Lossless analog compression

Authors

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

Reference

IEEE Transactions on Information Theory, Mar. 2018, submitted.

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Abstract

We establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary m-dimensional real random vectors x from the noiseless linear measurements y=Ax with n x m measurement matrix A. Our theory is inspired by the groundbreaking work of Wu and Verdú (2010) on almost lossless analog compression, but applies to the nonasymptotic, i.e., fixed-m case, and considers zero error probability. Specifically, our achievability result states that, for Lebesgue-almost all A, the random vector x can be recovered with zero error probability provided that n > K(x), where the description complexity K(x) is given by the infimum of the lower modified Minkowski dimensions over all support sets U of x (i.e., sets U such that x is in U with probability one). We then particularize this achievability result to the class of s-rectifiable random vectors as introduced in Koliander et al. (2016); these are random vectors of absolutely continuous distribution---with respect to the s-dimensional Hausdorff measure---supported on countable unions of s-dimensional differentiable submanifolds of the m-dimensional real coordinate space. Countable unions of differentiable submanifolds include essentially all signal models used in the compressed sensing literature such as the standard union of subspaces model underlying much of compressed sensing theory and spectrum-blind sampling, smooth manifolds, block-sparsity, and low-rank matrices as considered in the matrix completion problem. Specifically, we prove that, for Lebesgue-almost all A, s-rectifiable random vectors x can be recovered with zero error probability from n > s linear measurements. This threshold is, however, found not to be tight as exemplified by the construction of an s-rectifiable random vector that can be recovered with zero error probability from n < s linear measurements. This leads us to the introduction of the new class of s-analytic random vectors, which admit a strong converse in the sense of n >= s being necessary for recovery with probability of error smaller than one. The central conceptual tool in the development of our theory is geometric measure theory.

Keywords

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


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Copyright Notice: © 2018 G. Alberti, H. Bölcskei, C. De Lellis, G. Koliander, and E. Riegler.

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