Efficient Sparsification of Simplicial Complexes via Local Densities of States

Simplicial complexes (SCs), a generalization of graph models for relational data that account for higher-order relations between data items, have become a popular abstraction for analyzing complex data using tools from topological data analysis or topological signal processing. However, the analysis of many real-world datasets leads to dense SCs with a large number of higher-order interactions. Unfortunately, analyzing such large SCs often has a prohibitive cost in terms of computation time and memory consumption. The sparsification of such complexes, i.e., the approximation of an original SC with a sparser simplicial complex with only a log-linear number of high-order simplices while maintaining a spectrum close to the original SC, is of broad interest. In this work, we develop a novel method for a probabilistic sparsifaction of SCs. At its core lies the efficient computation of sparsifying sampling probability through local densities of states as functional descriptors of the spectral information. To avoid pathological structures in the spectrum of the corresponding Hodge Laplacian operators, we suggest a "kernel-ignoring" decomposition for approximating the sampling probability; additionally, we exploit error estimates to show asymptotically prevailing algorithmic complexity of the developed method. The performance of the framework is demonstrated on the family of Vietoris--Rips filtered simplicial complexes.

Rethinking Oversmoothing in Graph Neural Networks: A Rank-Based Perspective

Oversmoothing is a fundamental challenge in graph neural networks (GNNs): as the number of layers increases, node embeddings become increasingly similar, and model performance drops sharply. Traditionally, oversmoothing has been quantified using metrics that measure the similarity of neighbouring node features, such as the Dirichlet energy. While these metrics are related to oversmoothing, we argue they have critical limitations and fail to reliably capture oversmoothing in realistic scenarios. For instance, they provide meaningful insights only for very deep networks and under somewhat strict conditions on the norm of network weights and feature representations. As an alternative, we propose measuring oversmoothing by examining the numerical or effective rank of the feature representations. We provide theoretical support for this approach, demonstrating that the numerical rank of feature representations converges to one for a broad family of nonlinear activation functions under the assumption of nonnegative trained weights. To the best of our knowledge, this is the first result that proves the occurrence of oversmoothing without assumptions on the boundedness of the weight matrices. Along with the theoretical findings, we provide extensive numerical evaluation across diverse graph architectures. Our results show that rank-based metrics consistently capture oversmoothing, whereas energy-based metrics often fail. Notably, we reveal that a significant drop in the rank aligns closely with performance degradation, even in scenarios where energy metrics remain unchanged.

Solaris: A Foundation Model of the Sun

Foundation models have demonstrated remarkable success across various scientific domains, motivating our exploration of their potential in solar physics. In this paper, we present Solaris, the first foundation model for forecasting the Sun's atmosphere. We leverage 13 years of full-disk, multi-wavelength solar imagery from the Solar Dynamics Observatory, spanning a complete solar cycle, to pre-train Solaris for 12-hour interval forecasting. Solaris is built on a large-scale 3D Swin Transformer architecture with 109 million parameters. We demonstrate Solaris' ability to generalize by fine-tuning on a low-data regime using a single wavelength (1700 {\AA}), that was not included in pre-training, outperforming models trained from scratch on this specific wavelength. Our results indicate that Solaris can effectively capture the complex dynamics of the solar atmosphere and transform solar forecasting.

GeoLoRA: Geometric integration for parameter efficient fine-tuning

Low-Rank Adaptation (LoRA) has become a widely used method for parameter-efficient fine-tuning of large-scale, pre-trained neural networks. However, LoRA and its extensions face several challenges, including the need for rank adaptivity, robustness, and computational efficiency during the fine-tuning process. We introduce GeoLoRA, a novel approach that addresses these limitations by leveraging dynamical low-rank approximation theory. GeoLoRA requires only a single backpropagation pass over the small-rank adapters, significantly reducing computational cost as compared to similar dynamical low-rank training methods and making it faster than popular baselines such as AdaLoRA. This allows GeoLoRA to efficiently adapt the allocated parameter budget across the model, achieving smaller low-rank adapters compared to heuristic methods like AdaLoRA and LoRA, while maintaining critical convergence, descent, and error-bound theoretical guarantees. The resulting method is not only more efficient but also more robust to varying hyperparameter settings. We demonstrate the effectiveness of GeoLoRA on several state-of-the-art benchmarks, showing that it outperforms existing methods in both accuracy and computational efficiency.

