This paper considers the problem of compressive sensing over a finite alphabet, where the finite alphabet may be inherent to the nature of the data or a result of quantization. There are multiple examples of finite alphabet based static as well as time-series data with inherent sparse structure; and quantizing real values is an essential step while handling real data in practice. We show that there are significant benefits to analyzing the problem while incorporating its finite alphabet nature, versus ignoring it and employing a conventional real alphabet based toolbox. Specifically, when the alphabet is finite, our techniques (a) have a lower sample complexity compared to real-valued compressive sensing for sparsity levels below a threshold; (b) facilitate constructive designs of sensing matrices based on coding-theoretic techniques; (c) enable one to solve the exact @math -minimization problem in polynomial time rather than a approach of convex relaxation followed by sufficient conditions for when the relaxation matches the original problem; and finally, (d) allow for smaller amount of data storage (in bits).
The fact that real-valued compressive sensing allows for recovery of sparse signals based on linear measurements is reminiscent of error correction in linear channel codes and compression by lossless source codes over finite alphabet or fields @cite_11 @cite_28 . Such similarities have been identified in existing literature to serve varied goals. For example, the use of bipartite expander graphs to design real-valued sensing matrices is investigated in @cite_12 . The connection between real-valued compressive sensing and linear channel codes is explored in @cite_20 , by viewing sparse signal compression as syndrome-based source coding over real numbers and making use of linear codes over finite fields of large sizes. The design of real-valued sensing matrices based on LDPC codes is examined in @cite_27 and @cite_4 . The connection between sparse learning problems and coding theory is studied in @cite_0 . For real-valued compressive sensing over finite alphabet, the sparse signal recovery approaches that have been examined include approximate message passing @cite_14 , sphere decoding and semi-definite relaxation @cite_1 . However, an algebraic understanding of compressive sensing, particularly over finite fields, is yet limited, which is the main contribution of this paper.
@cite_14 @cite_4 @cite_28 @cite_1 @cite_0 @cite_27 @cite_12 @cite_20 @cite_11
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Compressive Sensing theory details how a sparsely represented signal in a known basis can be reconstructed with an underdetermined linear measurement model. However, in reality there is a mismatch between the assumed and the actual bases due to factors such as discretization of the parameter space defining basis components, sampling jitter in A D conversion, and model errors. Due to this mismatch, a signal may not be sparse in the assumed basis, which causes significant performance degradation in sparse reconstruction algorithms. To eliminate the mismatch problem, this paper presents a novel perturbed orthogonal matching pursuit (POMP) algorithm that performs controlled perturbation of selected support vectors to decrease the orthogonal residual at each iteration. Based on detailed mathematical analysis, conditions for successful reconstruction are derived. Simulations show that robust results with much smaller reconstruction errors in the case of perturbed bases can be obtained as compared to standard sparse reconstruction techniques. Compressive Sensing (CS) combines sampling and compression into a single sub-Nyquist linear measurement process for sparse and compressible signals. In this paper, we extend the theory of CS to include signals that are concisely represented in terms of a graphical model. In particular, we use Markov Random Fields (MRFs) to represent sparse signals whose nonzero coefficients are clustered. Our new model-based recovery algorithm, dubbed Lattice Matching Pursuit (LaMP), stably recovers MRF-modeled signals using many fewer measurements and computations than the current state-of-the-art algorithms. Compressive sensing (CS) has recently emerged as a powerful framework for acquiring sparse signals. The bulk of the CS literature has focused on the case where the acquired signal has a sparse or compressible representation in an orthonormal basis. In practice, however, there are many signals that cannot be sparsely represented or approximated using an orthonormal basis, but that do have sparse representations in a redundant dictionary. Standard results in CS can sometimes be extended to handle this case provided that the dictionary is sufficiently incoherent or well conditioned, but these approaches fail to address the case of a truly redundant or overcomplete dictionary. In this paper, we describe a variant of the iterative recovery algorithm CoSaMP for this more challenging setting. We utilize the D-RIP, a condition on the sensing matrix analogous to the well-known restricted isometry property. In contrast to prior work, the method and analysis are “signal-focused”; that is, they are oriented around recovering the signal rather than its dictionary coefficients. Under the assumption that we have a near-optimal scheme for projecting vectors in signal space onto the model family of candidate sparse signals, we provide provable recovery guarantees. Developing a practical algorithm that can provably compute the required near-optimal projections remains a significant open problem, but we include simulation results using various heuristics that empirically exhibit superior performance to traditional recovery algorithms. Compressive sensing (CS) has recently emerged as a powerful framework for acquiring sparse signals. The bulk of the CS literature has focused on the case where the acquired signal has a sparse or compressible representation in an orthonormal basis. In practice, however, there are many signals that cannot be sparsely represented or approximated using an orthonormal basis, but that do have sparse representations in a redundant dictionary. Standard results in CS can sometimes be extended to handle this case provided that the dictionary is sufficiently incoherent or well-conditioned, but these approaches fail to address the case of a truly redundant or overcomplete dictionary. In this paper we describe a variant of the iterative recovery algorithm CoSaMP for this more challenging setting. We utilize the D-RIP, a condition on the sensing matrix analogous to the well-known restricted isometry property. In contrast to prior work, the method and analysis are "signal-focused"; that is, they are oriented around recovering the signal rather than its dictionary coefficients. Under the assumption that we have a near-optimal scheme for projecting vectors in signal space onto the model family of candidate sparse signals, we provide provable recovery guarantees. Developing a practical algorithm that can provably compute the required near-optimal projections remains a significant open problem, but we include simulation results using various heuristics that empirically exhibit superior performance to traditional recovery algorithms.