diff --git "a/PNFJT4oBgHgl3EQfIiwq/content/tmp_files/load_file.txt" "b/PNFJT4oBgHgl3EQfIiwq/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/PNFJT4oBgHgl3EQfIiwq/content/tmp_files/load_file.txt" @@ -0,0 +1,1629 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf,len=1628 +page_content='Graph Scattering beyond Wavelet Shackles Christian Koke Technical University of Munich & Ludwig Maximilian University Munich christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='koke@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='de Gitta Kutyniok Ludwig Maximilian University Munich & University of Tromsø kutyniok@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='de Abstract This work develops a flexible and mathematically sound framework for the design and analysis of graph scattering networks with variable branching ratios and generic functional calculus filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Spectrally-agnostic stability guarantees for node- and graph-level perturbations are derived;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' the vertex-set non-preserving case is treated by utilizing recently developed mathematical-physics based tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Energy propaga- tion through the network layers is investigated and related to truncation stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' New methods of graph-level feature aggregation are introduced and stability of the resulting composite scattering architectures is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Finally, scattering transforms are extended to edge- and higher order tensorial input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Theoretical results are complemented by numerical investigations: Suitably chosen scattering networks conforming to the developed theory perform better than traditional graph- wavelet based scattering approaches in social network graph classification tasks and significantly outperform other graph-based learning approaches to regression of quantum-chemical energies on QM7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 1 Introduction Euclidean wavelet scattering networks [22, 4] are deep convolutional architectures where output- features are generated in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Employed filters are designed rather than learned and derive from a fixed (tight) wavelet frame, resulting in a tree structured network with constant branching ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Such networks provide state of the art methods in settings with limited data availability and serve as a mathematically tractable model of standard convolutional neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Rigorous investigations — establishing remarkable invariance- and stability properties of wavelet scattering networks — were initially carried out in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The extensive mathematical analysis [38] generalized the term ’scattering network’ to include tree structured networks with varying branching rations and frames of convolutional filters, thus significantly narrowing the conceptual gap to general CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' With increasing interest in data on graph-structured domains, well performing networks generalizing Euclidean CNNs to this geometric setting emerged [18, 5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If efficiently implemented, such graph convolutional networks (GCNs) replace Euclidean convolutional filters by functional calculus filters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' scalar functions applied to a suitably chosen graph-shift-oprator capturing the geometry of the underlying graph [18, 14, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Almost immediately, proposals aimed at extending the success story of Euclidean scattering networks to the graph convolutional setting began appearing: In [48], the authors utilize dyadic graph wavelets (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [14]) based on the non-normalized graph Laplacian resulting in a norm preserving graph wavelet scattering transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In [10], diffusion wavelets (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [8]) are used to construct a graph scattering transform enjoying spectrum-dependent stability guarantees to graph level perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For scattering transforms with N layers and K distinct functional calculus filters, the work [11] derives node-level stability bounds of OpKN{2q and conducts corresponding numerical experiments choosing diffusion wavelets, monic cubic wavelets [14] and tight Hann wavelets [35] as filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In [12] the authors, following [8], construct so called geometric wavelets and establish the expressivity of a scattering transform based on such a frame through extensive numerical Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='11456v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='LG] 26 Jan 2023 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' A theoretical analysis of this and a closely related wavelet based scattering transform is the main focus of [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Additionally, graph-wavelet based scattering transforms have been extended to the spatio-temporal domain [27], utilized to overcome the problem of oversmoothing in GCNs [25] and pruned to deal with their exponential (in network depth) increase in needed resources [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Common among all these contributions is the focus on graph wavelets, which are generically understood to derive in a scale-sampling procedure from a common wavelet generating kernel function g : R Ñ R satisfying various properties [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Established stability or expressivity properties — especially to structural perturbations — are then generally linked to the specific choice of the wavelet kernel g and utilized graph shift operator [10, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This severely limits the diversity of available filter banks in the design of scattering networks and draws into question their validity as models for more general GCNs whose filters generically do not derive from a wavelet kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' A primary focus of this work is to provide alleviation in this situation: After reviewing the graph signal processing setting in Section 2, we introduce a general framework for the construction of (generalized) graph scattering transforms beyond the wavelet setting in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Section 4 establishes spectrum- agnostic stability guarantees on the node signal level and for the first time also for graph-level perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To handle the vertex-set non-preserving case, a new ’distance measure’ for operators capturing the geometry of varying graphs is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' After providing conditions for energy decay (with the layers) and relating it to truncation stability, we consider graph level feature aggregation and higher order inputs in Sections 5 and 6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In Section 7 we then provide numerical results indicating that general functional calculus filter based scattering is at least as expressive as standard wavelet based scattering in graph classification tasks and outperforms leading graph neural network approaches to regression of quantum chemical energies on QM7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2 Graph Signal Processing Taking a signal processing approach, we consider signals on graphs as opposed to graph embeddings: Node-Signals: Given a graph pG, Eq, we are primarily interested in node-signals, which are functions from the node-set G to the complex numbers, modelled as elements of C|G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We equip this space with an inner product according to xf, gy “ ř|G| i“1 figiµi (with all vertex weights µi ě 1) and denote the resulting inner product space by ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We forego considering arbitrary inner products on C|G| solely in the interest of increased readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Functional Calculus Filters: Our fundamental objects in investigating node-signals will be func- tional calculus filters based on a normal operator ∆ : ℓ2pGq Ñ ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Prominent examples include the adjacency matrix W, the degree matrix D, normalized p1 ´ D´ 1 2 WD´ 1 2 q or un-normalized (L :“ D´W) graph Laplacians Writing normalized eigenvalue-eigenvector pairs of ∆ as pλi, φiq|G| i“1, the filter obtained from applying g : C Ñ C is given by gp∆qf “ ř|G| i“1 gpλiqxφi, fyℓ2pV qφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The operator we utilize in our numerical investigations of Section 6, is given by L :“ L{λmaxpLq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We divide by the largest eigenvalue to ensure that the spectrum σpL q is contained in the interval r0, 1s, which aids in the choice of functions from which filters are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Generalized Frames: We are most interested in filters that arise from a collection of functions adequately covering the spectrum of the operator to which they are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To this end we call a collection tgip¨quiPI of functions a generalized frame if it satisfies the generalized frame condition A ď ř iPI |gipcq|2 ď B for any c in C for constants A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' B ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As proved in Appendix B, this condition is sufficient to guarantee that the associated operators form a frame: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let ∆ : ℓ2pGq Ñ ℓ2pGq be normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If the family tgip¨quiPI of bounded functions satisfies A ď ř iPI |gipcq|2 ď B for all c in the spectrum σp∆q, we have (@f P ℓ2pGq) A}f}2 ℓ2pGq ď ÿ iPI }gip∆qf}2 ℓ2pGq ď B}f}2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Notably, the functions tgiuiPI need not be continuous: In fact, in our numerical implementations, we will – among other mappings – utilize the function δ0p¨q, defined by δ0p0q “ 1 and δ0pcq “ 0 for c ‰ 0 as well as a modified cosine, defined by cosp0q “ 0 and cospcq “ cospcq for c ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2 3 The Generalized Graph Scattering Transform A generalized graph scattering transform is a non-linear map Φ based on a tree structured multilayer graph convolutional network with constant branching factor in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For an input signal f P ℓ2pGq, outputs are generated in each layer of such a scattering network, and then concatenated to form a feature vector in a feature space F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The network is built up from three ingredients: Connecting Operators: To allow intermediate signal representations in the ’hidden’ network layers to be further processed with functional calculus filters based on varying operators, which might not all be normal for the same choice of node-weights, we allow these intermediate representations to live in varying graph signal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In fact, we do not even assume that these signal spaces are based on a common vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This is done to allow for modelling of recently proposed networks where input- and ’processing’ graphs are decoupled (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [1, 36]), as well as architectures incorporating graph pooling [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Instead, we associate one signal space ℓ2pGnq to each layer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Connecting operators are then (not necessarily linear) operators Pn : ℓ2pGn´1q Ñ ℓ2pGnq connecting the signal spaces of subsequent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We assume them to be Lipschitz continuous (}Ppfq ´ Ppgq}ℓ2pGn´1q ď R`}f ´ g}ℓ2pGnqq and triviality preserving (Pp0q “ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For our original node-signal space we also write ℓ2pGq ” ℓ2pG0q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Non-Linearities: To each layer, we also associate a (possibly) non-linear function ρn : C Ñ C acting poinwise on signals in ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Similar to connecting operators, we assume ρn preserves zero and is Lipschitz-continuous with Lipschitz constant denoted by L` n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This definition allows for the absolute value non-linearity, but also ReLu or – trivially – the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Operator Frames: Beyond these ingredients, the central building block of our scattering architecture is comprised of a family of functional calculus filters in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' That is, we assume that in each layer, the node signal space ℓ2pGnq carries a normal operator ∆n and an associated collection of functions comprised of an output generating function χnp¨q as well as a filter bank tgγnp¨quγnPΓn indexed by an index set Γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As the network layer n varies (and in contrast to wavelet-scattering networks) we allow the index set Γn as well as the collection tχnp¨qu Ťtgγnp¨quγnPΓn of functions to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We only demand that in each layer the functions in the filter bank together with the output generating function constitute a generalized frame with frame constants An, Bn ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We refer to the collection of functions ΩN :“ pρn, tχnp¨qu Ťtgγnp¨quγnPΓnqN n“1 as a mod- ule sequence and call DN :“ pPn, ∆nqN n“1 our operator collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The generalized scattering transform is then constructed iteratively: Figure 1: Schematic Scattering Architecture To our initial signal f P ℓ2pGq we first apply the connecting operator P1, yielding a signal rep- resentation in ℓ2pG1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Subsequently, we apply the pointwise non-linearity ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we apply our graph filters tχ1p∆1qu Ťtgγ1p∆1quγ1PΓ1 to ρ1pP1pfqq yielding the output V1pfq :“ χ1p∆1qρ1pP1pfqq as well as the intermedi- ate hidden representations tU1rγ1spfq :“ gγ1p∆1qρ1pP1pfqquγ1PΓ1 obtained in the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Here we have introduced the one-step scattering propagator Unrγns : ℓ2pGn´1q Ñ ℓ2pGnq mapping f ÞÑ gγnp∆nqρnpPnpfqq as well as the output generating operator Vn : ℓ2pGn´1q Ñ ℓ2pGnq mapping f to χnp∆nqρnpPnpfqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Upon defining the set ΓN´1 :“ ΓN´1 ˆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' ˆ Γ1 of paths of length pN ´ 1q terminating in layer N ´ 1 (with Γ0 taken to be the one-element set) and iterating the above procedure, we see that the outputs gener- ated in the N th-layer are indexed by paths ΓN´1 terminating in the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 3 p3(P3() (△2) p2(P2()) qa2 9b2 (△2) P3(P3() X2(△2) gal P3(P3()) (△2) IP1(Pi())/gbr (△1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [p2(P2()) ga2 9b2 (2) m P3(P3()) l2(G) gc1 (△1 X2(△2) p3(P3()) (△2) p2(P2()) qa2 9b2 (△2 p3(P3()) X1(△1) X2(△2) l2(G1) l2(G2) l2(G3)Outputs generated in the N th layer are thus given by tVN ˝UrγN´1s˝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝Urγ1spfqupγN´1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',γ1qPΓN´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Concatenating the features obtained in the various layers of a network with depth N, our full feature vectors thus live in the feature space FN “ ‘N n“1 ` ℓ2pGnq ˘|Γn´1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (1) The associated canonical norm is denoted } ¨ }FN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For convenience, a brief review of direct sums of spaces, their associated norms and a discussion of corresponding direct sums of maps is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We denote the hence constructed generalized scattering transform of length N, based on a module sequence ΩN and operator collection DN by ΦN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In our numerical experiments in Section 7, we consider two particular instantiations of the above general architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In both cases the utilized shift-operator is L :“ L{λmaxpLq, node weights satisfy µi “ 1, the branching ratio in each layer is chosen as 4 and the depth is set to N “ 4 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The connecting operators are set to the identity and non-linearities are set to the modulus (|¨|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The two archi- tectures differ in the utilized filters, which are repeated in each layer and depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Postponing a discussion of other parameter- choices, we note here that the filters tsinpπ{2¨q, cospπ{2¨qu provide a high and a low pass filter on the spectrum σpL q Ď r0, 1s, while tsinpπ¨q, cospπ¨qu provides a spectral refinement of the former two filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The inner two elements of the filter bank in Architecture II thus separate an input signal into high- and low-lying spectral Figure 2: Filters of tested Ar- chitectures components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The outer two act similarly at a higher spectral scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Additionally Architecture I – utilizing cos and δ0 as introduced Section 2 – prevents the lowest lying spectral information from propagating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Instead it is extracted via δ0p¨q