Related papers: Relocation of compact sets in $\mathbb{R}^n$ by di…
Kernel-based non-linear dimensionality reduction methods, such as Local Linear Embedding (LLE) and Laplacian Eigenmaps, rely heavily upon pairwise distances or similarity scores, with which one can construct and study a weighted graph…
We improve recently published results about resources of Restricted Boltzmann Machines (RBM) and Deep Belief Networks (DBN) required to make them Universal Approximators. We show that any distribution p on the set of binary vectors of…
Deep neural networks (DNNs) are powerful tools for compressing and distilling information. Their scale and complexity, often involving billions of inter-dependent parameters, render direct microscopic analysis difficult. Under such…
Recently, versions of neural networks with infinite-dimensional affine operators inside the computational units (``neural operator'' networks) have been applied to learn solutions to differential equations. To enable practical computations,…
Although Convolutional Neural Networks (CNNs) have achieved promising results in image classification, they still are vulnerable to affine transformations including rotation, translation, flip and shuffle. The drawback motivates us to…
This work proposes an algorithm for explicitly constructing a pair of neural networks that linearize and reconstruct an embedded submanifold, from finite samples of this manifold. Our such-generated neural networks, called Flattening…
Transport through generalized trees is considered. Trees contain the simple nodes and supernodes, either well-structured regular subgraphs or those with many triangles. We observe a superdiffusion for the highly connected nodes while it is…
Despite the superior performance of deep learning in many applications, challenges remain in the area of regression on function spaces. In particular, neural networks are unable to encode function inputs compactly as each node encodes just…
Deep neural networks, particularly those employing Rectified Linear Units (ReLU), are often perceived as complex, high-dimensional, non-linear systems. This complexity poses a significant challenge to understanding their internal learning…
We propose a variation of the self organizing map algorithm by considering the random placement of neurons on a two-dimensional manifold, following a blue noise distribution from which various topologies can be derived. These topologies…
Deep metric learning maps visually similar images onto nearby locations and visually dissimilar images apart from each other in an embedding manifold. The learning process is mainly based on the supplied image negative and positive training…
This paper presents a novel machine-hearing system that exploits deep neural networks (DNNs) and head movements for robust binaural localisation of multiple sources in reverberant environments. DNNs are used to learn the relationship…
Under the data manifold hypothesis, high-dimensional data are concentrated near a low-dimensional manifold. We study the problem of Riemannian optimization over such manifolds when they are given only implicitly through the data…
This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free…
This paper proves an abstract theorem addressing in a unified manner two important problems in function approximation: avoiding curse of dimensionality and estimating the degree of approximation for out-of-sample extension in manifold…
Localization of light is the photon analog of electron localization in disordered lattices for whose discovery Anderson received the Nobel prize in 1977. The question about its existence in open three-dimensional materials has eluded an…
We present Latent Diffeomorphic Dynamic Mode Decomposition (LDDMD), a new data reduction approach for the analysis of non-linear systems that combines the interpretability of Dynamic Mode Decomposition (DMD) with the predictive power of…
We show that there is an infinite group of special automorphisms of the deformed group of diffeomorphisms, which describes parallel transports in Riemannian spaces of any variable curvature. Generators of translations of such group contain…
A scheme is presented for accurately propagating the gravitational field constraints in finite difference implementations of numerical relativity. The method is based on similar techniques used in astrophysical magnetohydrodynamics and…
Convolutional neural networks are ubiquitous in Machine Learning applications for solving a variety of problems. They however can not be used in their native form when the domain of the data is commonly encountered manifolds such as the…