Daily Papers

Training deep neural networks (DNNs) using traditional backpropagation (BP) presents challenges in terms of computational complexity and energy consumption, particularly for on-device learning where computational resources are limited. Various alternatives to BP, including random feedback alignment, forward-forward, and local classifiers, have been explored to address these challenges. These methods have their advantages, but they can encounter difficulties when dealing with intricate visual tasks or demand considerable computational resources. In this paper, we propose a novel Local Learning rule inspired by neural activity Synchronization phenomena (LLS) observed in the brain. LLS utilizes fixed periodic basis vectors to synchronize neuron activity within each layer, enabling efficient training without the need for additional trainable parameters. We demonstrate the effectiveness of LLS and its variations, LLS-M and LLS-MxM, on multiple image classification datasets, achieving accuracy comparable to BP with reduced computational complexity and minimal additional parameters. Furthermore, the performance of LLS on the Visual Wake Word (VWW) dataset highlights its suitability for on-device learning tasks, making it a promising candidate for edge hardware implementations.

Strategies for the generation of periodic discrete structures with identical two-point correlation are developed. Starting from a pair of root structures, which are not related by translation, phase inversion or axis reflections, child structures of arbitrary resolution (i.e., pixel or voxel numbers) and number of phases (i.e., material phases/species) can be generated by means of trivial embedding based phase extension, application of kernels and/or phase coalescence, such that the generated structures inherit the two-point-correlation equivalence. Proofs of the inheritance property are provided by means of the Discrete Fourier Transform theory. A Python 3 implementation of the results is offered by the authors through the Github repository https://github.com/DataAnalyticsEngineering/EQ2PC in order to make the provided results reproducible and useful for all interested readers. Examples for the generation of structures are demonstrated, together with applications in the homogenization theory of periodic media.

Periodic signals play an important role in daily lives. Although conventional sequential models have shown remarkable success in various fields, they still come short in modeling periodicity; they either collapse, diverge or ignore details. In this paper, we introduce a novel framework inspired by Fourier series to generate periodic signals. We first decompose the given signals into multiple sines and cosines and then conditionally generate periodic signals with the output components. We have shown our model efficacy on three tasks: reconstruction, imputation and conditional generation. Our model outperforms baselines in all tasks and shows more stable and refined results.

Crystal structures are characterized by atomic bases within a primitive unit cell that repeats along a regular lattice throughout 3D space. The periodic and infinite nature of crystals poses unique challenges for geometric graph representation learning. Specifically, constructing graphs that effectively capture the complete geometric information of crystals and handle chiral crystals remains an unsolved and challenging problem. In this paper, we introduce a novel approach that utilizes the periodic patterns of unit cells to establish the lattice-based representation for each atom, enabling efficient and expressive graph representations of crystals. Furthermore, we propose ComFormer, a SE(3) transformer designed specifically for crystalline materials. ComFormer includes two variants; namely, iComFormer that employs invariant geometric descriptors of Euclidean distances and angles, and eComFormer that utilizes equivariant vector representations. Experimental results demonstrate the state-of-the-art predictive accuracy of ComFormer variants on various tasks across three widely-used crystal benchmarks. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

Generating the periodic structure of stable materials is a long-standing challenge for the material design community. This task is difficult because stable materials only exist in a low-dimensional subspace of all possible periodic arrangements of atoms: 1) the coordinates must lie in the local energy minimum defined by quantum mechanics, and 2) global stability also requires the structure to follow the complex, yet specific bonding preferences between different atom types. Existing methods fail to incorporate these factors and often lack proper invariances. We propose a Crystal Diffusion Variational Autoencoder (CDVAE) that captures the physical inductive bias of material stability. By learning from the data distribution of stable materials, the decoder generates materials in a diffusion process that moves atomic coordinates towards a lower energy state and updates atom types to satisfy bonding preferences between neighbors. Our model also explicitly encodes interactions across periodic boundaries and respects permutation, translation, rotation, and periodic invariances. We significantly outperform past methods in three tasks: 1) reconstructing the input structure, 2) generating valid, diverse, and realistic materials, and 3) generating materials that optimize a specific property. We also provide several standard datasets and evaluation metrics for the broader machine learning community.

The stable periodic patterns present in time series data serve as the foundation for conducting long-horizon forecasts. In this paper, we pioneer the exploration of explicitly modeling this periodicity to enhance the performance of models in long-term time series forecasting (LTSF) tasks. Specifically, we introduce the Residual Cycle Forecasting (RCF) technique, which utilizes learnable recurrent cycles to model the inherent periodic patterns within sequences, and then performs predictions on the residual components of the modeled cycles. Combining RCF with a Linear layer or a shallow MLP forms the simple yet powerful method proposed in this paper, called CycleNet. CycleNet achieves state-of-the-art prediction accuracy in multiple domains including electricity, weather, and energy, while offering significant efficiency advantages by reducing over 90% of the required parameter quantity. Furthermore, as a novel plug-and-play technique, the RCF can also significantly improve the prediction accuracy of existing models, including PatchTST and iTransformer. The source code is available at: https://github.com/ACAT-SCUT/CycleNet.

Previous literature offers limited clues on how to learn a periodic function using modern neural networks. We start with a study of the extrapolation properties of neural networks; we prove and demonstrate experimentally that the standard activations functions, such as ReLU, tanh, sigmoid, along with their variants, all fail to learn to extrapolate simple periodic functions. We hypothesize that this is due to their lack of a "periodic" inductive bias. As a fix of this problem, we propose a new activation, namely, x + sin^2(x), which achieves the desired periodic inductive bias to learn a periodic function while maintaining a favorable optimization property of the ReLU-based activations. Experimentally, we apply the proposed method to temperature and financial data prediction.

We provide a full characterisation of all of the possible alternating group (A_n) equivariant neural networks whose layers are some tensor power of R^{n}. In particular, we find a basis of matrices for the learnable, linear, A_n-equivariant layer functions between such tensor power spaces in the standard basis of R^{n}. We also describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.

Time series analysis is a fundamental task in various application domains, and deep learning approaches have demonstrated remarkable performance in this area. However, many real-world time series data exhibit significant periodic or quasi-periodic dynamics that are often not adequately captured by existing deep learning-based solutions. This results in an incomplete representation of the underlying dynamic behaviors of interest. To address this gap, we propose an unsupervised method called Floss that automatically regularizes learned representations in the frequency domain. The Floss method first automatically detects major periodicities from the time series. It then employs periodic shift and spectral density similarity measures to learn meaningful representations with periodic consistency. In addition, Floss can be easily incorporated into both supervised, semi-supervised, and unsupervised learning frameworks. We conduct extensive experiments on common time series classification, forecasting, and anomaly detection tasks to demonstrate the effectiveness of Floss. We incorporate Floss into several representative deep learning solutions to justify our design choices and demonstrate that it is capable of automatically discovering periodic dynamics and improving state-of-the-art deep learning models.

Despite the remarkable success achieved by neural networks, particularly those represented by MLP and Transformer, we reveal that they exhibit potential flaws in the modeling and reasoning of periodicity, i.e., they tend to memorize the periodic data rather than genuinely understanding the underlying principles of periodicity. However, periodicity is a crucial trait in various forms of reasoning and generalization, underpinning predictability across natural and engineered systems through recurring patterns in observations. In this paper, we propose FAN, a novel network architecture based on Fourier Analysis, which empowers the ability to efficiently model and reason about periodic phenomena. By introducing Fourier Series, the periodicity is naturally integrated into the structure and computational processes of the neural network, thus achieving a more accurate expression and prediction of periodic patterns. As a promising substitute to multi-layer perceptron (MLP), FAN can seamlessly replace MLP in various models with fewer parameters and FLOPs. Through extensive experiments, we demonstrate the effectiveness of FAN in modeling and reasoning about periodic functions, and the superiority and generalizability of FAN across a range of real-world tasks, including symbolic formula representation, time series forecasting, and language modeling.

Recently, universal waveform generation tasks have been investigated conditioned on various out-of-distribution scenarios. Although GAN-based methods have shown their strength in fast waveform generation, they are vulnerable to train-inference mismatch scenarios such as two-stage text-to-speech. Meanwhile, diffusion-based models have shown their powerful generative performance in other domains; however, they stay out of the limelight due to slow inference speed in waveform generation tasks. Above all, there is no generator architecture that can explicitly disentangle the natural periodic features of high-resolution waveform signals. In this paper, we propose PeriodWave, a novel universal waveform generation model. First, we introduce a period-aware flow matching estimator that can capture the periodic features of the waveform signal when estimating the vector fields. Additionally, we utilize a multi-period estimator that avoids overlaps to capture different periodic features of waveform signals. Although increasing the number of periods can improve the performance significantly, this requires more computational costs. To reduce this issue, we also propose a single period-conditional universal estimator that can feed-forward parallel by period-wise batch inference. Additionally, we utilize discrete wavelet transform to losslessly disentangle the frequency information of waveform signals for high-frequency modeling, and introduce FreeU to reduce the high-frequency noise for waveform generation. The experimental results demonstrated that our model outperforms the previous models both in Mel-spectrogram reconstruction and text-to-speech tasks. All source code will be available at https://github.com/sh-lee-prml/PeriodWave.

We show how the Schur-Weyl duality that exists between the partition algebra and the symmetric group results in a stronger theoretical foundation for characterising all of the possible permutation equivariant neural networks whose layers are some tensor power of the permutation representation M_n of the symmetric group S_n. In doing so, we unify two separate bodies of literature, and we correct some of the major results that are now widely quoted by the machine learning community. In particular, we find a basis of matrices for the learnable, linear, permutation equivariant layer functions between such tensor power spaces in the standard basis of M_n by using an elegant graphical representation of a basis of set partitions for the partition algebra and its related vector spaces. Also, we show how we can calculate the number of weights that must appear in these layer functions by looking at certain paths through the McKay quiver for M_n. Finally, we describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.

Vector Symbolic Architectures (VSAs) give a way to represent a complex object as a single fixed-length vector, so that similar objects have similar vector representations. These vector representations then become easy to use for machine learning or nearest-neighbor search. We review a previously proposed VSA method, MBAT (Matrix Binding of Additive Terms), which uses multiplication by random matrices for binding related terms. However, multiplying by such matrices introduces instabilities which can harm performance. Making the random matrices be orthogonal matrices provably fixes this problem. With respect to larger scale applications, we see how to apply MBAT vector representations for any data expressed in JSON. JSON is used in numerous programming languages to express complex data, but its native format appears highly unsuited for machine learning. Expressing JSON as a fixed-length vector makes it readily usable for machine learning and nearest-neighbor search. Creating such JSON vectors also shows that a VSA needs to employ binding operations that are non-commutative. VSAs are now ready to try with full-scale practical applications, including healthcare, pharmaceuticals, and genomics. Keywords: MBAT (Matrix Binding of Additive Terms), VSA (Vector Symbolic Architecture), HDC (Hyperdimensional Computing), Distributed Representations, Binding, Orthogonal Matrices, Recurrent Connections, Machine Learning, Search, JSON, VSA Applications

Designing effective positional encodings for graphs is key to building powerful graph transformers and enhancing message-passing graph neural networks. Although widespread, using Laplacian eigenvectors as positional encodings faces two fundamental challenges: (1) Non-uniqueness: there are many different eigendecompositions of the same Laplacian, and (2) Instability: small perturbations to the Laplacian could result in completely different eigenspaces, leading to unpredictable changes in positional encoding. Despite many attempts to address non-uniqueness, most methods overlook stability, leading to poor generalization on unseen graph structures. We identify the cause of instability to be a "hard partition" of eigenspaces. Hence, we introduce Stable and Expressive Positional Encodings (SPE), an architecture for processing eigenvectors that uses eigenvalues to "softly partition" eigenspaces. SPE is the first architecture that is (1) provably stable, and (2) universally expressive for basis invariant functions whilst respecting all symmetries of eigenvectors. Besides guaranteed stability, we prove that SPE is at least as expressive as existing methods, and highly capable of counting graph structures. Finally, we evaluate the effectiveness of our method on molecular property prediction, and out-of-distribution generalization tasks, finding improved generalization compared to existing positional encoding methods.

Crystalline materials are a fundamental component in next-generation technologies, yet modeling their distribution presents unique computational challenges. Of the plausible arrangements of atoms in a periodic lattice only a vanishingly small percentage are thermodynamically stable, which is a key indicator of the materials that can be experimentally realized. Two fundamental tasks in this area are to (a) predict the stable crystal structure of a known composition of elements and (b) propose novel compositions along with their stable structures. We present FlowMM, a pair of generative models that achieve state-of-the-art performance on both tasks while being more efficient and more flexible than competing methods. We generalize Riemannian Flow Matching to suit the symmetries inherent to crystals: translation, rotation, permutation, and periodic boundary conditions. Our framework enables the freedom to choose the flow base distributions, drastically simplifying the problem of learning crystal structures compared with diffusion models. In addition to standard benchmarks, we validate FlowMM's generated structures with quantum chemistry calculations, demonstrating that it is about 3x more efficient, in terms of integration steps, at finding stable materials compared to previous open methods.

We present a novel type of neural fields that uses general radial bases for signal representation. State-of-the-art neural fields typically rely on grid-based representations for storing local neural features and N-dimensional linear kernels for interpolating features at continuous query points. The spatial positions of their neural features are fixed on grid nodes and cannot well adapt to target signals. Our method instead builds upon general radial bases with flexible kernel position and shape, which have higher spatial adaptivity and can more closely fit target signals. To further improve the channel-wise capacity of radial basis functions, we propose to compose them with multi-frequency sinusoid functions. This technique extends a radial basis to multiple Fourier radial bases of different frequency bands without requiring extra parameters, facilitating the representation of details. Moreover, by marrying adaptive radial bases with grid-based ones, our hybrid combination inherits both adaptivity and interpolation smoothness. We carefully designed weighting schemes to let radial bases adapt to different types of signals effectively. Our experiments on 2D image and 3D signed distance field representation demonstrate the higher accuracy and compactness of our method than prior arts. When applied to neural radiance field reconstruction, our method achieves state-of-the-art rendering quality, with small model size and comparable training speed.

Time series analysis is of immense importance in extensive applications, such as weather forecasting, anomaly detection, and action recognition. This paper focuses on temporal variation modeling, which is the common key problem of extensive analysis tasks. Previous methods attempt to accomplish this directly from the 1D time series, which is extremely challenging due to the intricate temporal patterns. Based on the observation of multi-periodicity in time series, we ravel out the complex temporal variations into the multiple intraperiod- and interperiod-variations. To tackle the limitations of 1D time series in representation capability, we extend the analysis of temporal variations into the 2D space by transforming the 1D time series into a set of 2D tensors based on multiple periods. This transformation can embed the intraperiod- and interperiod-variations into the columns and rows of the 2D tensors respectively, making the 2D-variations to be easily modeled by 2D kernels. Technically, we propose the TimesNet with TimesBlock as a task-general backbone for time series analysis. TimesBlock can discover the multi-periodicity adaptively and extract the complex temporal variations from transformed 2D tensors by a parameter-efficient inception block. Our proposed TimesNet achieves consistent state-of-the-art in five mainstream time series analysis tasks, including short- and long-term forecasting, imputation, classification, and anomaly detection. Code is available at this repository: https://github.com/thuml/TimesNet.