in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Note that Id arises from applying the constant-1 function to L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Normalizations are chosen to generate frames with upper bounds B ž 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 4 Stability Guarantees In order to produce meaningful signal representations, a small change in input signal should produce a small change in the output of our generalized scattering transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This property is captured in the result below, which is proved in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' With the notation of Section 3, we have for all f, h P ℓ2pGq: }ΦNpfq ´ ΦNphq}FN ď ˜ 1 ` N ÿ n“1 maxtrBn ´ 1s, rBnpL` n R` n q2 ´ 1s, 0u n´1 ź k“1 Bk ¸ 1 2 }f ´ h}ℓ2pGq In the case where upper frame bounds Bn and Lipschitz constants L` n and R` n are all smaller than or equal one, this statement reduces to the much nicer inequality: }ΦNpfq ´ ΦNphq}FN ď }f ´ h}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (2) Below, we always assume R` n , L` n ď 1 as this easily achievable through rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We will keep Bn variable to demonstrate how filter size influences stability results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As for our experimentally tested architectures (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2), we note for Architecture I that Bn “ 1{2 for all n, so that (2) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For Architecture II we have Bn “ 3, which yields a stability constant of ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 1 ` 2 ¨ 3 ` 2 ¨ 32 ` 2 ¨ 33 “ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Similar to other constants derived in this section, this bound is however not necessarily tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Operators capturing graph geometries might only be known approximately in real world tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' if edge weights are only known to a certain level of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Hence it is important that our scattering representation be insensitive to small perturbations in the underlying normal operators in each layer, which is captured by our next result, proved in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Smallness here is measured in Frobenius norm } ¨ }F , which for convenience is briefly reviewed in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let ΦN and rΦN be two scattering transforms based on the same module sequence ΩN and operator sequences DN, r DN with the same connecting operators (Pn “ rPn) in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume R` n , L` n ď 1 and Bn ď B for some B and n ď N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume that the respective normal operators satisfy }∆n ´ r∆n}F ď δ for some δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Further assume that the functions 4 () ()SO0 cOs() cos() sin() sin() sin(π) () Id Architecture I Architecture IItgγnuγnPΓn and χn in each layer are Lipschitz continuous with associated Lipschitz constants satisfying L2 χn ` ř γnPΓn L2 gγn ď D2 for all n ď N and some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we have }rΦNpfq ´ ΦNpfq}FN ď b 2p2N ´ 1q ¨ b pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq for all f P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If B ď 1{2, the stability constant improves to a 2p1 ´ BNq{p1 ´ Bq ¨ D ď 2 ¨ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The condition B ď 1 2 is e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' satisfied by our Architecture I, but –strictly speaking– we may not apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2, since not all utilized filters are Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Remark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3 in Appendix D however shows, that the above stability result remains applicable for this architecture as long as we demand that ∆ and r∆ are (potentially rescaled) graph Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For Architecture II we note that D “ π ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 10{2 and thus the stability constant is given by a 2p24 ´ 1q ¨ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 33 ¨ π ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 10{2 “ 45π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We are also interested in perturbations that change the vertex set of the graphs in our architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This is important for example in the context of social networks, when passing from nodes representing individuals to nodes representing (close knit) groups of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To investigate this setting, we utilize tools originally developed within the mathematical physics community [29]: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let H and r H be two finite dimensional Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let ∆ and r∆ be normal operators on these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let J : H Ñ r H and rJ : r H Ñ H be linear maps — called identification operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We call the two spaces δ-quasi-unitarily-equivalent (with δ ě 0) if for any f P H and u P r H we have }Jf} r H ď 2}f}H, }pJ ´ rJ˚qf} r H ď δ}f}H, }f ´ rJJf}H ď δ b }f}2 H ` xf, |∆| fyH, }u ´ J rJu} r H ď δ b }u}2 r H ` xu, |r∆| uy r H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If, for some w P C the resolvent R :“ p∆ ´ ωq´1 satisfies }p rRJ ´ JRqf} r H ď δ}f}H for all f P H, we say that ∆ and r∆ are ω-δ-close with identification operator J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Absolute value |∆| and adjoint rJ˚ of operators are briefly reviewed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' While the above definition might seem fairly abstract at first, it is in fact a natural setting to investigate structural perturbations as Figure 3 exemplifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In our current setting, the Hilbert spaces in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3 are node-signal spaces H “ ℓ2pGq, r H “ ℓ2p rGq of different graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The notion of ω-δ-closeness is then useful, as it allows to compare filters defined on different graphs but obtained from applying the same function to the respective graph-operators: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In the setting of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3 let ∆ and r∆ be ω-δ- close and satisfy }∆}op, }r∆}op ď K for some K ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If g : C Ñ C is holomorphic on the disk BK`1p0q of radius pK ` 1q, there is a constant Cg ě 0 so that }gpr∆qJ ´ Jgp∆q}op ď Cg ¨ δ with Cg depending on g, ω and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' An explicit characterization of Cg together with a proof of this result is presented in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4 is our main tool in establish- ing our next result, proved in Appendix G, which captures stability under vertex-set non-preserving perturbations: Figure 3: Prototypical Exam- ple of δ-unitary-equivalent Node Signal Spaces with p´1q-12δ-close Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Details in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let ΦN, rΦN be scattering transforms based on a common module sequence ΩN and differing operator sequences DN, r DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume R` n , L` n ď 1 and Bn ď B for some B and n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume that there are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq, rJn : ℓ2p rGnq Ñ ℓ2pGnq (0 ď n ď N) so that the respective signal spaces are δ-unitarily equivalent, the respective normal operators ∆n, r∆n are ω-δ-close as well as bounded (in norm) by K ą 0 and the connecting operators satisfy } rPnJn´1f ´ JnPnf}ℓ2p r Gnq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For the common module sequence ΩN assume that the non-linearities satisfy }ρnpJnfq ´ Jnρnpfq}ℓ2p r Gnq “ 0 and that the constants Cχn and 5 tCgγn uγnPΓN associated through Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4 to the functions of the generalized frames in each layer satisfy C2 χn ` ř γnPΓN C2 gγn ď D2 for some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Denote the operator that the family tJnun of identification operators induce on FN through concatenation by JN : FN Ñ Ă FN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then, with KN “ a p2N ´ 1q2D2 ¨ BN´1 if B ą 1{2 and KN “ a 2D2 ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{2: }rΦNpJ0fq ´ JNΦNpfq} Ă FN ď KN ¨ δ ¨ }f}ℓ2pG, @f P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The stability result persists with slightly altered stability constants, if identification operators only almost commute with non-linearities and/or connecting operators, as Appendix G further elucidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5 is not applicable to Architecture I, where filters are not all holomorphic, but is directly applicable to Architecture II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Stability constants can be calculated in terms of D and B as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Beyond these results, stability under truncation of the scattering transform is equally desirable: Given the energy WN :“ ř pγN,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',γ1qPΓN }UrγNs ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' ˝ Urγ1spfq}2 ℓ2pGNq stored in the network at layer N, it is not hard to see that after extending ΦNpfq by zero to match dimensions with ΦN`1pfq we have }ΦNpfq ´ ΦN`1pfq}2 FN`1 ď ` R` N`1L` N`1 ˘2 BN`1 ¨ WN (see Appendix H for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' A bound for WN is then given as follows: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let Φ8 be a generalized graph scattering transform based on a an operator sequence D8 “ pPn, ∆nq8 n“1 and a module sequence Ω8 with each ρnp¨q ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume in each layer n ě 1 that there is an eigenvector ψn of ∆n with solely positive entries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' denote the smallest entry by mn :“ miniPGn ψnris and the eigenvalue corresponding to ψn by λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Quantify the ’spectral-gap’ opened up at this eigenvalue through neglecting the output-generating function by ηn :“ ř γnPΓn |gγnpλnq|2 and assume Bnmn ě ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We then have (with C` N :“ śN i“1 max ␣ 1, BipL` i R` i q2( ) WNpfq ď C` N ¨ « N ź n“1 ˆ 1 ´ ˆ mn ´ ηn Bn ˙˙ff ¨ }f}2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (3) The product in (3) decays if C` N Ñ C` converges and řN n“1pmn ´ ηn{Bnq Ñ 8 diverges as N Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The positivity-assumptions on the eigenvectors ψn can e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' always be ensured if they are chosen to lie in the lowest lying eigenspace of a graph Laplacian or normalized graph Laplacian (irrespective of the connectedness of the underlying graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As an example, we note that if we extend our Architecture I to infinite depth (recall from Section 3 that we are using the same filters, operators, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' in each layer) we have upon choosing λn “ 0 and ψn to be the constant normalized vector that ηn “ 0, CN “ 1 and mn “ 1{ a |G|, for a graph with |G| vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' On a graph with 16 vertices, we then e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' have WN ď p3{4qN}f}2 ℓ2pGq and thus }ΦNpfq ´ ΦN`1pfq}FN`1 ď p3{4qN ¨ }f}2 ℓ2pGq{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As detailed in Appendix H, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='6 also implies that under the given assumptions the scattering transform has trivial ’kernel’ for N Ñ 8, mapping only 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 5 Graph-Level Feature Aggregation To solve tasks such as graph classification or regression over multiple graphs, we need to represent graphs of varying sizes in a common feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Given a scattering transform ΦN, we thus need to find a stability preserving map from the feature space FN to some Euclidean space that is independent of any vertex set cardinalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Since FN is a large direct sum of smaller spaces (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (1)), we simply construct such maps on each summand independently and then concatenate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' General non-linear feature aggregation: Our main tool in passing to graph-level features is a non-linear map N G p : ℓ2pGq Ñ Rp given as N G p pfq “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='pp}f}ℓ1pGq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='µG, }f}ℓ2pGq, }f}ℓ3pGq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', }f}ℓppGqqJ, (4) with µG :“ ř iPG µi and }f}ℓqpGq :“ př iPG |fi|qµiq1{q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Our inspiration to use this map stems from the standard case where all µi “ 1: For p ě |G|, the vector |f| “ pp|f1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', |fG|qJ can then be recovered from N G p pfq up to permutation of indices [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Hence, employing N G p (with p ě |G|) to aggregate node-information into graph-level information, we lose the minimal necessary information 6 about node permutation (clearly N G p pfq “ N G p pΠfq for any permutation matrix Π) and beyond that only information about the complex phase (respectively the sign in the real case) in each entry of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Figure 4: Graph Level Scattering Given a scattering transform ΦN mapping from ℓ2pGq to the feature space FN “ ‘N n“1 ` ℓ2pGnq ˘|Γn´1|, we ob- tain a corresponding map ΨN mapping from ℓ2pGq to RN “ ‘N n“1 pRpnq|Γn´1| by concatenating the feature map ΦN with the operator that the family of non-linear maps tN pn GnuN n“1 induces on FN by concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Simi- larly we obtain the map rΨN : ℓ2p rGq Ñ RN by concatenat- ing the map rΦN : ℓ2p rGq Ñ ‘N n“1 ´ ℓ2p rGnq ¯|Γn´1| with the operator induced by the family tN pn r GnuN n“1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The feature space RN is completely determined by path-sets ΓN and used maximal p-norm indices pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It no longer depends on cardinalities of vertex sets of any graphs, allowing to compare (signals on) varying graphs with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Most of the results of the previous sections then readily transfer to the graph-level-feature setting (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Low-pass feature aggregation: The spectrum-free aggregation scheme of the previous paragraph is especially adapted to settings where there are no high-level spectral properties remaining constant under graph perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' However, many commonly utilized operators, such as normalized and un-normalized graph Laplacians, have a somewhat ’stable’ spectral theory: Eigenvalues are always real, non-negative, the lowest-lying eigenvalue equals zero and simple (if the graph is connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In this section we shall thus assume that each mentioned normal operator ∆n (r∆n) has these spectral properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We denote the lowest lying normalized eigenvector (which is generically determined up to a complex phase) by ψ∆n and denote by M |x¨,¨y| Gn : ℓ2pGnq Ñ C the map given by M |x¨,¨y| Gn pfq “ |xψ∆n, fyℓ2pGnq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The absolute value around the inner product is introduced to absorb the phase- ambiguity in the choice of ψ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Given a scattering transform ΦN mapping from ℓ2pGq to the feature space FN, we obtain a corresponding map Ψ|x¨,¨y| N mapping from ℓ2pGq to CN “ ‘N n“1C|Γn´1| by concatenating the feature map ΦN with the operator that the family of maps tM |x¨,¨y| Gn uN n“1 induces on FN by concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As detailed in Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2, this map inherits stability properties in complete analogy to the discussion of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 6 Higher Order Scattering Node signals capture information about nodes in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' However, one might be interested in binary, ternary or even higher order relations between nodes such as distances or angles in graphs representing molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In this section we focus on binary relations – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' edge level input – as this is the instantiation we also test in our regression experiment in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 provides more details and extends these considerations beyond the binary setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We equip the space of edge inputs with an inner product according to xf, gy “ ř|G| i,j“1 fijgijµij and denote the resulting inner-product space by ℓ2pEq with E “ GˆG the set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Setting e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' node-weights µi and edge weights µik to one, the adjacency matrix W as well as normalized or un-normalized graph Laplacians constitute self- adjoint operators on ℓ2pEq, where they act by matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Replacing the Gn of Section 3 by En, we can then follow the recipe laid out there in constructing 2nd-order scattering transforms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' all that we need are