Low-Rank Adversarial PGD Attack

Adversarial attacks on deep neural network models have seen rapid development and are extensively used to study the stability of these networks. Among various adversarial strategies, Projected Gradient Descent (PGD) is a widely adopted method in computer vision due to its effectiveness and quick implementation, making it suitable for adversarial training. In this work, we observe that in many cases, the perturbations computed using PGD predominantly affect only a portion of the singular value spectrum of the original image, suggesting that these perturbations are approximately low-rank. Motivated by this observation, we propose a variation of PGD that efficiently computes a low-rank attack. We extensively validate our method on a range of standard models as well as robust models that have undergone adversarial training. Our analysis indicates that the proposed low-rank PGD can be effectively used in adversarial training due to its straightforward and fast implementation coupled with competitive performance. Notably, we find that low-rank PGD often performs comparably to, and sometimes even outperforms, the traditional full-rank PGD attack, while using significantly less memory.

Subhomogeneous Deep Equilibrium Models

Implicit-depth neural networks have grown as powerful alternatives to traditional networks in various applications in recent years. However, these models often lack guarantees of existence and uniqueness, raising stability, performance, and reproducibility issues. In this paper, we present a new analysis of the existence and uniqueness of fixed points for implicit-depth neural networks based on the concept of subhomogeneous operators and the nonlinear Perron-Frobenius theory. Compared to previous similar analyses, our theory allows for weaker assumptions on the parameter matrices, thus yielding a more flexible framework for well-defined implicit networks. We illustrate the performance of the resulting subhomogeneous networks on feed-forward, convolutional, and graph neural network examples.

Neural Rank Collapse: Weight Decay and Small Within-Class Variability Yield Low-Rank Bias

Recent work in deep learning has shown strong empirical and theoretical evidence of an implicit low-rank bias: weight matrices in deep networks tend to be approximately low-rank and removing relatively small singular values during training or from available trained models may significantly reduce model size while maintaining or even improving model performance. However, the majority of the theoretical investigations around low-rank bias in neural networks deal with oversimplified deep linear networks. In this work, we consider general networks with nonlinear activations and the weight decay parameter, and we show the presence of an intriguing neural rank collapse phenomenon, connecting the low-rank bias of trained networks with networks' neural collapse properties: as the weight decay parameter grows, the rank of each layer in the network decreases proportionally to the within-class variability of the hidden-space embeddings of the previous layers. Our theoretical findings are supported by a range of experimental evaluations illustrating the phenomenon.

Robust low-rank training via approximate orthonormal constraints

With the growth of model and data sizes, a broad effort has been made to design pruning techniques that reduce the resource demand of deep learning pipelines, while retaining model performance. In order to reduce both inference and training costs, a prominent line of work uses low-rank matrix factorizations to represent the network weights. Although able to retain accuracy, we observe that low-rank methods tend to compromise model robustness against adversarial perturbations. By modeling robustness in terms of the condition number of the neural network, we argue that this loss of robustness is due to the exploding singular values of the low-rank weight matrices. Thus, we introduce a robust low-rank training algorithm that maintains the network's weights on the low-rank matrix manifold while simultaneously enforcing approximate orthonormal constraints. The resulting model reduces both training and inference costs while ensuring well-conditioning and thus better adversarial robustness, without compromising model accuracy. This is shown by extensive numerical evidence and by our main approximation theorem that shows the computed robust low-rank network well-approximates the ideal full model, provided a highly performing low-rank sub-network exists.

Learning the effective order of a hypergraph dynamical system

Dynamical systems on hypergraphs can display a rich set of behaviours not observable for systems with pairwise interactions. Given a distributed dynamical system with a putative hypergraph structure, an interesting question is thus how much of this hypergraph structure is actually necessary to faithfully replicate the observed dynamical behaviour. To answer this question, we propose a method to determine the minimum order of a hypergraph necessary to approximate the corresponding dynamics accurately. Specifically, we develop an analytical framework that allows us to determine this order when the type of dynamics is known. We utilize these ideas in conjunction with a hypergraph neural network to directly learn the dynamics itself and the resulting order of the hypergraph from both synthetic and real data sets consisting of observed system trajectories.

Learning the Right Layers: a Data-Driven Layer-Aggregation Strategy for Semi-Supervised Learning on Multilayer Graphs

Clustering (or community detection) on multilayer graphs poses several additional complications with respect to standard graphs as different layers may be characterized by different structures and types of information. One of the major challenges is to establish the extent to which each layer contributes to the cluster assignment in order to effectively take advantage of the multilayer structure and improve upon the classification obtained using the individual layers or their union. However, making an informed a-priori assessment about the clustering information content of the layers can be very complicated. In this work, we assume a semi-supervised learning setting, where the class of a small percentage of nodes is initially provided, and we propose a parameter-free Laplacian-regularized model that learns an optimal nonlinear combination of the different layers from the available input labels. The learning algorithm is based on a Frank-Wolfe optimization scheme with inexact gradient, combined with a modified Label Propagation iteration. We provide a detailed convergence analysis of the algorithm and extensive experiments on synthetic and real-world datasets, showing that the proposed method compares favourably with a variety of baselines and outperforms each individual layer when used in isolation.