相关论文: A Note on the Representation Power of GHHs

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…

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…

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.…

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…

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…

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…

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…

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…

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…

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…

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…

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…

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…

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…

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…

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…

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…

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…

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…

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,…