a module sequence ΩN and an operator sequence D2 N :“ pP 2 n, ∆2 nqN n“1, where now P 2 n : ℓ2pEn´1q Ñ ℓ2pEnq and ∆2 n : ℓ2pEnq Ñ ℓ2pEnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We denote the resulting feature map by Φ2 N and write F 2 N for the corresponding feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The map N G p introduced in (4) can also be adapted to aggregate higher-order features into graph level features: With }f}q :“ př ijPG |fij|qµijq1{q and µE :“ ř|G| ij“1 µij, we define N E p pfq “ p}f}ℓ1pEq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='µE, }f}ℓ2pEq, }f}ℓ3pEq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', }f}ℓppEqqJ{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Given a feature map Φ2 N with feature space F 2 N “ ‘N n“1 ` ℓ2pEnq ˘|Γn´1|, we obtain a corresponding 7 f pi(Pi() qc1 (△2) X1(△1) p2(P2()) 9b2 (△2 X2(△2) 12(G1) NG1 NG2 P1 P2 RP1map Ψ2 N mapping from ℓ2pEq to RN “ ‘N n“1 pRpnq|Γn´1| by concatenating ΦE N with the map that the family of non-linear maps tN En pn uN n“1 induces on F N by concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The stability results of the preceding sections then readily translate to Φ2 N and Ψ2 N (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Appendix J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 7 Experimental Results We showcase that even upon selecting the fairly simple Architectures I and II introduced in Section 3 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2), our generalized graph scattering networks are able to outperform both wavelet- based scattering transforms and leading graph-networks under different circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To aid visual clarity when comparing results, we colour-code the best-performing method in green, the second-best performing in yellow and the third-best performing method in orange respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Social Network Graph Classification: To facilitate contact between our generalized graph scat- tering networks, and the wider literature, we combine a network conforming to our general theory namely Architecture I in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2 (as discussed in Section 3 with depth N “ 4, identity as connect- ing operators and | ¨ |-non-linearities) with the low pass aggregation scheme of Section 5 and a Euclidean support vector machine with RBF-kernel (GGSN+EK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The choice N “ 4 was made to keep computation-time palatable, while aggregation scheme and non-linearities were chosen to facilitate comparison with standard wavelet-scattering approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For this hybrid architecture (GGSN+EK), classification accuracies under the standard choice of 10-fold cross validation on five common social network graph datasets are compared with performances of popular graph kernel approaches, leading deep learning methods as well as geometric wavelet scattering (GS-SVM) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' More details are provided in Appendix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As evident from Table 1, our network consistently achieves higher accuracies than the geometric wavelet scattering transform of [12], with the performance gap becoming significant on the more complex REDDIT datasets, reaching a relative mean performance increase of more than 10% on REDDIT-12K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This indicates the liberating power of transcending the graph wavelet setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' While on comparatively smaller and somewhat simpler datasets there is a performance gap between our static architecture and fully trainable networks, this gap closes on more complex datasets: While P-Poinc e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' outperforms our method on IMDB datasets, the roles are reversed on REDDIT datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' On REDDIT-B our approach trails only GIN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' with difference in accuracies insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' On REDDIT-5K our method comes in third, with the gap to the second best method (GIN) being statistically insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' On REDDIT-12K we generate state of the art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Table 1: Classification Accuracies on Social Network Datasets Method Classification Accuracies r%s COLLAB IMDB- B IMDB-M REDDIT-B REDDIT-5K REDDIT-12K WL [33] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='82˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='45 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='60˘5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='16 N/A 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='52˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='01 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='77 ˘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='02 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='57 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='32 Graphlet [34] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='42˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='43 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='40˘5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='95 N/A 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='26˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='34 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='75 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='36 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='98 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='29 DGK [42] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='00˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='20 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='90˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='50 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='50˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='50 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='00˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='30 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='20 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='10 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='20 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='10 DGCNN [46] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='76˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='49 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='03˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='86 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='83˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='85 N/A 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='70 ˘ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='54 N/A PSCN [26] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='60˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='15 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='00˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='29 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='23˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='84 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='30˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='58 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='10 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='70 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='32 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='42 P-Poinc [19] N/A 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='86˘4.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='60˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='97 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='89 ˘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='24 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='03 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='58 Regression of Quantum Chemical Energies: In order to showcase the prowess of both our higher order scattering scheme and our spectrum-agnostic aggregation method of Section 5, we combine these building blocks into a hybrid architecture which we then apply in combination with kernel methods (2GGST + EK) to the task of atomization energy regression on QM7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This is a comparatively small dataset of 7165 molecular graphs, taken from the 970 million strong molecular database GDB- 13 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Each graph in QM7 represents an organic molecule, with nodes corresponding to individual atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Beyond the node-level information of atomic charge, there is also edge level information characterising interaction strengths between individual nodes/atoms available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This is encoded into so called Coulomb matrices (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [31] or Appendix K) of molecular graphs, which for us serve a dual purpose: On the one hand we consider a Coulomb matrix as an edge-level input signal on a given graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 8 On the other hand, we also treat it as an adjacency matrix from which we build up a graph Laplacian L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Our normal operator is then chosen as L “ L{λmaxpLq again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Connecting operators are set to the identity, while non-linearities are fixed to ρně1p¨q “ | ¨ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Filters are chosen as psinpπ{2 ¨ L q, cospπ{2 ¨ L q, sinpπ ¨ L q, cospπ ¨ L qq acting through matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Output generating functions are set to the identity and depth is N “ 4, so that we essentially recover Architecture II of Figure 5: Atomization Energy as a Function of pri- mary Principal Components of Scattering Features Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' now applied to edge-level input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Graph level features are aggregated via the map N E 5 of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We chose p “ 5 (and not p " 5) for N E p to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Generated feature vectors are combined with node level scattering features obtained from applying Architecture II of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2 to the in- put signal of atomic charge into composite feature vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' plotted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As is visually evident, even when reduced to the low-dimensional subspace of their first three principal components, the generated scatter- ing features are able to aptly resolve the atom- ization energy of the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This aptitude is also reflected in Table 2, comparing our approach with leading graph-based learning methods trained with ten-fold cross validation on node and (depending on the model) edge level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Our method is the best performing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We significantly outperform the next best model (DTNN), producing less than half of its mean absolute error (MAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Errors of other methods are at least one — sometimes two — orders of magnitude greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In part, this performance discrepancy might be explained by the hightened suitability of our scattering transform for environ- ments with somewhat limited training-data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Here we speculate that the additional performance gap might be ex- plained by the fact that our graph shift operator ∆ carries the same information as the Coulomb matrix (a proven molecular graph descriptor in itself [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Additionally, our filters being infinite series’ in powers of the underlying normal operator allows for rapid dispersion of information across underlying molecular graphs, as opposed to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' the filters in GraphConv Table 2: Comparison of Methods Method MAE [kcal/mol] AttentiveFP [40] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 ˘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='8 DMPNN [44] 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='8 ˘ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 DTNN [39] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 ˘ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='9 GraphConv [18] 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='9 ˘ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 GROVER (base)[30] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5 ˘ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='9 MPNN [13] 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='0 ˘ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 N-GRAM[21] 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='6 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5 PAGTN (global) [6] 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='8 ˘ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='0 PhysChem [45] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='6 ˘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3 SchNet [32] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 ˘ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='0 Weave [17] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='6 ˘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3 GGST+EK [OURS] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='6 2GGST+EK [OURS] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3 or SchNet, which do not incorporate such higher powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To quantify the effect of including second order scattering coefficients, we also include the result of performing kernel-regression solely on first order features generated through Architecture II of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2 (GGST + EK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' While results are still better than those of all but one leading approach, incorporating higher order scattering improves performance significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 8 Discussion Leaving behind the traditional reliance on graph wavelets, we developed a theoretically well founded framework for the design and analysis of (generalized) graph scattering networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' allowing for varying branching rations, non-linearities and filter banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We provided spectrum independent stability guarantees, covering changes in input signals and for the first time also arbitrary normal perturbations in the underlying graph-shift-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' After introducing a new framework to quantify vertex-set non-preserving changes in graph domains, we obtained spectrum-independent stability guarantees for this setting too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We provided conditions for energy decay and discussed implications for truncation stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we introduced a new method of graph-level feature aggregation and extended scattering networks to higher order input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Our numerical experiments showed that a simple scattering transform conforming to our framework is able to outperform the traditional graph-wavelet based approach to graph scattering in social network graph classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' On complex datasets our method is also competitive with current fully trainable methods, ouperforming all competitors on REDDIT-12K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Additionally, higher order graph scattering transforms significantly outperform current leading graph-based learning methods in predicting atomization energies on QM7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' A reasonable critique of scattering networks as tractable models for general graph convolutional 9 kcal mo] 100 600 800 50 3rd eigenvector 1000 0 1200 8 50 1400 ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 100 1600 1800 150 2000 100 400 0 200 0 100 200 200 1st eigenvector 400 600 300 800 2nd eigenvectornetworks is their inability to emulate non-tree-structured network topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' While transcending the wavelet setting has arguably diminished the conceptual gap between the two architectures, this structural difference persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Additionally we note that despite a provided promising example, it is not yet clear whether the newly introduced graph-perturbation framework can aptly provide stability guarantees to all reasonable coarse-graining procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Exploring this question is the subject of ongoing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Broader Impact We caution against an over-interpretation of established mathematical guarantees: Such guarantees do not negate biases that may be inherent to utilized datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Disclosure of Funding Christian Koke acknowledges support from the German Research Foundation through the MIMO II-project (DFG SPP 1798, KU 1446/21-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Gitta Kutyniok acknowledges support from the ONE Munich Strategy Forum (LMU Munich, TU Munich, and the Bavarian Ministery for Science and Art), the Konrad Zuse School of Excellence in Reliable AI (DAAD), the Munich Center for Machine Learning (BMBF) as well as the German Research Foundation under Grants DFG-SPP-2298, KU 1446/31-1 and KU 1446/32-1 and under Grant DFG-SFB/TR 109, Project C09 and the Federal Ministry of Education and Research under Grant