Almost lossless analog signal separation

Abstract

We propose an information-theoretic framework for analog signal separation. Specifically, we consider the problem of recovering two analog signals from a noiseless sum of linear measurements of the signals. Our framework is inspired by the groundbreaking work of Wu and Verdú (2010) on almost lossless analog compression. The main results of the present paper are a general achievability bound for the compression rate in the analog signal separation problem, an exact expression for the optimal compression rate in the case of signals that have mixed discrete-continuous distributions, and a new technique for showing that the intersection of generic subspaces with subsets of sufficiently small Minkowski dimension is empty. This technique can also be applied to obtain a simplified proof of a key result in Wu and Verdú (2010).

Information-theoretic analysis of support recovery from sparsely corrupted measurements

“…The lower bound in (22) has an additional term (1−Ä+ÄD )a log[1+P(D ,p X )snr] on the right-hand side when compared with the result in Corollary 1 of [11]. When D → 0, the sampling rate-distortion function in (22) …”

“…e is usually called the gross error and its entries can have arbitrarily large magnitude. This problem and its variants have gained increasing attentions recently in terms of theoretical analysis [15][16][17][18][19][20][21][22][23][24][25][26]. [15,16] investigated the sparse signal recovery from y = Ax + e, and provided the recovery guarantees when x is recovered with high probability.…”

“…Later in [21], Studer and Baraniuk extended the results in [17][18][19] to the case of approximately sparse signals and noisy measurements. To investigate the recovery from y = Ax + Be, [22] derived a general achievability bound for the compression rate in separating signals x and e. In addition, [23,24] showed that the combination matrix [A, B] obeys the restricted isometry property when A is a Gaussian distributed matrix, where B is an identity matrix in [24]. Nguyen et al studied the problem of recovering sparse signal x from y = Ax + e + w [25,26].…”

“…[25] derived the recovery guarantees for random sub-sampled unitary matrices, and [26] proposed an extended optimization method and discussed its corresponding performance. In these existing works, [17][18][19][20][21] considered deterministic matrices A, [15,16,[23][24][25][26] considered random matrices A, and [22] dealt with almost all matrices A.…”

“…Though some papers have studied the related model [20][21][22] or even the same model [25,26], they mainly focused on the recovery guarantees or the performance of some reconstruction algorithms. By contrast, we discuss the relationship between the sampling rate and the recovery error.…”