S. Lange, A. Grigorjew, T. Zinner, P. Tran-Gia, M. Jarschel, A multi-
+objective heuristic for the optimization of virtual network function chain
+placement, in: 2017 29th International Teletraffic Congress (ITC 29),
+Vol. 1, IEEE, 2017, pp. 152–160.
+[5] M. Karimzadeh-Farshbafan, V. Shah-Mansouri, D. Niyato, A dynamic
+reliability-aware service placement for network function virtualization
+(nfv), IEEE Journal on Selected Areas in Communications 38 (2) (2020)
+318–333. doi:10.1109/JSAC.2019.2959196.
+[6] N. He, S. Yang, F. Li, S. Trajanovski, F. A. Kuipers, X. Fu, A-ddpg:
+Attention mechanism-based deep reinforcement learning for nfv, in:
+2021 IEEE/ACM 29th International Symposium on Quality of Service
+(IWQOS), 2021, pp. 1–10. doi:10.1109/IWQOS52092.2021.9521285.
+[7] P. Sun, J. Lan, J. Li, Z. Guo, Y. Hu, Combining deep reinforcement
+learning with graph neural networks for optimal VNF placement, IEEE
+Communications Letters 25 (1) (2021) 176–180. doi:10.1109/LCOMM.
+2020.3025298.
+[8] N. Seo, D. Heo, J. Hong, H.-G. Kim, J.-H. Yoo, J. W.-K. Hong, H. Choi,
+Updating VNF deployment with scaling actions using reinforcement al-
+gorithms (2022).
+[9] K. Cho, B. van Merrienboer, C¸. G¨ul¸cehre, F. Bougares, H. Schwenk,
+Y. Bengio, Learning phrase representations using RNN encoder-decoder
+for statistical machine translation, CoRR abs/1406.1078 (2014). arXiv:
+1406.1078.
+URL http://arxiv.org/abs/1406.1078
+[10] D. Bahdanau, K. Cho, Y. Bengio, Neural machine translation by jointly
+learning to align and translate (2014).
+doi:10.48550/ARXIV.1409.
+0473.
+URL https://arxiv.org/abs/1409.0473
+24
+
+[11] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez,
+L. Kaiser, I. Polosukhin, Attention is all you need, in: I. Guyon, U. V.
+Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, R. Garnett
+(Eds.), Advances in Neural Information Processing Systems, Vol. 30,
+Curran Associates, Inc., 2017.
+[12] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai,
+T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly,
+J. Uszkoreit, N. Houlsby, An image is worth 16x16 words:
+Trans-
+formers for image recognition at scale, CoRR abs/2010.11929 (2020).
+arXiv:2010.11929.
+URL https://arxiv.org/abs/2010.11929
+[13] H. Lee, H. Choi, Partitioning image representation in contrastive learn-
+ing (2022). doi:10.48550/ARXIV.2203.10454.
+URL https://arxiv.org/abs/2203.10454
+[14] P. Veliˇckovi´c, G. Cucurull, A. Casanova, A. Romero, P. Li`o, Y. Bengio,
+Graph Attention Networks, International Conference on Learning Rep-
+resentations (2018).
+URL https://openreview.net/forum?id=rJXMpikCZ
+[15] K. Cobbe, J. Hilton, O. Klimov, J. Schulman, Phasic policy gradient
+(2020). doi:10.48550/ARXIV.2009.04416.
+URL https://arxiv.org/abs/2009.04416
+[16] F. Scarselli, M. Gori, A. C. Tsoi, M. Hagenbuchner, G. Monfardini, The
+graph neural network model, IEEE Transactions on Neural Networks
+20 (1) (2009) 61–80.
+[17] J. Zhou, G. Cui, S. Hu, Z. Zhang, C. Yang, Z. Liu, L. Wang, C. Li,
+M. Sun, Graph neural networks: A review of methods and applications,
+AI Open 1 (2020) 57–81. doi:https://doi.org/10.1016/j.aiopen.
+2021.01.001.
+[18] T. N. Kipf, M. Welling, Semi-supervised classification with graph con-
+volutional networks (2016). doi:10.48550/ARXIV.1609.02907.
+URL https://arxiv.org/abs/1609.02907
+25
+
+[19] Y. Li, D. Tarlow, M. Brockschmidt, R. Zemel, Gated graph sequence
+neural networks (2015). doi:10.48550/ARXIV.1511.05493.
+URL https://arxiv.org/abs/1511.05493
+[20] J. Gilmer, S. S. Schoenholz, P. F. Riley, O. Vinyals, G. E. Dahl, Neural
+message passing for quantum chemistry (2017). doi:10.48550/ARXIV.
+1704.01212.
+URL https://arxiv.org/abs/1704.01212
+[21] X. Wang, H. Ji, C. Shi, B. Wang, P. Cui, P. Yu, Y. Ye, Heterogeneous
+graph attention network (2019). doi:10.48550/ARXIV.1903.07293.
+URL https://arxiv.org/abs/1903.07293
+[22] R. S. Sutton, D. McAllester, S. Singh, Y. Mansour, Policy gradient
+methods for reinforcement learning with function approximation, in:
+Proceedings of the 12th International Conference on Neural Informa-
+tion Processing Systems, NIPS’99, MIT Press, Cambridge, MA, USA,
+1999, p. 1057–1063.
+[23] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, O. Klimov, Proximal
+policy optimization algorithms (2017).
+doi:10.48550/ARXIV.1707.
+06347.
+URL https://arxiv.org/abs/1707.06347
+[24] V. Konda, J. Tsitsiklis, Actor-critic algorithms, in: S. Solla, T. Leen,
+K. M¨uller (Eds.), Advances in Neural Information Processing Systems,
+Vol. 12, MIT Press, 1999.
+[25] A. Conn, N. Gould, P. Toint, Trust Region Methods, MPS-SIAM Series
+on Optimization, Society for Industrial and Applied Mathematics, 2000.
+URL https://books.google.co.kr/books?id=5kNC4fqssYQC
+[26] J. Schulman, S. Levine, P. Moritz, M. I. Jordan, P. Abbeel, Trust region
+policy optimization (2015). doi:10.48550/ARXIV.1502.05477.
+URL https://arxiv.org/abs/1502.05477
+[27] D. Heo, S. Lange, H.-G. Kim, H. Choi, Graph neural network based
+service function chaining for automatic network control (2020). doi:
+10.48550/ARXIV.2009.05240.
+URL https://arxiv.org/abs/2009.05240
+26
+
+[28] Internet2 research network topology and traffic matrix.
+URL
+https://www.cs.utexas.edu/$\sim$yzhang/research/
+AbileneTM/
+[29] L. van der Maaten, G. Hinton, Visualizing data using t-SNE, Journal of
+Machine Learning Research 9 (86) (2008) 2579–2605.
+URL http://jmlr.org/papers/v9/vandermaaten08a.html
+27
+