相关论文: Spherical Analysis of Learning Nonlinear Functiona…

Semantic segmentation for spherical data is a challenging problem in machine learning since conventional planar approaches require projecting the spherical image to the Euclidean plane. Representing the signal on a fundamentally different…

Recent years have witnessed a hot wave of deep neural networks in various domains; however, it is not yet well understood theoretically. A theoretical characterization of deep neural networks should point out their approximation ability and…

The success of Neural networks in providing miraculous results when applied to a wide variety of tasks is astonishing. Insight in the working can be obtained by studying the universal approximation property of neural networks. It is proved…

We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces, which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are…

We construct spherical wavelets based on approximate identities that are directional, i.e. not rotation-invariant, and have an adaptive angular selectivity. The problem of how to find a proper representation of distinct kinds of details of…

Unsupervised semantic segmentation is a long-standing challenge in computer vision with great significance. Spectral clustering is a theoretically grounded solution to it where the spectral embeddings for pixels are computed to construct…

Neural networks have become an increasingly popular tool for solving many real-world problems. They are a general framework for differentiable optimization which includes many other machine learning approaches as special cases. In this…

The foundations of deep learning are supported by the seemingly opposing perspectives of approximation or learning theory. The former advocates for large/expressive models that need not generalize, while the latter considers classes that…

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…

The classical development of neural networks has been primarily for mappings between a finite-dimensional Euclidean space and a set of classes, or between two finite-dimensional Euclidean spaces. The purpose of this work is to generalize…

Convolutional Neural Networks are very efficient at processing signals defined on a discrete Euclidean space (such as images). However, as they can not be used on signals defined on an arbitrary graph, other models have emerged, aiming to…

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning…

We contribute to a better understanding of the class of functions that can be represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical…

Omnidirectional images and spherical representations of $3D$ shapes cannot be processed with conventional 2D convolutional neural networks (CNNs) as the unwrapping leads to large distortion. Using fast implementations of spherical and…

We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise…

Deep Neural Networks (DNNs) are universal function approximators providing state-of- the-art solutions on wide range of applications. Common perceptual tasks such as speech recognition, image classification, and object tracking are now…

Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by…

Models of neural networks have proven their utility in the development of learning algorithms in computer science and in the theoretical study of brain dynamics in computational neuroscience. We propose in this paper a spatial neural…

Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to…

Function approximation from input and output data pairs constitutes a fundamental problem in supervised learning. Deep neural networks are currently the most popular method for learning to mimic the input-output relationship of a general…