We propose Equiangular Basis Vectors (EBVs) for classification tasks. In deep neural networks, models usually end with a k-way fully connected layer with softmax to handle different classification tasks. The learning objective of these methods can be summarized as mapping the learned feature representations to the samples' label space. While in metric learning approaches, the main objective is to learn a transformation function that maps training data points from the original space to a new space where similar points are closer while dissimilar points become farther apart. Different from previous methods, our EBVs generate normalized vector embeddings as "predefined classifiers" which are required to not only be with the equal status between each other, but also be as orthogonal as possible. By minimizing the spherical distance of the embedding of an input between its categorical EBV in training, the predictions can be obtained by identifying the categorical EBV with the smallest distance during inference. Various experiments on the ImageNet-1K dataset and other downstream tasks demonstrate that our method outperforms the general fully connected classifier while it does not introduce huge additional computation compared with classical metric learning methods. Our EBVs won the first place in the 2022 DIGIX Global AI Challenge, and our code is open-source and available at https://github.com/NJUST-VIPGroup/Equiangular-Basis-Vectors.

The Schr\"odinger Bridge (SB) is a powerful framework for solving generative modeling tasks such as unpaired domain translation. Most SB-related research focuses on continuous data space R^{D} and leaves open theoretical and algorithmic questions about applying SB methods to discrete data, e.g, on finite spaces S^{D}. Notable examples of such sets S are codebooks of vector-quantized (VQ) representations of modern autoencoders, tokens in texts, categories of atoms in molecules, etc. In this paper, we provide a theoretical and algorithmic foundation for solving SB in discrete spaces using the recently introduced Iterative Markovian Fitting (IMF) procedure. Specifically, we theoretically justify the convergence of discrete-time IMF (D-IMF) to SB in discrete spaces. This enables us to develop a practical computational algorithm for SB which we call Categorical Schr\"odinger Bridge Matching (CSBM). We show the performance of CSBM via a series of experiments with synthetic data and VQ representations of images.

It is well noted that coordinate based MLPs benefit -- in terms of preserving high-frequency information -- through the encoding of coordinate positions as an array of Fourier features. Hitherto, the rationale for the effectiveness of these positional encodings has been solely studied through a Fourier lens. In this paper, we strive to broaden this understanding by showing that alternative non-Fourier embedding functions can indeed be used for positional encoding. Moreover, we show that their performance is entirely determined by a trade-off between the stable rank of the embedded matrix and the distance preservation between embedded coordinates. We further establish that the now ubiquitous Fourier feature mapping of position is a special case that fulfills these conditions. Consequently, we present a more general theory to analyze positional encoding in terms of shifted basis functions. To this end, we develop the necessary theoretical formulae and empirically verify that our theoretical claims hold in practice. Codes available at https://github.com/osiriszjq/Rethinking-positional-encoding.

We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of R^{n} for three symmetry groups that are missing from the machine learning literature: O(n), the orthogonal group; SO(n), the special orthogonal group; and Sp(n), the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of R^{n} when the group is O(n) or SO(n), and in the symplectic basis of R^{n} when the group is Sp(n).

Real-world multichannel time series prediction faces growing demands for efficiency across edge and cloud environments, making channel compression a timely and essential problem. Motivated by success of Multiple-Input Multiple-Output (MIMO) methods, we propose a predictability-aware compression-decompression framework to reduce runtime, lower communication cost, and maintain prediction accuracy across diverse predictors. The core idea involves using a circular periodicity key matrix with orthogonality to capture underlying time series predictability during compression and to mitigate reconstruction errors during decompression by relaxing oversimplified data assumptions. Theoretical and empirical analyses show that the proposed framework is both time-efficient and scalable under a large number of channels. Extensive experiments on six datasets across various predictors demonstrate that the proposed method achieves superior overall performance by jointly considering prediction accuracy and runtime, while maintaining strong compatibility with diverse predictors.

Modeling dynamic, large-scale urban scenes is challenging due to their highly intricate geometric structures and unconstrained dynamics in both space and time. Prior methods often employ high-level architectural priors, separating static and dynamic elements, resulting in suboptimal capture of their synergistic interactions. To address this challenge, we present a unified representation model, called Periodic Vibration Gaussian (PVG). PVG builds upon the efficient 3D Gaussian splatting technique, originally designed for static scene representation, by introducing periodic vibration-based temporal dynamics. This innovation enables PVG to elegantly and uniformly represent the characteristics of various objects and elements in dynamic urban scenes. To enhance temporally coherent representation learning with sparse training data, we introduce a novel flow-based temporal smoothing mechanism and a position-aware adaptive control strategy. Extensive experiments on Waymo Open Dataset and KITTI benchmarks demonstrate that PVG surpasses state-of-the-art alternatives in both reconstruction and novel view synthesis for both dynamic and static scenes. Notably, PVG achieves this without relying on manually labeled object bounding boxes or expensive optical flow estimation. Moreover, PVG exhibits 50/6000-fold acceleration in training/rendering over the best alternative.

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

Projection maintenance is one of the core data structure tasks. Efficient data structures for projection maintenance have led to recent breakthroughs in many convex programming algorithms. In this work, we further extend this framework to the Kronecker product structure. Given a constraint matrix {sf A} and a positive semi-definite matrix Win R^{ntimes n} with a sparse eigenbasis, we consider the task of maintaining the projection in the form of {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}, where {sf B}={sf A}(Wotimes I) or {sf B}={sf A}(W^{1/2}otimes W^{1/2}). At each iteration, the weight matrix W receives a low rank change and we receive a new vector h. The goal is to maintain the projection matrix and answer the query {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}h with good approximation guarantees. We design a fast dynamic data structure for this task and it is robust against an adaptive adversary. Following the beautiful and pioneering work of [Beimel, Kaplan, Mansour, Nissim, Saranurak and Stemmer, STOC'22], we use tools from differential privacy to reduce the randomness required by the data structure and further improve the running time.

Generative modeling of crystal data distribution is an important yet challenging task due to the unique periodic physical symmetry of crystals. Diffusion-based methods have shown early promise in modeling crystal distribution. More recently, Bayesian Flow Networks were introduced to aggregate noisy latent variables, resulting in a variance-reduced parameter space that has been shown to be advantageous for modeling Euclidean data distributions with structural constraints (Song et al., 2023). Inspired by this, we seek to unlock its potential for modeling variables located in non-Euclidean manifolds e.g. those within crystal structures, by overcoming challenging theoretical issues. We introduce CrysBFN, a novel crystal generation method by proposing a periodic Bayesian flow, which essentially differs from the original Gaussian-based BFN by exhibiting non-monotonic entropy dynamics. To successfully realize the concept of periodic Bayesian flow, CrysBFN integrates a new entropy conditioning mechanism and empirically demonstrates its significance compared to time-conditioning. Extensive experiments over both crystal ab initio generation and crystal structure prediction tasks demonstrate the superiority of CrysBFN, which consistently achieves new state-of-the-art on all benchmarks. Surprisingly, we found that CrysBFN enjoys a significant improvement in sampling efficiency, e.g., ~100x speedup 10 v.s. 2000 steps network forwards) compared with previous diffusion-based methods on MP-20 dataset. Code is available at https://github.com/wu-han-lin/CrysBFN.

In this article, we introduce a new parameterized family of topological invariants, taking the form of candidate decompositions, for multi-parameter persistence modules. We prove that our candidate decompositions are controllable approximations: when restricting to modules that can be decomposed into interval summands, we establish theoretical results about the approximation error between our candidate decompositions and the true underlying module in terms of the standard interleaving and bottleneck distances. Moreover, even when the underlying module does not admit such a decomposition, our candidate decompositions are nonetheless stable invariants; small perturbations in the underlying module lead to small perturbations in the candidate decomposition. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm for computing stable instances of such invariants, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Finally, we present empirical evidence validating the generalization capabilities and running time speed-ups of MMA on several data sets.

In this note we establish a 1-to-1 correspondence between the class of generalized quadrangles with ovoids and the class of balanced incomplete block designs that posses a non-triangular local resolution system and have the appropriate parameters. We present a non-triangular local resolution system for a difference family BIBD construction of Sprott.

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.

Time-series forecasting is crucial for numerous real-world applications including weather prediction and financial market modeling. While temporal-domain methods remain prevalent, frequency-domain approaches can effectively capture multi-scale periodic patterns, reduce sequence dependencies, and naturally denoise signals. However, existing approaches typically train model components for all frequencies under a unified training objective, often leading to mismatched learning speeds: high-frequency components converge faster and risk overfitting, while low-frequency components underfit due to insufficient training time. To deal with this challenge, we propose BEAT (Balanced frEquency Adaptive Tuning), a novel framework that dynamically monitors the training status for each frequency and adaptively adjusts their gradient updates. By recognizing convergence, overfitting, or underfitting for each frequency, BEAT dynamically reallocates learning priorities, moderating gradients for rapid learners and increasing those for slower ones, alleviating the tension between competing objectives across frequencies and synchronizing the overall learning process. Extensive experiments on seven real-world datasets demonstrate that BEAT consistently outperforms state-of-the-art approaches.

For any graph G having n vertices and its automorphism group Aut(G), we provide a full characterisation of all of the possible Aut(G)-equivariant neural networks whose layers are some tensor power of R^{n}. In particular, we find a spanning set of matrices for the learnable, linear, Aut(G)-equivariant layer functions between such tensor power spaces in the standard basis of R^{n}.

Quantized neural networks employ reduced precision representations for both weights and activations. This quantization process significantly reduces the memory requirements and computational complexity of the network. Binary Neural Networks (BNNs) are the extreme quantization case, representing values with just one bit. Since the sign function is typically used to map real values to binary values, smooth approximations are introduced to mimic the gradients during error backpropagation. Thus, the mismatch between the forward and backward models corrupts the direction of the gradient, causing training inconsistency problems and performance degradation. In contrast to current BNN approaches, we propose to employ a binary periodic (BiPer) function during binarization. Specifically, we use a square wave for the forward pass to obtain the binary values and employ the trigonometric sine function with the same period of the square wave as a differentiable surrogate during the backward pass. We demonstrate that this approach can control the quantization error by using the frequency of the periodic function and improves network performance. Extensive experiments validate the effectiveness of BiPer in benchmark datasets and network architectures, with improvements of up to 1% and 0.69% with respect to state-of-the-art methods in the classification task over CIFAR-10 and ImageNet, respectively. Our code is publicly available at https://github.com/edmav4/BiPer.

We study the matrix models pi:C(S_N^+)to M_N(C(X)) which are flat, in the sense that the standard generators of C(S_N^+) are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at N=4, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of C(S_N^+), inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful.

Learning physical simulations has been an essential and central aspect of many recent research efforts in machine learning, particularly for Navier-Stokes-based fluid mechanics. Classic numerical solvers have traditionally been computationally expensive and challenging to use in inverse problems, whereas Neural solvers aim to address both concerns through machine learning. We propose a general formulation for continuous convolutions using separable basis functions as a superset of existing methods and evaluate a large set of basis functions in the context of (a) a compressible 1D SPH simulation, (b) a weakly compressible 2D SPH simulation, and (c) an incompressible 2D SPH Simulation. We demonstrate that even and odd symmetries included in the basis functions are key aspects of stability and accuracy. Our broad evaluation shows that Fourier-based continuous convolutions outperform all other architectures regarding accuracy and generalization. Finally, using these Fourier-based networks, we show that prior inductive biases, such as window functions, are no longer necessary. An implementation of our approach, as well as complete datasets and solver implementations, is available at https://github.com/tum-pbs/SFBC.

Structure-Based Drug Design (SBDD) is crucial for identifying bioactive molecules. Recent deep generative models are faced with challenges in geometric structure modeling. A major bottleneck lies in the twisted probability path of multi-modalities -- continuous 3D positions and discrete 2D topologies -- which jointly determine molecular geometries. By establishing the fact that noise schedules decide the Variational Lower Bound (VLB) for the twisted probability path, we propose VLB-Optimal Scheduling (VOS) strategy in this under-explored area, which optimizes VLB as a path integral for SBDD. Our model effectively enhances molecular geometries and interaction modeling, achieving state-of-the-art PoseBusters passing rate of 95.9% on CrossDock, more than 10% improvement upon strong baselines, while maintaining high affinities and robust intramolecular validity evaluated on held-out test set. Code is available at https://github.com/AlgoMole/MolCRAFT.

Implicitly defined, continuous, differentiable signal representations parameterized by neural networks have emerged as a powerful paradigm, offering many possible benefits over conventional representations. However, current network architectures for such implicit neural representations are incapable of modeling signals with fine detail, and fail to represent a signal's spatial and temporal derivatives, despite the fact that these are essential to many physical signals defined implicitly as the solution to partial differential equations. We propose to leverage periodic activation functions for implicit neural representations and demonstrate that these networks, dubbed sinusoidal representation networks or Sirens, are ideally suited for representing complex natural signals and their derivatives. We analyze Siren activation statistics to propose a principled initialization scheme and demonstrate the representation of images, wavefields, video, sound, and their derivatives. Further, we show how Sirens can be leveraged to solve challenging boundary value problems, such as particular Eikonal equations (yielding signed distance functions), the Poisson equation, and the Helmholtz and wave equations. Lastly, we combine Sirens with hypernetworks to learn priors over the space of Siren functions.

Given a matrix Ain R^{ntimes d} and a vector bin R^n, we consider the regression problem with ell_infty guarantees: finding a vector x'in R^d such that |x'-x^*|_infty leq epsilon{d}cdot |Ax^*-b|_2cdot |A^dagger| where x^*=argmin_{xin R^d}|Ax-b|_2. One popular approach for solving such ell_2 regression problem is via sketching: picking a structured random matrix Sin R^{mtimes n} with mll n and SA can be quickly computed, solve the ``sketched'' regression problem argmin_{xin R^d} |SAx-Sb|_2. In this paper, we show that in order to obtain such ell_infty guarantee for ell_2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m=epsilon^{-2}dlog^3(n/delta) such that solving the sketched regression problem gives the ell_infty guarantee, with probability at least 1-delta. Moreover, the matrix SA can be computed in time O(ndlog n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in [Price, Song and Woodruff, ICALP'17], in which a super-linear in d rows, m=Omega(epsilon^{-2}d^{1+gamma}) for gamma=Theta(frac{loglog n{log d}}) is required. We also develop a novel analytical framework for ell_infty guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in [Song and Yu, ICML'21]. Our analysis is arguably much simpler and more general than [Price, Song and Woodruff, ICALP'17], and it extends to dense sketches for tensor product of vectors.

We obtain (small-parameter) well-posedness for the (space-time periodic) Phi^4 equation in the full subcritical regime in the context of regularity structures based on multi-indices. As opposed to Hairer's more extrinsic tree-based setting, due to the intrinsic description encoded by multi-indices, it is not possible to obtain a solution theory via the standard fixed-point argument. Instead, we develop a more intrinsic approach for existence using a variant of the continuity method from classical PDE theory based on a priori estimates for a new `robust' formulation of the equation. This formulation also allows us to obtain uniqueness of solutions and continuity of the solution map in the model norm even at the limit of vanishing regularisation scale. Since our proof relies on the structure of the nonlinearity in only a mild way, we expect the same ideas to be sufficient to treat a more general class of equations.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

Polynomial filters, a kind of Graph Neural Networks, typically use a predetermined polynomial basis and learn the coefficients from the training data. It has been observed that the effectiveness of the model is highly dependent on the property of the polynomial basis. Consequently, two natural and fundamental questions arise: Can we learn a suitable polynomial basis from the training data? Can we determine the optimal polynomial basis for a given graph and node features? In this paper, we propose two spectral GNN models that provide positive answers to the questions posed above. First, inspired by Favard's Theorem, we propose the FavardGNN model, which learns a polynomial basis from the space of all possible orthonormal bases. Second, we examine the supposedly unsolvable definition of optimal polynomial basis from Wang & Zhang (2022) and propose a simple model, OptBasisGNN, which computes the optimal basis for a given graph structure and graph signal. Extensive experiments are conducted to demonstrate the effectiveness of our proposed models.

