Stable recovery and the coordinate small-ball behaviour of random vectors
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Recovery procedures in various application in Data Science are based on stable point separation. In its simplest form, stable point separation implies that if f is "far away" from 0, and one is given a random sample (f(Z_i))_i=1^m where a proportional number of the sample points may be corrupted by noise, that information is still enough to exhibit that f is far from 0. Stable point separation is well understood in the context of iid sampling, and to explore it for general sampling methods we introduce a new notion---the coordinate small-ball of a random vector X. Roughly put, this feature captures the number of "relatively large coordinates" of (|<TX,u_i>|)_i=1^m, where T:R^n →R^m is an arbitrary linear operator and (u_i)_i=1^m is any fixed orthonormal basis of R^m. We show that under the bare-minimum assumptions on X, and with high probability, many of the values |<TX,u_i>| are at least of the order T_S_2/√(m). As a result, the "coordinate structure" of TX exhibits the typical Euclidean norm of TX and does so in a stable way. One outcome of our analysis is that random sub-sampled convolutions satisfy stable point separation under minimal assumptions on the generating random vector---a fact that was known previously only in a highly restrictive setup, namely, for random vectors with iid subgaussian coordinates.
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