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.

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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


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