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We derive bounds on the error, in high-order Sobolev norms, incurred in the approximation of Sobolev-regular as well as analytic functions by neural networks with the hyperbolic tangent activation function. These bounds provide explicit…

Numerical Analysis · Mathematics 2021-12-09 Tim De Ryck , Samuel Lanthaler , Siddhartha Mishra

Recently, deep learning has achieved huge successes in many important applications. In our previous studies, we proposed quadratic/second-order neurons and deep quadratic neural networks. In a quadratic neuron, the inner product of a vector…

Machine Learning · Computer Science 2019-08-29 Fenglei Fan , Jinjun Xiong , Ge Wang

In this paper, we present generalization bounds for the unsupervised risk in the Deep Contrastive Representation Learning framework, which employs deep neural networks as representation functions. We approach this problem from two angles.…

Machine Learning · Statistics 2024-12-20 Nong Minh Hieu , Antoine Ledent , Yunwen Lei , Cheng Yeaw Ku

We show the universality of depth-2 group convolutional neural networks (GCNNs) in a unified and constructive manner based on the ridgelet theory. Despite widespread use in applications, the approximation property of (G)CNNs has not been…

Machine Learning · Computer Science 2022-10-14 Sho Sonoda , Isao Ishikawa , Masahiro Ikeda

The results presented in this paper are refinements of some results presented in a previous paper. Three such refined results are presented. The first one relaxes one of the basic hypotheses assumed in the previous paper, and thus extends…

Complex Variables · Mathematics 2015-05-06 Jorge L. deLyra

Representation theory is shown to be incomplete in terms of enumerating all integrable limits of quantum systems. As a consequence, one can find exactly solvable Hamiltonians which have apparently strongly broken symmetry. The number of…

Nuclear Theory · Physics 2009-10-30 Dimitri Kusnezov

We analyze the universal approximation constraints of narrow Residual Neural Networks (ResNets) both theoretically and numerically. For deep neural networks without input space augmentation, a central constraint is the inability to…

Dynamical Systems · Mathematics 2026-03-31 Christian Kuehn , Sara-Viola Kuntz , Tobias Wöhrer

In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as the activation function and continuous piecewise linear (CPWL) functions, especially CPWL functions from the…

Numerical Analysis · Mathematics 2020-06-02 Juncai He , Lin Li , Jinchao Xu , Chunyue Zheng

The stunning empirical successes of neural networks currently lack rigorous theoretical explanation. What form would such an explanation take, in the face of existing complexity-theoretic lower bounds? A first step might be to show that…

Machine Learning · Computer Science 2017-07-18 Le Song , Santosh Vempala , John Wilmes , Bo Xie

This invention addresses fixed-point representations of convolutional neural networks (CNN) in integrated circuits. When quantizing a CNN for a practical implementation there is a trade-off between the precision used for operations between…

Neural and Evolutionary Computing · Computer Science 2018-07-27 Mo'taz Al-Hami , Marcin Pietron , Rishi Kumar , Raul A. Casas , Samer L. Hijazi , Chris Rowen

We develop a corrective mechanism for neural network approximation: the total available non-linear units are divided into multiple groups and the first group approximates the function under consideration, the second group approximates the…

Machine Learning · Computer Science 2020-06-23 Guy Bresler , Dheeraj Nagaraj

Learning representations of sets of nodes in a graph is crucial for applications ranging from node-role discovery to link prediction and molecule classification. Graph Neural Networks (GNNs) have achieved great success in graph…

Machine Learning · Computer Science 2020-10-30 Pan Li , Yanbang Wang , Hongwei Wang , Jure Leskovec

The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for geometric graphs embedded in…

Machine Learning · Computer Science 2024-03-05 Chaitanya K. Joshi , Cristian Bodnar , Simon V. Mathis , Taco Cohen , Pietro Liò

We propose semi-random features for nonlinear function approximation. The flexibility of semi-random feature lies between the fully adjustable units in deep learning and the random features used in kernel methods. For one hidden layer…

Machine Learning · Computer Science 2017-11-22 Kenji Kawaguchi , Bo Xie , Vikas Verma , Le Song

We study the problem of approximating compactly-supported integrable functions while implementing their support set using feedforward neural networks. Our first main result transcribes this "structured" approximation problem into a…

Machine Learning · Computer Science 2022-08-02 Anastasis Kratsios , Behnoosh Zamanlooy

Graph Neural Networks (GNN) are inherently limited in their expressive power. Recent seminal works (Xu et al., 2019; Morris et al., 2019b) introduced the Weisfeiler-Lehman (WL) hierarchy as a measure of expressive power. Although this…

Machine Learning · Computer Science 2023-06-06 Omri Puny , Derek Lim , Bobak T. Kiani , Haggai Maron , Yaron Lipman

We provide a counterexample to the minimal unimodal conjecture for polynomial neural networks (PNNs) with power activation functions. Fixing the input and output widths, the conjecture states that any minimal filling architecture has…

Machine Learning · Computer Science 2026-05-12 Kevin Dao , Jose Israel Rodriguez

Message passing Graph Neural Networks (GNNs) are known to be limited in expressive power by the 1-WL color-refinement test for graph isomorphism. Other more expressive models either are computationally expensive or need preprocessing to…

Machine Learning · Computer Science 2024-02-01 Mohammed Haroon Dupty , Yanfei Dong , Wee Sun Lee

Graph neural networks (GNNs) have led to major breakthroughs in a variety of domains such as drug discovery, social network analysis, and travel time estimation. However, they lack interpretability which hinders human trust and thereby…

Machine Learning · Computer Science 2023-12-05 Jonas Jürß , Lucie Charlotte Magister , Pietro Barbiero , Pietro Liò , Nikola Simidjievski

An approach to construct explicit integral representations for two-layer ReLU networks is presented, which provides relatively simple representations for any multivariate polynomial. Quantitative bounds are provided for a particular,…

Machine Learning · Statistics 2026-05-13 Anthony Lee
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