Related papers: Small Width, Low Distortions: Quantized Random Emb…
Recently, many works have focused on the characterization of non-linear dimensionality reduction methods obtained by quantizing linear embeddings, e.g., to reach fast processing time, efficient data compression procedures, novel…
In this paper we show that for the purposes of dimensionality reduction certain class of structured random matrices behave similarly to random Gaussian matrices. This class includes several matrices for which matrix-vector multiply can be…
We consider the vector embedding problem. We are given a finite set of items, with the goal of assigning a representative vector to each one, possibly under some constraints (such as the collection of vectors being standardized, i.e.,…
Quantized compressive sensing (QCS) deals with the problem of coding compressive measurements of low-complexity signals with quantized, finite precision representations, i.e., a mandatory process involved in any practical sensing model.…
For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are…
This paper provides new error bounds on "consistent" reconstruction methods for signals observed from quantized random projections. Those signal estimation techniques guarantee a perfect matching between the available quantized data and a…
The Fr\'echet distance is a popular distance measure for curves which naturally lends itself to fundamental computational tasks, such as clustering, nearest-neighbor searching, and spherical range searching in the corresponding metric…
This note is concerned with deterministic constructions of $m \times N$ matrices satisfying a restricted isometry property from $\ell_2$ to $\ell_1$ on $s$-sparse vectors. Similarly to the standard ($\ell_2$ to $\ell_2$) restricted isometry…
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…
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…
Quantised random embeddings are an efficient dimensionality reduction technique which preserves the distances of low-complexity signals up to some controllable additive and multiplicative distortions. In this work, we instead focus on…
In this paper we study {\em terminal embeddings}, in which one is given a finite metric $(X,d_X)$ (or a graph $G=(V,E)$) and a subset $K \subseteq X$ of its points are designated as {\em terminals}. The objective is to embed the metric into…
We consider the task of lossy compression of high-dimensional vectors through quantization. We propose the approach that learns quantization parameters by minimizing the distortion of scalar products and squared distances between pairs of…
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,…
We study embedding a subset $K$ of the unit sphere to the Hamming cube $\{-1,+1\}^m$. We characterize the tradeoff between distortion and sample complexity $m$ in terms of the Gaussian width $\omega(K)$ of the set. For subspaces and several…
Modeling data as being sampled from a union of independent subspaces has been widely applied to a number of real world applications. However, dimensionality reduction approaches that theoretically preserve this independence assumption have…
In this paper, we investigate a trade-off between the number of radar observations (or measurements) and their resolution in the context of radar range estimation. To this end, we introduce a novel estimation scheme that can deal with…
We introduce average-distortion sketching for metric spaces. As in (worst-case) sketching, these algorithms compress points in a metric space while approximately recovering pairwise distances. The novelty is studying average-distortion: for…
This paper is concerned with the lossy compression of general random variables, specifically with rate-distortion theory and quantization of random variables taking values in general measurable spaces such as, e.g., manifolds and fractal…
We consider mappings satisfying an upper bound for the distortion of families of curves. We establish lower bounds for the distortion of distances under such mappings. As applications, we obtain theorems on the discreteness of the limit…