相关论文: Neural Polytopes

Due to the over-parameterization nature, neural networks are a powerful tool for nonlinear function approximation. In order to achieve good generalization on unseen data, a suitable inductive bias is of great importance for neural networks.…

Despite their prevalence in neural networks we still lack a thorough theoretical characterization of ReLU layers. This paper aims to further our understanding of ReLU layers by studying how the activation function ReLU interacts with the…

We study the approximation properties of random ReLU features through their reproducing kernel Hilbert space (RKHS). We first prove a universality theorem for the RKHS induced by random features whose feature maps are of the form of nodes…

Although neural networks traditionally are typically used to approximate functions defined over $\mathbb{R}^n$, the successes of graph neural networks, point-cloud neural networks, and manifold deep learning among other methods have…

In a neural network with ReLU activations, the number of piecewise linear regions in the output can grow exponentially with depth. However, this is highly unlikely to happen when the initial parameters are sampled randomly, which therefore…

Neural networks are complex functions of both their inputs and parameters. Much prior work in deep learning theory analyzes the distribution of network outputs at a fixed a set of inputs (e.g. a training dataset) over random initializations…

Neural nets are known to be universal approximators. In particular, formal neurons implementing wavelets have been shown to build nets able to approximate any multidimensional task. Such very specialized formal neurons may be, however,…

While classic studies proved that wide networks allow universal approximation, recent research and successes of deep learning demonstrate the power of deep networks. Based on a symmetric consideration, we investigate if the design of…

We study the expressivity of rational neural networks (RationalNets) through the lens of algebraic geometry. We consider rational functions that arise from a given RationalNet to be tuples of fractions of homogeneous polynomials of fixed…

Successfully training deep neural networks often requires either batch normalization, appropriate weight initialization, both of which come with their own challenges. We propose an alternative, geometrically motivated method for training.…

Many tasks require mapping continuous input data (e.g. images) to discrete task outputs (e.g. class labels). Yet, how neural networks learn to perform such discrete computations on continuous data manifolds remains poorly understood. Here,…

Neural networks are popular and useful in many fields, but they have the problem of giving high confidence responses for examples that are away from the training data. This makes the neural networks very confident in their prediction while…

This study explores the number of neurons required for a Rectified Linear Unit (ReLU) neural network to approximate multivariate monomials. We establish an exponential lower bound on the complexity of any shallow network approximating the…

We discovered that the neural networks, especially the deep ReLU networks, demonstrate an `over-generalization' phenomenon. That is, the output values for the inputs that were not seen during training are mapped close to the output range…

We discuss the expressive power of neural networks which use the non-smooth ReLU activation function $\varrho(x) = \max\{0,x\}$ by analyzing the approximation theoretic properties of such networks. The existing results mainly fall into two…

In this paper, we have extended the well-established universal approximator theory to neural networks that use the unbounded ReLU activation function and a nonlinear softmax output layer. We have proved that a sufficiently large neural…

Two networks are equivalent if they produce the same output for any given input. In this paper, we study the possibility of transforming a deep neural network to another network with a different number of units or layers, which can be…

Motivated by problems in the neural networks setting, we study moduli spaces of double framed quiver representations and give both a linear algebra description and a representation theoretic description of these moduli spaces. We define a…

We derive upper bounds on the complexity of ReLU neural networks approximating the solution of a linear system given the matrix and the right-hand side. We focus on matrices which are symmetric positive definite and sparse, as they appear…

Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as ``reverse engineering," or ``parameter identifiability", has led to the natural question of parameter…