Many real-world problems deal with collections of high-dimensional data, such as images, videos, text, and web documents, DNA microarray data, and more. Often, such high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories to which the data belong. In this paper, we propose and study an algorithm, called sparse subspace clustering, to cluster data points that lie in a union of low-dimensional subspaces. The key idea is that, among the infinitely many possible representations of a data point in terms of other points, a sparse representation corresponds to selecting a few points from the same subspace. This motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of the data into subspaces. Since solving the sparse optimization program is in general NP-hard, we consider a convex relaxation and show that, under appropriate conditions on the arrangement of the subspaces and the distribution of the data, the proposed minimization program succeeds in recovering the desired sparse representations. The proposed algorithm is efficient and can handle data points near the intersections of subspaces. Another key advantage of the proposed algorithm with respect to the state of the art is that it can deal directly with data nuisances, such as noise, sparse outlying entries, and missing entries, by incorporating the model of the data into the sparse optimization program. We demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real-world problems of motion segmentation and face clustering.

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Sparse Subspace Clustering: Algorithm, Theory, and Applications

In many real-world problems, we are dealing with collections of high-dimensional data, such as images, videos, text and web documents, DNA microarray data, and more. Often, high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories the data belongs to. In this paper, we propose and study an algorithm, called Sparse Subspace Clustering (SSC), to cluster data points that lie in a union of low-dimensional subspaces. The key idea is that, among infinitely many possible representations of a data point in terms of other points, a sparse representation corresponds to selecting a few points from the same subspace. This motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of data into subspaces. Since solving the sparse optimization program is in general NP-hard, we consider a convex relaxation and show that, under appropriate conditions on the arrangement of subspaces and the distribution of data, the proposed minimization program succeeds in recovering the desired sparse representations. The proposed algorithm can be solved efficiently and can handle data points near the intersections of subspaces. Another key advantage of the proposed algorithm with respect to the state of the art is that it can deal with data nuisances, such as noise, sparse outlying entries, and missing entries, directly by incorporating the model of the data into the sparse optimization program. We demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real-world problems of motion segmentation and face clustering.

tions. Bootstrap has found many applications in engineering field, including artificial neural networks, biomedical engineering, environmental engineering, image processing, and radar and sonar signal processing. Basic concepts of the bootstrap are summarized in each section as a step-by-step algorithm for ease of implementation. Most of the applications are taken from the signal processing literature. The principles of the bootstrap are introduced in Chapter 2. Both the nonparametric and parametric bootstrap procedures are explained. Babu and Singh (1984) have demonstrated that in general, these two procedures behave similarly for pivotal (Studentized) statistics. The fact that the bootstrap is not the solution for all of the problems has been known to statistics community for a long time; however, this fact is rarely touched on in the manuscripts meant for practitioners. It was first observed by Babu (1984) that the bootstrap does not work in the infinite variance case. Bootstrap Techniques for Signal Processing explains the limitations of bootstrap method with an example. I especially liked the presentation style. The basic results are stated without proofs; however, the application of each result is presented as a simple step-by-step process, easy for nonstatisticians to follow. The bootstrap procedures, such as moving block bootstrap for dependent data, along with applications to autoregressive models and for estimation of power spectral density, are also presented in Chapter 2. Signal detection in the presence of noise is generally formulated as a testing of hypothesis problem. Chapter 3 introduces principles of bootstrap hypothesis testing. The topics are introduced with interesting real life examples. Flow charts, typical in engineering literature, are used to aid explanations of the bootstrap hypothesis testing procedures. The bootstrap leads to second-order correction due to pivoting; this improvement in the results due to pivoting is also explained. In the second part of Chapter 3, signal processing is treated as a regression problem. The performance of the bootstrap for matched filters as well as constant false-alarm rate matched filters is also illustrated. Chapters 2 and 3 focus on estimation problems. Chapter 4 introduces bootstrap methods used in model selection. Due to the inherent structure of the subject matter, this chapter may be difficult for nonstatisticians to follow. Chapter 5 is the most impressive chapter in the book, especially from the standpoint of statisticians. It provides real data bootstrap applications to illustrate the theory covered in the earlier chapters. These include applications to optimal sensor placement for knock detection and land-mine detection. The authors also provide a MATLAB toolbox comprising frequently used routines. Overall, this is a very useful handbook for engineers, especially those working in signal processing.