MaGriDo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' References [1] Uri Alon and Eran Yahav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' On the bottleneck of graph neural networks and its practical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In International Conference on Learning Representations, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' C.' metadata={'source': 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2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 12 [43] Pinar Yanardag and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Vishwanathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Deep graph kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’15, page 1365–1374, New York, NY, USA, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [44] Kevin Yang, Kyle Swanson, Wengong Jin, Connor Coley, Philipp Eiden, Hua Gao, Angel Guzman-Perez, Timothy Hopper, Brian Kelley, Miriam Mathea, Andrew Palmer, Volker Settels, Tommi Jaakkola, Klavs Jensen, and Regina Barzilay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Analyzing learned molecular representa- tions for property prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Journal of Chemical Information and Modeling, 59(8):3370–3388, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' PMID: 31361484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [45] Shuwen Yang, Ziyao Li, Guojie Song, and Lingsheng Cai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Deep molecular representation learning via fusing physical and chemical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [46] Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' An end-to-end deep learning architecture for graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In Sheila A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' McIlraith and Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Weinberger, editors, Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), the 30th innovative Applications of Artificial Intelligence (IAAI-18), and the 8th AAAI Symposium on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA, February 2-7, 2018, pages 4438–4445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' AAAI Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [47] Qi Zhao and Yusu Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Learning metrics for persistence-based summaries and applications for graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [48] Dongmian Zou and Gilad Lerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Graph convolutional neural networks via scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Applied and Computational Harmonic Analysis, 49(3):1046–1074, nov 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Checklist 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] A discussion of how and in which order the main claims made in Abstract and Introduction were substantiated within the paper is a main focus of Section 8 (b) Did you describe the limitations of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] This is a second focus of Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (c) Did you discuss any potential negative societal impacts of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] This is part of Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If you are including theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (a) Did you state the full set of assumptions of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] Every Theorem and Lemma is stated in a mathematically precise way, with all underlying assumptions included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Additionally, Appendix A briefly reviews some terminology that might not be immediately present in every readers mind, but is utilized in order to be able to state theoretical results in a precise and concise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (b) Did you include complete proofs of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] This is the Focus of Appendices B, C, D, F, G, H and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Results in Appendix J are merely stated, as statements and corresponding proofs are in complete analogy (in fact almost verbatim the same) to previously discussed statements and proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If you ran experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' please see the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (b) Did you specify all the training details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', data splits, hyperparameters, how they were chosen)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] This is the main focus of Appendix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 13 (c) Did you report error bars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', with respect to the random seed after running experi- ments multiple times)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] Errors for results of conducted experiments are included in Table 1 and Table 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (d) Did you include the total amount of compute and the type of resources used (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', type of GPUs, internal cluster, or cloud provider)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] This is described at the beginning of Appendix K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If you are using existing assets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', code, data, models) or curating/releasing new assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (a) If your work uses existing assets, did you cite the creators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] All utilized datasets were matched to the papers that introduced them (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Section K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Additionally, we partially built on code corresponding to [12] which we mentioned in Appendix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (b) Did you mention the license of the assets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] We mentioned in Appendix K that the code corresponding to [12] is freely available under an Apache License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (c) Did you include any new assets either in the supplemental material or as a URL?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] Please see supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [Yes] For the utilized social network datasets, we mention in Appendix K that neither personally identifyable data nor content that might be considered offensive is utilised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If you used crowdsourcing or conducted research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (a) Did you include the full text of instructions given to participants and screenshots, if applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' [N/A] A Some Concepts in Linear Algebra In the interest of self-containedness, we provide a brief review of some concepts from linear algebra utilized in this work that might potentially be considered more advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Presented results are all standard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' a very thorough reference is [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Hilbert Spaces: To us, a Hilbert space — often denoted by H — is a vector space over the complex numbers which also has an inner product — often denoted by x¨, ¨yH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Prototypical examples are given by the Euclidean spaces Cd with inner product xx, yyCd :“ řd i“1 xiyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Associated to an inner product is a norm, denoted by } ¨ }H and defined by }x}H :“ a xx, xyH for x P H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Direct Sums of Spaces: Given two potentially different Hilbert spaces H and p H, one can form their direct sum H ‘ p H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Elements of H ‘ p H are vectors of the form pa, bq, with a P H and b P p H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Addition and scalar multiplication are defined in the obvious way by pa, bq ` λpc, dq :“ pa ` λc, b ` λdq for a, c P H, b, d P p H and λ P C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The inner product on the direct sum is defined by xpa, bq, pc, dqyH‘ p H :“ xa, cyH ` xb, dy p H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As is readily checked, this implies that the norm } ¨ }H‘ p H on the direct sum is given by }pa, bq}2 H‘ p H :“ }a}2 H ` }b}2 p H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Standard examples of direct sums are again the Euclidean spaces, where one has Cd “ Cn ‘ Cm if m ` n “ d, as is easily checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' One might also consider direct sums with more than two summands, writing Cd “ ‘d i“1C for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In fact, one might also consider infinite sums of Hilbert spaces: 14 The space ‘8 i“1Hi is made up of those elements a “ pa1, a2, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='q with ai P Hi for which the norm }a}2 ‘8 i“1Hi :“ 8 ÿ i“1 }ai}2 Hi is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This means for example that the vector p1, 0, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='q is in ‘8 i“1C, while p1, 1, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='q is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Direct Sums of Maps: Suppose we have two collections of Hilbert spaces tHiuΓ i“1, t r HiuΓ i“1 with Γ P N or Γ “ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Suppose further that for each i ď Γ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' i ă Γ) we have a (not necessarily linear) map Ji : Hi Ñ r Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then the collection tJiuΓ i“1 of these ’component’ maps induce a ’composite’ map J : ‘Γ i“1Hi ÝÑ ‘Γ i“1 r Hi between the direct sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Its value on an element a “ pa1, a2, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='q P ‘Γ i“1Hi is defined by J paq “ pJ1pa1q, J2pa2q, J3pa3q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='q P ‘Γ i“1 r Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Strictly speaking, one has to be a bit more careful in the case where Γ “ 8 to ensure that }J paq}‘8 i“1 r Hi ‰ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This can however be ensured if we have }Jipaiq} r Hi ď C}ai}Hi for all 1 ď i and some C independent of all i, since then }J paq}‘8 i“1 r Hi ď C}a}‘8 i“1Hi ď 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If each Ji is a linear operator, such a C exists precisely if the operator norms (defined below) of all Ji are smaller than some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Operator Norm: Let J : H Ñ r H be a linear operator between Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We measure its ’size’ by what is called the operator norm, denoted by } ¨ }op and defined by }J}op :“ sup ψPH,}ψ}H“1 }Aψ} r H }ψ}H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Adjoint Operators Let J : H Ñ r H be a linear operator from the Hilbert space H to the Hilbert space r H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Its adjoint J˚ : r H Ñ H is an operator mapping in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It is uniquely determined by demanding that xJf, uy r H “ xf, J˚uyH holds true for arbitrary f P H and u P r H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Normal Operators: If a linear operator ∆ : H Ñ H maps from and to the same Hilbert space, we can compare it directly with its adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If ∆∆˚ “ ∆˚∆, we say that the operator ∆ is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Special instances of normal operators are self-adjoint operators, for which we have the stronger property ∆ “ ∆˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If an operator is normal, there are unitary maps U : H Ñ H diagonalizing ∆ as U ˚∆U “ diagpλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='λnq, with eigenvalues in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We call the collection of eigenvalues the spectrum σp∆q of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If dim H “ d, we may write σp∆q “ tλud i“1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It is a standard exercise to verify that each eigenvalue satisfies |λi| ď }∆}op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Associated to each eigenvalue is an eigenvector φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The collection of all (normalized) eigenvectors forms an orthonormal basis of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We may then write ∆f “ dÿ i“1 λi xφi, fyHφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Resolvent of a (normal) Operator: Given a normal operator ∆ on some Hilbert space H, we have that the operator p∆ ´ zq : H Ñ H is invertible precisely if z ‰ σp∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In this case we write Rpz, ∆q “ p∆ ´ zq´1 and call this operator the resolvent of ∆ at z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It can be proved that the norm of the resolvent satisfies }Rpz, ∆q}op “ 1 distpz, σp∆qq, where distpz, σp∆qq denotes the minimal distance between z and any eigenvalue of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 15 Functional Calculus: Given a normal operator ∆ : H Ñ H on a Hilbert space of dimension d and a complex function g : C Ñ C, we can define another normal operator obtained from applying the function g to ∆ by gp∆qf “ fÿ i“1 gpλiqxφi, fyHφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For example if gp¨q “ | ¨ |, we obtain the absolute value |∆| of ∆ by specifying for all f P H that |∆|f “ dÿ i“1 |λi|xφi, fyHφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Similarly we find (if z R σp∆q and for f P H) 1 ∆ ´ z “ dÿ i“1 1 λi ´ z xφi, fyHφi “ p∆ ´ zq´1 “ Rpz, ∆q where we think of the left-hand-side as applying a function to ∆, while we think of the right-hand-side as inverting the operator p∆ ´ zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This now allows us to apply tools from complex analysis also to operators: If a function g is analytic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' can be expanded into a power series), we have gpλq “ ´ 1 2πi ¿ S gpzq λ ´ z dz for any circle S Ď C encircling λ by Cauchy’s integral formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Thus, if we chose S large enough to encircle the entire spectrum σp∆q, we have gp∆qf “ ´ dÿ i“1 1 2πi ¿ S gpzq λi ´ z dzxφi, fyHφi “ ´ 1 2πi ¿ S gpzqRpz, λqdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Frobenius Norm: Given a finite dimensional Hilbert space H with inner product x¨, ¨yH, and an orthonormal basis tφiud i“1, we define the trace of an operator A : H Ñ H as TrpAq :“ dÿ k“1 xφk, AφkyH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It is a standard exercise to show that this is independent of the choice of orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The associated Frobenius inner product on the space of operators is then given as xB, AyF :“ TrpB˚Aq dÿ k“1 xφk, B˚AφkyH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Hence the Frobenius norm of an operator is determined by }A}2 F “ TrpA˚Aq “ dÿ k“1 xφk, A˚AφkyH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It is a standard exercise to verify that we have }A}op ď }A}F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Since the trace is independent of the choice of orthonormal basis, the Frobenius norm is invariant under unitary transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' More precisely, if U, V : H Ñ H are unitary, we have }UAV }2 F “ }A}2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Frobenius norms can be used to transfer Lipschitz continuity properties of complex functions to the setting of functions applied to normal operators: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let g : C Ñ C be Lipschitz continuous with Lipschitz constant Dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This implies }gpXq ´ gpY q}F ď Dg ¨ }X ´ Y }F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' for normal operators X, Y on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This proof is taken (almost) verbatim from [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For an operator A : H Ñ H denote by Aij its matrix representation with respect to the orthonormal basis tφiud i“1: Aij :“ xφi, AφjyH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We then have }A}2 F “ dÿ i,j“1 |Aij|2 as a quick calculation shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let now U, W be unitary (with respect to the inner product x¨, ¨yH) operators diagonalizing the normal operators X and Y as V ˚XV “ diagpλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='λnq “: DpXq W ˚Y W “ diagpµ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='µnq “: DpY q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Since the Frobenius norm is invariant under unitary transformations we find }gpXq ´ gpY q||2 F “ ||gpV DpXqV ˚q ´ gpWDpY qW ˚q}2 F “ }V gpDpXqqV ˚ ´ WgpDpY qqW ˚}2 F “ }W ˚V gpDpXqq ´ gpDpY qqW ˚V }2 F “ dÿ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='j“1 |pW ˚V gpDpXqq ´ gpDpY qqW ˚V qij|2 “ dÿ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='j“1 ˇˇˇˇˇ n ÿ k“1 rW ˚V sikrgpDpXqqskj ´ rgpDpY qqsikrW ˚V skj ˇˇˇˇˇ 2 “ dÿ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='j“1 |rW ˚V sij|2 |gpλjq ´ gpµiq|2 ď dÿ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='j“1 |rW ˚V sij|2 D2 g|λj ´ µi|2 “ D2 g dÿ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='j“1 ˇˇˇˇˇ n ÿ k“1 rW ˚V sikrDpXqskj ´ rDpY qsikrW ˚V skj ˇˇˇˇˇ 2 “ D2 g}X ´ Y }2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' B Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1 Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let ∆ : ℓ2pGq Ñ ℓ2pGq be normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If the family tgip¨quiPI of bounded functions satisfies A ď ř iPI |gipcq|2 ď B for all c in the spectrum σp∆q, we have (@f P ℓ2pGq) A}f}2 ℓ2pGq ď ÿ iPI }gip∆qf}2 ℓ2pGq ď B}f}2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Writing the normalized eigenvalue-eigenvector sequence of ∆ as pλi, φiq|G| i“1, we simply note ÿ iPI |G| ÿ k“1 |xgipλkqφk, fyℓ2pGq|2 “ |G| ÿ k“1 ˜ÿ iPI |gipλkq|2 ¸ |xφk, fyℓ2pGq|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Now under the assumption, we can estimate the sum in brackets by A from below and by B from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we need only use Bessel’s (in)equality to prove A||f||2 ď ÿ iPpI |G| ÿ k“1 |xgipλkqφk, fyℓ2pGq|2 ď B||f||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 17 C Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1 Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' With the notation of Section 3 and setting B0 “ 1, we have: }ΦNpfq ´ ΦNphq}2 FN ď ˜ 1 ` N ÿ n“1 maxtrBnpL` n R` n q2 ´ 1s, 0u n´1 ź k“0 BkpR` k L` k q2 ¸ }f ´ h}2 ℓ2pGq To streamline the argumentation let us first introduce some notation: Notation C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let us denote paths in ΓN as q :“ pγN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', γ1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For f P ℓ2pGq let us write fq :“ UrγNs ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' ˝ Urγ1spfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' By Definition, we have }ΦNpfq ´ ΦNpgq}2 FN “ N ÿ n“1 ¨ ˝ ÿ qPΓn´1 }Vnpfqq ´ Vnphqq}2 ℓ2pGnq ˛ ‚ “ N ÿ n“1 ¨ ˝ ÿ qPΓn´1 }χnp∆nqρnpPnpfqqq ´ χnp∆nqρnpPnphqqq}2 ℓ2pGnq ˛ ‚ looooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooon “:an .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We proceed in two steps: Our initial goal is to upper bound an as an ď BnpL` n R` n q2 ¨ bn´1 ´ bn ” pbn´1 ´ bnq ` “ BnpL` n R` n q2 ´ 1 ‰ ¨ bn´1 (5) for bn :“ ř qPΓn }fq ´ hq}2 ℓ2pGnq with b0 “ }f ´ h}2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To achieve this we note that (5) is equivalent to an ` bn ď BnpL` n R` n q2 ¨ bn´1 which upon unraveling definitions may be written as ÿ qPΓn´1 }χnp∆nqρnpPnppfqqqq ´ χnp∆nqρnpPnphqq}2 ℓ2pGnq ` ÿ pqPΓn }fpq ´ hpq}2 ℓ2pGnq ďBnpL` n R` n q2 ÿ qPΓn´1 }fq ´ hq}2 ℓ2pGn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (6) To establish (6), we note, that in the sum over paths of length n, any pq P Γn can uniquely be written as pq “ pγn, qq, with the path q P Γn´1 of length pn ´ 1q determined by pq “ pγn, γn´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', γ1 looooomooooon “:q q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' With this we find ÿ pqPΓn }fpq ´ hpq}2 ℓ2pGnq “ ÿ γnPΓn ÿ qPΓn´1 }gγnp∆nqρnpPnppfqqqq ´ gγnp∆nqρnpPnphqqq}2 ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Thus we can rewrite the left hand side of (6) as ÿ qPΓn´1 }χnp∆nqρnpPnppfqqqq ´ χnp∆nqρnpPnphqq}2 ℓ2pGnq ` ÿ pqPΓn }fpq ´ hpq}2 ℓ2pGnq “ ÿ qPΓn´1 ˆ }χnp∆nqρnpPnpfqq ´ χnp∆nqρnpPnphqq}2 ℓ2pGnq ` ÿ γnPΓn }gγnp∆nqρnpPnppfqqqq ´ gγnp∆nqρnpPnphqqq}2 ℓ2pGnq ¸ “:‹ 18 The fact that in each layer the function tχnp¨qu Ťtgγnp¨quγnPΓn form a generalized frame with upper frame constant Bn implies by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1, that we can further bound this as ‹ ď Bn ÿ qPΓn´1 }ρnpPnpfqq ´ ρnpPnphqq}2 ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Using the Lipschitz continuity of ρn and Pn, we arrive at the desired expression (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Having established that an ď pbn´1 ´ bnq ` “ BnpL` n R` n q2 ´ 1 ‰ ¨ bn´1 holds true, we note that we can establish bn´1 ď n´1 ź k“1 BkpL` k R` k q2bn´2 arguing similarly as in the case of (6) by using (for f P ℓ2pGn´1q) ÿ γn´1PΓn´1 }gγn´1p∆n´1qf}2 ℓ2pGn´1q ď }χn´1p∆n´1qf}2 ℓ2pGn´1q ` ÿ γPΓ }gγn´1p∆n´1qf}2 ℓ2pGn´1q together with the frame property and Lipschitz continuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We then iterate this inequality and recall that b0 “ }f ´ h}2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Using the fact that N ÿ n“1 pbn´1 ´ bnq “ b0 ´ bN ď b0, we finally find }ΦNpfq ´ ΦNphq}2 FN ď ˜ 1 ` N ÿ n“1 maxtrBnpL` n R` n q2 ´ 1s, 0u n´1 ź k“0 BkpR` k L` k q2 ¸ }f ´ h}2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' D Proof or Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let ΦN and rΦN be two scattering transforms based on the same module sequence ΩN and operator sequences DN, r DN with the same connecting operators (Pn “ rPn) in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume R` n , L` n ď 1 and Bn ď B for some B and n ď N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume that the respective normal operators satisfy }∆n ´ r∆n}F ď δ for some δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Further assume that the functions tgγnuγnPΓn and χn in each layer are Lipschitz continuous with associated Lipschitz constants satisfying L2 χn ` ř γnPΓn L2 gγn ď D2 for all n ď N and some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we have }rΦNpfq ´ ΦNpfq}FN ď b 2p2N ´ 1q ¨ b pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq for all f P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If B ď 1{2, the stability constant improves to a 2p1 ´ BNq{p1 ´ Bq ¨ D ď 2 ¨ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Notation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let us denote scattering propagators based on operators ∆n and connecting operators Pn by Un and scattering propagators based on operators r∆n by rUn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Similarly, to Notation C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2, let us then write (with q “ pγN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', γ1q) rfq :“ rUnrγns ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' ˝ rU1rγ1spfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' By definition we have }ΦNpfq ´ rΦN}2 FN “ N ÿ n“1 ¨ ˝ ÿ qPΓn´1 }χnp∆nqρnpPnppfqqqq ´ χnpr∆nqρnpPnp rfqqq}2 ℓ2pGnq ˛ ‚ loooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooon “:an .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 19 We define bn :“ ř qPΓn }fq ´ rfq}2 ℓ2pGnq, with b0 “ }f ´ h}2 ℓ2pGq “ 0 and note an ` bn “ ÿ qPΓn´1 ˆ }χnp∆nqρnpPnpfqq ´ χnpr∆nqρnpPnp rfqq}2 ℓ2pGnq ` ÿ γnPΓn }gγnp∆nqρnpPnppfqqqq ´ gγnpr∆nqρnpPnp rfqqq}2 ℓ2pGnq ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Using (with |a ` b|2 ď 2p|a|2 ` |b|2q) 1 2}gγnp∆nqρnpPnpfqqq ´ gγnpr∆nqρnpPnp rfqqq}2 ℓ2pGnq ď}rgγnp∆nq ´ gγnpr∆nqsρnpPnpfqqq}2 ℓ2pGnq `}gγnpr∆nqrρnpPnppfqqqq ´ ρnpPnp rfqqqs}2 ℓ2pGnq ď}rgγnp∆nq ´ gγnpr∆nqs}2 8 ¨ }ρnpPnpfqqq}2 ℓ2pGnq `}gγnpr∆nqrρnpPnpfqqq ´ ρnpPnp rfqqqs}2 ℓ2pGnq, and }rgγnp∆nq ´ gγnpr∆nqs}2 8 ď }rgγnp∆nq ´ gγnpr∆nqs}2 F ď L2 gγ ¨ δ2 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1 ), we find an ` bn ď2 ÿ qPΓn´1 ˜ L2 χn ` ÿ γnPΓn Lg2γn ¸ pL` n R` n q2δ2||ρnpPnpfqqq||2 ℓ2pGnq `2 ÿ qPΓn´1 Bn||ρnpPnpfqqq ´ ρnpPnp rfqqq||2 ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Using L2 χn ` ř γnPΓn L2 γn ď D2, we then infer (using the assumption L` n , R` n ď 1) an ď pbn´1 ´ bnq ` r2B ´ 1sbn´1 ` Bn´12D2δ2||f||ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Now if B ď 1 2, we have an ď pbn´1 ´ bnq ` Bn´12D2δ2||f||ℓ2pGq and results of geometric sums leads to the desired bound after summing over n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Hence let us assume B ą 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Using similar arguments as before, we find bn´1 ďBn´22D2δ2||f||2 ℓ2pGq ` 2Bbn´2 ď Bn´22D2δ2||f||2 ℓ2pGq ` Bn´24D2δ2||f||2 ℓ2pGq ` 4bn´3 ďBn´2 ˜n´1 ÿ k“1 2k ¸ D2δ2||f||2 ℓ2pGq “ Bn´2p2n ´ 2qD2δ2||f||2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Thus we now know an ď 2D2δ2Bn´1||f||2 ℓ2pGq ` r2B ´ 1sp2n ´ 2qD2δ2Bn´2||f||2 ℓ2pGq ` pbn´1 ´ bnq In total we find g f f e N ÿ n“1 an ď b 2p2N ´ 1q ¨ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' BN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq, where we have estimated the sum over pbn´1 ´ bnq by zero from above again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This establishes the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Remark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To see that this also holds for our Architecture I of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2, we note that the critical step is establishing that Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1 also applies to δ0 and cos, as defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Here we establish that }δ0p∆q ´ δ0pr∆q}F “ 0 20 and }cosp∆q ´ cospr∆q}F ď Dcos}∆ ´ r∆}F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Indeed, since ∆ and r∆ are (possibly) rescaled graph Laplacians on the same graph, the spectral projections to their lowest lying eigen space, associated to the eigenvalue λmin “ 0 agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Denoting this spectral projection by P, we have cosp∆q ´ cospr∆q “ rcosp∆q ´ Ps ´ rcospr∆q ´ Ps “ cosp∆q ´ cospr∆q and we can apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Similar considerations apply to δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' E Prototypical Example illustrating ω-δ Closeness and δ-Unitary Equivalence To investigate the example of Figure 3, we label the vertices of the respective graphs as depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We denote the left graph by G and the right graph by rG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The node-weights on rG are given as rµi “ 1 for 1 ď i ď 7, while on G the weights are given as µi “ 1 for 1 ď i ď 5 while µ6 “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We then consider the respective un-normalized graph Laplacians ∆ : ℓ2pGq Ñ ℓ2pGq and r∆ : ℓ2p rGq Ñ ℓ2p rGq, which for a given adjacency matrix W on a graph signal space ℓ2pGq with node weights tµiui is given as p∆fqi “ 1 µi ÿ j Wijpfi ´ fjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Such operators are positive and hence |∆| “ ∆ (similarly for r∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We now need to find operators J : ℓ2pGq Ñ ℓ2p rGq and rJ : ℓ2p rGq Ñ ℓ2pGq satisfying the conditions of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To construct J, we define a family tψiu6 i“1 of vectors on ℓ2p rGq as ψ1 “ p1, 0, 0, 0, 0, 0, 0q, ψ2 “ p0, 1, 0, 0, 0, 0, 0q, ψ3 “ p0, 0, 1, 0, 0, 0, 0q, ψ4 “ p0, 0, 0, 1, 0, 0, 0q, ψ5 “ p0, 0, 0, 0, 1, 0, 0q, ψ6 “ p0, 0, 0, 0, 0, 1, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Figure 6: Indexing on the re- spective graphs The map J : ℓ2pGq Ñ ℓ2p rGq is then defined as Jf :“ 6ÿ i“1 fiψi, for any f P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We take rJ : ℓ2p rGq Ñ ℓ2pGq to be its adjoint ( rJ :“ J˚), which determined explicitly by p rJuqi “ 1 µi xψi, uyℓ2p r Gq for any u P ℓ2p rGq We shall now first check the conditions for δ-quasi unitary equivalence, which we list again for convenience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' now adapted to our current setting: }Jf}ℓ2p r Gq ď 2}f}ℓ2pGq, }pJ ´ rJ˚qf}ℓ2p r Gq ď δ}f}ℓ2pGq, }f ´ rJJf}2 ℓ2pGq ď δ2 ´ }f}2 ℓ2pGq ` xf, ∆, fyℓ2pGq ¯ , }u ´ J rJu}2 ℓ2p r Gq ď δ2 ´ }u}2 ℓ2p r Gq ` xu, r∆ uyℓ2p r Gq ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We first note that since rJ “ J˚, we have }pJ ´ rJ˚qf}ℓ2p r Gq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Next we note }Jf}2 ℓ2p r Gq “ 7ÿ i“1 |pJfqi|2 “ |f6|2 ` 6ÿ i“1 |fi|2 “ 6ÿ i“1 µi “ }f}2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 21 Furthermore we note p rJJfqi “ 6ÿ k“1 fk 1 µi xψi, ψkyℓ2p r Gq loooooooomoooooooon “δik “ fi and hence }f ´ rJJf}2 ℓ2pGq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It remains to control }u ´ J rJu}2 ℓ2p r Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We note rJu “ pu1, u2, u3, u4, u5, pu5 ` u6q{2qJ and thus J rJu “ pu1, u2, u3, u4, u5, pu6 ` u7q{2, pu6 ` u7q{2qJ, Which implies u ´ J rJu “ p0, 0, 0, 0, 0, pu7 ´ u6q{2, pu6 ´ u7q{2qJ, and thus }u ´ J rJu}2 ℓ2p r Gq “ 2|u6 ´ u7|2 4 “ |u6 ´ u7|2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We have xu, r∆ uyℓ2p r Gq “ 1 2 dÿ i,j“1 Ă Wij|ui ´ uj|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Since Ă W67 “ 1{δ2 by assumption, we have }u ´ J rJu}2 ℓ2p r Gq “ 1 2|u6 ´ u7|2 “ 1 2 δ2 δ2 |u6 ´ u7|2 “ 1 2δ2Ă W67|u6 ´ u7|2 ď 1 2δ2 dÿ i,j“1 Ă Wij|ui ´ uj|2 “ δ2xu, r∆ uyℓ2p r Gq ď δ2 ´ }u}2 ℓ2p r Gq ` xu, r∆ uyℓ2p r Gq ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Thus we have proven δ-unitary-equivalence and it remains to establish p´1q-12δ closeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Com- bining Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='15 of [29], instead of bounding }p rRJ ´ JRqf}ℓ2p r Gq ď 12δ}f}ℓ2pGq directly, we may instead establish that there are operators J1 : ℓ2pGq Ñ ℓ2p rGq, Ă J1 : ℓ2p rGq Ñ ℓ2pGq satisfying }J1f ´ Jf}ℓ2p r Gq ď δ2 ` }f}ℓ2pGq ` xf, ∆, fyℓ2pGq ˘ , (7) }Ă J1u ´ rJu}ℓ2pGq ď δ2 ´ }u}ℓ2p r Gq ` xf, r∆, uyℓ2p r Gq ¯ , (8) and xJ1f, r∆ uyℓ2p r Gq “ xf, ∆ Ă J1uyℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' (9) We chose J1 “ J and determine Ă J1 by setting (for (1 ď i ď 6)) pĂ J1uqi “ ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Thus (7) is clearly satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For (8) we note that we have p rJu ´ Ă J1uq “ p0, 0, 0, 0, 0, pu7 ´ u6q{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Thus we have }Ă J1u ´ rJu}ℓ2pGq “ 1 2|u6 ´ u7|2 ď δ2 ´ }u}2 ℓ2p r Gq ` xu, r∆ uyℓ2p r Gq ¯ as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It remains to establish (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We have xf, ∆ Ă J1uyℓ2pGq “ 6ÿ i,j“1 fiWijpui ´ ujq, 22 while we have xJ1f, r∆ uyℓ2p r Gq “ 6ÿ i“1 fi ¨ xψi, r∆ uyℓ2p r Gq “ 5ÿ i,j“1 fiWijpUj ´ uiq ` f6 ¨ xψ6, r∆ uyℓ2p r Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We have (with all node-weights on ℓ2pGq equal to unity) xψ6, r∆ uyℓ2p r Gq “ p∆ uq6 ` p∆ uq7 “ ˜ÿ j W6jpf6 ´ fjq ` 1 δ2 pf6 ´ f7q ¸ ` ˆ 1 δ2 pf7 ´ f6q ˙ “ ˜ÿ j W6jpf6 ´ fjq ` 1 δ2 pf6 ´ f7q ¸ And thus xJ1f, r∆ uyℓ2p r Gq “ 6ÿ i,j“1 fiWijpui ´ ujq “ xf, ∆ Ă J1uyℓ2pGq which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' F Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4 Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In the setting of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3 let ∆ and r∆ be ω-δ-close and satisfy }∆}op, }r∆}op ď K for some K ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If g : C Ñ C is holomorphic on the disk BK`1p0q of radius pK ` 1q, there is a constant Cg ě 0 so that }gpr∆qJ ´ Jgp∆q}op ď Cg ¨ δ with Cg depending on g , ω and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Without loss of generality, let us assume that K ą |ω|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let us denote the circle of radius r in C by Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For any holomorphic function g and (normal) operator ∆ whose spectrum is enclosed by the circle Sr, we can express the operator gp∆q as gp∆q “ ´ 1 2πi ¿ Sr gpzq ∆ ´ z dz as discussed in Appendix A (see also [7] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Note that in our case the resolvent Rpz, ∆q “ p∆ ´ zq´1 is well defined for |z| ě K, since with our assumptions all eigenvalues are within the circle of radius K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Additionally note that we have distpz, σp∆qq ě distpz, SKq “ |z| ´ K if |z| ě K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The same holds true after replacing ∆ with r∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Since for any normal operator ∆ we have }Rpz, ∆q}op “ 1{distpz, σp∆qq, we find |Rpz, r∆q}op, }Rpz, ∆q}op ď 1{p|z| ´ Kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To quantify the difference }Rpz, r∆qJ ´ JRpz, ∆q}op in terms of the difference } rRpωqJ ´ JRpωq}op ď δ, we define the function γ0pzq :“ 1 ` |z ´ ω| |z| ´ K , for which }Rpz, r∆qJ ´ JRpz, ∆q}op ď γ0pzq2}Rpω, r∆qJ ´ JRpω, r∆q}op 23 holds, as proved (in more general form) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='9 in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Since on SK`1 we have and |z ´ ω| ď 2K ` 1 hence γ0pzq ď 2pK ` 1q, we find }gpr∆qJ ´ Jgp∆q}op “ ››››››› 1 2πi ¿ SK`1 gpzq ´ Rpz, r∆q ´ Rpz, ∆q ¯ dz ››››››› op ď 1 2π ¿ SK`1 |gpzq| ›››Rpz, r���q ´ Rpz, ∆q ››› op dz ď2pK ` 1q2 π ¨ ˚ ˝ ¿ SK`1 |gpzq|dz ˛ ‹‚¨ }Rpω, r∆qJ ´ JRpω, r∆q}op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Thus we may set Cg :“ 2pK ` 1q2 π ¿ SK`1 |gpzq|dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' G Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5 We state and prove a somewhat more general theorem, incorporating also the case where the identifi- cation operators only almost commute with connecting operators or non-linearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We also would like to point out that the constant 2 in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3 is arbitrary and any constant larger than one would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Much more details are provided in Chapter IV of [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Theorem G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let ΦN, rΦN be scattering transforms based on a common module sequence ΩN and differing operator sequences DN, r DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume R` n , L` n ď 1 and Bn ď B for some B and n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume that there are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq, rJn : ℓ2p rGnq Ñ ℓ2pGnq (0 ď n ď N) so that the respective signal spaces are δ-unitarily equivalent, the respective normal operators ∆n, r∆n are ω-δ-close as well as bounded (in norm) by K ą 0 and the connecting operators satisfy } rPnJn´1f ´ JnPnf}ℓ2p r Gnq ď δ}f}ℓ2pGn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For the common module sequence ΩN assume that the non-linearities satisfy }ρnpJnfq ´ Jnρnpfq}ℓ2p r Gnq ď δ}f}ℓ2pGnq and that the constants Cχn and tCgγn uγnPΓN associated through Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4 to the functions of the generalized frames in each layer satisfy C2 χn ` ř γnPΓN C2 gγn ď D2 for some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Denote the operator that the family tJnun of identification operators induce on FN through concatenation by JN : FN Ñ Ă FN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we have with KN “ a p8N ´ 1qp2D2 ` 12Bq{7 ¨ BN´1 if B ą 1{8 and KN “ a p2D2 ` 12Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{8 that }rΦNpJ0fq ´ JNΦNpfq} Ă FN ď KN ¨ δ ¨ }f}ℓ2pG, @f P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If additionally } rPnJn´1f ´ JnPnf}ℓ2p r Gnq “ 0 or }ρnpJnfq ´ Jnρnpfq}ℓ2p r Gnq “ 0 holds in each layer, then we have KN “ a p4N ´ 1qp2D2 ` 4Bq{3 ¨ BN´1 if B ą 1{4 and KN “ a p2D2 ` 4Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If both additional equations hold, we have KN “ a p2N ´ 1q2D2 ¨ BN´1 if B ą 1{2 and KN “ a 2D2 ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Notation G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let us denote scattering propagators based on operators ∆n and Pn by Un and scattering propagators based on operators r∆n and rPn by rUn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Similarly, to Notation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 and , let us then write (with q “ pγN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', γ1q) rfq :“ rUnrγns ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' ˝ rU1rγ1spJ0fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 24 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' By definition we have }J ΦNpfq ´ rΦNpJ0fq}2 Ă FN “ N ÿ n“1 ¨ ˝ ÿ qPΓn´1 }Jnχnp∆nqρnpPnpfqqq ´ χnpr∆nqρnpPnp rfqqq}2 ℓ2p r Gnq ˛ ‚ looooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooon “:an .