Lossless linear analog compression
AuthorsGiovanni Alberti, Helmut Bölcskei, Camillo De Lellis, Günther Koliander, and Erwin Riegler
ReferenceProc. of IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, pp. 2789-2793, July 2016.
AbstractWe 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.
KeywordsAnalog compression, lossless compression, box counting dimension, geometric measure theory, compressed sensing
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