Related papers: Low Dimensional Euclidean Volume Preserving Embedd…

An oblivious subspace embedding is a random $m\times n$ matrix $\Pi$ such that, for any $d$-dimensional subspace, with high probability $\Pi$ preserves the norms of all vectors in that subspace within a $1\pm\epsilon$ factor. In this work,…

Let (N,g) be a Riemannian manifold. For a compact, connected and oriented submanifold M of N. we define the space of volume preserving embeddings Emb_{\mu}(M,N) as the set of smooth embeddings f:M \rightarrow N such that f*\mu^{f}=\mu,…

We develop a new method to estimate the area, and more generally the intrinsic volumes, of a compact subset $X$ of $\mathbb{R}^d$ from a set $Y$ that is close in the Hausdorff distance. This estimator enjoys a linear rate of convergence as…

We introduce and study the notion of an outer bi-Lipschitz extension of a map between Euclidean spaces. The notion is a natural analogue of the notion of a Lipschitz extension of a Lipschitz map. We show that for every map $f$ there exists…

The study of (minimally) rigid graphs is motivated by numerous applications, mostly in robotics and bioinformatics. A major open problem concerns the number of embeddings of such graphs, up to rigid motions, in Euclidean space. We capture…

A point set $M$ in $m$-dimensional Euclidean space is called an integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on an $(m-1)$-dimensional hyperplane. We improve the linear lower…

Let $(\mathcal{X},\rho)$ be a metric space and $\lambda$ be a Borel measure on this space defined on the $\sigma$-algebra generated by open subsets of $\mathcal{X}$; this measure $\lambda$ defines volumes of Borel subsets of $\mathcal{X}$.…

Two themes associated with invariant measures on the matrix groups ${\rm SL}_N(\mathbb F)$, with $\mathbb F = \mathbb R, \mathbb C$ or $\mathbb H$, and their corresponding lattices parametrised by ${\rm SL}_N(\mathbb F)/{\rm SL}_N(\mathbb…

The Whitney embedding theorem gives an upper bound on the smallest embedding dimension of a manifold. If a data set lies on a manifold, a random projection into this reduced dimension will retain the manifold structure. Here we present an…

For a measure mu supported on a compact connected subset of a Euclidean space which satisfies a uniform d-dimensional decay of the volume of balls we show that the maximal edge in the minimum spanning tree of n indepndent samples from mu…

We study representations of data from an arbitrary metric space $\mathcal{X}$ in the space of univariate Gaussian mixtures with a transport metric (Delon and Desolneux 2020). We derive embedding guarantees for feature maps implemented by…

Suppose that we have the unit Euclidean ball in $\R^n$ and construct new bodies using three operations - linear transformations, closure in the radial metric and multiplicative summation defined by $\|x\|_{K+_0L} = \sqrt{\|x\|_K\|x\|_L}.$…

The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. It supports embedding a set of $n$ points in $\mathbb{R}^d$ into $m=O(\varepsilon^{-2} \lg n)$ dimensions while preserving all pairwise…

Let $Q_n$ be the cube of side length one centered at the origin in $\mathbb{R}^n$, and let $F$ be an affine $(n-d)$-dimensional subspace of $\mathbb{R}^n$ having distance to the origin less than or equal to $\frac 1 2$, where $0<d<n$. We…

In this note we present a construction which improves the best known bound on the minimal dispersion of large volume boxes in the unit cube. Let $d>1$. The dispersion of $T \subset [0,1]^d$ is defined as the supremum of the volume taken…

Random projections are random linear maps, sampled from appropriate distributions, that approx- imately preserve certain geometrical invariants so that the approximation improves as the dimension of the space grows. The well-known…

We consider a connected smooth $n$-dimensional manifold $M$ endowed with a volume form $\Omega$, and we show that an open subset $U$ of $R^n$ of Lebesgue measure $\Vol (U)$ embeds into $M$ by a smooth volume preserving embedding whenever…

We develop an algorithm that determines, for a given squarefree binary form $F$ with real coefficients, a smallest representative of its orbit under $\operatorname{SL}(2,\mathbb Z)$, either with respect to the Euclidean norm or with respect…

The Borsuk number $b(n)$ of $n$-dimensional Euclidean space $\mathbb{R}^n$ is the smallest integer such that any set $F \subset \mathbb{R}^n$ of unit diameter can be partitioned into $b(n)$ subsets of strictly smaller diameter. For $n=4$,…

We consider {\em monotone} embeddings of a finite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean…