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Despite significant advances in the field of deep learning in ap-plications to various areas, an explanation of the learning pro-cess of neural network models remains an important open ques-tion. The purpose of this paper is a comprehensive…
We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of…
Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed…
We use a tensor unfolding technique to prove a new identifiability result for discrete bipartite graphical models, which have a bipartite graph between an observed and a latent layer. This model family includes popular models such as…
Multilevel methods are among the most efficient numerical methods for solving large-scale systems of equations that arise from discretized partial differential equations. Two-level convergence theory plays a fundamental role in the analysis…
We present several modifications to the previously proposed MSPP algorithm that can speed-up its execution considerably. The MSPP algorithm leverages a multiscale representation of the environment in $n$ dimensions. The information of the…
The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished function spaces on $\mathbb{R}^n$. The degree of compactness will be measured in terms of related entropy numbers. We are more…
The unordered configuration space of $n$ points on a graph $\Gamma,$ denoted here by $UC^n(\Gamma),$ can be viewed as the space of all configurations of $n$ unlabeled robots on a system of one-dimensional tracks, which is interpreted as a…
Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\geq 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. Our main result asserts that every…
We consider feed-forward neural networks with one hidden layer, tree architecture and a fixed hidden-to-output Boolean function. Focusing on the saturation limit of the storage problem the influence of replica symmetry breaking on the…
We show that both Lusternik-Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be…
In this paper, we introduce relative LS category of a map and study some of its properties. Then we introduce `higher topological complexity' of a map, a homotopy invariant. We give a cohomological lower bound and compare it with previously…
In a sequence of seminal results in the 80's, Kaltofen showed that the complexity class VP is closed under taking factors. A natural question in this context is to understand if other natural classes of multivariate polynomials, for…
The article concerns low-rank approximation of matrices generated by sampling a smooth function of two $m$-dimensional variables. We identify several misconceptions surrounding a claim that, for a specific class of analytic functions, such…
For $n\in \mathbb{N}$, let $Y_n$ denote the linear span of the first $n+1$ levels of the Haar system in a Haar system Hardy space $Y$ (this class contains all separable rearrangement-invariant function spaces and also related spaces such as…
Given a rational dominant map $\phi: Y \dashrightarrow X$ between two generic hypersurfaces $Y,X \subset \mathbb{P}^n$ of dimension $\ge 3$, we prove (under an addition assumption on $\phi$) a "Noether-Fano type" inequality $m_Y \ge m_X$…
The relationship among three correlated variables could be very sophisticated, as a result, we may not be able to find their hidden causality and model their relationship explicitly. However, we still can make our best guess for possible…
This research is motivated by studying image processing algorithms through a topological lens. The images we focus on here are those that have been segmented by digital Jordan curves as a means of image compression. The algorithms of…
Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and…
Multidimensional network data can have different levels of complexity, as nodes may be characterized by heterogeneous individual-specific features, which may vary across the networks. This paper introduces a class of models for…