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Applications · Physics

Rafael Pastrana, Eder Medina, Isabel M. de Oliveira, Sigrid Adriaenssens, Ryan P Adams

Designing mechanically efficient geometry for architectural structures like shells, towers, and bridges, is an expensive iterative process.Existing techniques for solving such inverse problems rely on traditional optimization methods, which are slow and computationally expensive, limiting iteration speed and design exploration.Neural networks would seem to offer a solution via data-driven amortized optimization, but they often require extensive fine-tuning and cannot ensure that important design criteria, such as mechanical integrity, are met.In this work, we combine neural networks with a differentiable mechanics simulator to develop a model that accelerates the solution of shape approximation problems for architectural structures represented as bar systems.This model explicitly guarantees compliance with mechanical constraints while generating designs that closely match target geometries.We validate our approach in two tasks, the design of masonry shells and cable-net towers.Our model achieves better accuracy and generalization than fully neural alternatives, and comparable accuracy to direct optimization but in real time, enabling fast and reliable design exploration.We further demonstrate its advantages by integrating it into 3D modeling software and fabricating a physical prototype.Our work opens up new opportunities for accelerated mechanical design enhanced by neural networks for the built environment.

YUJIE MO, Runpeng Yu, Xiaofeng Zhu, Xinchao Wang

The "pre-train, prompt-tuning'' paradigm has demonstrated impressive performance for tuning pre-trained heterogeneous graph neural networks (HGNNs) by mitigating the gap between pre-trained models and downstream tasks. However, most prompt-tuning-based works may face at least two limitations: (i) the model may be insufficient to fit the graph structures well as they are generally ignored in the prompt-tuning stage, increasing the training error to decrease the generalization ability; and (ii) the model may suffer from the limited labeled data during the prompt-tuning stage, leading to a large generalization gap between the training error and the test error to further affect the model generalization. To alleviate the above limitations, we first derive the generalization error bound for existing prompt-tuning-based methods, and then propose a unified framework that combines two new adapters with potential labeled data extension to improve the generalization of pre-trained HGNN models. Specifically, we design dual structure-aware adapters to adaptively fit task-related homogeneous and heterogeneous structural information. We further design a label-propagated contrastive loss and two self-supervised losses to optimize dual adapters and incorporate unlabeled nodes as potential labeled data. Theoretical analysis indicates that the proposed method achieves a lower generalization error bound than existing methods, thus obtaining superior generalization ability. Comprehensive experiments demonstrate the effectiveness and generalization of the proposed method on different downstream tasks.

Deep Learning · Large Language Models

Xingyi Yang, Xinchao Wang

Transformers stand as the cornerstone of mordern deep learning. Traditionally, these models rely on multi-layerperceptron (MLP) layers to mix the information between channels. In this paper, we introduce the Kolmogorov–ArnoldTransformer (KAT), a novel architecture that replaces MLP layers with Kolmogorov-Arnold Network (KAN) layers toenhance the expressiveness and performance of the model. Integrating KANs into transformers, however, is no easyfeat, especially when scaled up. Specifically, we identify three key challenges: (C1) Base function. The standard B-splinefunction used in KANs is not optimized for parallel computing on modern hardware, resulting in slower inference speeds.(C2) Parameter and Computation Inefficiency. KAN requires a unique function for each input-output pair, making thecomputation extremely large. (C3) Weight initialization. The initialization of weights in KANs is particularly challengingdue to their learnable activation functions, which are critical for achieving convergence in deep neural networks. Toovercome the aforementioned challenges, we propose three key solutions: (S1) Rational basis. We replace B-spline functionswith rational functions to improve compatibility with modern GPUs. By implementing this in CUDA, we achieve fastercomputations. (S2) Group KAN. We share the activation weights through a group of neurons, to reduce the computationalload without sacrificing performance. (S3) Variance-preserving initialization. We carefully initialize the activation weightsto make sure that the activation variance is maintained across layers. With these designs, KAT scales effectively and readilyoutperforms traditional MLP-based transformers. We demonstrate the advantages of KAT across various tasks, includingimage recognition, object detection, and semantic segmentation. It consistently enhances performance over the standardtransformer architectures of different model sizes.