Crystals are the foundation of numerous scientific and industrial applications. While various learning-based approaches have been proposed for crystal generation, existing methods seldom consider the space group constraint which is crucial in describing the geometry of crystals and closely relevant to many desirable properties. However, considering space group constraint is challenging owing to its diverse and nontrivial forms. In this paper, we reduce the space group constraint into an equivalent formulation that is more tractable to be handcrafted into the generation process. In particular, we translate the space group constraint into two parts: the basis constraint of the invariant logarithmic space of the lattice matrix and the Wyckoff position constraint of the fractional coordinates. Upon the derived constraints, we then propose DiffCSP++, a novel diffusion model that has enhanced a previous work DiffCSP by further taking space group constraint into account. Experiments on several popular datasets verify the benefit of the involvement of the space group constraint, and show that our DiffCSP++ achieves promising performance on crystal structure prediction, ab initio crystal generation and controllable generation with customized space groups.

Crystal Structure Prediction (CSP) is crucial in various scientific disciplines. While CSP can be addressed by employing currently-prevailing generative models (e.g. diffusion models), this task encounters unique challenges owing to the symmetric geometry of crystal structures -- the invariance of translation, rotation, and periodicity. To incorporate the above symmetries, this paper proposes DiffCSP, a novel diffusion model to learn the structure distribution from stable crystals. To be specific, DiffCSP jointly generates the lattice and atom coordinates for each crystal by employing a periodic-E(3)-equivariant denoising model, to better model the crystal geometry. Notably, different from related equivariant generative approaches, DiffCSP leverages fractional coordinates other than Cartesian coordinates to represent crystals, remarkably promoting the diffusion and the generation process of atom positions. Extensive experiments verify that our DiffCSP significantly outperforms existing CSP methods, with a much lower computation cost in contrast to DFT-based methods. Moreover, the superiority of DiffCSP is also observed when it is extended for ab initio crystal generation.

A sizeable fraction of gamma-ray burst (GRB) light curves (LCs) features a sequence of peaks, which holds information on the unknown way energy is dissipated into gamma-rays over time. Traditional searches for periodic signals in GRB LCs turned out to be inconclusive, partly because they are challenging as a consequence of the short-lived, coloured-noise, and non-stationary nature of the LCs themselves. Yet, recent claims have revived the issue. We searched for periodic components in GRB LCs through a new approach to GRBs, that avoids most of the issues faced by traditional techniques. We identified peaks through a well tested algorithm and selected GRBs with at least 10 peaks out of 5 GRB catalogues (Swift/BAT, CGRO/BATSE, Fermi/GBM, Insight-HXMT, BeppoSAX/GRBM). Each GRB was simply treated as a discrete point process, whose realisation coincides with the sequence of peak times. We searched for possible periodic recurrences based on the multinomial distribution, after accounting for the clustering of peaks due to the non-stationarity of the GRB signals. The best candidate has a p-value of 3e-4 that there is no periodic recurrence. However, accounting for the multiple trials of 555 searched GRBs, its statistical significance is demoted to 17%. The overall distribution of the p-values obtained for all GRBs is compatible with a uniform distribution in [0,1]. We found no robust evidence for multi-peaked GRBs with periodic recurrences. We can exclude that a sizeable fraction (>~ 0.75) of peaks of each GRB with at least 10 peaks are periodic. While our result does not necessarily clash with claimed periodicities based on Fourier techniques, it constrains the putative recurrent behaviour, which would not manifest itself through the sequence of peaks, but, evidently, in a more elusive way.

Neural architectures that learn potential energy surfaces from molecular data have undergone fast improvement in recent years. A key driver of this success is the Message Passing Neural Network (MPNN) paradigm. Its favorable scaling with system size partly relies upon a spatial distance limit on messages. While this focus on locality is a useful inductive bias, it also impedes the learning of long-range interactions such as electrostatics and van der Waals forces. To address this drawback, we propose Ewald message passing: a nonlocal Fourier space scheme which limits interactions via a cutoff on frequency instead of distance, and is theoretically well-founded in the Ewald summation method. It can serve as an augmentation on top of existing MPNN architectures as it is computationally inexpensive and agnostic to architectural details. We test the approach with four baseline models and two datasets containing diverse periodic (OC20) and aperiodic structures (OE62). We observe robust improvements in energy mean absolute errors across all models and datasets, averaging 10% on OC20 and 16% on OE62. Our analysis shows an outsize impact of these improvements on structures with high long-range contributions to the ground truth energy.

Objective: Electroencephalography signals are recorded as a multidimensional dataset. We propose a new framework based on the augmented covariance extracted from an autoregressive model to improve motor imagery classification. Methods: From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to extend the standard approach using these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search. Results: The augmented covariance matrix performed noticeably better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation. Conclusion: The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information, incorporating nonlinear components of the signal through an embedding procedure, which allows the leveraging of dynamical systems algorithms. Significance: These results extend the concepts and the results of the Riemannian distance based classification algorithm.

Riemannian submanifold optimization with momentum is computationally challenging because, to ensure that the iterates remain on the submanifold, we often need to solve difficult differential equations. Here, we simplify such difficulties for a class of structured symmetric positive-definite matrices with the affine-invariant metric. We do so by proposing a generalized version of the Riemannian normal coordinates that dynamically orthonormalizes the metric and locally converts the problem into an unconstrained problem in the Euclidean space. We use our approach to simplify existing approaches for structured covariances and develop matrix-inverse-free 2^nd-order optimizers for deep learning in low precision settings. Code: https://github.com/yorkerlin/StructuredNGD-DL

Linear time-invariant state space models (SSM) are a classical model from engineering and statistics, that have recently been shown to be very promising in machine learning through the Structured State Space sequence model (S4). A core component of S4 involves initializing the SSM state matrix to a particular matrix called a HiPPO matrix, which was empirically important for S4's ability to handle long sequences. However, the specific matrix that S4 uses was actually derived in previous work for a particular time-varying dynamical system, and the use of this matrix as a time-invariant SSM had no known mathematical interpretation. Consequently, the theoretical mechanism by which S4 models long-range dependencies actually remains unexplained. We derive a more general and intuitive formulation of the HiPPO framework, which provides a simple mathematical interpretation of S4 as a decomposition onto exponentially-warped Legendre polynomials, explaining its ability to capture long dependencies. Our generalization introduces a theoretically rich class of SSMs that also lets us derive more intuitive S4 variants for other bases such as the Fourier basis, and explains other aspects of training S4, such as how to initialize the important timescale parameter. These insights improve S4's performance to 86% on the Long Range Arena benchmark, with 96% on the most difficult Path-X task.

This paper makes a formal connection between two families of widely used matrix factorization algorithms in numerical linear algebra. One family consists of the Jacobi eigenvalue algorithm and its variants for computing the Hermitian eigendecomposition and singular value decomposition. The other consists of Gaussian elimination and the Gram-Schmidt procedure with various pivoting rules for computing the Cholesky decomposition and QR decomposition respectively. Both families are cast as special cases of a more general class of factorization algorithms. We provide a randomized pivoting rule that applies to this general class (which differs substantially from the usual pivoting rules for Gaussian elimination / Gram-Schmidt) which results in the same linear rate of convergence for each algorithm, irrespective of which factorization it computes. A second important consequence of this randomized pivoting rule is a provable, effective bound on the numerical stability of the Jacobi eigenvalue algorithm, which addresses a longstanding open problem of Demmel and Veseli\'c `92.

We present Simplex Random Features (SimRFs), a new random feature (RF) mechanism for unbiased approximation of the softmax and Gaussian kernels by geometrical correlation of random projection vectors. We prove that SimRFs provide the smallest possible mean square error (MSE) on unbiased estimates of these kernels among the class of weight-independent geometrically-coupled positive random feature (PRF) mechanisms, substantially outperforming the previously most accurate Orthogonal Random Features at no observable extra cost. We present a more computationally expensive SimRFs+ variant, which we prove is asymptotically optimal in the broader family of weight-dependent geometrical coupling schemes (which permit correlations between random vector directions and norms). In extensive empirical studies, we show consistent gains provided by SimRFs in settings including pointwise kernel estimation, nonparametric classification and scalable Transformers.

Developing equivariant neural networks for the E(3) group plays an important role in modeling 3D data across real-world applications. Enforcing this equivariance primarily involves the tensor products of irreducible representations (irreps). However, the computational complexity of such operations increases significantly as higher-order tensors are used. In this work, we propose a systematic approach to substantially accelerate the computation of the tensor products of irreps. We mathematically connect the commonly used Clebsch-Gordan coefficients to the Gaunt coefficients, which are integrals of products of three spherical harmonics. Through Gaunt coefficients, the tensor product of irreps becomes equivalent to the multiplication between spherical functions represented by spherical harmonics. This perspective further allows us to change the basis for the equivariant operations from spherical harmonics to a 2D Fourier basis. Consequently, the multiplication between spherical functions represented by a 2D Fourier basis can be efficiently computed via the convolution theorem and Fast Fourier Transforms. This transformation reduces the complexity of full tensor products of irreps from O(L^6) to O(L^3), where L is the max degree of irreps. Leveraging this approach, we introduce the Gaunt Tensor Product, which serves as a new method to construct efficient equivariant operations across different model architectures. Our experiments on the Open Catalyst Project and 3BPA datasets demonstrate both the increased efficiency and improved performance of our approach.

We present a new approach for understanding the periodicity structure and semantics of motion datasets, independently of the morphology and skeletal structure of characters. Unlike existing methods using an overly sparse high-dimensional latent, we propose a phase manifold consisting of multiple closed curves, each corresponding to a latent amplitude. With our proposed vector quantized periodic autoencoder, we learn a shared phase manifold for multiple characters, such as a human and a dog, without any supervision. This is achieved by exploiting the discrete structure and a shallow network as bottlenecks, such that semantically similar motions are clustered into the same curve of the manifold, and the motions within the same component are aligned temporally by the phase variable. In combination with an improved motion matching framework, we demonstrate the manifold's capability of timing and semantics alignment in several applications, including motion retrieval, transfer and stylization. Code and pre-trained models for this paper are available at https://peizhuoli.github.io/walkthedog.

We present a new regular grid search algorithm for quick fixed-radius nearest-neighbor lookup developed in Python. This module indexes a set of k-dimensional points in a regular grid, with optional periodic conditions, providing a fast approach for nearest neighbors queries. In this first installment we provide three types of queries: bubble, shell and the nth-nearest; as well as three different metrics of interest in astronomy: the euclidean and two distance functions in spherical coordinates of varying precision, haversine and Vincenty; and the possibility of providing a custom distance function. This package results particularly useful for large datasets where a brute-force search turns impractical.

This paper studies the problem of recovering cameras from a set of fundamental matrices. A set of fundamental matrices is said to be compatible if a set of cameras exists for which they are the fundamental matrices. We focus on the complete graph, where fundamental matrices for each pair of cameras are given. Previous work has established necessary and sufficient conditions for compatibility as rank and eigenvalue conditions on the n-view fundamental matrix obtained by concatenating the individual fundamental matrices. In this work, we show that the eigenvalue condition is redundant. We provide explicit homogeneous polynomials that describe necessary and sufficient conditions for compatibility in terms of the fundamental matrices and their epipoles. In this direction, we find that quadruple-wise compatibility is enough to ensure global compatibility for any number of cameras. We demonstrate that for four cameras, compatibility is generically described by triple-wise conditions and one additional equation involving all fundamental matrices.

Adaptive sparse coding methods learn a possibly overcomplete set of basis functions, such that natural image patches can be reconstructed by linearly combining a small subset of these bases. The applicability of these methods to visual object recognition tasks has been limited because of the prohibitive cost of the optimization algorithms required to compute the sparse representation. In this work we propose a simple and efficient algorithm to learn basis functions. After training, this model also provides a fast and smooth approximator to the optimal representation, achieving even better accuracy than exact sparse coding algorithms on visual object recognition tasks.

We introduce the concept of programmable feature engineering for time series modeling and propose a feature programming framework. This framework generates large amounts of predictive features for noisy multivariate time series while allowing users to incorporate their inductive bias with minimal effort. The key motivation of our framework is to view any multivariate time series as a cumulative sum of fine-grained trajectory increments, with each increment governed by a novel spin-gas dynamical Ising model. This fine-grained perspective motivates the development of a parsimonious set of operators that summarize multivariate time series in an abstract fashion, serving as the foundation for large-scale automated feature engineering. Numerically, we validate the efficacy of our method on several synthetic and real-world noisy time series datasets.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups G = GL^{+}(n, R) and G = SL(n, R), as well as their representation as affine transformations R^{n} rtimes G. Invariant integration as well as a global parametrization is realized by decomposing the `larger` groups into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

We consider a family of linear systems A_mu alpha=C with system matrix A_mu depending on a parameter mu and for simplicity parameter-independent right-hand side C. These linear systems typically result from the finite-dimensional approximation of a parameter-dependent boundary-value problem. We derive a procedure based on the Empirical Interpolation Method to obtain a separated representation of the system matrix in the form A_muapproxsum_{m}beta_m(mu)A_{mu_m} for some selected values of the parameter. Such a separated representation is in particular useful in the Reduced Basis Method. The procedure is called nonintrusive since it only requires to access the matrices A_{mu_m}. As such, it offers a crucial advantage over existing approaches that instead derive separated representations requiring to enter the code at the level of assembly. Numerical examples illustrate the performance of our new procedure on a simple one-dimensional boundary-value problem and on three-dimensional acoustic scattering problems solved by a boundary element method.

Popular parameter-efficient fine-tuning (PEFT) methods, such as LoRA and its variants, freeze pre-trained model weights \(W\) and inject learnable matrices \(\Delta W\). These \(\Delta W\) matrices are structured for efficient parameterization, often using techniques like low-rank approximations or scaling vectors. However, these methods typically show a performance gap compared to full fine-tuning. Although recent PEFT methods have narrowed this gap, they do so at the cost of additional learnable parameters. We propose SVFT, a simple approach that fundamentally differs from existing methods: the structure imposed on \(\Delta W\) depends on the specific weight matrix \(W\). Specifically, SVFT updates \(W\) as a sparse combination of outer products of its singular vectors, training only the coefficients (scales) of these sparse combinations. This approach allows fine-grained control over expressivity through the number of coefficients. Extensive experiments on language and vision benchmarks show that SVFT recovers up to 96% of full fine-tuning performance while training only 0.006 to 0.25% of parameters, outperforming existing methods that only recover up to 85% performance using 0.03 to 0.8% of the trainable parameter budget.

Classification of time series data is an important task for many application domains. One of the best existing methods for this task, in terms of accuracy and computation time, is MiniROCKET. In this work, we extend this approach to provide better global temporal encodings using hyperdimensional computing (HDC) mechanisms. HDC (also known as Vector Symbolic Architectures, VSA) is a general method to explicitly represent and process information in high-dimensional vectors. It has previously been used successfully in combination with deep neural networks and other signal processing algorithms. We argue that the internal high-dimensional representation of MiniROCKET is well suited to be complemented by the algebra of HDC. This leads to a more general formulation, HDC-MiniROCKET, where the original algorithm is only a special case. We will discuss and demonstrate that HDC-MiniROCKET can systematically overcome catastrophic failures of MiniROCKET on simple synthetic datasets. These results are confirmed by experiments on the 128 datasets from the UCR time series classification benchmark. The extension with HDC can achieve considerably better results on datasets with high temporal dependence without increasing the computational effort for inference.