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We define bn :“ ř qPΓn }Jnfq ´ rfq}2 ℓ2p r Gnq, with b0 “ }J0f ´ J0f}2 ℓ2p r Gq “ 0 and note an ` bn “ ÿ qPΓn´1 ˆ }Jnχnp∆nqρnpPnpfqq ´ χnpr∆nqρnpPnp rfqq}2 ℓ2p r Gnq ` ÿ γnPΓn }Jngγnp∆nqρnpPnppfqqqq ´ gγnpr∆nqρnpPnp rfqqq}2 ℓ2p r Gnq ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Using 1 2}Jngγnp∆nqρnpPnppfqqqq ´ gγnpr∆nqρnpPnp rfqqq}2 ℓ2p r Gnq ď}rJngγnp∆nq ´ gγnpr∆nqJnsρnpPnpfqqq}2 ℓ2p r Gnq `}gγnpr∆nqrJnρnpPnppfqqqq ´ ρnpPnp rfqqqs}2 ℓ2p r Gnq ď}rJngγnp∆nq ´ gγnpr∆nqJns}op ¨ }ρnpPnpfqqq}2 ℓ2p r Gnq `}gγnpr∆nqrJnρnpPnppfqqqq ´ ρnpPnp rfqqqs}2 ℓ2p r Gnq, and }rgγnp∆nq ´ gγnpr∆nqs}8 ď C2 gγ ¨ δ2 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4), we find an ` bn ď2 ÿ qPΓn´1 ˜ C2 χn ` ÿ γnPΓn Cg2γn ¸ pL` n R` n q2δ2||ρnpPnp rfqqq||2 ℓ2p r Gnq `2 ÿ qPΓn´1 Bn||JnρnpPnpfqqq ´ ρnpPnp rfqqq||2 ℓ2p r Gnq ď2 ÿ qPΓn´1 δ2 ˜ C2 χn ` ÿ γnPΓn Cg2γn ¸ pL` n R` n q2||ρnpPnp rfqqq||2 ℓ2p r Gnq `4B ¨ Bn´1||f||2 ℓ2pGqδ2 ` 8B ¨ Bn´1||f||2 ℓ2pGqδ2 ` 8Bbn´1, where the second inequality arises from permuting the identification operator Jn through non-linearity and connecting operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Using C2 χn ` ř γnPΓn C2 γn ď D2, we then infer an ď pbn´1 ´ bnq ` r8B ´ 1sbn´1 ` p2D2 ` 12BqBn´1δ2||f||2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If B ď 1 8, summing over n and using a geometric sum argument yields the desired stability constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Hence let us assume B ą 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Using similar arguments as before, we find bn´1 ďp2D2 ` 12Bqδ2Bn´2||f||2 ℓ2pGq ` 8Bbn´2 ď ˜n´1 ÿ k“1 8k´1 ¸ Bn´2p2D2 ` 12Bqδ2||f||2 ℓ2pGq “ 1 56p8n ´ 8qp2D2 ` 12qδ2||f||2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In total we find N ÿ n“1 an ď pb0 ´ bNq loooomoooon ď0 `p2D2 ` 12BqBn´1δ2||f||2 ℓ2pGq ` p8B ´ 1qp8n´1 ´ 1q{7Bn´2 ¨ p2D2 ` 12Bqδ2||f||2 ℓ2pGq ďp8N ´ 1qp2D2 ` 12Bq{7 ¨ BN´1||f||2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 25 If one of the additional equations holds, we find an ` bn ď pbn´1 ´ bnq ` r4B ´ 1sbn´1 ` p2D2 ` 4Bqδ2||f||2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' and bn´1 ďp2D2 ` 4Bqδ2Bn´2||f||2 ℓ2pGq ` 4Bbn´2 ď ˜n´1 ÿ k“1 4k´1 ¸ Bn´2p2D2 ` 4qδ2||f||2 ℓ2pGq “ 1 12p4n ´ 4qBn´2p2D2 ` 4qδ2||f||2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Arguing as previously yields the desired stability bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If both additional equations are satisfied the proof is virtually the same as the one for Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' H Details on Energy Decay and Truncation Stability We first prove the statement made about the relation between truncation stability and energy: Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Given the energy WN :“ ř pγN,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',γ1qPΓN }UrγNs ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' ˝ Urγ1spfq}2 ℓ2pGNq stored in the network at layer N, we have after extending ΦNpfq by zero to match dimensions with ΦN`1pfq that }ΦNpfq ´ ΦN`1pfq}2 FN`1 ď ` R` N`1L` N`1 ˘2 BN`1 ¨ WN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We note }ΦNpfq ´ ΦN`1pfq}2 FN`1 “ ÿ pγN´1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',γ1qPΓN }VN`1 ˝ UrγNs ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' ˝ Urγ1spfq}2 ℓ2pGN`1q ď ` R` N`1L` N`1 ˘2 BN`1 ÿ pγN´1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',γ1qPΓN´1 }UrγNs ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' ˝ Urγ1spfq}2 ℓ2pGN`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In fact one can prove even more: Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The energy WN stored in layer N satisfies C´ N}f}2 ℓ2pGq ď }ΦNpfq}FN ` WNpfq ď C` N}f}2 ℓ2pGq, with constants C´ N :“ Nś i“1 min ␣ 1, AipL´ i R´ i q2( and C` N :“ Nś i“1 max ␣ 1, BipL` i R` i q2( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' min ␣ 1, A1pL´ 1 R´ 1 q2( ||f||2 ℓ2pGq “A1pL´ 1 R´ 1 q2||f||2 ℓ2pGq “A1||ρ1pP1pfqq||2 ℓ2pG1q ď ÿ γ1PΓ1 ||gγ1p∆1qρ1pP1pfqq||2 ℓ2pG1q ` ||χ1p∆1qρ1pP1pfqq||2 ℓ2pG1q “ ÿ qPΓ1 ||Urqspfq||2 ℓ2pG1q ` ||χ1p∆1qρ1pP1pfqq||2 ℓ2pG1q “||χ1p∆1qρ1pP1pfqq||2 ℓ2pG1q ` W1pfq, and similarly ||χ1p∆1qρ1pP1pfqq||2 ℓ2pG1q ` W1pfq “ ÿ qPΓ1 ||Urqspfq||2 ℓ2pG1q ` ||χ1p∆1qρ1pP1pfqq||2 ℓ2pG1q ďB1pL` 1 R` 1 q2||f||2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 26 This yields the starting point for our induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Now for the inductive step assume the claim holds up until layer N ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we have C´ N´1||f||2 ℓ2pGq ď N´1 ÿ n“1 ¨ ˝ ÿ qPΓn´1 ||χnp∆nqfq||2 ℓ2pGnq ˛ ‚` WN´1pfq ď C` N´1||f||2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' using Notation C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We note N ÿ n“1 ¨ ˝ ÿ qPΓn´1 ||χnp∆nqρnpPnpfqqq||2 ℓ2pGnq ˛ ‚` WN “ N´1 ÿ n“1 ¨ ˝ ÿ qPΓn´1 ||χnp∆nqρnpPnpfqqq||2 ℓ2pGnq ˛ ‚` ÿ qPΓN´1 ||χNp∆NqρNpPNpfqqq||2 ℓ2pGNq ` ÿ qPΓN ||fq||2 ℓ2pGNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Every path rq P ΓN may be written as q “ pγn, qq, for some γn P Γn and q P ΓN´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Thus we have ÿ qPΓN ||fq||2 ℓ2pGNq “ ÿ qPΓN´1 ÿ γNPΓN ||gγN p∆NqPNpρNpfqqq||2 ℓ2pGNq Inserting this in the above equation yields N ÿ n“1 ¨ ˝ ÿ qPΓn´1 ||χnp∆nqρnpPnpfqqq||2 ℓ2pGnq ˛ ‚` WN “ N´1 ÿ n“1 ¨ ˝ ÿ qPΓn´1 ||χnp∆nqρnpPnpfqqq||2 ℓ2pGnq ˛ ‚ ` ÿ qPΓN´1 ˜ ||χNp∆NqρNpPNpfqqq||2 ℓ2pGn´1q ` ÿ γnPΓN ||gγN p∆NqPNpρNpfqq||2 ℓ2pGNq ¸ looooooooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooooooon “:βpfqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We have pL´ NR´ Nq2AN||fq||2 ℓ2pGn´1q ď βpfqq ď pL` NR` Nq2BN||fq||2 ℓ2pGn´1q, by the operator frame property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' With this we find: mint1, pL´ NR´ Nq2ANu ¨ ˝ N´1 ÿ n“1 ¨ ˝ ÿ qPΓn´1 ||χnp∆nqρnpPnpfqqq||2 ℓ2pGnq ˛ ‚` WN´1 ˛ ‚ ď N ÿ n“1 ÿ qPΓn´1 ||χnp∆nqρnpPnpfqqq||2 ℓ2pGnq ` WN ď maxt1, pL´ NR´ Nq2BNu ˜N´1 ÿ n“1 ˜ ÿ qPΓn ||χnp∆nqUrqspfq||2 ℓ2pGnq ¸ ` WN´1 ¸ , after unravelling the definition WN´1pfq ” ÿ qPΓN ||fq||2 ℓ2pGn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The induction hypothesis together with the definition of C˘ N now yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 27 With this we now prove our main theorem concerning energy decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Theorem H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let Φ8 be a generalized graph scattering transform based on a an operator sequence D8 “ pPn, ∆nq8 n“1 and a module sequence Ω8 with each ρnp¨q ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume in each layer n ě 1 that there is an eigenvector ψn of ∆n with solely positive entries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' denote the smallest entry by mn :“ miniPGn ψnris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Denote the eigenvalue corresponding to ψn by λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Quantify the ’spectral-gap’ opened up at this eigenvalue through neglecting the output-generating function by ηn :“ ř γnPΓn |gγnpλnq|2 and assume Bnmn ě ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We then have WNpfq ď C` N ¨ « N ź n“1 ˆ 1 ´ ˆ m2 n ´ ηn Bn ˙˙ff ¨ }f}2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Denote the spectral projection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' the orthogonal projection projecting to the space of eigenvectors) onto the eigenspace corresponding to λn by P n c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we have WNpfq “ ÿ qPΓN´1 ÿ γNPΓN ||gγN p∆NqρNpPNpfqqq||2 ℓ2pGNq “ ÿ qPΓN´1 ÿ γNPΓN ||gγN p∆Nqp1 ´ P N c qρNpPNpfqqq||2 ℓ2pGNq ` ÿ qPΓN´1 ÿ γNPΓN ||gγN p∆NqP N c ρNpPNpfqqq||2 ℓ2pGNq ď ÿ qPΓN´1 BN||p1 ´ P N c qρNpPNpfqqq||2 ℓ2pGNq ` ÿ qPΓN´1 ηN||P N c ρNpPNpfqqq||2 ℓ2pGNq ď ÿ qPΓN´1 BN||p1 ´ P N c qρNpPNpfqqq||2 ℓ2pGNq ` ÿ qPΓN´1 ηN||ρNpPNpfqqq||2 ℓ2pGNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' By orthogonality of the spectral projection, we then have ||p1 ´ P N c qρNpPnpfqqq||2 ℓ2pGNq “ ||ρNpPNpfqqq||2 ℓ2pGNq ´ ||P N c ρNpPnpfqqq||2 ℓ2pGNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Furthermore, we have |xψN, ρNpPnpfqqqyℓ2pGNq|2 ď ||P N c ρNpPnpfqqq||2 ℓ2pGNq with equality if the multiplicity of λN is exactly one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' With this we find ||p1 ´ P N c qρNpPNpfqqq||2 ℓ2pGNq “ ||ρNpPNpfqqq||2 ℓ2pGNq ´ ||P N c ρNpPNpfqqq||2 ℓ2pGNq ď ||ρNpPNpfqqq||2 ℓ2pGNq ´ |xψN, ρNpPNpfqqqyℓ2pGNq|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 28 Since the image of ρN is contained in R` by assumption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' we have |xψN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' ρNpPNpfqqqyℓ2pGNq|2 “ ˇˇˇˇˇˇ |GN| ÿ i“1 ρNpPNpfqqqipψNqiµi ˇˇˇˇˇˇ 2 ě ˇˇˇˇˇˇ |GN| ÿ i“1 |ρNpPNpfqqqi|µi ˇˇˇˇˇˇ 2 ¨ m2 N ě ˇˇˇˇˇˇ |GN| ÿ i“1 |ρNpPNpfqqqi|2µ2 i ˇˇˇˇˇˇ ¨ m2 N ě ˇˇˇˇˇˇ |GN| ÿ i“1 |ρNpPNpfqqqi|2µi ˇˇˇˇˇˇ ¨ m2 N ě ||ρNpPNpfqqq||2 ℓ2pGNq ¨ m2 N Here the second to last inequality follows since in any finite dimensional vector space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' the 1-norm is larger than the 2-norm (||f||1 ě ||f||2) and all weights are assumed to satisfy µi ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Thus we now know ||p1 ´ P N c qρNpPNpfqqq||2 ℓ2pGNq ď ` 1 ´ m2 N ˘ ||ρNpPNpfqqq||2 ℓ2pGNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Inserting this in our estimate for WNpfq we find WNpfq ď ˆ 1 ´ ˆ m2 N ´ ηn Bn ˙˙ L` NR` NBN ¨ WN´1pfq ď C` N N ź n“1 ˆ 1 ´ ˆ m2 N ´ ηn Bn ˙˙ ||f||2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Taking N to infinity, we know that C` N converges to something larger than zero by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For products of the form Nś n“0 p1 ´ qnq with 0 ď qn ă 1 it is a standard exercise to prove that the limit is non-zero precisely if the sum over the qn converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Combining the above result with Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2, we obtain as an immediate Corollary: Corollary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In the setting of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='6, the generalized scattering transform satisfies Φ´1 8 p0q “ t0u if C˘ N Ñ C˘ for some positive constants C˘ and řN n“1pmn ´ ηn{Bnq Ñ 8 as N Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' I Stability of Graph Level Feature Aggregation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1 General non-linear feature aggregation: Our main stability theorem for non-linear feature aggregation is as follows: Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We have }ΨNpfq´ΨNpgq}RN ď ˜ 1 ` N ÿ n“1 maxtrBn ´ 1s, rBnpL` n R` n q2 ´ 1s, 0u n´1 ź k“1 Bk ¸ 1 2 }f ´h}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' With the conditions and notation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 we have }ΨNpfq ´ rΨNpfq}RN ď b 2p2N ´ 1q ¨ b pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 29 Additionally, in the setting of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5, assuming that for each n ď N the identification operator Jn satisfies ˇˇ}Jnf}ℓ1p r Gnq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='µ r Gn ´}f}ℓ1pGnq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='µGn ˇˇ, ˇˇ}Jnf}ℓkp r Gnq´}f}ℓkpGnq ˇˇ ď δ¨K ¨}f}ℓ2pGnq (2 ď k ď pn) implies (@f P ℓ2pGq) }rΨNpJ0fq ´ ΨNpfq}RN ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2 ¨ b K2 N ¨ `K2 ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Furhermore, under the assumptions of Corollary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4 Ψ8pfq “ 0 implies f “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let f, h P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To prove the first two claims, it suffices to prove }ΨNpfq ´ ΨNphq}RN ď }ΦNpfq ´ ΦNphq}FN , and }ΨNpfq ´ rΨNpfq}RN ď }ΦNpfq ´ rΦNpfq}FN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Both statements follow immediately, as soon as we have proved }N G p pfq ´ N G p phq}Rp ď }f ´ h}ℓ2pGq for arbitrary choices of p and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To this end we note that for p ě 2 we have }f}ℓppGq ď }f}ℓ2pGq by the monotonicity of p-norms, while we have }f}ℓ1pGq ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='µG ¨ }f}ℓ2pGq by Hölder’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' With this we find }N G p pfq ´ N G p phq}2 Rp “ 1 p ˜ 1 µG |}f}ℓ1pGq ´ }h}ℓ1pGq|2 ` pÿ i“2 |}f}ℓipGq ´ }h}ℓipGq|2 ¸ ď 1 p ˜ 1 µG |}f ´ h}ℓ1pGq|2 ` pÿ i“2 |}f ´ h}ℓipGq|2 ¸ ď 1 p ¨ p ¨ |}f ´ h}ℓ2pGq|2 “ }f ´ h}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' where we have employed the reverse triangle inequality in the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To prove the second claim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' we note that we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='}ΨNpfq ´ rΨNpJ0fq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='RN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='“:xq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='q ´ N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='pn pχnpr∆nqρnpPnp rfqqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='loooooooooomoooooooooon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='“:rxq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='q}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Rpn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‹‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ď2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='}N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='pn pJnxqq ´ N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='pn prxqq}Rpn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='`2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='}N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='pn pJnxqq ´ N Gn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='pn pxqq}Rpn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='“2}J ΦNpfq ´ rΦNpJ0fq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='FN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='`2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='}N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='pn pJnxqq ´ N Gn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='pn pxqq}Rpn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Thus it remains to bound the last expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We have }N r Gn p pJnxqq ´ N Gn pn pxqq}Rpn “ 1 pn ¨ ˝ ˇˇˇˇˇ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='µGn }f}ℓ1pGq ´ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='µ r Gn }Jnf}ℓ1p r Gq ˇˇˇˇˇ 2 ` pn ÿ i“2 |}f}ℓipGq ´ }Jnf}ℓip r Gq|2 ˛ ‚ ďK2 ¨ δ2 ¨ }xq}2 ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 30 By our results of Appendix C and since we assume admissibility, we have N ÿ n“1 ÿ qPΓn´1 }xq}2 ℓ2pGnq ď }f}2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Thus in total }ΨNpfq ´ rΨNpJ0fq}2 FN ď 2}J ΦNpfq ´ rΦNpJ0fq}2 FN ` 2Kδ}f}ℓ2pGq, from which our stability claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It remains to prove that the assumptions of Corollary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4 Ψ8pfq “ 0 imply f “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' But since N G p pfq “ 0 implies f “ 0, this is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 Low-Pass feature Aggregation The main assumption we have in this section is that each operator ∆n (and r∆n) has a simple lowest lying eigenvalue equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We denote the associated eigenvector (determined up to a complex phase) by ψ∆n and the associated spectral projection to the lowest lying eigenvalue by P∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It acts as P∆nf ” ψ∆nxψ∆n, fyℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Now we are ready to state our main stability result under these circumstances: Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We have }Ψ|x¨,¨y| N pfq´Ψ|x¨,¨y| N pgq}CN ď ˜ 1 ` N ÿ n“1 maxtrBn ´ 1s, rBnpL` n R` n q2 ´ 1s, 0u n´1 ź k“1 Bk ¸ 1 2 }f´h}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' With the conditions and notation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 and under the additional assumption }pP∆n ´ P r∆nq}op ď K ¨ δ for n ď N and some K ě 0, we have }Ψ|x¨,¨y| N pfq ´ rΨ|x¨,¨y| N pfq}CN ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2 ¨ b 2p2N ´ 1qpmaxtB, 1{2uqN´1 ` K2 ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In the setting of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5 and under the additional assumption |}P∆nf}ℓ2pGnq ´ }P r∆nJnf}ℓ2p r Gnq| ď Kδ||f||ℓ2pGnq for all f P ℓ2pGnq (n ď N), we have }rΨ|x¨,¨y| N pJ0fq ´ Ψ|x¨,¨y| N pfq}CN ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2 ¨ b K2 N ¨ `K2 ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let f, h P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To prove the first claim, it suffices to prove }Ψ|x¨,¨y| N pfq ´ Ψ|x¨,¨y| N phq}CN ď }ΦNpfq ´ ΦNphq}FN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This immediately follows from the fact that for all f P ℓ2pGnq |xψ∆n, fyℓ2pGnq|2 ď }ψ∆n}2 ℓ2pGnq ¨ }f}2 ℓ2pGnq by Hölder’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The next claim we want to prove is that we have for all f P ℓ2pGq }Ψ|x¨,¨y| N pfq ´ rΨ|x¨,¨y| N pfq}CN ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2 ¨ b 2p2N ´ 1q ` K2 ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We note 31 }Ψ|x¨,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨y| N pfq ´ rΨ|x¨,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨y| N pfq}2 CN “ N ÿ n“1 ¨ ˚ ˝ ÿ qPΓn´1 ˇˇˇˇˇˇˇ |xψ∆n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' χnp∆nqρnpPnpfqqq loooooooooomoooooooooon “:xq yℓ2pGnq| ´ |xψ r∆n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' χnpr∆nqρnpPnp rfqqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='loooooooooomoooooooooon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='rxq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='yℓ2pGnq| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‹‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nrxq}ℓ2pGnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='}P∆nxq ´ P r∆nrxq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ℓ2pGnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ď2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='}P r∆npxq ´ rxqq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ℓ2pGnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚` 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='}pP∆n ´ P r∆nqxq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ℓ2pGnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ď2}ΦNpfq ´ ΦNphq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='FN ` 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='}pP∆n ´ P r∆nqxq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ℓ2pGnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Hence we need to bound the expression "}pP∆n ´ P r∆nqxq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ℓ2pGnq".