Social Aspects · Trustworthy Machine Learning

Steve Azzolin, Antonio Longa, Stefano Teso, Andrea Passerini

As Graph Neural Networks (GNNs) become more pervasive, it becomes paramount to build reliable tools for explaining their predictions.A core desideratum is that explanations are *faithful*, i.e., that they portray an accurate picture of the GNN's reasoning process.However, a number of different faithfulness metrics exist, begging the question of what is faithfulness exactly and how to achieve it.We make three key contributions.We begin by showing that *existing metrics are not interchangeable* -- i.e., explanations attaining high faithfulness according to one metric may be unfaithful according to others -- and can *systematically ignore important properties of explanations*.We proceed to show that, surprisingly, *optimizing for faithfulness is not always a sensible design goal*. Specifically, we prove that for injective regular GNN architectures, perfectly faithful explanations are completely uninformative.This does not apply to modular GNNs, such as self-explainable and domain-invariant architectures, prompting us to study the relationship between architectural choices and faithfulness.Finally, we show that *faithfulness is tightly linked to out-of-distribution generalization*, in that simply ensuring that a GNN can correctly recognize the domain-invariant subgraph, as prescribed by the literature, does not guarantee that it is invariant unless this subgraph is also faithful.All our code can be found in the supplementary material.

Applications · Chemistry and Drug Discovery

Mohamed Amine Ketata, Nicholas Gao, Johanna Sommer, Tom Wollschläger, Stephan Günnemann

We introduce a new framework for 2D molecular graph generation using 3D molecule generative models. Our Synthetic Coordinate Embedding (SyCo) framework maps 2D molecular graphs to 3D Euclidean point clouds via synthetic coordinates and learns the inverse map using an E($n$)-Equivariant Graph Neural Network (EGNN). The induced point cloud-structured latent space is well-suited to apply existing 3D molecule generative models. This approach simplifies the graph generation problem into a point cloud generation problem followed by node and edge classification tasks, without relying on molecular fragments nor autoregressive decoding. Further, we propose a novel similarity-constrained optimization scheme for 3D diffusion models based on inpainting and guidance. As a concrete implementation of our framework, we develop EDM-SyCo based on the E(3) Equivariant Diffusion Model (EDM). EDM-SyCo achieves state-of-the-art performance in distribution learning of molecular graphs, outperforming the best non-autoregressive methods by more than 26\% on ZINC250K and 16\% on the GuacaMol dataset while improving conditional generation by up to 3.9 times.

Applications · Computer Vision

Peimeng Guan, Naveed Iqbal, Mark Davenport, Mudassir Masood

Model-based deep learning methods such as loop unrolling (LU) and deep equilibrium model (DEQ) extensions offer outstanding performance in solving inverse problems (IP). These methods unroll the optimization iterations into a sequence of neural networks that in effect learn a regularization function from data. While these architectures are currently state-of-the-art in numerous applications, their success heavily relies on the accuracy of the forward model. This assumption can be limiting in many physical applications due to model simplifications or uncertainties in the apparatus. To address forward model mismatch, we introduce an untrained forward model residual block within the model-based architecture to match the data consistency in the measurement domain for each instance. We propose two variants in well-known model-based architectures (LU and DEQ) and prove convergence under mild conditions. Our approach offers a unified solution that is less parameter-sensitive, requires no additional data, and enables simultaneous fitting of the forward model and reconstruction in a single pass, benefiting both linear and nonlinear inverse problems. The experiments show significant quality improvement in removing artifacts and preserving details across three distinct applications, encompassing both linear and nonlinear inverse problems. Moreover, we highlight reconstruction effectiveness in intermediate steps and showcase robustness to random initialization of the residual block and a higher number of iterations during evaluation.

Miscellaneous Aspects of Machine Learning · General Machine Learning Techniques

Hanling Tian, Yuhang Liu, Mingzhen He, Zhengbao He, Zhehao Huang, Ruikai Yang, Xiaolin Huang

Whether deep neural networks (DNNs) memorize the training data is a fundamental open question in understanding deep learning. A direct way to verify the memorization of DNNs is to reconstruct training data from DNNs’ parameters. Since parameters are gradually determined by data throughout training, characterizing training dynamics is important for reconstruction. Pioneering works rely on the linear training dynamics of shallow NNs with large widths, but cannot be extended to more practical DNNs which have non-linear dynamics. We propose Simulation of training Dynamics (SimuDy) to reconstruct training data from DNNs. Specifically, we simulate the training dynamics by training the model from the initial parameters with a dummy dataset, then optimize this dummy dataset so that the simulated dynamics reach the same final parameters as the true dynamics. By incorporating dummy parameters in the simulated dynamics, SimuDy effectively describes non-linear training dynamics. Experiments demonstrate that SimuDy significantly outperforms previous approaches when handling non-linear training dynamics, and for the first time, most training samples can be reconstructed from a trained ResNet’s parameters.