Applications of machine learning techniques for materials modeling typically involve functions known to be equivariant or invariant to specific symmetries. While graph neural networks (GNNs) have proven successful in such tasks, they enforce symmetries via the model architecture, which often reduces their expressivity, scalability and comprehensibility. In this paper, we introduce (1) a flexible framework relying on stochastic frame-averaging (SFA) to make any model E(3)-equivariant or invariant through data transformations. (2) FAENet: a simple, fast and expressive GNN, optimized for SFA, that processes geometric information without any symmetrypreserving design constraints. We prove the validity of our method theoretically and empirically demonstrate its superior accuracy and computational scalability in materials modeling on the OC20 dataset (S2EF, IS2RE) as well as common molecular modeling tasks (QM9, QM7-X). A package implementation is available at https://faenet.readthedocs.io.

Spectral analysis provides one of the most effective paradigms for information-preserving dimensionality reduction, as simple descriptions of naturally occurring signals are often obtained via few terms of periodic basis functions. In this work, we study deep neural networks designed to harness the structure in frequency domain for efficient learning of long-range correlations in space or time: frequency-domain models (FDMs). Existing FDMs are based on complex-valued transforms i.e. Fourier Transforms (FT), and layers that perform computation on the spectrum and input data separately. This design introduces considerable computational overhead: for each layer, a forward and inverse FT. Instead, this work introduces a blueprint for frequency domain learning through a single transform: transform once (T1). To enable efficient, direct learning in the frequency domain we derive a variance-preserving weight initialization scheme and investigate methods for frequency selection in reduced-order FDMs. Our results noticeably streamline the design process of FDMs, pruning redundant transforms, and leading to speedups of 3x to 10x that increase with data resolution and model size. We perform extensive experiments on learning the solution operator of spatio-temporal dynamics, including incompressible Navier-Stokes, turbulent flows around airfoils and high-resolution video of smoke. T1 models improve on the test performance of FDMs while requiring significantly less computation (5 hours instead of 32 for our large-scale experiment), with over 20% reduction in average predictive error across tasks.

We derive a Fast Multipole Method (FMM) where a low-rank approximation of the kernel is obtained using the Empirical Interpolation Method (EIM). Contrary to classical interpolation-based FMM, where the interpolation points and basis are fixed beforehand, the EIM is a nonlinear approximation method which constructs interpolation points and basis which are adapted to the kernel under consideration. The basis functions are obtained using evaluations of the kernel itself. We restrict ourselves to translation-invariant kernels, for which a modified version of the EIM approximation can be used in a multilevel FMM context; we call the obtained algorithm Empirical Interpolation Fast Multipole Method (EIFMM). An important feature of the EIFMM is a built-in error estimation of the interpolation error made by the low-rank approximation of the far-field behavior of the kernel: the algorithm selects the optimal number of interpolation points required to ensure a given accuracy for the result, leading to important gains for inhomogeneous kernels.

We provide a new theoretical framework for the variable-step deferred correction (DC) methods based on the well-known BDF2 formula. By using the discrete orthogonal convolution kernels, some high-order BDF2-DC methods are proven to be stable on arbitrary time grids according to the recent definition of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The associated sharp error estimates are established by taking the numerical effects of the starting approximations into account, and they suggest that the BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme for the BDF2 scheme will not cause a loss in the accuracy of the high-order BDF2-DC methods. Extensive tests on the graded and random time meshes are presented to support the new theory.

Predicting physical properties of materials from their crystal structures is a fundamental problem in materials science. In peripheral areas such as the prediction of molecular properties, fully connected attention networks have been shown to be successful. However, unlike these finite atom arrangements, crystal structures are infinitely repeating, periodic arrangements of atoms, whose fully connected attention results in infinitely connected attention. In this work, we show that this infinitely connected attention can lead to a computationally tractable formulation, interpreted as neural potential summation, that performs infinite interatomic potential summations in a deeply learned feature space. We then propose a simple yet effective Transformer-based encoder architecture for crystal structures called Crystalformer. Compared to an existing Transformer-based model, the proposed model requires only 29.4% of the number of parameters, with minimal modifications to the original Transformer architecture. Despite the architectural simplicity, the proposed method outperforms state-of-the-art methods for various property regression tasks on the Materials Project and JARVIS-DFT datasets.

Irregularly sampled time series data arise naturally in many application domains including biology, ecology, climate science, astronomy, and health. Such data represent fundamental challenges to many classical models from machine learning and statistics due to the presence of non-uniform intervals between observations. However, there has been significant progress within the machine learning community over the last decade on developing specialized models and architectures for learning from irregularly sampled univariate and multivariate time series data. In this survey, we first describe several axes along which approaches to learning from irregularly sampled time series differ including what data representations they are based on, what modeling primitives they leverage to deal with the fundamental problem of irregular sampling, and what inference tasks they are designed to perform. We then survey the recent literature organized primarily along the axis of modeling primitives. We describe approaches based on temporal discretization, interpolation, recurrence, attention and structural invariance. We discuss similarities and differences between approaches and highlight primary strengths and weaknesses.

We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable to encode such symmetries in the neural operator architecture for better performance and easier learning. While encoding symmetries in the physical domain using group theory has been studied extensively, how to capture symmetries in the frequency domain is under-explored. In this work, we extend group convolutions to the frequency domain and design Fourier layers that are equivariant to rotations, translations, and reflections by leveraging the equivariance property of the Fourier transform. The resulting G-FNO architecture generalizes well across input resolutions and performs well in settings with varying levels of symmetry. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

Designing machine learning architectures for processing neural networks in their raw weight matrix form is a newly introduced research direction. Unfortunately, the unique symmetry structure of deep weight spaces makes this design very challenging. If successful, such architectures would be capable of performing a wide range of intriguing tasks, from adapting a pre-trained network to a new domain to editing objects represented as functions (INRs or NeRFs). As a first step towards this goal, we present here a novel network architecture for learning in deep weight spaces. It takes as input a concatenation of weights and biases of a pre-trained MLP and processes it using a composition of layers that are equivariant to the natural permutation symmetry of the MLP's weights: Changing the order of neurons in intermediate layers of the MLP does not affect the function it represents. We provide a full characterization of all affine equivariant and invariant layers for these symmetries and show how these layers can be implemented using three basic operations: pooling, broadcasting, and fully connected layers applied to the input in an appropriate manner. We demonstrate the effectiveness of our architecture and its advantages over natural baselines in a variety of learning tasks.

Many machine learning tasks involve learning functions that are known to be invariant or equivariant to certain symmetries of the input data. However, it is often challenging to design neural network architectures that respect these symmetries while being expressive and computationally efficient. For example, Euclidean motion invariant/equivariant graph or point cloud neural networks. We introduce Frame Averaging (FA), a general purpose and systematic framework for adapting known (backbone) architectures to become invariant or equivariant to new symmetry types. Our framework builds on the well known group averaging operator that guarantees invariance or equivariance but is intractable. In contrast, we observe that for many important classes of symmetries, this operator can be replaced with an averaging operator over a small subset of the group elements, called a frame. We show that averaging over a frame guarantees exact invariance or equivariance while often being much simpler to compute than averaging over the entire group. Furthermore, we prove that FA-based models have maximal expressive power in a broad setting and in general preserve the expressive power of their backbone architectures. Using frame averaging, we propose a new class of universal Graph Neural Networks (GNNs), universal Euclidean motion invariant point cloud networks, and Euclidean motion invariant Message Passing (MP) GNNs. We demonstrate the practical effectiveness of FA on several applications including point cloud normal estimation, beyond 2-WL graph separation, and n-body dynamics prediction, achieving state-of-the-art results in all of these benchmarks.

A challenging problem in many modern machine learning tasks is to process weight-space features, i.e., to transform or extract information from the weights and gradients of a neural network. Recent works have developed promising weight-space models that are equivariant to the permutation symmetries of simple feedforward networks. However, they are not applicable to general architectures, since the permutation symmetries of a weight space can be complicated by recurrence or residual connections. This work proposes an algorithm that automatically constructs permutation equivariant models, which we refer to as universal neural functionals (UNFs), for any weight space. Among other applications, we demonstrate how UNFs can be substituted into existing learned optimizer designs, and find promising improvements over prior methods when optimizing small image classifiers and language models. Our results suggest that learned optimizers can benefit from considering the (symmetry) structure of the weight space they optimize. We open-source our library for constructing UNFs at https://github.com/AllanYangZhou/universal_neural_functional.

Dynamical systems (DS) theory is fundamental for many areas of science and engineering. It can provide deep insights into the behavior of systems evolving in time, as typically described by differential or recursive equations. A common approach to facilitate mathematical tractability and interpretability of DS models involves decomposing nonlinear DS into multiple linear DS separated by switching manifolds, i.e. piecewise linear (PWL) systems. PWL models are popular in engineering and a frequent choice in mathematics for analyzing the topological properties of DS. However, hand-crafting such models is tedious and only possible for very low-dimensional scenarios, while inferring them from data usually gives rise to unnecessarily complex representations with very many linear subregions. Here we introduce Almost-Linear Recurrent Neural Networks (AL-RNNs) which automatically and robustly produce most parsimonious PWL representations of DS from time series data, using as few PWL nonlinearities as possible. AL-RNNs can be efficiently trained with any SOTA algorithm for dynamical systems reconstruction (DSR), and naturally give rise to a symbolic encoding of the underlying DS that provably preserves important topological properties. We show that for the Lorenz and R\"ossler systems, AL-RNNs discover, in a purely data-driven way, the known topologically minimal PWL representations of the corresponding chaotic attractors. We further illustrate on two challenging empirical datasets that interpretable symbolic encodings of the dynamics can be achieved, tremendously facilitating mathematical and computational analysis of the underlying systems.

Dynamical systems provide a comprehensive way to study complex and changing behaviors across various sciences. Many modern systems are too complicated to analyze directly or we do not have access to models, driving significant interest in learning methods. Koopman operators have emerged as a dominant approach because they allow the study of nonlinear dynamics using linear techniques by solving an infinite-dimensional spectral problem. However, current algorithms face challenges such as lack of convergence, hindering practical progress. This paper addresses a fundamental open question: When can we robustly learn the spectral properties of Koopman operators from trajectory data of dynamical systems, and when can we not? Understanding these boundaries is crucial for analysis, applications, and designing algorithms. We establish a foundational approach that combines computational analysis and ergodic theory, revealing the first fundamental barriers -- universal for any algorithm -- associated with system geometry and complexity, regardless of data quality and quantity. For instance, we demonstrate well-behaved smooth dynamical systems on tori where non-trivial eigenfunctions of the Koopman operator cannot be determined by any sequence of (even randomized) algorithms, even with unlimited training data. Additionally, we identify when learning is possible and introduce optimal algorithms with verification that overcome issues in standard methods. These results pave the way for a sharp classification theory of data-driven dynamical systems based on how many limits are needed to solve a problem. These limits characterize all previous methods, presenting a unified view. Our framework systematically determines when and how Koopman spectral properties can be learned.

In large scale machine learning, random sampling is a popular way to approximate datasets by a small representative subset of examples. In particular, sensitivity sampling is an intensely studied technique which provides provable guarantees on the quality of approximation, while reducing the number of examples to the product of the VC dimension d and the total sensitivity mathfrak S in remarkably general settings. However, guarantees going beyond this general bound of mathfrak S d are known in perhaps only one setting, for ell_2 subspace embeddings, despite intense study of sensitivity sampling in prior work. In this work, we show the first bounds for sensitivity sampling for ell_p subspace embeddings for pneq 2 that improve over the general mathfrak S d bound, achieving a bound of roughly mathfrak S^{2/p} for 1leq p<2 and mathfrak S^{2-2/p} for 2<p<infty. For 1leq p<2, we show that this bound is tight, in the sense that there exist matrices for which mathfrak S^{2/p} samples is necessary. Furthermore, our techniques yield further new results in the study of sampling algorithms, showing that the root leverage score sampling algorithm achieves a bound of roughly d for 1leq p<2, and that a combination of leverage score and sensitivity sampling achieves an improved bound of roughly d^{2/p}mathfrak S^{2-4/p} for 2<p<infty. Our sensitivity sampling results yield the best known sample complexity for a wide class of structured matrices that have small ell_p sensitivity.

Existing vector quantization (VQ) based autoregressive models follow a two-stage generation paradigm that first learns a codebook to encode images as discrete codes, and then completes generation based on the learned codebook. However, they encode fixed-size image regions into fixed-length codes and ignore their naturally different information densities, which results in insufficiency in important regions and redundancy in unimportant ones, and finally degrades the generation quality and speed. Moreover, the fixed-length coding leads to an unnatural raster-scan autoregressive generation. To address the problem, we propose a novel two-stage framework: (1) Dynamic-Quantization VAE (DQ-VAE) which encodes image regions into variable-length codes based on their information densities for an accurate and compact code representation. (2) DQ-Transformer which thereby generates images autoregressively from coarse-grained (smooth regions with fewer codes) to fine-grained (details regions with more codes) by modeling the position and content of codes in each granularity alternately, through a novel stacked-transformer architecture and shared-content, non-shared position input layers designs. Comprehensive experiments on various generation tasks validate our superiorities in both effectiveness and efficiency. Code will be released at https://github.com/CrossmodalGroup/DynamicVectorQuantization.

Machine learning potentials (MLPs) trained on accurate quantum chemical data can retain the high accuracy, while inflicting little computational demands. On the downside, they need to be trained for each individual system. In recent years, a vast number of MLPs has been trained from scratch because learning additional data typically requires to train again on all data to not forget previously acquired knowledge. Additionally, most common structural descriptors of MLPs cannot represent efficiently a large number of different chemical elements. In this work, we tackle these problems by introducing element-embracing atom-centered symmetry functions (eeACSFs) which combine structural properties and element information from the periodic table. These eeACSFs are a key for our development of a lifelong machine learning potential (lMLP). Uncertainty quantification can be exploited to transgress a fixed, pre-trained MLP to arrive at a continuously adapting lMLP, because a predefined level of accuracy can be ensured. To extend the applicability of an lMLP to new systems, we apply continual learning strategies to enable autonomous and on-the-fly training on a continuous stream of new data. For the training of deep neural networks, we propose the continual resilient (CoRe) optimizer and incremental learning strategies relying on rehearsal of data, regularization of parameters, and the architecture of the model.

The Reduced Basis Method can be exploited in an efficient way only if the so-called affine dependence assumption on the operator and right-hand side of the considered problem with respect to the parameters is satisfied. When it is not, the Empirical Interpolation Method is usually used to recover this assumption approximately. In both cases, the Reduced Basis Method requires to access and modify the assembly routines of the corresponding computational code, leading to an intrusive procedure. In this work, we derive variants of the EIM algorithm and explain how they can be used to turn the Reduced Basis Method into a nonintrusive procedure. We present examples of aeroacoustic problems solved by integral equations and show how our algorithms can benefit from the linear algebra tools available in the considered code.

In this paper, we introduce a mesh-free two-level hybrid Tucker tensor format for approximation of multivariate functions, which combines the product Chebyshev interpolation with the ALS-based Tucker decomposition of the tensor of Chebyshev coefficients. It allows to avoid the expenses of the rank-structured approximation of function-related tensors defined on large spacial grids, while benefiting from the Tucker decomposition of the rather small core tensor of Chebyshev coefficients. This leads to nearly optimal Tucker rank parameters which are close to the results for well established Tucker-ALS algorithm applied to the large grid-based tensors. These rank parameters inherited from the Tucker-ALS decomposition of the coefficient tensor can be much less than the polynomial degrees of the initial Chebyshev interpolant via function independent basis set. Furthermore, the tensor product Chebyshev polynomials discretized on a tensor grid leads to a low-rank two-level orthogonal algebraic Tucker tensor that approximates the initial function with controllable accuracy. It is shown that our techniques could be gainfully applied to the long-range part of the electrostatic potential of multi-particle systems approximated in the range-separated tensor format. Error and complexity estimates of the proposed methods are presented. We demonstrate the efficiency of the suggested method numerically on examples of the long-range components of multi-particle interaction potentials generated by 3D Newton kernel for large bio-molecule systems and lattice-type compounds.

