We establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary random vectors x ∈ Rmfrom the noiseless linear measurements y = Ax with measurement matrix A ∈ Rn×m. 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 K(x) is given by the infimum of the lower modified Minkowski dimension over all support sets U of x (i.e., sets U ⊆ Rmwith P[x ∈ U] = 1). 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 C1-submanifolds of Rm. Countable unions of C1-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 submanifolds, 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. Motivated by this observation, we introduce 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 tools in the development of our theory are geometric measure theory and the theory of real analytic functions.

We establish the fundamental limits of lossless linear analog compression by considering the recovery of random vectors x ∈ ℝmfrom the noiseless linear measurements y = Ax with measurement matrix A ∈ ℝn×m. Specifically, for a random vector x ∈ ℝmof 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 ⊆ ℝmwith P[x ∈ 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.