Optimization · Non-Convex

Ryuichi Kanoh, Mahito Sugiyama

Linear Mode Connectivity (LMC) refers to the phenomenon that performance remains consistent for linearly interpolated models in the parameter space. For independently optimized model pairs from different random initializations, achieving LMC is considered crucial for understanding the stable success of the non-convex optimization in modern machine learning models and for facilitating practical parameter-based operations such as model merging. While LMC has been achieved for neural networks by considering the permutation invariance of neurons in each hidden layer, its attainment for other models remains an open question. In this paper, we first achieve LMC for soft tree ensembles, which are tree-based differentiable models extensively used in practice. We show the necessity of incorporating two invariances: subtree flip invariance and splitting order invariance, which do not exist in neural networks but are inherent to tree architectures, in addition to permutation invariance of trees. Moreover, we demonstrate that it is even possible to exclude such additional invariances while keeping LMC by designing decision list-based tree architectures, where such invariances do not exist by definition. Our findings indicate the significance of accounting for architecture-specific invariances in achieving LMC.

Deep Learning · Generative Models and Autoencoders

Ségolène Martin, Anne Gagneux, Paul Hagemann, Gabriele Steidl

In this paper, we introduce Plug-and-Play (PnP) Flow Matching, an algorithm for solving imaging inverse problems. PnP methods leverage the strength of pre-trained denoisers, often deep neural networks, by integrating them in optimization schemes. While they achieve state-of-the-art performance on various inverse problems in imaging, PnP approaches face inherent limitations on more generative tasks like inpainting. On the other hand, generative models such as Flow Matching pushed the boundary in image sampling yet lack a clear method for efficient use in image restoration. We propose to combine the PnP framework with Flow Matching (FM) by defining a time-dependent denoiser using a pre-trained FM model. Our algorithm alternates between gradient descent steps on the data-fidelity term, reprojections onto the learned FM path, and denoising. Notably, our method is computationally efficient and memory-friendly, as it avoids backpropagation through ODEs and trace computations. We evaluate its performance on denoising, super-resolution, deblurring, and inpainting tasks, demonstrating superior results compared to existing PnP algorithms and Flow Matching based state-of-the-art methods. Code available at https://github.com/annegnx/PnP-Flow.

Mert Pilanci

In this paper, we introduce a novel analysis of neural networks based on geometric (Clifford) algebra and convex optimization. We show that optimal weights of deep ReLU neural networks are given by the wedge product of training samples when trained with standard regularized loss. Furthermore, the training problem reduces to convex optimization over wedge product features, which encode the geometric structure of the training dataset. This structure is given in terms of signed volumes of triangles and parallelotopes generated by data vectors. The convex problem finds a small subset of samples via $\ell_1$ regularization to discover only relevant wedge product features. Our analysis provides a novel perspective on the inner workings of deep neural networks and sheds light on the role of the hidden layers.

Deep Learning · Theory

Levi Rauchwerger, Stefanie Jegelka, Ron Levie

We analyze the universality and generalization of graph neural networks (GNNs) on attributed graphs, i.e., with node attributes. To this end, we propose pseudometrics over the space of all attributed graphs that describe the fine-grained expressivity of GNNs. Namely, GNNs are both Lipschitz continuous with respect to our pseudometrics and can separate attributed graphs that are distant in the metric. Moreover, we prove that the space of all attributed graphs is relatively compact with respect to our metrics. Based on these properties, we prove a universal approximation theorem for GNNs and generalization bounds for GNNs on any data distribution of attributed graphs. The proposed metrics compute the similarity between the structures of attributed graphs via a hierarchical optimal transport between computation trees. Our work extends and unites previous approaches which either derived theory only for graphs with no attributes, derived compact metrics under which GNNs are continuous but without separation power, or derived metrics under which GNNs are continuous and separate points but the space of graphs is not relatively compact, which prevents universal approximation and generalization analysis.

Theory · Optimization

Steffen Schotthöfer, Emanuele Zangrando, Gianluca Ceruti, Francesco Tudisco, Jonas Kusch

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 bothaccuracy and computational efficiency

Deep Learning · Everything Else

Chris Kolb, Tobias Weber, Bernd Bischl, David Rügamer

Sparse regularization techniques are well-established in machine learning, yet their application in neural networks remains challenging due to the non-differentiability of penalties like the $L_1$ norm, which is incompatible with stochastic gradient descent. A promising alternative is shallow weight factorization, where weights are decomposed into two factors, allowing for smooth optimization of $L_1$-penalized neural networks by adding differentiable $L_2$ regularization to the factors. In this work, we introduce deep weight factorization, extending previous shallow approaches to more than two factors. We theoretically establish equivalence of our deep factorization with non-convex sparse regularization and analyze its impact on training dynamics and optimization. Due to the limitations posed by standard training practices, we propose a tailored initialization scheme and identify important learning rate requirements necessary for training factorized networks.We demonstrate the effectiveness of our deep weight factorization through experiments on various architectures and datasets, consistently outperforming its shallow counterpart and widely used pruning methods.