Nonconvex optimization is central in solving many machine learning problems, in which block-wise structure is commonly encountered. In this work, we propose cyclic block coordinate methods for nonconvex optimization problems with non-asymptotic gradient norm guarantees. Our convergence analysis is based on a gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a recent progress on cyclic block coordinate methods. In deterministic settings, our convergence guarantee matches the guarantee of (full-gradient) gradient descent, but with the gradient Lipschitz constant being defined w.r.t.~a Mahalanobis norm. In stochastic settings, we use recursive variance reduction to decrease the per-iteration cost and match the arithmetic operation complexity of current optimal stochastic full-gradient methods, with a unified analysis for both finite-sum and infinite-sum cases. We prove a faster linear convergence result when a Polyak-{\L}ojasiewicz (P{\L}) condition holds. To our knowledge, this work is the first to provide non-asymptotic convergence guarantees -- variance-reduced or not -- for a cyclic block coordinate method in general composite (smooth + nonsmooth) nonconvex settings. Our experimental results demonstrate the efficacy of the proposed cyclic scheme in training deep neural nets.

We introduce meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nystr\"{o}m method, and the CUR decomposition are revealed here as an illustration. Finally, meta-factorization offers hints on the structure of new factorizations and provides the potential of creating them.

Dimensionality reduction in vector databases is pivotal for streamlining AI data management, enabling efficient storage, faster computation, and improved model performance. This paper explores the benefits of reducing vector database dimensions, with a focus on computational efficiency and overcoming the curse of dimensionality. We introduce a novel application of Fast Fourier Transform (FFT) to dimensionality reduction, a method previously underexploited in this context. By demonstrating its utility across various AI domains, including Retrieval-Augmented Generation (RAG) models and image processing, this FFT-based approach promises to improve data retrieval processes and enhance the efficiency and scalability of AI solutions. The incorporation of FFT may not only optimize operations in real-time processing and recommendation systems but also extend to advanced image processing techniques, where dimensionality reduction can significantly improve performance and analysis efficiency. This paper advocates for the broader adoption of FFT in vector database management, marking a significant stride towards addressing the challenges of data volume and complexity in AI research and applications. Unlike many existing approaches, we directly handle the embedding vectors produced by the model after processing a test input.

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s geq 3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young et al., Phys. Rev. E 96, 032312 (2017)], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analysis of higher-order phenomena based on local structures.

Computing the numerical solution to high-dimensional tensor differential equations can lead to prohibitive computational costs and memory requirements. To reduce the memory and computational footprint, dynamical low-rank approximation (DLRA) has proven to be a promising approach. DLRA represents the solution as a low-rank tensor factorization and evolves the resulting low-rank factors in time. A central challenge in DLRA is to find time integration schemes that are robust to the arising small singular values. A robust parallel basis update & Galerkin integrator, which simultaneously evolves all low-rank factors, has recently been derived for matrix differential equations. This work extends the parallel low-rank matrix integrator to Tucker tensors and general tree tensor networks, yielding an algorithm in which all bases and connecting tensors are evolved in parallel over a time step. We formulate the algorithm, provide a robust error bound, and demonstrate the efficiency of the new integrators for problems in quantum many-body physics, uncertainty quantification, and radiative transfer.

Covariance matrices have proven highly effective across many scientific fields. Since these matrices lie within the Symmetric Positive Definite (SPD) manifold - a Riemannian space with intrinsic non-Euclidean geometry, the primary challenge in representation learning is to respect this underlying geometric structure. Drawing inspiration from the success of Euclidean deep learning, researchers have developed neural networks on the SPD manifolds for more faithful covariance embedding learning. A notable advancement in this area is the implementation of Riemannian batch normalization (RBN), which has been shown to improve the performance of SPD network models. Nonetheless, the Riemannian metric beneath the existing RBN might fail to effectively deal with the ill-conditioned SPD matrices (ICSM), undermining the effectiveness of RBN. In contrast, the Bures-Wasserstein metric (BWM) demonstrates superior performance for ill-conditioning. In addition, the recently introduced Generalized BWM (GBWM) parameterizes the vanilla BWM via an SPD matrix, allowing for a more nuanced representation of vibrant geometries of the SPD manifold. Therefore, we propose a novel RBN algorithm based on the GBW geometry, incorporating a learnable metric parameter. Moreover, the deformation of GBWM by matrix power is also introduced to further enhance the representational capacity of GBWM-based RBN. Experimental results on different datasets validate the effectiveness of our proposed method.

One-hot vectors, a common method for representing discrete/categorical data, in machine learning are widely used because of their simplicity and intuitiveness. However, one-hot vectors suffer from a linear increase in dimensionality, posing computational and memory challenges, especially when dealing with datasets containing numerous categories. In this paper, we focus on tabular data generation, and reveal the multinomial diffusion faces the mode collapse phenomenon when the cardinality is high. Moreover, due to the limitations of one-hot vectors, the training phase takes time longer in such a situation. To address these issues, we propose Residual Bit Vectors (ResBit), a technique for densely representing categorical data. ResBit is an extension of analog bits and overcomes limitations of analog bits when applied to tabular data generation. Our experiments demonstrate that ResBit not only accelerates training but also maintains performance when compared with the situations before applying ResBit. Furthermore, our results indicate that many existing methods struggle with high-cardinality data, underscoring the need for lower-dimensional representations, such as ResBit and latent vectors.

Pre-trained foundation models (FMs) have shown exceptional performance in univariate time series forecasting tasks. However, several practical challenges persist, including managing intricate dependencies among features and quantifying uncertainty in predictions. This study aims to tackle these critical limitations by introducing adapters; feature-space transformations that facilitate the effective use of pre-trained univariate time series FMs for multivariate tasks. Adapters operate by projecting multivariate inputs into a suitable latent space and applying the FM independently to each dimension. Inspired by the literature on representation learning and partially stochastic Bayesian neural networks, we present a range of adapters and optimization/inference strategies. Experiments conducted on both synthetic and real-world datasets confirm the efficacy of adapters, demonstrating substantial enhancements in forecasting accuracy and uncertainty quantification compared to baseline methods. Our framework, AdaPTS, positions adapters as a modular, scalable, and effective solution for leveraging time series FMs in multivariate contexts, thereby promoting their wider adoption in real-world applications. We release the code at https://github.com/abenechehab/AdaPTS.

Low-rank adaptation~(LoRA) has recently gained much interest in fine-tuning foundation models. It effectively reduces the number of trainable parameters by incorporating low-rank matrices A and B to represent the weight change, i.e., Delta W=BA. Despite LoRA's progress, it faces storage challenges when handling extensive customization adaptations or larger base models. In this work, we aim to further compress trainable parameters by enjoying the powerful expressiveness of the Fourier transform. Specifically, we introduce FourierFT, which treats Delta W as a matrix in the spatial domain and learns only a small fraction of its spectral coefficients. With the trained spectral coefficients, we implement the inverse discrete Fourier transform to recover Delta W. Empirically, our FourierFT method shows comparable or better performance with fewer parameters than LoRA on various tasks, including natural language understanding, natural language generation, instruction tuning, and image classification. For example, when performing instruction tuning on the LLaMA2-7B model, FourierFT surpasses LoRA with only 0.064M trainable parameters, compared to LoRA's 33.5M. Our code is released at https://github.com/Chaos96/fourierft.

State-space models (SSMs) have recently emerged as a framework for learning long-range sequence tasks. An example is the structured state-space sequence (S4) layer, which uses the diagonal-plus-low-rank structure of the HiPPO initialization framework. However, the complicated structure of the S4 layer poses challenges; and, in an effort to address these challenges, models such as S4D and S5 have considered a purely diagonal structure. This choice simplifies the implementation, improves computational efficiency, and allows channel communication. However, diagonalizing the HiPPO framework is itself an ill-posed problem. In this paper, we propose a general solution for this and related ill-posed diagonalization problems in machine learning. We introduce a generic, backward-stable "perturb-then-diagonalize" (PTD) methodology, which is based on the pseudospectral theory of non-normal operators, and which may be interpreted as the approximate diagonalization of the non-normal matrices defining SSMs. Based on this, we introduce the S4-PTD and S5-PTD models. Through theoretical analysis of the transfer functions of different initialization schemes, we demonstrate that the S4-PTD/S5-PTD initialization strongly converges to the HiPPO framework, while the S4D/S5 initialization only achieves weak convergences. As a result, our new models show resilience to Fourier-mode noise-perturbed inputs, a crucial property not achieved by the S4D/S5 models. In addition to improved robustness, our S5-PTD model averages 87.6% accuracy on the Long-Range Arena benchmark, demonstrating that the PTD methodology helps to improve the accuracy of deep learning models.

We identify seven new 192-periodic infinite families of elements in the 2-primary stable homotopy groups of spheres. Although their Hurewicz image is trivial for topological modular forms, they remain nontrivial after T(2)- as well as K(2)-localization. We also obtain new information about 2-torsion and 2-divisibility of some of the previously known 192-periodic infinite families in the stable stems.

Updating a truncated Singular Value Decomposition (SVD) is crucial in representation learning, especially when dealing with large-scale data matrices that continuously evolve in practical scenarios. Aligning SVD-based models with fast-paced updates becomes increasingly important. Existing methods for updating truncated SVDs employ Rayleigh-Ritz projection procedures, where projection matrices are augmented based on original singular vectors. However, these methods suffer from inefficiency due to the densification of the update matrix and the application of the projection to all singular vectors. To address these limitations, we introduce a novel method for dynamically approximating the truncated SVD of a sparse and temporally evolving matrix. Our approach leverages sparsity in the orthogonalization process of augmented matrices and utilizes an extended decomposition to independently store projections in the column space of singular vectors. Numerical experiments demonstrate a remarkable efficiency improvement of an order of magnitude compared to previous methods. Remarkably, this improvement is achieved while maintaining a comparable precision to existing approaches.

We propose the convergent graph solver (CGS), a deep learning method that learns iterative mappings to predict the properties of a graph system at its stationary state (fixed point) with guaranteed convergence. CGS systematically computes the fixed points of a target graph system and decodes them to estimate the stationary properties of the system without the prior knowledge of existing solvers or intermediate solutions. The forward propagation of CGS proceeds in three steps: (1) constructing the input dependent linear contracting iterative maps, (2) computing the fixed-points of the linear maps, and (3) decoding the fixed-points to estimate the properties. The contractivity of the constructed linear maps guarantees the existence and uniqueness of the fixed points following the Banach fixed point theorem. To train CGS efficiently, we also derive a tractable analytical expression for its gradient by leveraging the implicit function theorem. We evaluate the performance of CGS by applying it to various network-analytic and graph benchmark problems. The results indicate that CGS has competitive capabilities for predicting the stationary properties of graph systems, irrespective of whether the target systems are linear or non-linear. CGS also shows high performance for graph classification problems where the existence or the meaning of a fixed point is hard to be clearly defined, which highlights the potential of CGS as a general graph neural network architecture.

Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions R^3, position and orientations R^3 {times} S^2, and the group SE(3) itself. Among these, R^3 {times} S^2 is an optimal choice due to the ability to represent directional information, which R^3 methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full SE(3) group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.

Diagnosing deep neural networks (DNNs) through the eigenspectrum of weight matrices has been an active area of research in recent years. At a high level, eigenspectrum analysis of DNNs involves measuring the heavytailness of the empirical spectral densities (ESD) of weight matrices. It provides insight into how well a model is trained and can guide decisions on assigning better layer-wise training hyperparameters. In this paper, we address a challenge associated with such eigenspectrum methods: the impact of the aspect ratio of weight matrices on estimated heavytailness metrics. We demonstrate that matrices of varying sizes (and aspect ratios) introduce a non-negligible bias in estimating heavytailness metrics, leading to inaccurate model diagnosis and layer-wise hyperparameter assignment. To overcome this challenge, we propose FARMS (Fixed-Aspect-Ratio Matrix Subsampling), a method that normalizes the weight matrices by subsampling submatrices with a fixed aspect ratio. Instead of measuring the heavytailness of the original ESD, we measure the average ESD of these subsampled submatrices. We show that measuring the heavytailness of these submatrices with the fixed aspect ratio can effectively mitigate the aspect ratio bias. We validate our approach across various optimization techniques and application domains that involve eigenspectrum analysis of weights, including image classification in computer vision (CV) models, scientific machine learning (SciML) model training, and large language model (LLM) pruning. Our results show that despite its simplicity, FARMS uniformly improves the accuracy of eigenspectrum analysis while enabling more effective layer-wise hyperparameter assignment in these application domains. In one of the LLM pruning experiments, FARMS reduces the perplexity of the LLaMA-7B model by 17.3% when compared with the state-of-the-art method.

We present a new class of fast polylog-linear algorithms based on the theory of structured matrices (in particular low displacement rank) for integrating tensor fields defined on weighted trees. Several applications of the resulting fast tree-field integrators (FTFIs) are presented, including (a) approximation of graph metrics with tree metrics, (b) graph classification, (c) modeling on meshes, and finally (d) Topological Transformers (TTs) (Choromanski et al., 2022) for images. For Topological Transformers, we propose new relative position encoding (RPE) masking mechanisms with as few as three extra learnable parameters per Transformer layer, leading to 1.0-1.5%+ accuracy gains. Importantly, most of FTFIs are exact methods, thus numerically equivalent to their brute-force counterparts. When applied to graphs with thousands of nodes, those exact algorithms provide 5.7-13x speedups. We also provide an extensive theoretical analysis of our methods.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

Time series foundation models have demonstrated impressive performance as zero-shot forecasters. However, achieving effectively unified training on time series remains an open challenge. Existing approaches introduce some level of model specialization to account for the highly heterogeneous nature of time series data. For instance, Moirai pursues unified training by employing multiple input/output projection layers, each tailored to handle time series at a specific frequency. Similarly, TimesFM maintains a frequency embedding dictionary for this purpose. We identify two major drawbacks to this human-imposed frequency-level model specialization: (1) Frequency is not a reliable indicator of the underlying patterns in time series. For example, time series with different frequencies can display similar patterns, while those with the same frequency may exhibit varied patterns. (2) Non-stationarity is an inherent property of real-world time series, leading to varied distributions even within a short context window of a single time series. Frequency-level specialization is too coarse-grained to capture this level of diversity. To address these limitations, this paper introduces Moirai-MoE, using a single input/output projection layer while delegating the modeling of diverse time series patterns to the sparse mixture of experts (MoE) within Transformers. With these designs, Moirai-MoE reduces reliance on human-defined heuristics and enables automatic token-level specialization. Extensive experiments on 39 datasets demonstrate the superiority of Moirai-MoE over existing foundation models in both in-distribution and zero-shot scenarios. Furthermore, this study conducts comprehensive model analyses to explore the inner workings of time series MoE foundation models and provides valuable insights for future research.

