Related papers: Sparsity and non-Euclidean embeddings

We study graph spanners for point-set in the high-dimensional Euclidean space. On the one hand, we prove that spanners with stretch <\sqrt{2} and subquadratic size are not possible, even if we add Steiner points. On the other hand, if we…

We present an instructive example of using Banach spaces of solutions to (linear, generally, non-scalar) elliptic operator $A$ to investigate variational inverse problems related to neural networks and/or to regularization of solutions to…

We construct differential algebras in which spaces of (one-dimensional) periodic ultradistributions are embedded. By proving a Schwartz impossibility type result, we show that our embeddings are optimal in the sense of being consistent with…

In 2010, Olson \& Robinson [Transactions of the American Mathematical Society, 362(1), 145-168] introduced the notion of an almost homogeneous metric space and showed that if $X$ is a subset of a Hilbert space such that $X-X$ is almost…

This article extends the known restricted isometric projection of sparse datasets in Euclidean spaces $\mathbb{R}^N$ down into low-dimensional subspaces $\mathbb{R}^k, k \ll N,$ to the case of low-dimensional varieties $\mathcal{M} \subset…

We investigate different possiblities of subspaces of the space $\ell_{\infty}$ in terms of whether the subspaces are polyhedral or not. We further study finite-dimensional subspaces of $\ell_{\infty}$ which are of the form $\ell_\infty^n$…

Goemans showed that any $n$ points $x_1, \dotsc x_n$ in $d$-dimensions satisfying $\ell_2^2$ triangle inequalities can be embedded into $\ell_{1}$, with worst-case distortion at most $\sqrt{d}$. We extend this to the case when the points…

We give two different and simple constructions for dimensionality reduction in $\ell_2$ via linear mappings that are sparse: only an $O(\varepsilon)$-fraction of entries in each column of our embedding matrices are non-zero to achieve…

In this paper we study constrained subspace approximation problem. Given a set of $n$ points $\{a_1,\ldots,a_n\}$ in $\mathbb{R}^d$, the goal of the {\em subspace approximation} problem is to find a $k$ dimensional subspace that best…

It is shown that every Banach space either contains $\ell ^1$ or it has an infinite dimensional closed subspace which is a quotient of a H.I. Banach space.Further on, $L^p(\lambda )$, $1<p<\infty $, is a quotient of a H.I Banach space.

We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into $c_0$ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the…

Let $\M$ be a semi-finite von Neumann algebra equipped with a faithful normal trace $\tau$. We study the subspace structures of non-commutative Lorentz spaces $L_{p,q}(\M, \tau)$, extending results of Carothers and Dilworth to the…

Let $X$ be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces $E$ of $X$-valued sequences, the sets $E-\bigcup _{q\in\Gamma}\ell_{q}(X)$, where $\Gamma$ is any subset of $(0,\infty]$, and $E-c_{0}(X)$ contain…

In this note, we obtain new obstructions to symplectic embeddings of a product of disks (a polydisk) into a 4-dimensional ball. The polydisk P(r,s) is the product of the disk of area r with the disk of area s. The ball of capacity a,…

The recent work of \cite{CRT,DonohoPol} rigorously proved (in a large dimensional and statistical context) that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of…

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the…

This article contains two rigidity type results for $\mathrm{SL}(n,\mathbb{Z})$ for large $n$ that share the same proof. Firstly, we prove that for every $p \in [1,\infty]$ different from $2$, the noncommutative $L^p$-space associated with…

Group sparsity combines the underlying sparsity and group structure of the data in problems. We develop a proximally linearized algorithm InISSAPL for the non-Lipschitz group sparse $\ell_{p,q}$-$\ell_r$ optimization problem.

We consider low-distortion embeddings for subspaces under \emph{entrywise nonlinear transformations}. In particular we seek embeddings that preserve the norm of all vectors in a space $S = \{y: y = f(x)\text{ for }x \in Z\}$, where $Z$ is a…

Given a separable Banach space $E$, we construct an extremely non-complex Banach space (i.e. a space satisfying that $\|Id + T^2\|=1+\|T^2\|$ for every bounded linear operator $T$ on it) whose dual contains $E^*$ as an $L$-summand. We also…