Optimization · Large Scale, Parallel and Distributed

Pedro Valdeira, Shiqiang Wang, Yuejie Chi

Vertical federated learning trains models from feature-partitioned datasets across multiple clients, who collaborate without sharing their local data. Standard approaches assume that all feature partitions are available during both training and inference. Yet, in practice, this assumption rarely holds, as for many samples only a subset of the clients observe their partition. However, not utilizing incomplete samples during training harms generalization, and not supporting them during inference limits the utility of the model. Moreover, if any client leaves the federation after training, its partition becomes unavailable, rendering the learned model unusable. Missing feature blocks are therefore a key challenge limiting the applicability of vertical federated learning in real-world scenarios. To address this, we propose LASER-VFL, a vertical federated learning method for efficient training and inference of split neural network-based models that is capable of handling arbitrary sets of partitions. Our approach is simple yet effective, relying on the sharing of model parameters and on task-sampling to train a family of predictors. We show that LASER-VFL achieves a $\mathcal{O}({1}/{\sqrt{T}})$ convergence rate for nonconvex objectives and, under the Polyak-Łojasiewicz inequality, it achieves linear convergence to a neighborhood of the optimum. Numerical experiments show improved performance of LASER-VFL over the baselines. Remarkably, this is the case even in the absence of missing features. For example, for CIFAR-100, we see an improvement in accuracy of $19.3$\% when each of four feature blocks is observed with a probability of 0.5 and of $9.5$\% when all features are observed. The code for this work is available at https://github.com/Valdeira/LASER-VFL.

Deep Learning · Everything Else

Jie Ren, Pin-Yu Chen, Ren Wang

Understanding the loss landscapes of neural networks (NNs) is critical for optimizing model performance. Previous research has identified the phenomenon of mode connectivity on curves, where two well-trained NNs can be connected by a continuous path in parameter space where the path maintains nearly constant loss. In this work, we extend the concept of mode connectivity to explore connectivity on surfaces, significantly broadening its applicability and unlocking new opportunities. While initial attempts to connect models via linear surfaces in parameter space were unsuccessful, we propose a novel optimization technique that consistently discovers Bézier surfaces with low-loss and high-accuracy connecting multiple NNs in a nonlinear manner. We further demonstrate that even without optimization, mode connectivity exists in certain cases of Bézier surfaces, where the models are carefully selected and combined linearly. This approach provides a deeper and more comprehensive understanding of the loss landscape and offers a novel way to identify models with enhanced performance for model averaging and output ensembling. We demonstrate the effectiveness of our method on CIFAR-10, CIFAR-100, and Tiny-ImageNet datasets using VGG16, ResNet18, and ViT architectures.

Deep Learning · Attention Mechanisms

Quoc-Vinh Lai-Dang, Taemin Kang, Seungah Son

Balancing high performance with interpretability in increasingly powerful Transformer-based models remains a challenge. While mechanistic interpretability aims to specify neural network computations in explicit, pseudocode-like formats, existing methods often involve laborious manual analysis or struggle to fully elucidate learned internal algorithms. Recent efforts to build intrinsically interpretable models have introduced considerable expressivity and optimization challenges. This work introduces Adaptive Transformer Programs, an enhanced framework building upon RASP language and Transformer Programs to create more robust and interpretable models. The proposed method increases expressivity by redesigning two primary attention modules to improve categorical and numerical reasoning capabilities. To overcome optimization hurdles, we introduce a novel reparameterization scheme that enhances the exploration-exploitation trade-off during training. We validate our approach through extensive experiments on diverse tasks, including in-context learning, algorithmic problems (e.g., sorting and Dyck languages), and NLP benchmarks such as named entity recognition and text classification. Results demonstrate that Adaptive Transformer Programs substantially narrow the performance gap between black-box Transformers and interpretable models, enhancing transparency. This work advances the development of high-performing, transparent AI systems for critical applications, addressing crucial ethical concerns in AI development.

