In this paper, we address the subspace clustering problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to cluster the samples into their respective subspaces and remove possible outliers as well. To this end, we propose a novel objective function named Low-Rank Representation (LRR), which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that the convex program associated with LRR solves the subspace clustering problem in the following sense: When the data is clean, we prove that LRR exactly recovers the true subspace structures; when the data are contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for data corrupted by arbitrary sparse errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace clustering and error correction in an efficient and effective way.

Robust Recovery of Subspace Structures by Low-Rank Representation

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

Machine generated contents note: 1. Introduction to compressed sensing Mark A. Davenport, Marco F. Duarte, Yonina C. Eldar and Gitta Kutyniok; 2. Second generation sparse modeling: structured and collaborative signal analysis Alexey Castrodad, Ignacio Ramirez, Guillermo Sapiro, Pablo Sprechmann and Guoshen Yu; 3. Xampling: compressed sensing of analog signals Moshe Mishali and Yonina C. Eldar; 4. Sampling at the rate of innovation: theory and applications Jose Antonia Uriguen, Yonina C. Eldar, Pier Luigi Dragotta and Zvika Ben-Haim; 5. Introduction to the non-asymptotic analysis of random matrices Roman Vershynin; 6. Adaptive sensing for sparse recovery Jarvis Haupt and Robert Nowak; 7. Fundamental thresholds in compressed sensing: a high-dimensional geometry approach Weiyu Xu and Babak Hassibi; 8. Greedy algorithms for compressed sensing Thomas Blumensath, Michael E. Davies and Gabriel Rilling; 9. Graphical models concepts in compressed sensing Andrea Montanari; 10. Finding needles in compressed haystacks Robert Calderbank, Sina Jafarpour and Jeremy Kent; 11. Data separation by sparse representations Gitta Kutyniok; 12. Face recognition by sparse representation Arvind Ganesh, Andrew Wagner, Zihan Zhou, Allen Y. Yang, Yi Ma and John Wright.

Compressed sensing : theory and applications

Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for the acquisition of sparse or compressible signals that can be well approximated by just K ? N elements from an N -dimensional basis. Instead of taking periodic samples, CS measures inner products with M < N random vectors and then recovers the signal via a sparsity-seeking optimization or greedy algorithm. Standard CS dictates that robust signal recovery is possible from M = O(K log(N/K)) measurements. It is possible to substantially decrease M without sacrificing robustness by leveraging more realistic signal models that go beyond simple sparsity and compressibility by including structural dependencies between the values and locations of the signal coefficients. This paper introduces a model-based CS theory that parallels the conventional theory and provides concrete guidelines on how to create model-based recovery algorithms with provable performance guarantees. A highlight is the introduction of a new class of structured compressible signals along with a new sufficient condition for robust structured compressible signal recovery that we dub the restricted amplification property, which is the natural counterpart to the restricted isometry property of conventional CS. Two examples integrate two relevant signal models-wavelet trees and block sparsity-into two state-of-the-art CS recovery algorithms and prove that they offer robust recovery from just M = O(K) measurements. Extensive numerical simulations demonstrate the validity and applicability of our new theory and algorithms.

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Model-Based Compressive Sensing

We propose low-rank representation (LRR) to segment data drawn from a union of multiple linear (or affine) subspaces. Given a set of data vectors, LRR seeks the lowest-rank representation among all the candidates that represent all vectors as the linear combination of the bases in a dictionary. Unlike the well-known sparse representation (SR), which computes the sparsest representation of each data vector individually, LRR aims at finding the lowest-rank representation of a collection of vectors jointly. LRR better captures the global structure of data, giving a more effective tool for robust subspace segmentation from corrupted data. Both theoretical and experimental results show that LRR is a promising tool for subspace segmentation.

https://icml.cc/Conferences/2010/papers/521.pdf

Robust Subspace Segmentation by Low-Rank Representation

Conventional sub-Nyquist sampling methods for analog signals exploit prior information about the spectral support. In this paper, we consider the challenging problem of blind sub-Nyquist sampling of multiband signals, whose unknown frequency support occupies only a small portion of a wide spectrum. Our primary design goals are efficient hardware implementation and low computational load on the supporting digital processing. We propose a system, named the modulated wideband converter, which first multiplies the analog signal by a bank of periodic waveforms. The product is then low-pass filtered and sampled uniformly at a low rate, which is orders of magnitude smaller than Nyquist. Perfect recovery from the proposed samples is achieved under certain necessary and sufficient conditions. We also develop a digital architecture, which allows either reconstruction of the analog input, or processing of any band of interest at a low rate, that is, without interpolating to the high Nyquist rate. Numerical simulations demonstrate many engineering aspects: robustness to noise and mismodeling, potential hardware simplifications, real-time performance for signals with time-varying support and stability to quantization effects. We compare our system with two previous approaches: periodic nonuniform sampling, which is bandwidth limited by existing hardware devices, and the random demodulator, which is restricted to discrete multitone signals and has a high computational load. In the broader context of Nyquist sampling, our scheme has the potential to break through the bandwidth barrier of state-of-the-art analog conversion technologies such as interleaved converters.

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From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals

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.

/pdf/robust-recovery-of-signals-from-a-structured-union-of-7wxfy5abte.pdf

Robust Recovery of Signals From a Structured Union of Subspaces

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 modelled 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 non-zero elements appear in fixed blocks. We then propose a mixed $\ell_2/\ell_1$ 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 modelling 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.

We address the problem of reconstructing a multiband signal from its sub-Nyquist pointwise samples, when the band locations are unknown. Our approach assumes an existing multi-coset sampling. To date, recovery methods for this sampling strategy ensure perfect reconstruction either when the band locations are known, or under strict restrictions on the possible spectral supports. In this paper, only the number of bands and their widths are assumed without any other limitations on the support. We describe how to choose the parameters of the multi-coset sampling so that a unique multiband signal matches the given samples. To recover the signal, the continuous reconstruction is replaced by a single finite-dimensional problem without the need for discretization. The resulting problem is studied within the framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate required for blind signal reconstruction, which is twice the minimal rate of known-spectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal rate, and provides a first systematic study of compressed sensing in a truly analog setting. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate.

Blind Multiband Signal Reconstruction: Compressed Sensing for Analog Signals

We address the problem of reconstructing a multi-band signal from its sub-Nyquist point-wise samples. To date, all reconstruction methods proposed for this class of signals assumed knowledge of the band locations. In this paper, we develop a non-linear blind perfect reconstruction scheme for multi-band signals which does not require the band locations. Our approach assumes an existing blind multi-coset sampling method. The sparse structure of multi-band signals in the continuous frequency domain is used to replace the continuous reconstruction with a single finite dimensional problem without the need for discretization. The resulting problem can be formulated within the framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate required for blind signal reconstruction, which is twice the minimal rate of known-spectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal rate. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate.

Moshe Mishali

Papers

From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals

Robust Recovery of Signals From a Structured Union of Subspaces

Robust Recovery of Signals From a Structured Union of Subspaces

Blind Multiband Signal Reconstruction: Compressed Sensing for Analog Signals

Blind Multi-Band Signal Reconstruction: Compressed Sensing for Analog Signals