Learning feature representations of geographical space is vital for any machine learning model that integrates geolocated data, spanning application domains such as remote sensing, ecology, or epidemiology. Recent work mostly embeds coordinates using sine and cosine projections based on Double Fourier Sphere (DFS) features -- these embeddings assume a rectangular data domain even on global data, which can lead to artifacts, especially at the poles. At the same time, relatively little attention has been paid to the exact design of the neural network architectures these functional embeddings are combined with. This work proposes a novel location encoder for globally distributed geographic data that combines spherical harmonic basis functions, natively defined on spherical surfaces, with sinusoidal representation networks (SirenNets) that can be interpreted as learned Double Fourier Sphere embedding. We systematically evaluate the cross-product of positional embeddings and neural network architectures across various classification and regression benchmarks and synthetic evaluation datasets. In contrast to previous approaches that require the combination of both positional encoding and neural networks to learn meaningful representations, we show that both spherical harmonics and sinusoidal representation networks are competitive on their own but set state-of-the-art performances across tasks when combined. We provide source code at www.github.com/marccoru/locationencoder

Recent works have demonstrated reasonable success of representation learning in hypercomplex space. Specifically, "fully-connected layers with Quaternions" (4D hypercomplex numbers), which replace real-valued matrix multiplications in fully-connected layers with Hamilton products of Quaternions, both enjoy parameter savings with only 1/4 learnable parameters and achieve comparable performance in various applications. However, one key caveat is that hypercomplex space only exists at very few predefined dimensions (4D, 8D, and 16D). This restricts the flexibility of models that leverage hypercomplex multiplications. To this end, we propose parameterizing hypercomplex multiplications, allowing models to learn multiplication rules from data regardless of whether such rules are predefined. As a result, our method not only subsumes the Hamilton product, but also learns to operate on any arbitrary nD hypercomplex space, providing more architectural flexibility using arbitrarily 1/n learnable parameters compared with the fully-connected layer counterpart. Experiments of applications to the LSTM and Transformer models on natural language inference, machine translation, text style transfer, and subject verb agreement demonstrate architectural flexibility and effectiveness of the proposed approach.

Implicit neural representations (INRs) have arisen as useful methods for representing signals on Euclidean domains. By parameterizing an image as a multilayer perceptron (MLP) on Euclidean space, INRs effectively represent signals in a way that couples spatial and spectral features of the signal that is not obvious in the usual discrete representation, paving the way for continuous signal processing and machine learning approaches that were not previously possible. Although INRs using sinusoidal activation functions have been studied in terms of Fourier theory, recent works have shown the advantage of using wavelets instead of sinusoids as activation functions, due to their ability to simultaneously localize in both frequency and space. In this work, we approach such INRs and demonstrate how they resolve high-frequency features of signals from coarse approximations done in the first layer of the MLP. This leads to multiple prescriptions for the design of INR architectures, including the use of complex wavelets, decoupling of low and band-pass approximations, and initialization schemes based on the singularities of the desired signal.

We present Symphony, an E(3)-equivariant autoregressive generative model for 3D molecular geometries that iteratively builds a molecule from molecular fragments. Existing autoregressive models such as G-SchNet and G-SphereNet for molecules utilize rotationally invariant features to respect the 3D symmetries of molecules. In contrast, Symphony uses message-passing with higher-degree E(3)-equivariant features. This allows a novel representation of probability distributions via spherical harmonic signals to efficiently model the 3D geometry of molecules. We show that Symphony is able to accurately generate small molecules from the QM9 dataset, outperforming existing autoregressive models and approaching the performance of diffusion models.

Fourier PlenOctrees have shown to be an efficient representation for real-time rendering of dynamic Neural Radiance Fields (NeRF). Despite its many advantages, this method suffers from artifacts introduced by the involved compression when combining it with recent state-of-the-art techniques for training the static per-frame NeRF models. In this paper, we perform an in-depth analysis of these artifacts and leverage the resulting insights to propose an improved representation. In particular, we present a novel density encoding that adapts the Fourier-based compression to the characteristics of the transfer function used by the underlying volume rendering procedure and leads to a substantial reduction of artifacts in the dynamic model. Furthermore, we show an augmentation of the training data that relaxes the periodicity assumption of the compression. We demonstrate the effectiveness of our enhanced Fourier PlenOctrees in the scope of quantitative and qualitative evaluations on synthetic and real-world scenes.

Time series classification (TSC) on multivariate time series is a critical problem. We propose a novel multi-view approach integrating frequency-domain and time-domain features to provide complementary contexts for TSC. Our method fuses continuous wavelet transform spectral features with temporal convolutional or multilayer perceptron features. We leverage the Mamba state space model for efficient and scalable sequence modeling. We also introduce a novel tango scanning scheme to better model sequence relationships. Experiments on 10 standard benchmark datasets demonstrate our approach achieves an average 6.45% accuracy improvement over state-of-the-art TSC models.

Model order reduction has been extensively studied over the last two decades. Projection-based methods such as the Proper Orthogonal Decomposition and the Reduced Basis Method enjoy the important advantages of Galerkin methods in the derivation of the reduced problem, but are limited to linear data compression for which the reduced solution is sought as a linear combination of spatial modes. Nonlinear data compression must be used when the solution manifold is not embedded in a low-dimensional subspace. Early methods involve piecewise linear data compression, by constructing a dictionary of reduced-order models tailored to a partition of the solution manifold. In this work, we introduce the concept of optimal partition of the solution manifold in terms of normalized Kolmogorov widths, and prove that the optimal partitions can be found by means of a representative-based clustering algorithm using the sine dissimilarity measure on the solution manifold.

We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.

Harmonic model predictive control (HMPC) is a recent model predictive control (MPC) formulation for tracking piece-wise constant references that includes a parameterized artificial harmonic reference as a decision variable, resulting in an increased performance and domain of attraction with respect to other MPC formulations. This article presents an extension of the HMPC formulation to track periodic harmonic/sinusoidal references and discusses its use for tracking arbitrary trajectories. The proposed formulation inherits the benefits of its predecessor, namely its good performance and large domain of attraction when using small prediction horizons, and that the complexity of its optimization problem does not depend on the period of the reference. We show closed-loop results discussing its performance and comparing it to other MPC formulations.

We present a novel Parameter-Efficient Fine-Tuning (PEFT) method, dubbed as Adaptive Freezing of Low Rank Adaptation (AFLoRA). Specifically, for each pre-trained frozen weight tensor, we add a parallel path of trainable low-rank matrices, namely a down-projection and an up-projection matrix, each of which is followed by a feature transformation vector. Based on a novel freezing score, we the incrementally freeze these projection matrices during fine-tuning to reduce the computation and alleviate over-fitting. Our experimental results demonstrate that we can achieve state-of-the-art performance with an average improvement of up to 0.85% as evaluated on GLUE benchmark while yeilding up to 9.5times fewer average trainable parameters. While compared in terms of runtime, AFLoRA can yield up to 1.86times improvement as opposed to similar PEFT alternatives. Besides the practical utility of our approach, we provide insights on the trainability requirements of LoRA paths at different modules and the freezing schedule for the different projection matrices. Code will be released.

Given a matrix Min R^{mtimes n}, the low rank matrix completion problem asks us to find a rank-k approximation of M as UV^top for Uin R^{mtimes k} and Vin R^{ntimes k} by only observing a few entries specified by a set of entries Omegasubseteq [m]times [n]. In particular, we examine an approach that is widely used in practice -- the alternating minimization framework. Jain, Netrapalli and Sanghavi~jns13 showed that if M has incoherent rows and columns, then alternating minimization provably recovers the matrix M by observing a nearly linear in n number of entries. While the sample complexity has been subsequently improved~glz17, alternating minimization steps are required to be computed exactly. This hinders the development of more efficient algorithms and fails to depict the practical implementation of alternating minimization, where the updates are usually performed approximately in favor of efficiency. In this paper, we take a major step towards a more efficient and error-robust alternating minimization framework. To this end, we develop an analytical framework for alternating minimization that can tolerate moderate amount of errors caused by approximate updates. Moreover, our algorithm runs in time widetilde O(|Omega| k), which is nearly linear in the time to verify the solution while preserving the sample complexity. This improves upon all prior known alternating minimization approaches which require widetilde O(|Omega| k^2) time.

Previously, methods for learning marginally stable linear dynamical systems either required the transition matrix to be symmetric or incurred regret bounds that scale polynomially with the system's hidden dimension. In this work, we introduce a novel method that overcomes this trade-off, achieving dimension-free regret despite the presence of asymmetric matrices and marginal stability. Our method combines spectral filtering with linear predictors and employs Chebyshev polynomials in the complex plane to construct a novel spectral filtering basis. This construction guarantees sublinear regret in an online learning framework, without relying on any statistical or generative assumptions. Specifically, we prove that as long as the transition matrix has eigenvalues with complex component bounded by 1/poly log T, then our method achieves regret O(T^{9/10}) when compared to the best linear dynamical predictor in hindsight.

Four-dimensional computed tomography (4D CT) reconstruction is crucial for capturing dynamic anatomical changes but faces inherent limitations from conventional phase-binning workflows. Current methods discretize temporal resolution into fixed phases with respiratory gating devices, introducing motion misalignment and restricting clinical practicality. In this paper, We propose X^2-Gaussian, a novel framework that enables continuous-time 4D-CT reconstruction by integrating dynamic radiative Gaussian splatting with self-supervised respiratory motion learning. Our approach models anatomical dynamics through a spatiotemporal encoder-decoder architecture that predicts time-varying Gaussian deformations, eliminating phase discretization. To remove dependency on external gating devices, we introduce a physiology-driven periodic consistency loss that learns patient-specific breathing cycles directly from projections via differentiable optimization. Extensive experiments demonstrate state-of-the-art performance, achieving a 9.93 dB PSNR gain over traditional methods and 2.25 dB improvement against prior Gaussian splatting techniques. By unifying continuous motion modeling with hardware-free period learning, X^2-Gaussian advances high-fidelity 4D CT reconstruction for dynamic clinical imaging. Project website at: https://x2-gaussian.github.io/.

We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the Pin(p,q,r) group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.

Let F be a field, and n geq r>0 be integers, with r even. Denote by A_n(F) the space of all n-by-n alternating matrices with entries in F. We consider the problem of determining the greatest possible dimension for an affine subspace of A_n(F) in which every matrix has rank equal to r (or rank at least r). Recently Rubei has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that n geq r+3 and |F|geq minbigl(r-1,r{2}+2bigr), we also determine the affine subspaces of rank r matrices in A_n(F) that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case nleq r+2.

We revisit the well-studied problem of approximating a matrix product, A^TB, based on small space sketches S(A) and S(B) of A in R^{n times d} and Bin R^{n times m}. We are interested in the setting where the sketches must be computed independently of each other, except for the use of a shared random seed. We prove that, when A and B are sparse, methods based on coordinated random sampling can outperform classical linear sketching approaches, like Johnson-Lindenstrauss Projection or CountSketch. For example, to obtain Frobenius norm error epsilon|A|_F|B|_F, coordinated sampling requires sketches of size O(s/epsilon^2) when A and B have at most s leq d,m non-zeros per row. In contrast, linear sketching leads to sketches of size O(d/epsilon^2) and O(m/epsilon^2) for A and B. We empirically evaluate our approach on two applications: 1) distributed linear regression in databases, a problem motivated by tasks like dataset discovery and augmentation, and 2) approximating attention matrices in transformer-based language models. In both cases, our sampling algorithms yield an order of magnitude improvement over linear sketching.

Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this classic problem in the fully dynamic setting, where elements can be both inserted and deleted in real-time. Our main result is a randomized algorithm that maintains an efficient data structure with an O(k^2) amortized update time (in the number of additions and deletions) and yields a 4-approximate solution, where k is the rank of the matroid.

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.

Equivariant neural networks have shown improved performance, expressiveness and sample complexity on symmetrical domains. But for some specific symmetries, representations, and choice of coordinates, the most common point-wise activations, such as ReLU, are not equivariant, hence they cannot be employed in the design of equivariant neural networks. The theorem we present in this paper describes all possible combinations of finite-dimensional representations, choice of coordinates and point-wise activations to obtain an exactly equivariant layer, generalizing and strengthening existing characterizations. Notable cases of practical relevance are discussed as corollaries. Indeed, we prove that rotation-equivariant networks can only be invariant, as it happens for any network which is equivariant with respect to connected compact groups. Then, we discuss implications of our findings when applied to important instances of exactly equivariant networks. First, we completely characterize permutation equivariant networks such as Invariant Graph Networks with point-wise nonlinearities and their geometric counterparts, highlighting a plethora of models whose expressive power and performance are still unknown. Second, we show that feature spaces of disentangled steerable convolutional neural networks are trivial representations.

Although Kolmogorov-Arnold based interpretable networks (KAN) have strong theoretical expressiveness, they face significant parameter explosion and high-frequency feature capture challenges in high-dimensional tasks. To address this issue, we propose the Kolmogorov-Arnold-Fourier Network (KAF), which effectively integrates trainable Random Fourier Features (RFF) and a novel hybrid GELU-Fourier activation mechanism to balance parameter efficiency and spectral representation capabilities. Our key technical contributions include: (1) merging KAN's dual-matrix structure through matrix association properties to substantially reduce parameters; (2) introducing learnable RFF initialization strategies to eliminate spectral distortion in high-dimensional approximation tasks; (3) implementing an adaptive hybrid activation function that progressively enhances frequency representation during the training process. Comprehensive experiments demonstrate the superiority of our KAF across various domains including vision, NLP, audio processing, and differential equation-solving tasks, effectively combining theoretical interpretability with practical utility and computational efficiency.

Within the graph learning community, conventional wisdom dictates that spectral convolutional networks may only be deployed on undirected graphs: Only there could the existence of a well-defined graph Fourier transform be guaranteed, so that information may be translated between spatial- and spectral domains. Here we show this traditional reliance on the graph Fourier transform to be superfluous and -- making use of certain advanced tools from complex analysis and spectral theory -- extend spectral convolutions to directed graphs. We provide a frequency-response interpretation of newly developed filters, investigate the influence of the basis used to express filters and discuss the interplay with characteristic operators on which networks are based. In order to thoroughly test the developed theory, we conduct experiments in real world settings, showcasing that directed spectral convolutional networks provide new state of the art results for heterophilic node classification on many datasets and -- as opposed to baselines -- may be rendered stable to resolution-scale varying topological perturbations.

Recent advancements in Spectral Graph Convolutional Networks (SpecGCNs) have led to state-of-the-art performance in various graph representation learning tasks. To exploit the potential of SpecGCNs, we analyze corresponding graph filters via polynomial interpolation, the cornerstone of graph signal processing. Different polynomial bases, such as Bernstein, Chebyshev, and monomial basis, have various convergence rates that will affect the error in polynomial interpolation. Although adopting Chebyshev basis for interpolation can minimize maximum error, the performance of ChebNet is still weaker than GPR-GNN and BernNet. We point out it is caused by the Gibbs phenomenon, which occurs when the graph frequency response function approximates the target function. It reduces the approximation ability of a truncated polynomial interpolation. In order to mitigate the Gibbs phenomenon, we propose to add the Gibbs damping factor with each term of Chebyshev polynomials on ChebNet. As a result, our lightweight approach leads to a significant performance boost. Afterwards, we reorganize ChebNet via decoupling feature propagation and transformation. We name this variant as ChebGibbsNet. Our experiments indicate that ChebGibbsNet is superior to other advanced SpecGCNs, such as GPR-GNN and BernNet, in both homogeneous graphs and heterogeneous graphs.