Deep Learning · Generative Models and Autoencoders

Maosheng Yang

Given two boundary distributions, the \emph{Schrödinger Bridge} (SB) problem seeks the “most likely” random evolution between them with respect to a reference process. It has revealed rich connections to recent machine learning methods for generative modeling and distribution matching. While these methods perform well in Euclidean domains, they are not directly applicable to topological domains such as graphs and simplicial complexes, which are crucial for data defined over network entities, such as node signals and edge flows. In this work, we propose the \emph{Topological Schrödinger Bridge problem} ($\mathcal{T}$SBP) for matching signal distributions on a topological domain. We set the reference process to follow some linear tractable \emph{topology-aware} stochastic dynamics such as topological heat diffusion. For the case of Gaussian boundary distributions, we derive a \emph{closed-form} topological SB ($\mathcal{T}$SB) in terms of its time-marginal and stochastic differential. In the general case, leveraging the well-known result, we show that the optimal process follows the forward-backward topological dynamics governed by some unknowns. Building on these results, we develop $\mathcal{T}$SB-based models for matching topological signals by parameterizing the unknowns in the optimal process as \emph{(topological) neural networks} and learning them through \emph{likelihood training}. We validate the theoretical results and demonstrate the practical applications of $\mathcal{T}$SB-based models on both synthetic and real-world networks, emphasizing the role of topology. Additionally, we discuss the connections of $\mathcal{T}$SB-based models to other emerging models, and outline future directions for topological signal matching.

Optimization · Learning for Optimization

Fu Luo, Xi Lin, Yaoxin Wu, Zhenkun Wang, Tong Xialiang, Mingxuan Yuan, Qingfu Zhang

Neural Combinatorial Optimization (NCO) methods have exhibited promising performance in solving Vehicle Routing Problems (VRPs). However, most NCO methods rely on the conventional self-attention mechanism that induces excessive computational complexity, thereby struggling to contend with large-scale VRPs and hindering their practical applicability. In this paper, we propose a lightweight cross-attention mechanism with linear complexity, by which a Transformer network is developed to learn efficient and favorable solutions for large-scale VRPs. We also propose a Self-Improved Training (SIT) algorithm that enables direct model training on large-scale VRP instances, bypassing extensive computational overhead for attaining labels. By iterating solution reconstruction, the Transformer network itself can generate improved partial solutions as pseudo-labels to guide the model training. Experimental results on the Travelling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) with up to 100K nodes indicate that our method consistently achieves superior performance for synthetic and real-world benchmarks, significantly boosting the scalability of NCO methods.

Deep Learning · Large Language Models

Pengwei Tang, Xiaolin Hu, Yong Liu

Prompt Tuning (PT) enables the adaptation of Pre-trained Large Language Models (PLMs) to downstream tasks by optimizing a small amount of soft virtual tokens, which are prepended to the input token embeddings. Recently, Decomposed Prompt Tuning (DePT) has demonstrated superior adaptation capabilities by decomposing the soft prompt into a shorter soft prompt and a pair of low-rank matrices. The product of the pair of low-rank matrices is added to the input token embeddings to offset them. Additionally, DePT achieves faster inference compared to PT due to the shorter soft prompt. However, in this paper, we find that the position-based token embedding offsets of DePT restricts its ability to generalize across diverse model inputs, and that the shared embedding offsets across many token embeddings result in sub-optimization. To tackle these issues, we introduce \textbf{A}daptive \textbf{De}composed \textbf{P}rompt \textbf{T}uning (ADePT), which is composed of a short soft prompt and a shallow token-shared feed-forward neural network. ADePT utilizes the token-shared feed-forward neural network to learn the embedding offsets for each token, enabling adaptive embedding offsets that vary according to the model input and better optimization of token embedding offsets. This enables ADePT to achieve superior adaptation performance without requiring more inference time or additional trainable parameters compared to vanilla PT and its variants. In comprehensive experiments across 23 natural language processing tasks and 4 typical PLMs of different scales, ADePT consistently surpasses the leading parameter-efficient fine-tuning methods, and even outperforms the full fine-tuning in certain scenarios. We also provide a theoretical analysis towards ADePT. Code is available at https://github.com/HungerPWAY/ADePT.

Deep Learning · Other Representation Learning

Can Pouliquen, Mathurin Massias, Titouan Vayer

Estimating matrices in the symmetric positive-definite (SPD) cone is of interest for many applications ranging from computer vision to graph learning. While there exist various convex optimization-based estimators, they remain limited in expressivity due to their model-based approach. The success of deep learning motivates the use of learning-based approaches to estimate SPD matrices with neural networks in a data-driven fashion. However, designing effective neural architectures for SPD learning is challenging, particularly when the task requiresadditional structural constraints, such as element-wise sparsity. Current approaches either do not ensure that the output meets all desired properties or lack expressivity. In this paper, we introduce SpodNet, a novel and generic learning module that guarantees SPD outputs and supports additional structural constraints. Notably, it solves the challenging task of learning jointly SPD andsparse matrices. Our experiments illustrate the versatility and relevance of SpodNet layers for such applications.