Related papers: Non-Euclidean Universal Approximation
Various topological techniques and tools have been applied to neural networks in terms of network complexity, explainability, and performance. One fundamental assumption of this line of research is the existence of a global (Euclidean)…
Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging,…
Deep neural networks (DNN) with a huge number of adjustable parameters remain largely black boxes. To shed light on the hidden layers of DNN, we study supervised learning by a DNN of width $N$ and depth $L$ consisting of $NL$ perceptrons…
This paper investigates the relationship between convolutional neural network (CNN) topology and image recognition performance through a comparative study of the VGG, ResNet, and GoogLeNet architectural families. Utilizing a unified…
This work develops the global equations of neural networks through stacked piecewise manifolds, fixed-point theory, and boundary-conditioned iteration. Once fixed coordinates and operators are removed, a neural network appears as a…
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…
While convolutional neural networks (CNNs) have recently made great strides in supervised classification of data structured on a grid (e.g. images composed of pixel grids), in several interesting datasets, the relations between features can…
One of the most influential results in neural network theory is the universal approximation theorem [1, 2, 3] which states that continuous functions can be approximated to within arbitrary accuracy by single-hidden-layer feedforward neural…
Three important properties of a classification machinery are: (i) the system preserves the core information of the input data; (ii) the training examples convey information about unseen data; and (iii) the system is able to treat…
We present, for the first time, a novel deep neural network architecture called \dcn with a dual-path connection between the input image and output class label for mammogram image processing. This architecture is built upon U-Net, which…
We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with…
In the context of classification problems, Deep Learning (DL) approaches represent state of art. Many DL approaches are based on variations of standard multi-layer feed-forward neural networks. These are also referred to as deep networks.…
This work is concerned with a representation of shapes that disentangles fine, local and possibly repeating geometry, from global, coarse structures. Achieving such disentanglement leads to two unrelated advantages: i) a significant…
We show that deep neural networks trained across diverse tasks exhibit remarkably similar low-dimensional parametric subspaces. We provide the first large-scale empirical evidence that demonstrates that neural networks systematically…
The k-Nearest Neighbors (kNN) classifier is a fundamental non-parametric machine learning algorithm. However, it is well known that it suffers from the curse of dimensionality, which is why in practice one often applies a kNN classifier on…
Advancements in modern science have led to the increasing availability of non-Euclidean data in metric spaces. This paper addresses the challenge of modeling relationships between non-Euclidean responses and multivariate Euclidean…
In this paper we consider a new class of RBF (Radial Basis Function) neural networks, in which smoothing factors are replaced with shifts. We prove under certain conditions on the activation function that these networks are capable of…
In the era of foundation models and Large Language Models (LLMs), Euclidean space has been the de facto geometric setting for machine learning architectures. However, recent literature has demonstrated that this choice comes with…
Recent advances in deep learning from probability distributions successfully achieve classification or regression from distribution samples, thus invariant under permutation of the samples. The first contribution of the paper is to extend…
Deep Convolutional Neural Networks (CNNs) have significantly advanced deep learning, driving breakthroughs in computer vision, natural language processing, medical diagnosis, object detection, and speech recognition. Architectural…