The performance of transformers for time-series forecasting has improved significantly. Recent architectures learn complex temporal patterns by segmenting a time-series into patches and using the patches as tokens. The patch size controls the ability of transformers to learn the temporal patterns at different frequencies: shorter patches are effective for learning localized, high-frequency patterns, whereas mining long-term seasonalities and trends requires longer patches. Inspired by this observation, we propose a novel framework, Multi-resolution Time-Series Transformer (MTST), which consists of a multi-branch architecture for simultaneous modeling of diverse temporal patterns at different resolutions. In contrast to many existing time-series transformers, we employ relative positional encoding, which is better suited for extracting periodic components at different scales. Extensive experiments on several real-world datasets demonstrate the effectiveness of MTST in comparison to state-of-the-art forecasting techniques.

Generative models have advanced significantly in sampling material systems with continuous variables, such as atomistic structures. However, their application to discrete variables, like atom types or spin states, remains underexplored. In this work, we introduce a Boltzmann generator built on discrete flow matching, specifically tailored for systems with discrete phase-space coordinates (e.g., the Ising model or crystalline compounds). This approach enables a single model to sample free energy surfaces over a wide temperature range with minimal training overhead. In addition, the model generation is scalable to larger lattice sizes than those in the training set. We demonstrate the effectiveness of our approach on the 2D Ising model, showing efficient and reliable free energy sampling. This framework provides a scalable and computationally efficient solution for discrete coordinate systems and can be extended to sample the alchemical degrees of freedom in crystalline compounds.

We determine the 4-point correlation function and amplitude in planar, maximally supersymmetric Yang-Mills theory to 12 loops. We find that the recently-introduced 'double-triangle' rule in fact implies the previously described square and pentagon rules; and when applied to 12 loops, it fully determines the 11-loop correlator and fixes all but 3 of the (22,024,902) 12-loop coefficients; these remaining coefficients can be subsequently fixed using the '(single-)triangle' rule. Not only do we confirm the Catalan conjecture for anti-prism graphs, but we discover evidence for a greatly generalized Catalan conjecture for the coefficients of all polygon-framed fishnet graphs. We provide all contributions through 12 loops as ancillary files to this work.

Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD) matrices and Grassmann manifolds, appear in many applications. Recently, by applying the theory of gyrogroups and gyrovector spaces that is a powerful framework for studying hyperbolic geometry, some works have attempted to build principled generalizations of Euclidean neural networks on matrix manifolds. However, due to the lack of many concepts in gyrovector spaces for the considered manifolds, e.g., the inner product and gyroangles, techniques and mathematical tools provided by these works are still limited compared to those developed for studying hyperbolic geometry. In this paper, we generalize some notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new models and layers for building neural networks on these manifolds. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.

The inference of topological principles is a key problem in structured reconstruction. We observe that wrongly predicted topological relationships are often incurred by the lack of holistic geometry clues in low-level features. Inspired by the fact that massive signals can be compactly described with frequency analysis, we experimentally explore the efficiency and tendency of learning structure geometry in the frequency domain. Accordingly, we propose a frequency-domain feature learning strategy (F-Learn) to fuse scattered geometric fragments holistically for topology-intact structure reasoning. Benefiting from the parsimonious design, the F-Learn strategy can be easily deployed into a deep reconstructor with a lightweight model modification. Experiments demonstrate that the F-Learn strategy can effectively introduce structure awareness into geometric primitive detection and topology inference, bringing significant performance improvement to final structured reconstruction. Code and pre-trained models are available at https://github.com/Geo-Tell/F-Learn.

Hyperdimensional computing (HDC) is a biologically-inspired framework which represents symbols with high-dimensional vectors, and uses vector operations to manipulate them. The ensemble of a particular vector space and a prescribed set of vector operations (including one addition-like for "bundling" and one outer-product-like for "binding") form a *vector symbolic architecture* (VSA). While VSAs have been employed in numerous applications and have been studied empirically, many theoretical questions about VSAs remain open. We analyze the *representation capacities* of four common VSAs: MAP-I, MAP-B, and two VSAs based on sparse binary vectors. "Representation capacity' here refers to bounds on the dimensions of the VSA vectors required to perform certain symbolic tasks, such as testing for set membership i in S and estimating set intersection sizes |X cap Y| for two sets of symbols X and Y, to a given degree of accuracy. We also analyze the ability of a novel variant of a Hopfield network (a simple model of associative memory) to perform some of the same tasks that are typically asked of VSAs. In addition to providing new bounds on VSA capacities, our analyses establish and leverage connections between VSAs, "sketching" (dimensionality reduction) algorithms, and Bloom filters.

In this paper, we study the problem of principal component analysis with generative modeling assumptions, adopting a general model for the observed matrix that encompasses notable special cases, including spiked matrix recovery and phase retrieval. The key assumption is that the underlying signal lies near the range of an L-Lipschitz continuous generative model with bounded k-dimensional inputs. We propose a quadratic estimator, and show that it enjoys a statistical rate of order frac{klog L{m}}, where m is the number of samples. We also provide a near-matching algorithm-independent lower bound. Moreover, we provide a variant of the classic power method, which projects the calculated data onto the range of the generative model during each iteration. We show that under suitable conditions, this method converges exponentially fast to a point achieving the above-mentioned statistical rate. We perform experiments on various image datasets for spiked matrix and phase retrieval models, and illustrate performance gains of our method to the classic power method and the truncated power method devised for sparse principal component analysis.

Time series foundation models have shown impressive performance on a variety of tasks, across a wide range of domains, even in zero-shot settings. However, most of these models are designed to handle short univariate time series as an input. This limits their practical use, especially in domains such as healthcare with copious amounts of long and multivariate data with strong temporal and intra-variate dependencies. Our study bridges this gap by cataloging and systematically comparing various context expansion techniques from both language and time series domains, and introducing a novel compressive memory mechanism to allow encoder-only TSFMs to effectively model intra-variate dependencies. We demonstrate the benefits of our approach by imbuing MOMENT, a recent family of multi-task time series foundation models, with the multivariate context.

We propose a distributed bundle adjustment (DBA) method using the exact Levenberg-Marquardt (LM) algorithm for super large-scale datasets. Most of the existing methods partition the global map to small ones and conduct bundle adjustment in the submaps. In order to fit the parallel framework, they use approximate solutions instead of the LM algorithm. However, those methods often give sub-optimal results. Different from them, we utilize the exact LM algorithm to conduct global bundle adjustment where the formation of the reduced camera system (RCS) is actually parallelized and executed in a distributed way. To store the large RCS, we compress it with a block-based sparse matrix compression format (BSMC), which fully exploits its block feature. The BSMC format also enables the distributed storage and updating of the global RCS. The proposed method is extensively evaluated and compared with the state-of-the-art pipelines using both synthetic and real datasets. Preliminary results demonstrate the efficient memory usage and vast scalability of the proposed method compared with the baselines. For the first time, we conducted parallel bundle adjustment using LM algorithm on a real datasets with 1.18 million images and a synthetic dataset with 10 million images (about 500 times that of the state-of-the-art LM-based BA) on a distributed computing system.

This paper provides a non-asymptotic analysis of linear stochastic approximation (LSA) algorithms with fixed stepsize. This family of methods arises in many machine learning tasks and is used to obtain approximate solutions of a linear system Atheta = b for which A and b can only be accessed through random estimates {({bf A}_n, {bf b}_n): n in N^*}. Our analysis is based on new results regarding moments and high probability bounds for products of matrices which are shown to be tight. We derive high probability bounds on the performance of LSA under weaker conditions on the sequence {({bf A}_n, {bf b}_n): n in N^*} than previous works. However, in contrast, we establish polynomial concentration bounds with order depending on the stepsize. We show that our conclusions cannot be improved without additional assumptions on the sequence of random matrices {{bf A}_n: n in N^*}, and in particular that no Gaussian or exponential high probability bounds can hold. Finally, we pay a particular attention to establishing bounds with sharp order with respect to the number of iterations and the stepsize and whose leading terms contain the covariance matrices appearing in the central limit theorems.

Reliably reconstructing physical fields from sparse sensor data is a challenge that frequently arises in many scientific domains. In practice, the process generating the data often is not understood to sufficient accuracy. Therefore, there is a growing interest in using the deep neural network route to address the problem. This work presents a novel approach that learns a continuous representation of the physical field using implicit neural representations (INRs). Specifically, after factorizing spatiotemporal variability into spatial and temporal components using the separation of variables technique, the method learns relevant basis functions from sparsely sampled irregular data points to develop a continuous representation of the data. In experimental evaluations, the proposed model outperforms recent INR methods, offering superior reconstruction quality on simulation data from a state-of-the-art climate model and a second dataset that comprises ultra-high resolution satellite-based sea surface temperature fields.

Implicit neural representations (INRs) use neural networks to provide continuous and resolution-independent representations of complex signals with a small number of parameters. However, existing INR models often fail to capture important frequency components specific to each task. To address this issue, in this paper, we propose a Fourier Kolmogorov Arnold network (FKAN) for INRs. The proposed FKAN utilizes learnable activation functions modeled as Fourier series in the first layer to effectively control and learn the task-specific frequency components. In addition, the activation functions with learnable Fourier coefficients improve the ability of the network to capture complex patterns and details, which is beneficial for high-resolution and high-dimensional data. Experimental results show that our proposed FKAN model outperforms three state-of-the-art baseline schemes, and improves the peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM) for the image representation task and intersection over union (IoU) for the 3D occupancy volume representation task, respectively.

Given a K-vertex simplex in a d-dimensional space, suppose we measure n points on the simplex with noise (hence, some of the observed points fall outside the simplex). Vertex hunting is the problem of estimating the K vertices of the simplex. A popular vertex hunting algorithm is successive projection algorithm (SPA). However, SPA is observed to perform unsatisfactorily under strong noise or outliers. We propose pseudo-point SPA (pp-SPA). It uses a projection step and a denoise step to generate pseudo-points and feed them into SPA for vertex hunting. We derive error bounds for pp-SPA, leveraging on extreme value theory of (possibly) high-dimensional random vectors. The results suggest that pp-SPA has faster rates and better numerical performances than SPA. Our analysis includes an improved non-asymptotic bound for the original SPA, which is of independent interest.

Tagged magnetic resonance imaging (tMRI) has been employed for decades to measure the motion of tissue undergoing deformation. However, registration-based motion estimation from tMRI is difficult due to the periodic patterns in these images, particularly when the motion is large. With a larger motion the registration approach gets trapped in a local optima, leading to motion estimation errors. We introduce a novel "momenta, shooting, and correction" framework for Lagrangian motion estimation in the presence of repetitive patterns and large motion. This framework, grounded in Lie algebra and Lie group principles, accumulates momenta in the tangent vector space and employs exponential mapping in the diffeomorphic space for rapid approximation towards true optima, circumventing local optima. A subsequent correction step ensures convergence to true optima. The results on a 2D synthetic dataset and a real 3D tMRI dataset demonstrate our method's efficiency in estimating accurate, dense, and diffeomorphic 2D/3D motion fields amidst large motion and repetitive patterns.

Crystal symmetry plays a fundamental role in determining its physical, chemical, and electronic properties such as electrical and thermal conductivity, optical and polarization behavior, and mechanical strength. Almost all known crystalline materials have internal symmetry. However, this is often inadequately addressed by existing generative models, making the consistent generation of stable and symmetrically valid crystal structures a significant challenge. We introduce WyFormer, a generative model that directly tackles this by formally conditioning on space group symmetry. It achieves this by using Wyckoff positions as the basis for an elegant, compressed, and discrete structure representation. To model the distribution, we develop a permutation-invariant autoregressive model based on the Transformer encoder and an absence of positional encoding. Extensive experimentation demonstrates WyFormer's compelling combination of attributes: it achieves best-in-class symmetry-conditioned generation, incorporates a physics-motivated inductive bias, produces structures with competitive stability, predicts material properties with competitive accuracy even without atomic coordinates, and exhibits unparalleled inference speed.

Symmetry learning has proven to be an effective approach for extracting the hidden structure of data, with the concept of equivariance relation playing the central role. However, most of the current studies are built on architectural theory and corresponding assumptions on the form of data. We propose Neural Fourier Transform (NFT), a general framework of learning the latent linear action of the group without assuming explicit knowledge of how the group acts on data. We present the theoretical foundations of NFT and show that the existence of a linear equivariant feature, which has been assumed ubiquitously in equivariance learning, is equivalent to the existence of a group invariant kernel on the dataspace. We also provide experimental results to demonstrate the application of NFT in typical scenarios with varying levels of knowledge about the acting group.

Vector Symbolic Architectures (VSAs) have emerged as a novel framework for enabling interpretable machine learning algorithms equipped with the ability to reason and explain their decision processes. The basic idea is to represent discrete information through high dimensional random vectors. Complex data structures can be built up with operations over vectors such as the "binding" operation involving element-wise vector multiplication, which associates data together. The reverse task of decomposing the associated elements is a combinatorially hard task, with an exponentially large search space. The main algorithm for performing this search is the resonator network, inspired by Hopfield network-based memory search operations. In this work, we introduce a new variant of the resonator network, based on self-attention based update rules in the iterative search problem. This update rule, based on the Hopfield network with log-sum-exp energy function and norm-bounded states, is shown to substantially improve the performance and rate of convergence. As a result, our algorithm enables a larger capacity for associative memory, enabling applications in many tasks like perception based pattern recognition, scene decomposition, and object reasoning. We substantiate our algorithm with a thorough evaluation and comparisons to baselines.

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f =(f_1, ldots, f_N)in C[x_1, ldots, x_n]^N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes the multiplicity structure at the root. We use $alpha$-theory test to certify the quadratic convergence, and togive bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.

We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization technique that takes into account the fundamental structures and operations of geometric algebras. Due to this technique, GLGENN architecture is parameter-light and has less tendency to overfitting than baseline equivariant models. GLGENN outperforms or matches competitors on several benchmarking equivariant tasks, including estimation of an equivariant function and a convex hull experiment, while using significantly fewer optimizable parameters.

State-space models (SSMs) offer a powerful framework for dynamical system analysis, wherein the temporal dynamics of the system are assumed to be captured through the evolution of the latent states, which govern the values of the observations. This paper provides a selective review of recent advancements in deep neural network-based approaches for SSMs, and presents a unified perspective for discrete time deep state space models and continuous time ones such as latent neural Ordinary Differential and Stochastic Differential Equations. It starts with an overview of the classical maximum likelihood based approach for learning SSMs, reviews variational autoencoder as a general learning pipeline for neural network-based approaches in the presence of latent variables, and discusses in detail representative deep learning models that fall under the SSM framework. Very recent developments, where SSMs are used as standalone architectural modules for improving efficiency in sequence modeling, are also examined. Finally, examples involving mixed frequency and irregularly-spaced time series data are presented to demonstrate the advantage of SSMs in these settings.

Metric space magnitude, an active field of research in algebraic topology, is a scalar quantity that summarizes the effective number of distinct points that live in a general metric space. The {\em weighting vector} is a closely-related concept that captures, in a nontrivial way, much of the underlying geometry of the original metric space. Recent work has demonstrated that when the metric space is Euclidean, the weighting vector serves as an effective tool for boundary detection. We recast this result and show the weighting vector may be viewed as a solution to a kernelized SVM. As one consequence, we apply this new insight to the task of outlier detection, and we demonstrate performance that is competitive or exceeds performance of state-of-the-art techniques on benchmark data sets. Under mild assumptions, we show the weighting vector, which has computational cost of matrix inversion, can be efficiently approximated in linear time. We show how nearest neighbor methods can approximate solutions to the minimization problems defined by SVMs.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

Linear Recurrent Neural Networks (LRNNs) such as Mamba, RWKV, GLA, mLSTM, and DeltaNet have emerged as efficient alternatives to Transformers for long sequences. However, both Transformers and LRNNs struggle to perform state-tracking, which may impair performance in tasks such as code evaluation. In one forward pass, current architectures are unable to solve even parity, the simplest state-tracking task, which non-linear RNNs can handle effectively. Recently, Sarrof et al. (2024) demonstrated that the failure of LRNNs like Mamba to solve parity stems from restricting the value range of their diagonal state-transition matrices to [0, 1] and that incorporating negative values can resolve this issue. We extend this result to non-diagonal LRNNs such as DeltaNet. We prove that finite precision LRNNs with state-transition matrices having only positive eigenvalues cannot solve parity, while non-triangular matrices are needed to count modulo 3. Notably, we also prove that LRNNs can learn any regular language when their state-transition matrices are products of identity minus vector outer product matrices, each with eigenvalues in the range [-1, 1]. Our experiments confirm that extending the eigenvalue range of Mamba and DeltaNet to include negative values not only enables them to solve parity but consistently improves their performance on state-tracking tasks. We also show that state-tracking enabled LRNNs can be pretrained stably and efficiently at scale (1.3B parameters), achieving competitive performance on language modeling and showing promise on code and math tasks.

