Abstract — We consider the problem of automatically recognizing human faces from frontal views with varying expression and illumination, as well as occlusion and disguise. We cast the recognition problem as one of classifying among multiple linear regression models, and argue that new theory from sparse signal representation offers the key to addressing this problem. Based on a sparse representation computed by ℓ 1-minimization, we propose a general classification algorithm for (image-based) object recognition. This new framework provides new insights into two crucial issues in face recognition: feature extraction and robustness to occlusion. For feature extraction, we show that if sparsity in the recognition problem is properly harnessed, the choice of features is no longer critical. What is critical, however, is whether the number of features is sufficiently large and whether the sparse representation is correctly computed. Unconventional features such as downsampled images and random projections perform just as well as conventional features such as Eigenfaces and Laplacianfaces, as long as the dimension of the feature space surpasses certain threshold, predicted by the theory of sparse representation. This framework can handle errors due to occlusion and corruption uniformly, by exploiting the fact that these errors are often sparse w.r.t. to the standard (pixel) basis. The theory of sparse representation helps predict how much occlusion the recognition algorithm can handle and how to choose the training images to maximize robustness to occlusion. We conduct extensive experiments on publicly available databases to verify the efficacy of the proposed algorithm, and corroborate the above claims.

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Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies below the Nyquist rate. In this paper we consider a more general signal model and assume signals that live on or close to the union of linear subspaces of low dimension. We present sampling theorems for this model that are in the same spirit as the Nyquist-Shannon sampling theorem in that they connect the number of required samples to certain model parameters. Contrary to the Nyquist-Shannon sampling theorem, which gives a necessary and sufficient condition for the number of required samples as well as a simple linear algorithm for signal reconstruction, the model studied here is more complex. We therefore concentrate on two aspects of the signal model, the existence of one to one maps to lower dimensional observation spaces and the smoothness of the inverse map. We show that almost all linear maps are one to one when the observation space is at least of the same dimension as the largest dimension of the convex hull of the union of any two subspaces in the model. However, we also show that in order for the inverse map to have certain smoothness properties such as a given finite Lipschitz constant, the required observation dimension necessarily depends logarithmically

The paper considers the problem of detecting the sparsity pattern of a k-sparse vector in R n from m random noisy measurements. A new necessary condition on the number of measurements for asymptotically reliable detection with maximum likelihood (ML) estimation and Gaussian measurement matrices is derived. This necessary condition for ML detection is compared against a sufficient condition for simple maximum correlation (MC) or thresholding algorithms. The analysis shows that the gap between thresholding and ML can be described by a simple expression in terms of the total signal-to-noise ratio (SNR), with the gap growing with increasing SNR. Thresholding is also compared against the more sophisticated lasso and orthogonal matching pursuit (OMP) methods. At high SNRs, it is shown that the gap between lasso and OMP over thresholding is described by the range of powers of the nonzero component values of the unknown signals. Specifically, the key benefit of lasso and OMP over thresholding is the ability of lasso and OMP to detect signals with relatively small components.

The recently introduced theory of compressive sensing enables the recovery of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist-rate samples. Interestingly, it has been shown that random projections are a near-optimal measurement scheme. This has inspired the design of hardware systems that directly implement random measurement protocols. However, despite the intense focus of the community on signal recovery, many (if not most) signal processing problems do not require full signal recovery. In this paper, we take some first steps in the direction of solving inference problems—such as detection, classification, or estimation—and filtering problems using only compressive measurements and without ever reconstructing the signals involved. We provide theoretical bounds along with experimental results.

We consider the problem of recovering a matrix from its action on a known vector in the setting where the matrix can be represented efficiently in a known matrix dictionary. Connections with sparse signal recovery allows for the use of efficient reconstruction techniques such as Basis Pursuit (BP). Of particular interest is the dictionary of time-frequency shift matrices and its role for channel estimation and identification in communications engineering. We present recovery results for BP with the time-frequency shift dictionary and various dictionaries of random matrices.

In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation based on a mixed matrix norm. Typically, worst-case analysis is carried out in order to analyze conditions under which the algorithms are able to recover any jointly sparse set of vectors. However, such an approach is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Previous work considered an average case analysis of thresholding and SOMP by imposing a probability model on the measured signals. In this paper, our main focus is on analysis of convex relaxation techniques. In particular, we focus on the mixed ℓ2,1 approach to multichannel recovery. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. Our probability bounds are valid and meaningful even for a small number of signals. Using the tools we develop to analyze the convex relaxation method, we also tighten the previous bounds for thresholding and SOMP.

We propose an approximation framework for distributed target localization in sensor networks. We represent the unknown target positions on a location grid as a sparse vector, whose support encodes the multiple target locations. The location vector is linearly related to multiple sensor measurements through a sensing matrix, which can be locally estimated at each sensor. We show that we can successfully determine multiple target locations by using linear dimensionality-reducing projections of sensor measurements. The overall communication bandwidth requirement per sensor is logarithmic in the number of grid points and linear in the number of targets, ameliorating the communication requirements. Simulations results demonstrate the performance of the proposed framework. 1.

Abstract- Sparse signals representation, analysis, and sensing, has received a lot of attention in recent years from the signal processing, optimization, and learning communities. On one hand, the learning of overcomplete dictionaries that facilitate a sparse representation of the image as a liner combination of a few atoms from such dictionary, leads to state-of-the-art results in image and video restoration and image classification. On the other hand, the framework of compressed sensing (CS) has shown that sparse signals can be recovered from far less samples than those required by the classical Shannon-Nyquist Theorem. The goal of this paper is to present a framework that unifies the learning of overcomplete dictionaries for sparse image representation with the concepts of signal recovery from very few samples put forward by the CS theory. The samples used in CS correspond to linear projections defined by a sampling projection matrix. It has been shown that, for example, a non-adaptive random sampling matrix satisfies the fundamental theoretical requirements of CS, enjoying the additional benefit of universality. On the other hand, a projection sensing matrix that is optimally designed for a certain signal class can further improve the reconstruction accuracy or further reduce the necessary number of samples. In this work we introduce a framework for the joint design and optimization, from a set of training images, of the

Compressive sensing (CS) is a new approach to simultaneous sensing and compression of sparse and compressible signals. A great many applications feature smooth or modulated signals that can be modeled as a linear combination of a small number of sinusoids; such signals are sparse in the frequency domain. In practical applications, the standard frequency domain signal representation is the discrete Fourier transform (DFT). Unfortunately, the DFT coefficients of a frequency-sparse signal are themselves sparse only in the contrived case where the sinusoid frequencies are integer multiples of the DFT’s fundamental frequency. As a result, practical DFT-based CS acquisition and recovery of smooth signals does not perform nearly as well as one might expect. In this paper, we develop a new spectral compressive sensing (SCS) theory for general frequency-sparse signals. The key ingredients are an over-sampled DFT frame, a signal model that inhibits closely spaced sinusoids, and classical sinusoid parameter estimation algorithms from the field of spectrum estimation. Using peridogram and eigen-analysis based spectrum estimates (e.g., MUSIC), our new SCS algorithms significantly outperform the current state-of-the-art CS algorithms while providing provable bounds on the number of measurements required for stable recovery. I.