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Undersampling the k-space during MR acquisitions saves time, however results in an ill-posed inversion problem, leading to an infinite set of images as possible solutions. Traditionally, this is tackled as a reconstruction problem by…
Randomized dimensionality reduction has been recognized as one of the fundamental techniques in handling high-dimensional data. Starting with the celebrated Johnson-Lindenstrauss Lemma, such reductions have been studied in depth for the…
The arrangement of network nodes in hyperbolic spaces has become a widely studied problem, motivated by numerous results suggesting the existence of hidden metric spaces behind the structure of complex networks. Although several methods…
Recently, Su and Cook proposed a dimension reduction technique called the inner envelope which can be substantially more efficient than the original envelope or existing dimension reduction techniques for multivariate regression. However,…
We consider the problem of embedding a subset of $\mathbb{R}^n$ into a low-dimensional Hamming cube in an almost isometric way. We construct a simple, data-oblivious, and computationally efficient map that achieves this task with high…
Rod-based structures are commonly used in practical applications in science and engineering. However, in many design, analysis, and manufacturing tasks, handling the rod-based structures in three dimensions directly is generally…
The field of compressed sensing has become a major tool in high-dimensional analysis, with the realization that vectors can be recovered from relatively very few linear measurements as long as the vectors lie in a low-dimensional structure,…
The projective space of order $n$ over a finite field $\F_q$ is a set of all subspaces of the vector space $\F_q^{n}$. In this work, we consider error-correcting codes in the projective space, focusing mainly on constant dimension codes. We…
Let $\mathcal{M}$ be a smooth submanifold of $\mathbb{R}^n$ equipped with the Euclidean (chordal) metric. This note considers the smallest dimension $m$ for which there exists a bi-Lipschitz function $f: \mathcal{M} \mapsto \mathbb{R}^m$…
A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based…
A local expansion is proposed for two-point distributions involving an ultraviolet regularization in a four-dimensional globally hyperbolic space-time. The regularization is described by an infinite number of functions which can be computed…
We show that every cross ratio preserving homeomorphism between boundaries of Hadamard manifolds extends to a continuous map, called circumcenter extension, provided that the manifolds satisfy certain visibility conditions. We show that…
High-dimensional big data appears in many research fields such as image recognition, biology and collaborative filtering. Often, the exploration of such data by classic algorithms is encountered with difficulties due to `curse of…
We give a practical random mapping that takes any set of documents represented as vectors in Euclidean space and then maps them to a sparse subset of the Hamming cube while retaining ordering of inter-vector inner products. Once represented…
Most state-of-the-art deep geometric learning single-view reconstruction approaches rely on encoder-decoder architectures that output either shape parametrizations or implicit representations. However, these representations rarely preserve…
In this paper, we explore a volume-based stable embedding of multi-dimensional signals based on Grassmann manifold, via Gaussian random measurement matrices. The Grassmann manifold is a topological space in which each point is a linear…
<ENGLISH> Consider a closed, smooth manifold M of nonpositive sectional curvature. Write p:UM-> M for the unit tangent bundle over M and let R_> denote the subset consisting of all vectors of higher rank. This subset is closed and invariant…
Improvements in technology lead to increasing availability of large data sets which makes the need for data reduction and informative subsamples ever more important. In this paper we construct $ D $-optimal subsampling designs for…
We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework…
We consider the task of building compact deep learning pipelines suitable for deployment on storage and power constrained mobile devices. We propose a unified framework to learn a broad family of structured parameter matrices that are…