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

This paper studies sequence modeling for prediction tasks with long range dependencies. We propose a new formulation for state space models (SSMs) based on learning linear dynamical systems with the spectral filtering algorithm (Hazan et al. (2017)). This gives rise to a novel sequence prediction architecture we call a spectral state space model. Spectral state space models have two primary advantages. First, they have provable robustness properties as their performance depends on neither the spectrum of the underlying dynamics nor the dimensionality of the problem. Second, these models are constructed with fixed convolutional filters that do not require learning while still outperforming SSMs in both theory and practice. The resulting models are evaluated on synthetic dynamical systems and long-range prediction tasks of various modalities. These evaluations support the theoretical benefits of spectral filtering for tasks requiring very long range memory.

Chaotic dynamical systems (DS) are ubiquitous in nature and society. Often we are interested in reconstructing such systems from observed time series for prediction or mechanistic insight, where by reconstruction we mean learning geometrical and invariant temporal properties of the system in question (like attractors). However, training reconstruction algorithms like recurrent neural networks (RNNs) on such systems by gradient-descent based techniques faces severe challenges. This is mainly due to exploding gradients caused by the exponential divergence of trajectories in chaotic systems. Moreover, for (scientific) interpretability we wish to have as low dimensional reconstructions as possible, preferably in a model which is mathematically tractable. Here we report that a surprisingly simple modification of teacher forcing leads to provably strictly all-time bounded gradients in training on chaotic systems, and, when paired with a simple architectural rearrangement of a tractable RNN design, piecewise-linear RNNs (PLRNNs), allows for faithful reconstruction in spaces of at most the dimensionality of the observed system. We show on several DS that with these amendments we can reconstruct DS better than current SOTA algorithms, in much lower dimensions. Performance differences were particularly compelling on real world data with which most other methods severely struggled. This work thus led to a simple yet powerful DS reconstruction algorithm which is highly interpretable at the same time.

Block coordinate descent (BCD) methods are widely used for large-scale numerical optimization because of their cheap iteration costs, low memory requirements, amenability to parallelization, and ability to exploit problem structure. Three main algorithmic choices influence the performance of BCD methods: the block partitioning strategy, the block selection rule, and the block update rule. In this paper we explore all three of these building blocks and propose variations for each that can significantly improve the progress made by each BCD iteration. We (i) propose new greedy block-selection strategies that guarantee more progress per iteration than the Gauss-Southwell rule; (ii) explore practical issues like how to implement the new rules when using "variable" blocks; (iii) explore the use of message-passing to compute matrix or Newton updates efficiently on huge blocks for problems with sparse dependencies between variables; and (iv) consider optimal active manifold identification, which leads to bounds on the "active-set complexity" of BCD methods and leads to superlinear convergence for certain problems with sparse solutions (and in some cases finite termination at an optimal solution). We support all of our findings with numerical results for the classic machine learning problems of least squares, logistic regression, multi-class logistic regression, label propagation, and L1-regularization.

The concept class of low-degree polynomial threshold functions (PTFs) plays a fundamental role in machine learning. In this paper, we study PAC learning of K-sparse degree-d PTFs on R^n, where any such concept depends only on K out of n attributes of the input. Our main contribution is a new algorithm that runs in time ({nd}/{epsilon})^{O(d)} and under the Gaussian marginal distribution, PAC learns the class up to error rate epsilon with O(K^{4d}{epsilon^{2d}} cdot log^{5d} n) samples even when an eta leq O(epsilon^d) fraction of them are corrupted by the nasty noise of Bshouty et al. (2002), possibly the strongest corruption model. Prior to this work, attribute-efficient robust algorithms are established only for the special case of sparse homogeneous halfspaces. Our key ingredients are: 1) a structural result that translates the attribute sparsity to a sparsity pattern of the Chow vector under the basis of Hermite polynomials, and 2) a novel attribute-efficient robust Chow vector estimation algorithm which uses exclusively a restricted Frobenius norm to either certify a good approximation or to validate a sparsity-induced degree-2d polynomial as a filter to detect corrupted samples.

This paper discusses a simple and explicit toy-model example of the categorical Hopfield equations introduced in previous work of Manin and the author. These describe dynamical assignments of resources to networks, where resources are objects in unital symmetric monoidal categories and assignments are realized by summing functors. The special case discussed here is based on computational resources (computational models of neurons) as objects in a category of DNNs, with a simple choice of the endofunctors defining the Hopfield equations that reproduce the usual updating of the weights in DNNs by gradient descent.

We study the problem of designing models for machine learning tasks defined on sets. In contrast to traditional approach of operating on fixed dimensional vectors, we consider objective functions defined on sets that are invariant to permutations. Such problems are widespread, ranging from estimation of population statistics poczos13aistats, to anomaly detection in piezometer data of embankment dams Jung15Exploration, to cosmology Ntampaka16Dynamical,Ravanbakhsh16ICML1. Our main theorem characterizes the permutation invariant functions and provides a family of functions to which any permutation invariant objective function must belong. This family of functions has a special structure which enables us to design a deep network architecture that can operate on sets and which can be deployed on a variety of scenarios including both unsupervised and supervised learning tasks. We also derive the necessary and sufficient conditions for permutation equivariance in deep models. We demonstrate the applicability of our method on population statistic estimation, point cloud classification, set expansion, and outlier detection.

A central problem in learning from sequential data is representing cumulative history in an incremental fashion as more data is processed. We introduce a general framework (HiPPO) for the online compression of continuous signals and discrete time series by projection onto polynomial bases. Given a measure that specifies the importance of each time step in the past, HiPPO produces an optimal solution to a natural online function approximation problem. As special cases, our framework yields a short derivation of the recent Legendre Memory Unit (LMU) from first principles, and generalizes the ubiquitous gating mechanism of recurrent neural networks such as GRUs. This formal framework yields a new memory update mechanism (HiPPO-LegS) that scales through time to remember all history, avoiding priors on the timescale. HiPPO-LegS enjoys the theoretical benefits of timescale robustness, fast updates, and bounded gradients. By incorporating the memory dynamics into recurrent neural networks, HiPPO RNNs can empirically capture complex temporal dependencies. On the benchmark permuted MNIST dataset, HiPPO-LegS sets a new state-of-the-art accuracy of 98.3%. Finally, on a novel trajectory classification task testing robustness to out-of-distribution timescales and missing data, HiPPO-LegS outperforms RNN and neural ODE baselines by 25-40% accuracy.

Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single real-valued function (the one-parameter setting), there is a canonical choice of descriptor for persistent homology: the barcode. The operation mapping a real-valued function to its barcode is differentiable almost everywhere, and the convergence of gradient descent for losses using barcodes is relatively well understood. When optimizing a vector-valued function (the multiparameter setting), there is no unique choice of descriptor for multiparameter persistent homology, and many distinct descriptors have been proposed. This calls for the development of a general framework for differentiability and optimization that applies to a wide range of multiparameter homological descriptors. In this article, we develop such a framework and show that it encompasses well-known descriptors of different flavors, such as signed barcodes and the multiparameter persistence landscape. We complement the theory with numerical experiments supporting the idea that optimizing multiparameter homological descriptors can lead to improved performances compared to optimizing one-parameter descriptors, even when using the simplest and most efficiently computable multiparameter descriptors.

The trade-off between regret and computational cost is a fundamental problem for online kernel regression, and previous algorithms worked on the trade-off can not keep optimal regret bounds at a sublinear computational complexity. In this paper, we propose two new algorithms, AOGD-ALD and NONS-ALD, which can keep nearly optimal regret bounds at a sublinear computational complexity, and give sufficient conditions under which our algorithms work. Both algorithms dynamically maintain a group of nearly orthogonal basis used to approximate the kernel mapping, and keep nearly optimal regret bounds by controlling the approximate error. The number of basis depends on the approximate error and the decay rate of eigenvalues of the kernel matrix. If the eigenvalues decay exponentially, then AOGD-ALD and NONS-ALD separately achieves a regret of O(L(f)) and O(d_{eff}(mu)T) at a computational complexity in O(ln^2{T}). If the eigenvalues decay polynomially with degree pgeq 1, then our algorithms keep the same regret bounds at a computational complexity in o(T) in the case of p>4 and pgeq 10, respectively. L(f) is the cumulative losses of f and d_{eff}(mu) is the effective dimension of the problem. The two regret bounds are nearly optimal and are not comparable.

The motif-scaffolding problem is a central task in computational protein design: Given the coordinates of atoms in a geometry chosen to confer a desired biochemical function (a motif), the task is to identify diverse protein structures (scaffolds) that include the motif and maintain its geometry. Significant recent progress on motif-scaffolding has been made due to computational evaluation with reliable protein structure prediction and fixed-backbone sequence design methods. However, significant variability in evaluation strategies across publications has hindered comparability of results, challenged reproducibility, and impeded robust progress. In response we introduce MotifBench, comprising (1) a precisely specified pipeline and evaluation metrics, (2) a collection of 30 benchmark problems, and (3) an implementation of this benchmark and leaderboard at github.com/blt2114/MotifBench. The MotifBench test cases are more difficult compared to earlier benchmarks, and include protein design problems for which solutions are known but on which, to the best of our knowledge, state-of-the-art methods fail to identify any solution.

In "Backprop as functor", the authors show that the fundamental elements of deep learning -- gradient descent and backpropagation -- can be conceptualized as a strong monoidal functor Para(Euc)toLearn from the category of parameterized Euclidean spaces to that of learners, a category developed explicitly to capture parameter update and backpropagation. It was soon realized that there is an isomorphism LearncongPara(Slens), where Slens is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that Slens is a full subcategory of Poly, the category of polynomial functors in one variable, via the functor Amapsto Ay^A. Using the fact that (Poly,otimes) is monoidal closed, we show that a map Ato B in Para(Slens) has a natural interpretation in terms of dynamical systems (more precisely, generalized Moore machines) whose interface is the internal-hom type [Ay^A,By^B]. Finally, we review the fact that the category p-Coalg of dynamical systems on any p in Poly forms a topos, and consider the logical propositions that can be stated in its internal language. We give gradient descent as an example, and we conclude by discussing some directions for future work.

Despite their widespread success, the application of deep neural networks to functional data remains scarce today. The infinite dimensionality of functional data means standard learning algorithms can be applied only after appropriate dimension reduction, typically achieved via basis expansions. Currently, these bases are chosen a priori without the information for the task at hand and thus may not be effective for the designated task. We instead propose to adaptively learn these bases in an end-to-end fashion. We introduce neural networks that employ a new Basis Layer whose hidden units are each basis functions themselves implemented as a micro neural network. Our architecture learns to apply parsimonious dimension reduction to functional inputs that focuses only on information relevant to the target rather than irrelevant variation in the input function. Across numerous classification/regression tasks with functional data, our method empirically outperforms other types of neural networks, and we prove that our approach is statistically consistent with low generalization error. Code is available at: https://github.com/jwyyy/AdaFNN.

We present Manifold Diffusion Fields (MDF), an approach to learn generative models of continuous functions defined over Riemannian manifolds. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator. MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. Empirical results on several datasets and manifolds show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.

In this article, we study the consistency of the template estimation with the Fr\'echet mean in quotient spaces. The Fr\'echet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.

Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. In this work, we propose a novel algorithm to find efficient low-rank subnetworks. Remarkably, these subnetworks are determined and adapted already during the training phase and the overall time and memory resources required by both training and evaluating them are significantly reduced. The main idea is to restrict the weight matrices to a low-rank manifold and to update the low-rank factors rather than the full matrix during training. To derive training updates that are restricted to the prescribed manifold, we employ techniques from dynamic model order reduction for matrix differential equations. This allows us to provide approximation, stability, and descent guarantees. Moreover, our method automatically and dynamically adapts the ranks during training to achieve the desired approximation accuracy. The efficiency of the proposed method is demonstrated through a variety of numerical experiments on fully-connected and convolutional networks.

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance matrices or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular the kernel metrics introduced by Hiai and Petz. The class of kernel metrics interpolates between many classical O(n)-invariant metrics and it satisfies key results of stability and completeness. However, it does not contain all the classical O(n)-invariant metrics. Therefore in this work, we investigate super-classes of kernel metrics and we study which key results remain true. We also introduce an additional key result called cometric-stability, a crucial property to implement geodesics with a Hamiltonian formulation. Our method to build intermediate embedded classes between O(n)-invariant metrics and kernel metrics is to give a characterization of the whole class of O(n)-invariant metrics on SPD matrices and to specify requirements on metrics one by one until we reach kernel metrics. As a secondary contribution, we synthesize the literature on the main O(n)-invariant metrics, we provide the complete formula of the sectional curvature of the affine-invariant metric and the formula of the geodesic parallel transport between commuting matrices for the Bures-Wasserstein metric.

Innovations like protein diffusion have enabled significant progress in de novo protein design, which is a vital topic in life science. These methods typically depend on protein structure encoders to model residue backbone frames, where atoms do not exist. Most prior encoders rely on atom-wise features, such as angles and distances between atoms, which are not available in this context. Thus far, only several simple encoders, such as IPA, have been proposed for this scenario, exposing the frame modeling as a bottleneck. In this work, we proffer the Vector Field Network (VFN), which enables network layers to perform learnable vector computations between coordinates of frame-anchored virtual atoms, thus achieving a higher capability for modeling frames. The vector computation operates in a manner similar to a linear layer, with each input channel receiving 3D virtual atom coordinates instead of scalar values. The multiple feature vectors output by the vector computation are then used to update the residue representations and virtual atom coordinates via attention aggregation. Remarkably, VFN also excels in modeling both frames and atoms, as the real atoms can be treated as the virtual atoms for modeling, positioning VFN as a potential universal encoder. In protein diffusion (frame modeling), VFN exhibits an impressive performance advantage over IPA, excelling in terms of both designability (67.04% vs. 53.58%) and diversity (66.54% vs. 51.98%). In inverse folding (frame and atom modeling), VFN outperforms the previous SoTA model, PiFold (54.7% vs. 51.66%), on sequence recovery rate. We also propose a method of equipping VFN with the ESM model, which significantly surpasses the previous ESM-based SoTA (62.67% vs. 55.65%), LM-Design, by a substantial margin.

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

Bagging is an important technique for stabilizing machine learning models. In this paper, we derive a finite-sample guarantee on the stability of bagging for any model. Our result places no assumptions on the distribution of the data, on the properties of the base algorithm, or on the dimensionality of the covariates. Our guarantee applies to many variants of bagging and is optimal up to a constant. Empirical results validate our findings, showing that bagging successfully stabilizes even highly unstable base algorithms.

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539--542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the Empirical Interpolation Method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.