Compressive sampling (CoSa) has provided many methods for signal recovery of signals compressible with respect to an orthonormal basis. However, modern applications have sparked the emergence of approaches for signals not sparse in an orthonormal basis but in some arbitrary, perhaps highly overcomplete, dictionary. Recently, several "signal-space" greedy methods have been proposed to address signal recovery in this setting. However, such methods inherently rely on the existence of fast and accurate projections which allow one to identify the most relevant atoms in a dictionary for any given signal, up to a very strict accuracy. When the dictionary is highly overcomplete, no such projections are currently known; the requirements on such projections do not even hold for incoherent or well-behaved dictionaries. In this work, we provide an alternate analysis for signal space greedy methods which enforce assumptions on these projections which hold in several settings including those when the dictionary is incoherent or structurally coherent. These results align more closely with traditional results in the standard CoSa literature and improve upon previous work in the signal space setting.
The importance of using and analyzing two separate projection schemes in sparse recovery is also discussed in an independent line of work by Hegde et.al. @cite_8 . There, the authors call the two projections the head'' and tail'' projections, and analyze a variant of Iterative Hard Thresholding (IHT) for signal recovery under the Model-RIP, a generalization of the @math -RIP. In fact, they show that without a projection satisfying essentially the second inequality of , conventional IHT will fail.
@cite_8
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Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x lies in a union of subspaces. In this paper, we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a sum of k subspaces, chosen from a larger set of m possibilities. The samples are modeled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a block-sparse vector whose nonzero elements appear in fixed blocks. We then propose a mixed lscr2 lscr1 program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP, we also prove stability of our approach in the presence of noise and modeling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently. This paper presents a new analysis for the orthogonal matching pursuit (OMP) algorithm. It is shown that if the restricted isometry property (RIP) is satisfied at sparsity level O(k), then OMP can stably recover a k-sparse signal in 2-norm under measurement noise. For compressed sensing applications, this result implies that in order to uniformly recover a k-sparse signal in Rd, only O(k lnd) random projections are needed. This analysis improves some earlier results on OMP depending on stronger conditions that can only be satisfied with Ω(k2 lnd) or Ω(k1.6 lnd) random projections. Most of the existing sparse-recovery methods assume a static system: the signal is a finite-length vector for which a fixed set of measurements and sparse representation are available and an l1 problem is solved for the reconstruction. However, the same representation and reconstruction framework is not readily applicable in a streaming system: the signal changes over time, and it is measured and reconstructed sequentially over small intervals. This is particularly desired when dividing signals into disjoint blocks and processing each block separately is infeasible or inefficient. In this paper, we discuss two streaming systems and a new homotopy algorithm for quickly solving the associated l1 problems: 1) recovery of smooth, time-varying signals for which, instead of using block transforms, we use lapped orthogonal transforms for sparse representation and 2) recovery of sparse, time-varying signals that follows a linear dynamic model. For both systems, we iteratively process measurements over a sliding interval and solve a weighted l1-norm minimization problem for estimating sparse coefficients. Since we estimate overlapping portions of the signal while adding and removing measurements, instead of solving a new l1 program as the system changes, we use available signal estimates as starting point in a homotopy formulation and update the solution in a few simple steps. We demonstrate with numerical experiments that our proposed streaming recovery framework provides better reconstruction compared to the methods that represent and reconstruct signals as independent, disjoint blocks, and that our proposed homotopy algorithm updates the solution faster than the current state-of-the-art solvers. The problem of sparse signal recovery from a relatively small number of noisy measurements has been studied extensively in the recent literature on compressed sensing. However, the focus of those studies appears to be limited to the case of linear projections disturbed by Gaussian noise, and the sparse signal reconstruction problem is treated as linear regression with l 1 -norm regularization constraint. A natural question to ask is whether one can accurately recover sparse signals under different noise assumptions. Herein, we extend the results of [13] to the more general case of exponential-family noise that includes Gaussian noise as a particular case, and yields l 1 -regularized Generalized Linear Model (GLM) regression problem. We show that, under standard restricted isometry property (RIP) assumptions on the design matrix, l 1 -minimization can provide stable recovery of a sparse signal in presence of the exponential-family noise, provided that certain sufficient conditions on the noise distribution are satisfied.