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We note }pP∆n ´ P r∆nqxq}2 ℓ2pGnq ď }pP∆n ´ P r∆nq}op ¨ }xq}2 ℓ2pGnq ď K2 ¨ δ2 ¨ }xq}2 ℓ2pGnq and thus }Ψ|x¨,¨y| N pfq ´ rΨ|x¨,¨y| N pfq}2 CN ď2}ΦNpfq ´ ΦNphq}2 FN ` 2K2 ¨ δ2 ¨ N ÿ n“1 ¨ ˝ ÿ qPΓn´1 }χnp∆nqρnpPnppfqqqq}2 ℓ2pGnq ˛ ‚ ď2}ΦNpfq ´ ΦNphq}2 FN ` 2K2 ¨ δ2 ¨ }f}2 ℓ2pGq and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Finally we want to prove }rΨ|x¨,¨y| N pJ0fq ´ Ψ|x¨,¨y| N pfq}CN ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2 ¨ b K2 N ¨ `K2 ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We note 32 }Ψ|x¨,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨y|2 N pfq ´ rΨ|x¨,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨y| N pfq}CN “ N ÿ n“1 ¨ ˚ ˝ ÿ qPΓn´1 ˇˇˇˇˇˇˇ |xψ∆n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' χnp∆nqρnpPnppfqqqq loooooooooooomoooooooooooon “:xq yℓ2pGnq| ´ |xψ r∆n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' χnpr∆nqρnpPnp rfqqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='loooooooooomoooooooooon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='rxq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='yℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‹‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nrxq}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq ` }P r∆nJnxq}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq ´ }P r∆nrxq}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ď2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ}P r∆nJnxq}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq ´ }P r∆nrxq}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='`2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ď2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ}P r∆nJnxq}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq ´ }P r∆nrxq}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='`2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ď2}J ΦNpfq ´ rΦNpJ0fq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='FN ` 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Gnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ď2}J ΦNpfq ´ rΦNpJ0fq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='FN ` 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='n“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='qPΓn´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='K2 ¨ δ2}xq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ℓ2pGnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ď2}J ΦNpfq ´ rΦNpJ0fq}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='FN ` 2K2 ¨ δ2}f}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In establishing triviality of the ’kernel’, we have to be a tiny bit more careful: Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In the setting of of Corollary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4, assume that in each layer n, the output generating function χn of the underlying scattering transform satisfies χnp0q ‰ 0 and χnpλiq “ 0 for ordered non-zero eigenvalues λ2 ď .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' ď λ|Gn| of the operator ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then Ψ|x¨,¨y| 8 pfq “ 0 implies f “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Under these assumptions, we do not lose any information by projecting to ψ∆n in each ℓ2pGnq, since the image of χnp∆nq is already contained in the one-dimensional space generated by the lowest lying eigenvector ψ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' J Details on Higher Order Scattering Node signals capture information about nodes in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' However, one might also want to analyse or incorporate information about binary, ternary or even higher order relations between nodes, such as distances or angles between nodes representing atoms in a molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This can be formalized by considering tensorial input signals: 33 Tensorial input: A 2-tensor on a graph G, as it was already utilized in Section 6, is simply an element of C|G|ˆ|G| or – equivalently – a map from GˆG to C, since it associates a complex number to each element pg1, g2q P G ˆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Since G ˆ G is precisely the set of (possible) edges E, we can equivalently think of 2-tensors edge-signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' A 3-tensor an element of C|G|ˆ|G|ˆ|G| or equivalently a map from G ˆ G ˆ G ” G3 to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' A 4-tensor then is a map from G4 ” G ˆ G ˆ G ˆ G to C or equivalenlty an element of C|G|ˆ|G|ˆ|G|ˆ|G| and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Clearly the space of k-tensors forms a linear vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Addition and scalar multiplication by λ P C are given by pf ` λgqi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',ik :“ fi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',ik ` λgi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',ik with f and g being k-tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For fixed k, we equip the space of k-tensors with an inner product according to xf, gy “ |G| ÿ i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',ik“1 fi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',ikgi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',ikµi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',ik and denote the resulting inner-product space by ℓ2pGkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Operators on Spaces of Tensors: Since for fixed k the space ℓ2pGkq is simply a |G|k-dimensional complex inner product space, there are exist normal operators ∆k : ℓ2pGkq Ñ ℓ2pGkq on this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Note that the k in ∆k signifies on which space this operator acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It does not signify that an operator is raised to the kth power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Setting for example node-weights µi and edge weights µik to one, the adjacency matrix W as well as normalized or un-normalized graph Laplacians constitute self-adjoint operators on ℓ2pG2q, where they act by matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Higher order Scattering Transforms: We can then follow the recipe laid out Section 3 in con- structing kth-order scattering transforms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' all that we need are a module sequence ΩN and an operator sequence Dk N :“ pP k n, ∆k nqN n“1, where now P k n : ℓ2pGk n´1q Ñ ℓ2pGk nq and ∆k n : ℓ2pGk nq Ñ ℓ2pGk nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Figure 7: Schematic Higher Order Scattering Ar- chitecture To our initial signal f P ℓ2pGkq we first apply the connecting operator P k 1 , yielding a signal rep- resentation in ℓ2pGk 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Subsequently, we apply the pointwise non-linearity ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we apply our graph filters tχ1p∆k 1qu Ťtgγ1p∆k 1quγ1PΓ1 to ρ1pP k 1 pfqq yielding the output V1pfq :“ χ1p∆k 1qρ1pP k 1 pfqq as well as the interme- diate hidden representations tU1rγ1spfq :“ gγ1p∆k 1qρ1pP k 1 pfqquγ1PΓ1 obtained in the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Here we have introduced the one-step scattering propagator Unrγns : ℓ2pGk n´1q Ñ ℓ2pGk nq mapping f ÞÑ gγnp∆nqρnpPnpfqq as well as the output generating operator Vn : ℓ2pGk n´1q Ñ ℓ2pGk nq mapping f to χnp∆k nqρnpP k npfqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Upon defining the set ΓN´1 :“ ΓN´1 ˆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' ˆ Γ1 of paths of length pN ´ 1q terminating in layer N ´ 1 (with Γ0 taken to be the one-element set) and iterating the above procedure, we see that the outputs gener- ated in the N th-layer are indexed by paths ΓN´1 terminating in the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We denote the resulting feature map by Φk N and write F k N for the corresponding feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The node-level stability results of the preceding sections then readily translate to higher order scattering transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As the respective proofs are identical to the corresponding results for the node setting, we do not repeat them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' With the notation of Section 4, we have for all f, h P ℓ2pGkq: }Φk Npfq ´ Φk Nphq}2 F k N ď ˜ 1 ` N ÿ n“1 maxtrBn ´ 1s, rBnpL` n R` n q2 ´ 1s, 0u n´1 ź ℓ“1 Bℓ ¸ }f ´ h}2 ℓ2pGkq 34 p3(Pk(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=')) p2(Pk(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=')) ga2 9b2 (△) p3(Pk()) X2(△) ga p3(Pk()) P1(Pk())/gb1(△)、Ip2(Pk()) qa?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' T p3(Pk()) 6 JC1 X2(△5) p3(Pk()) (△) p2(Pk(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=')) qa2 9b2 (△k p3(Pk()) X1(△) X2(△))Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let ΦN and rΦN be two scattering transforms based on the same module sequence ΩN and operator sequences Dk N, r Dk N with the same connecting operators (P k n “ rP k n) in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume R` n , L` n ď 1 and Bn ď B for some B and n ď N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume that the respective normal operators satisfy }∆k n ´ r∆k n}F ď δ for some δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Further assume that the functions tgγnuγnPΓn and χn in each layer are Lipschitz continuous with associated Lipschitz constants satisfying L2 χn ` ř γnPΓn L2 gγn ď D2 for all n ď N and some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we have for all f P ℓ2pGkq }rΦk Npfq ´ Φk Npfq}FN ď b 2p2N ´ 1q ¨ b pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGkq Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Let Φk N, rΦk N be higher order scattering transforms based on a common module sequence ΩN and differing operator sequences Dk N, r Dk N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume R` n , L` n ď 1 and Bn ď B for some B and n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assume that there are identification operators Jn : ℓ2pGk nq Ñ ℓ2p rGk nq, rJn : ℓ2p rGk nq Ñ ℓ2pGk nq (0 ď n ď N) so that the respective signal spaces are δ-unitarily equivalent, the respective normal operators ∆k n, r∆k n are ω-δ-close as well as bounded (in norm) by K ą 0 and the connecting operators satisfy } rP k nJn´1f ´ JnP k nf}ℓ2p r Gknq ď δ}f}ℓ2pGk n´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For the common module sequence ΩN assume that the non-linearities satisfy }ρnpJnfq ´ Jnρnpfq}ℓ2p r Gknq ď δ}f}ℓ2pGknq and that the constants Cχn and tCgγn uγnPΓN associated through Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4 to the functions of the generalized frames in each layer satisfy C2 χn ` ř γnPΓN C2 gγn ď D2 for some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Denote the operator that the family tJnun of identification operators induce on F k N through concatenation by JN : F k N Ñ Ă F k N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we have with KN “ a p8N ´ 1qp2D2 ` 12Bq{7 ¨ BN´1 if B ą 1{8 and KN “ a p2D2 ` 12Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{8 that }rΦk NpJ0fq ´ JNΦk Npfq} Ă F k N ď KN ¨ δ ¨ }f}ℓ2pG, @f P ℓ2pGkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If additionally } rP k nJn´1f ´ JnP k nf}ℓ2p r Gnq “ 0 or }ρnpJnfq ´ Jnρnpfq}ℓ2p r Gknq “ 0 holds in each layer, then we have KN “ a p4N ´ 1qp2D2 ` 4Bq{3 ¨ BN´1 if B ą 1{4 and KN “ a p2D2 ` 4Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If both additional equations hold, we have KN “ a p2N ´ 1q2D2 ¨ BN´1 if B ą 1{2 and KN “ a 2D2 ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The map N G p introduced in (4) can also be adapted to aggregate higher-order tensorial features into graph level features: With }f}q :“ ˜ ÿ i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',ikPG |fi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',ik|qµi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',ik ¸1{q and µGk :“ ř|G| i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='ik“1 µi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=',ik, we define N Gk p pfq “ p}f}ℓ1pGkq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='µGk, }f}ℓ2pGkq, }f}ℓ3pGkq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=', }f}ℓppGkqqJ{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Given a feature map Φk N with feature space FN “ ‘N n“1 ` ℓ2pGk nq ˘|Γn´1| , we obtain a corresponding map Ψk N mapping from ℓ2pGkq to RN “ ‘N n“1 pRpnq|Γn´1| by concatenating Φk N with the map that the family of non-linear maps tN pn GknuN n“1 induces on F N by concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The resulting map Ψk N again has stability properties analogous to the node level case: Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Assuming admissibility, we have }Ψk Npfq ´ Ψk Nphq}RN ď ˜ 1 ` N ÿ n“1 maxtrBn ´ 1s, rBnpL` n R` n q2 ´ 1s, 0u n´1 ź ℓ“1 Bℓ ¸ }f ´ h}2 ℓ2pGkq 35 for all f, h P ℓ2pGq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' With the conditions and notation of Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 we have }Ψk Npfq ´ rΨk Npfq}RN ď b 2p2N ´ 1q ¨ b pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Additionally, in the setting of Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3, assuming that for each n ď N the identification operator Jn satisfies ˇˇ}Jnf}ℓ1p r Gknq{aµ r Gkn ´}f}ℓ1pGknq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='µGkn ˇˇ, ˇˇ}Jnf}ℓrp r Gknq´}f}ℓrpGknq ˇˇ ď δ¨K¨}f}ℓ2pGknq for 2 ď r ď pn implies (@f P ℓ2pGkq) }rΨNpJ0fq ´ ΨNpfq}RN ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2 ¨ b K2 N ` K2 ¨ δ ¨ }f}ℓ2pGkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As the proofs here are virtually the same as for the corresponding results in previous sections – essentially only replacing G by Gk, we omit a repetition of them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' K Additional Details on Experiments Here we provide additional details on utilized scattering architectures, training procedures, datasets and (performance of) other methods our approach is being compared to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Irrespective of task, our models are trained on an NVIDIA DGX A100 architecture utilizing between two and eight NVIDIA Tesla A100 GPUs with 80GB memory each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Running 10-fold cross validation for the respective experiments took at most 71 hours (which was needed for social network graph classification on REDDIT-12K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1 Social Network Graph Classification Datasets: The data we are working with is taken from [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' In particular this work introduced six social network datasets extracted from from scientific collaborations (COLLAB), movie collabora- tions (IMDB-B, IMDB-M) and Reddit discussion threads (REDDIT-B, REDDIT-5K, REDDIT-12K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Data is anonymised and contains no content that might be considered offensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Each graph carries a class label, and the goal is to predict this label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Some basic properties of these datasets are listed in Table 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Table 3: Social Network Dataset Characteristics Attributes: COLLAB IMDB- B IMDB-M REDDIT-B REDDIT-5K REDDIT-12K Graphs 5K 1K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5K 2K 5K 12K Nodes 372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5K 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='8K 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5K 859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='5M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='7M