# Lossless linear analog compression

### Authors

Giovanni Alberti, Helmut Bölcskei, Camillo De Lellis, Günther Koliander, and Erwin Riegler### Reference

*Proc. of IEEE International Symposium on Information Theory (ISIT)*, Barcelona, Spain, pp. 2789-2793, July 2016.

[BibTeX, LaTeX, and HTML Reference]

### Abstract

We establish the fundamental limits of lossless linear analog compression by considering the recovery of random vectors x in R^m from the noiseless linear measurements y = Ax with measurement matrix A in R^(nxm). Specifically, for a random vector x in R^m of arbitrary distribution we show that x can be recovered with zero error probability from n > inf dimMB(U) linear measurements, where dimMB() denotes the lower modified Minkowski dimension and the infimum is over all sets U in R^m with P[x in U] = 1. This achievability statement holds for Lebesgue almost all measurement matrices A. We then show that s-rectifiable random vectors—a stochastic generalization of s-sparse vectors—can be recovered with zero error probability from n > s linear measurements. From classical compressed sensing theory we would expect n >= s to be necessary for successful recovery of x. Surprisingly, certain classes of s-rectifiable random vectors can be recovered from fewer than s measurements. Imposing an additional regularity condition on the distribution of s-rectifiable random vectors x, we do get the expected converse result of s measurements being necessary. The resulting class of random vectors appears to be new and will be referred to as s-analytic random vectors.### Keywords

Analog compression, lossless compression, box counting dimension, geometric measure theory, compressed sensing

Download this document:

Copyright Notice: © 2016 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.