Cosparse Analysis Modeling

A ubiquitous problem that has found many applications, from signal processing to machine learning, is to estimate a highdimensional vector x0 ∈ R from a set of incomplete linear observations y = Mx0 ∈ R. This is an ill-posed problem which admits infinitely many solutions, hence solving it is hopeless unless we can use additional prior knowledge on x0. The assumption that x0 admits a sparse representation z0 in some synthesis dictionary D is known to be of significant help, and it is now well understood that under incoherence assumptions on the matrix MD, one can recover vectors x0 with sufficiently sparse representations by solving the optimization problem: x̂S := Dẑ; ẑ := arg min z ‖z‖τ subject to y = MDz (1) for 0 ≤ τ ≤ 1. An alternative to (1) which has been used successfully in practice is to consider the analysis `τ -optimization: x̂A := arg min x ‖Ωx‖τ subject to y = Mx, (2) where Ω : R → R is an analysis operator. Typically the dimensions are m ≤ d ≤ p, n. The focus of our work is the study of a data model that makes possible to identify a collection of signals x0 that can be recovered via the optimization (2). Roughly speaking, in the case of (1), the signals x0 that are sparse, in other words, satisfy the sparse synthesis model, can be recovered via (1). In the sparse synthesis model, we consider the number of the non-zeros ‖z‖0 of the representation z of x0 (meaning that x0 = Dz), and we say that x0 is sparse if there is a representation z0 of x0 with small ‖z0‖0. To the contrary, in the case of (2), we are more interested in the number of the zeros p − ‖Ωx0‖0 of the representation Ωx0 of x0. We call the quantity ` = p − ‖Ωx0‖0 the cosparsity of x0 and say that x0 is cosparse, or it satisfies cosparse analysis model, if ` is large. For the cosparse analysis model, we have the following uniqueness result: The authors acknowledge the support by the European Community’s FP7FET program, SMALL project, under grant agreement no. 225913. Proposition 1. Let Ω be an analysis operator in general position. Then, for almost all M (with respect to the Lebesgue measure), a necessary and sufficient condition for (2) with τ = 0 to have a unique minimum is