Edges 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1M 386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1K 395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='6 4M 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='9M 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='8M Maximum Degree 2k 540 352 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2K 8K 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2K Minimum Degree 4 4 4 4 4 4 Average Degree 263 39 40 9 9 9 Target Labels 3 2 3 2 5 11 Disconnected Graphs No No No Yes Yes Yes These datasets contain graph structures, however they don’t contain associated weights or graph signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Having unspecified weights simply means that the adjacency matrix W from which we construct the graph Laplacian L “ D ´ W on which our operator ∆ is based simply has each entry corresponding to an edge set to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If no edge is present between vertices i and j, the entry Wij is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It remains to solve the problem of the missing input signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Our strategy is to generate signals reflecting the geometry of the underlying graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We do this by utilizing features that associate to each node a number that characterizes its role or importance within its local environment or within the entire graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We briefly describe them here: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Degree: The degree of a node is the number of edges incident at this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Eccentricity: For a connected graph, the eccentricity of a node is the maximum distance from this node to all other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' On a disconnected graph it is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 36 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Clustering: For unweighted graphs the clustering cpuq of a node u is the fraction of possible triangles through that node that actually exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' It is calculated as cpuq “ 2Tpuq degpuqpdegpuq ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Number of triangles: The number of triangles containing the given node as a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Core number: A k-core is a maximal subgraph that only contains nodes of degree k or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The core number of a node is the largest value k of a k-core containing that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Clique number: A clique is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This input assigns the number of cliques the nodes participates in to each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Pagerank: This returns the PageRank of the respective nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' PageRank computes a ranking of the nodes in the graph based on the structure of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Originally it was designed as an algorithm to rank web pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For the first three datasets listed in Table 3 we utilize all listed input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For the latter three datasets we have to refrain from using eccentricity as an input signal, as these datasets contain graphs that have multiple non-connected graph components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Scattering Architecture: We chose a generalized scattering architecture of depth N “ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As normal-operators, we utilize in each layer the un-normalized graph Laplacian L “ D ´ W scaled by its largest eigenvalue (∆ “ L{λmaxpLq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Filters are chosen as 1 2psinpπ{2 ¨ ∆q, rcospπ{2 ¨ ∆q ´ ψ∆ψJ ∆s, sinpπ ¨ ∆q, rcospπ ¨ ∆q ´ ψ∆ψJ ∆sq, which allows to specify the output generating function solely by demanding χp0q “ 1 and χpλq “ 0 on all other eigenvalues of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Here ψ∆ is the normalized vector of all ones (satisfying ∆ψ∆ “ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Connecting operators are chosen as the identity, while we set ρně1p¨q “ | ¨ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We note that for connected graphs, this recovers Architecture I of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' On disconnected graphs (as they can appear in the REDDIT datasets), we however do not account for vectors other than ψ∆ in the lowest-lying eigenspace of the graph Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This scattering architecture is then applied to each of these input signal individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' For each input signal, this returns a feature vector with 1 ` 4 ` 16 ` 64 “ 85 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' These individual feature vectors are then concatenated into one final composite feature vector for each graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Concerning applicable theoretical results, we note the following: Training Procedure: We train RBF kernel support vector classifiers on our composite scattering features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We fix ϵ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The hyperparameter γ scaling the exponent is chosen from Gpool :“ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='00001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1, 1, 10, 100u, while we pick the C that controls the error our slack variables introduce among Cpool :“ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='1, 1, 10, 25, 50, 100, 1000u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We chose these parameters in agreement with the choices of [12] to facilitate comparison between the two works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We could simply implement the training of the RBF-classifier on our composite scattering features by dividing each social-network dataset into 10 folds, then iteratively choosing one fold for testing and among the remaining 9 folds randomly choosing one for validation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' for tuning the hyperpa- rameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Instead, following [12] (whose code is released under an Apache license and on which we partially built), we take a slightly different approach: We still randomly split our dataset into 10 folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Among the 10 folds, we iteratively pick one for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Say we have picked the nth fold for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then there are 9 remaining folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We iteratively pick the mth n (with 1 ď mn ď 9) of the remaining 9 folds for choosing hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This leaves 8 folds on which we train our model for each choice of hyper parameter in Cpool ˆ Gpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The resulting classifiers are all evaluated on the mth n fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The one that performs best is retained as classifier mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As mn varies between 1 and 9 (still for fixed n), this yields a set tfmn : 1 ď mn ď 9u of nine classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' From these we build the classifier fn, whose classification result is obtained from a majority vote among the nine classifiers in tfmn : 1 ď mn ď 9u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then we evaluate the performance of fn on the nth fold to obtain the nth estimation of how well our model performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As n varies from one to ten, we built the mean and variance of the performances of the classifiers fn on the nth fold expressed as the percentage of correct classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 37 Reference Methods: To allow for a comparison of our results to the literature, typical classification accuracies for graph algorithms on social network datasets are displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Following the standard format of reporting classification accuracies, they are presented in the format (Accuracy ˘ standard deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If results are not reported for a dataset, we denote this as not available (N/A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The first three rows of Table 1 display results for graph kernel methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' namely Weisfeiler-Lehman graph kernels (WL, [33]), Graphlet kernels (Graphlet, [34]) and deep graph kernels (DGK, [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The subsequent rows display results for geometric deep learning algorithms: Deep graph convolutional neural networks (DGCNN,[46]), Patchy-san (PSCN (with k=10), [26]), recurrent neural network autoencoders (S2S-N2N-PP, [16]) and graph isomorphism networks (GIN [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' These results are taken from [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Additionally we compare with P-Poinc [19], which embeds nodes into a hyperbolic space (the Poincare ball, to be precise), GSN-e [3] which combines message passing with structural features extracted via subgraph isomorphism and WKPI-kC [47] which utilizes a weighted kernel within its metric learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The second to last row (GS-SVM [12]) provides a result that is also based on a method that combines a static scattering architecture with a support vector machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Its filters are based on graph wavelets built from differences between lazy random walks that have propagated at different time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='2 Regression of Quantum Chemical Energies Dataset: The dataset we consider is the QM7 dataset, introduced in [2, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This dataset contains descriptions of 7165 organic molecules, each with up to seven heavy atoms, with all non-hydrogen atoms being considered heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' A molecule is represented by its Coulomb matrix CClmb, whose off-diagonal elements CClmb ij “ ZiZj |Ri ´ Rj| (10) correspond to the Coulomb-repulsion between atoms i and j, while diagonal elements encode a polynomial fit of atomic energies to nuclear charge [31]: CClmb ii “ 1 2Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4 i For each atom in any given molecular graph, the individual Cartesian coordinates Ri and the atomic charge Zi are also accessible individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To each molecule an atomization energy - calculated via density functional theory - is associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The objective is to predict this quantity, the performance metric is mean absolute error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Numerically, atomization energies are negative numbers in the range ´600 to ´2200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The associated unit is rkcal/mols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Scattering Architecture: Off-diagonal entries in the Coulomb Matrix clearly represent an inverse distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' A weight of zero can then heuristically be thought of as the inverse distance between two infinitely separated atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' After calculating the degree matrix D associated to C, we obtain the graph Laplacian once more as L “ D ´ C and set our normal operator to ∆ “ L λmaxpLq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' If we continuously vary the distances in (10), staying clear of zero, then the adjacency matrix and hence the graph Laplacian L varies continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As long as we avoid complete degeneracy, the largest eigenvalue λmaxpLq will remain positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This implies that our normal operator ∆ varies continuously under changes of the inter-atomic distances, which implies that our feature vector also varies continuously, as distances are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Connecting operators are set to the identity, while non-linearities are fixed to ρně1p¨q “ | ¨ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Filters are chosen as psinpπ{2 ¨ ∆q, cospπ{2 ¨ ∆q, sinpπ ¨ ∆q, cospπ ¨ ∆qq acting through matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The output generating functions are set to the identity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Graph level features are aggregated via the map N E 5 p¨q of Section 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' slightly modified to neglect the normalizing factor in the first entry for improved convenience in numerical implementability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As weights µij for our second-order feature space are set to unity and molecular graphs in QM7 contain at most 23-molecules, we note that ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='µG2 ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 232 “ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Going through the proofs of our graph-level stability results, we see that they remain valid after multiplying each stability constant by 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The Coulomb matrix (divided by a factor of 10 as this empirically improved performance) is then also utilized as an edge level input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Node level 38 features are obtained by applying the above architecture to the node level information provided by the respective atomic charges tZiu on each graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We aggregate to graph level features using N G 5 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Section 5), again neglecting the normalizing factor in the first entry for improved convenience in implementing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' The network depth is set to N “ 4 in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We then concatenate graph level features obtained from node- and edge level input into a composite scattering feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Training Procedure: The QM7 dataset comes with a precomputed partition into five subsets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' each containing a representative amount of heavy and light molecules covering the entire complexity range of QM7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' To allow for 10-fold cross validation, we further dissect each of these subsets into two smaller datasets, one containing graphs indexed by an even number, one containing graphs indexed by an odd number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' On these 10-subsets, we then perform 10-fold cross validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Among the 10 folds, we iteratively pick one for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Say we have picked the nth fold for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Then there are 9 remaining folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We iteratively pick the mth n (with 1 ď mn ď 9) of the remaining 9 folds for choosing hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This leaves 8 folds on which we train our model for each choice of hyper parameter in Cpool ˆ Gpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This yields 8 regression models, which we average to built our final predictor for the nth run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' This mean absolute error of this predictor is then evaluated on the nth fold which was retained for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' As n varies from one to ten, we built the mean and variance of the performances of the generated regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' We chose log-linear equidistant hyperparameters from Gpool :“ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='00003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='0003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='3, 3, 30u, and Cpool :“ t400000, 40000, 4000, 400, 40, 4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content='4u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Reference Methods: We comprehensively evaluate our method against 11 popular baselines and state of the art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Among these methods are graph convolutional methods such as GraphConv [18], Weave [17] or SchNet [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' MPNN [13] and its variant DMPNN [44] are models considering edge features during message passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' AttentiveFP [40] is an extension of the graph attention framework, while N-Gram [21] is a pretrained method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Results for these methods as well as for GROVER are taken from [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' PhysChem [45] learns molecular representations via fusing physical and chemical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Deep Tensor Neural Networks (DTNN [39]) are adaptable extensions of the Coulomb Matrix featurizer mapping atom numbers to trainable embeddings which are then updated based on distance information and other (node-level) atomic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' Finally Path-Augmented Graph Transformer Networks (PATGN, [6]) exploit the connectivity structure of the data in a global attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'} +page_content=' 39' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}