Probability and Random Processes

Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discrete-to-discrete measurement architectures using matrices of randomized nature and signal models based on standard sparsity. In recent years, CS has worked its way into several new application areas. This, in turn, necessitates a fresh look on many of the basics of CS. The random matrix measurement operator must be replaced by more structured sensing architectures that correspond to the characteristics of feasible acquisition hardware. The standard sparsity prior has to be extended to include a much richer class of signals and to encode broader data models, including continuous-time signals. In our overview, the theme is exploiting signal and measurement structure in compressive sensing. The prime focus is bridging theory and practice; that is, to pinpoint the potential of structured CS strategies to emerge from the math to the hardware. Our summary highlights new directions as well as relations to more traditional CS, with the hope of serving both as a review to practitioners wanting to join this emerging field, and as a reference for researchers that attempts to put some of the existing ideas in perspective of practical applications.

Structured Compressed Sensing: From Theory to Applications

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.

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Robust Recovery of Signals From a Structured Union of Subspaces

We consider efficient methods for the recovery of block-sparse signals-ie, sparse signals that have nonzero entries occurring in clusters-from an underdetermined system of linear equations An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce We then show that a block-version of the orthogonal matching pursuit algorithm recovers block -sparse signals in no more than steps if the block-coherence is sufficiently small The same condition on block-coherence is shown to guarantee successful recovery through a mixed -optimization approach This complements previous recovery results for the block-sparse case which relied on small block-restricted isometry constants The significance of the results presented in this paper lies in the fact that making explicit use of block-sparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem

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Block-Sparse Signals: Uncertainty Relations and Efficient Recovery

We investigate the recovery of signals exhibiting a sparse representation in a general (i.e., possibly redundant or incomplete) dictionary that are corrupted by additive noise admitting a sparse representation in another general dictionary. This setup covers a wide range of applications, such as image inpainting, super-resolution, signal separation, and recovery of signals that are impaired by, e.g., clipping, impulse noise, or narrowband interference. We present deterministic recovery guarantees based on a novel uncertainty relation for pairs of general dictionaries and we provide corresponding practicable recovery algorithms. The recovery guarantees we find depend on the signal and noise sparsity levels, on the coherence parameters of the involved dictionaries, and on the amount of prior knowledge about the signal and noise support sets.

/pdf/recovery-of-sparsely-corrupted-signals-e7sg6flm9m.pdf

Recovery of Sparsely Corrupted Signals

We present an uncertainty relation for the representation of signals in two different general (possibly redundant or incomplete) signal sets. This uncertainty relation is relevant for the analysis of signals containing two distinct features each of which can be described sparsely in a suitable general signal set. Furthermore, the new uncertainty relation is shown to lead to im- proved sparsity thresholds for recovery of signals that are sparse in general dictionaries. Specifically, our results improve on the well-known (1 + 1/d)/2-threshold for dictionaries with coherence d by up to a factor of two. Furthermore, we provide probabilistic recovery guarantees for pairs of general dictionaries that also allow us to understand which parts of a general dictionary one needs to randomize over to "weed out" the sparsity patterns that prohibit breaking the square-root bottleneck.

https://publications.lib.chalmers.se/records/fulltext/local_146195.pdf

Uncertainty Relations and Sparse Signal Recovery for Pairs of General Signal Sets

A signal for relay is received from a first node. From a local memory that stores K transform matrices (K being an integer greater than one) is selected a transform matrix that is associated with a relay rank order. The matrices define an optimal tradeoff of diversity gain versus spatial multiplexing. The received signal is transformed using the selected transform matrix, and the transformed signal is relayed over a wireless channel to a second node. In a particular embodiment the received signal is an input symbol vector of N consecutive symbols (N being an integer at least equal to two) and the transform linearly combines the N symbols of the input vector according to the relay rank order. In various embodiments the first node may give the rank order, it may be mutually agreed via signaling among the various relay nodes, or it may be based on an intrinsically generated random number.

Patrick Kuppinger

Papers

Block-Sparse Signals: Uncertainty Relations and Efficient Recovery

Recovery of Sparsely Corrupted Signals

Uncertainty Relations and Sparse Signal Recovery for Pairs of General Signal Sets

Recovery of Sparsely Corrupted Signals

Linear transformation matrices for distributed diversity