Abstract-This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry constants. Thus, signals that are sparse with respect to the dictionary can be recovered via Basis Pursuit from a small number of random measurements. Further, thresholding is investigated as recovery algorithm for compressed sensing and conditions are provided that guarantee reconstruction with high probability. The different schemes are compared by numerical experiments.

To cite this version:Rémi Gribonval, Holger Rauhut, Karin Schnass, Pierre Vandergheynst. Atoms of all channels, unite! Average case analysis of multi-channel sparse recovery using greedy algorithms. ABSTRACT. This paper provides new results on computing simultaneous sparse approximations of multichannel signals over redundant dictionaries using two greedy algorithms. The first one, p-thresholding, selects the S atoms that have the largest p-correlation while the second one, p-simultaneous matching pursuit (p-SOMP), is a generalisation of an algorithm studied by Tropp in [31]. We first provide exact recovery conditions as well as worst case analyses of all algorithms. The results, expressed using the standard cumulative coherence, are very reminiscent of the single channel case and, in particular, impose stringent restrictions on the dictionary. We unlock the situation by performing an average case analysis of both algorithms. First, we set up a general probabilistic signal model in which the coefficients of the atoms are drawn at random from the standard Gaussian distribution. Second, we show that under this model, and with mild conditions on the coherence, the probability that p-thresholding and p-SOMP fail to recover the correct components is overwhelmingly small and gets smaller as the number of channels increases.Furthermore, we consider an extended random signal model where the atoms involved in the ideal simultaneous sparse representation are drawn at random. We show that, if the dictionary satisfies a uniform uncertainty principle [6], the probability that simultaneous OMP fails to recover any sufficiently sparse set of atoms gets increasingly smaller as the number of channels increases.

Let us assume that f is a continuous function defined on the unit ball of R d , of the form f (x) = g(Ax), where A is a k × d matrix and g is a function of k variables for k d. We are given a budget m ∈ N of possible point evaluations f (x i ), i = 1, . . . , m, of f , which we are allowed to query in order to construct a uniform approximating function. Under certain smoothness and variation assumptions on the function g, and an arbitrary choice of the matrix A, we present in this paper 1. a sampling choice of the points {x i } drawn at random for each function approximation; 2. algorithms (Algorithm 1 and Algorithm 2) for computing the approximating function, whose complexity is at most polynomial in the dimension d and in the number m of points.Dedicated to Ronald A. DeVore on his 70th birthday.Communicated by Emmanuel Candès. M. Fornasier ( )Due to the arbitrariness of A, the sampling points will be chosen according to suitable random distributions, and our results hold with overwhelming probability. Our approach uses tools taken from the compressed sensing framework, recent Chernoff bounds for sums of positive semidefinite matrices, and classical stability bounds for invariant subspaces of singular value decompositions.Keywords High-dimensional function approximation · Compressed sensing · Chernoff bounds for sums of positive semidefinite matrices · Stability bounds for invariant subspaces of singular value decompositions Mathematics Subject Classification (2010) 65D15 · 03D32 · 68Q30 · 60B20 · 60G50

This article treats the problem of learning a dictionary providing sparse representations for a given signal class, via 1-minimisation. The problem can also be seen as factorising a d × N matrix Y = (y1. .. yN), yn ∈ R d of training signals into a d × K dictionary matrix Φ and a K × N coefficient matrix X = (x1. .. xN), xn ∈ R K , which is sparse. The exact question studied here is when a dictionary coefficient pair (Φ, X) can be recovered as local minimum of a (nonconvex) 1-criterion with input Y = ΦX. First, for general dictionaries and coefficient matrices, algebraic conditions ensuring local identifiability are derived, which are then specialised to the case when the dictionary is a basis. Finally, assuming a random Bernoulli-Gaussian sparse model on the coefficient matrix, it is shown that sufficiently incoherent bases are locally identifiable with high probability. The perhaps surprising result is that the typically sufficient number of training samples N grows up to a logarithmic factor only linearly with the signal dimension, i.e. N ≈ CK log K, in contrast to previous approaches requiring combinatorially many samples.