Convex variational formulation with smooth coupling for multicomponent signal decomposition and recovery

A convex variational formulation is proposed to solve multicomponent signal processing problems in Hilbert spaces. The cost function consists of a separable term, in which each component is modeled through its own potential, and of a coupling term, in which constraints on linear transformations of the components are penalized with smooth functionals. An algorithm with guaranteed weak convergence to a solution to the problem is provided. Various multicomponent signal decomposition and recovery applications are discussed. 1 Problem statement The processing of multicomponent signals has become increasingly important due, on the one hand, to the development of new imaging modalities and sensing devices, and, on the other hand, to the introduction of sophisticated mathematical models to represent complex signals. It is for instance required in applications dealing with the recovery of multichannel signals [8, 33, 34, 40], which arise in particular in color imaging and in the multiand hyperspectral imaging techniques used in astronomy and in satellite imaging. Another important instance of multicomponent processing is found 1 in signal decomposition problems, e.g., [2, 5, 6, 7, 15, 43, 44]. In such problems, the ideal signal is viewed as a mixture of elementary components that need to be identified individually. Mathematically, a multicomponent signal can be viewed as an m-tuple (xi)1≤i≤m, where each component xi lies in a real Hilbert space Hi. A generic convex variational formulation for solving multicomponent signal recovery or decomposition problems is minimize x1∈H1,..., xm∈Hm Φ(x1, . . . , xm), (1.1) where Φ: H1 ⊕ · · · ⊕Hm → ]−∞,+∞] is a convex cost function. At this level of generality, however, no algorithm exists to solve (1.1) reliably in the sense that it produces m sequences (x1,n)n∈N, . . . , (xm,n)n∈N converging (weakly or strongly) to points x1, . . . , xm, respectively, such that (xi)1≤i≤m minimizes Φ. Let us recall that, even in the elementary case when m = 2 and H1 = H2 = R, the basic Gauss-Seidel alternating minimization algorithm does not possess this property [28]. In this paper, we consider the following, more structured version of (1.1). Problem 1.1 Let m ≥ 2 and p ≥ 1 be integers, let (Hi)1≤i≤m and (Gk)1≤k≤p be real Hilbert spaces, and let (τk)1≤k≤p be in ]0,+∞[. For every i ∈ {1, . . . ,m}, let fi : Hi → ]−∞,+∞] be a proper lower semicontinuous convex function and, for every k ∈ {1, . . . , p}, let φk : Gk → R be convex and differentiable with a τk–Lipschitz continuous gradient, and let Lki : Hi → Gk be linear and bounded. It is assumed that min1≤k≤p ∑m i=1 ‖Lki‖ > 0. The problem is to minimize x1∈H1,..., xm∈Hm m ∑