Stable recovery of low-dimensional cones in Hilbert spaces: One RIP to rule them all

Abstract

Many inverse problems in signal processing deal with the robust estimation of unknown data from underdetermined linear observations. Low-dimensional models, when combined with appropriate regularizers, have been shown to be efficient at performing this task. Sparse models with the 1-norm or low rank models with the nuclear norm are examples of such successful combinations. Stable recovery guarantees in these settings have been established using a common tool adapted to each case: the notion of restricted isometry property (RIP). In this paper, we establish generic RIP-based guarantees for the stable recovery of cones (positively homogeneous model sets) with arbitrary regularizers. These guarantees are illustrated on selected examples. For block structured sparsity in the infinite-dimensional setting, we use the guarantees for a family of regularizers which efficiency in terms of RIP constant can be controlled, leading to stronger and sharper guarantees than the state of the art.

Breaking the Coherence Barrier: A New Theory for Compressed Sensing

“…There have also been several theoretical extensions. First, generalizations of the RIP for the setting of asymptotic sparsity, asymptotic incoherence and multilevel random subsampling have been introduced and analysed in [9,60,80]. These complement the results proved in this paper by establishing uniform recovery guarantees.…”

“…(Structured sparsity, especially multiscale-type sparsity, also predates CS by some yearssee, for example, the work of Donoho and Huo [33]-and finds use outside of CS-see, for example, the work of Donoho and Kutyniok on geometric separation [34].) These include group, block, weighted and tree sparsity, amongst others (see [8,11,38,72,80] and references therein). In most of these works, structured sparsity is exploited by the design of the recovery algorithm (for example, by replacing the thresholding step in an iterative algorithm or the regularization functional in an optimization approach), with the sensing being carried out by a standard, incoherent operator (for example, a Gaussian random matrix).…”

“…We note in passing that it is quite different to the notions of block sparsity [8], weighted sparsity [72] or tree-structured sparsity [8], the latter of which has been used previously in the context of model-based CS for the recovery of wavelet coefficients. For further discussion on different structured sparsity models in CS, we refer to [11,80].…”

Optimality of 1-norm regularization among weighted 1-norms for sparse recovery: a case study on how to find optimal regularizations

“…the n−1-dimensional Haussdorf measure of T R (Σ)∩S (1)). When maximizing this compliance measure with convex regularizers (proper, coercive and continuous), it has been shown that we can limit ourselves to atomic norms with atoms included in the model [11,Lemma 2.1]. When looking at non-uniform recovery for random Gaussian measurements, the quantity vol(T R (x 0 )∩S(1)) vol(S(1)) represents the probability that a randomly oriented kernel of dimension 1 intersects (non trivially) T R (x 0 ).…”

“…In [11], an explicit constant δ suf f Σ (R) is given, such that δ(M ) < δ suf f Σ (R) guarantees exact recovery of elements of Σ by minimization (2). This constant is only sufficient (and sharp in some sense for sparse and low rank recovery).…”