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Robust face recognition via sparse representation
 IEEE TRANS. PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2008
"... 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 signa ..."
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Cited by 936 (40 self)
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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 1minimization, we propose a general classification algorithm for (imagebased) 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.
Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ¹ minimization
 PROC. NATL ACAD. SCI. USA 100 2197–202
, 2002
"... Given a ‘dictionary’ D = {dk} of vectors dk, we seek to represent a signal S as a linear combination S = ∑ k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considered ..."
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Cited by 632 (38 self)
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Given a ‘dictionary’ D = {dk} of vectors dk, we seek to represent a signal S as a linear combination S = ∑ k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considered the special case where D is an overcomplete system consisting of exactly two orthobases, and has shown that, under a condition of mutual incoherence of the two bases, and assuming that S has a sufficiently sparse representation, this representation is unique and can be found by solving a convex optimization problem: specifically, minimizing the ℓ¹ norm of the coefficients γ. In this paper, we obtain parallel results in a more general setting, where the dictionary D can arise from two or several bases, frames, or even less structured systems. We introduce the Spark, ameasure of linear dependence in such a system; it is the size of the smallest linearly dependent subset (dk). We show that, when the signal S has a representation using less than Spark(D)/2 nonzeros, this representation is necessarily unique. We
Efficient Implementation of the KSVD Algorithm using Batch Orthogonal Matching Pursuit
"... The KSVD algorithm is a highly effective method of training overcomplete dictionaries for sparse signal representation. In this report we discuss an efficient implementation of this algorithm, which both accelerates it and reduces its memory consumption. The two basic components of our implementati ..."
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Cited by 52 (1 self)
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The KSVD algorithm is a highly effective method of training overcomplete dictionaries for sparse signal representation. In this report we discuss an efficient implementation of this algorithm, which both accelerates it and reduces its memory consumption. The two basic components of our implementation are the replacement of the exact SVD computation with a much quicker approximation, and the use of the BatchOMP method for performing the sparsecoding operations. BatchOMP, which we also present in this report, is an implementation of the Orthogonal Matching Pursuit (OMP) algorithm which is specifically optimized for sparsecoding large sets of signals over the same dictionary. The BatchOMP implementation is useful for a variety of sparsitybased techniques which involve coding large numbers of signals. In the report, we discuss the BatchOMP and KSVD implementations and analyze their complexities. The report is accompanied by Matlab Ⓡ toolboxes which implement these techniques, and can be downloaded at
Toward Fast Transform Learning
, 2013
"... The dictionary learning problem aims at finding a dictionary of atoms that best represents an image according to a given objective. The most usual objective consists of representing an image or a class of images sparsely. Most algorithms performing dictionary learning iteratively estimate the dictio ..."
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Cited by 3 (2 self)
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The dictionary learning problem aims at finding a dictionary of atoms that best represents an image according to a given objective. The most usual objective consists of representing an image or a class of images sparsely. Most algorithms performing dictionary learning iteratively estimate the dictionary and a sparse representation of images using this dictionary. Dictionary learning has led to many state of the art algorithms in image processing. However, its numerical complexity restricts its use to atoms with a small support since the computations using the constructed dictionaries require too much resources to be deployed for large scale applications. In order to alleviate these issues, this paper introduces a new strategy to learn dictionaries composed of atoms obtained as a composition of K convolutions with Ssparse kernels. The dictionary update step associated with this strategy is a nonconvex optimization problem. We reformulate the problem in order to reduce the number of its irrelevant stationary points and introduce a GaussSeidel type algorithm, referred to as Alternative Least Square Algorithm, for its resolution. The search space of the considered optimization problem is of dimension KS, which is typically smaller than the size of the target atom and is much smaller than the size of the image. The complexity of the algorithm is linear with regard to the size of the image. Our experiments show that we are able to approximate with a very high accuracy many atoms such as modified DCT, curvelets, sinc functions or cosines when K is large (say K = 10). We also argue empirically that, maybe surprisingly, the algorithm generally converges to a global minimum for large values of K and S.
Uncovering Structure . . . : Networks and Multitask Learning Problems
, 2013
"... Extracting knowledge and providing insights into complex mechanisms underlying noisy highdimensional data sets is of utmost importance in many scientific domains. Statistical modeling has become ubiquitous in the analysis of high dimensional functional data in search of better understanding of cogn ..."
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Extracting knowledge and providing insights into complex mechanisms underlying noisy highdimensional data sets is of utmost importance in many scientific domains. Statistical modeling has become ubiquitous in the analysis of high dimensional functional data in search of better understanding of cognition mechanisms, in the exploration of largescale gene regulatory networks in hope of developing drugs for lethal diseases, and in prediction of volatility in stock market in hope of beating the market. Statistical analysis in these highdimensional data sets is possible only if an estimation procedure exploits hidden structures underlying data. This thesis develops flexible estimation procedures with provable theoretical guarantees for uncovering unknown hidden structures underlying data generating process. Of particular interest are procedures that can be used on high dimensional data sets where the number of samplesnis much smaller than the ambient dimension p. Learning in highdimensions is difficult due to the curse of dimensionality, however,
1 Computing Sparse Representations of Multidimensional Signals Using Kronecker Bases
"... Recently, there is a great interest in sparse representations of signals under the assumption that signals (datasets) can be well approximated by a linear combination of few elements of a known basis (dictionary). Many algorithms have been developed to find such kind of representations for the case ..."
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Recently, there is a great interest in sparse representations of signals under the assumption that signals (datasets) can be well approximated by a linear combination of few elements of a known basis (dictionary). Many algorithms have been developed to find such kind of representations for the case of onedimensional signals (vectors) which involves to find the sparsest solution of an underdetermined linear system of algebraic equations. In this paper, we generalize the theory of sparse representations of vectors to multiway arrays (tensors), i.e. signals with a multidimensional structure, by using the Tucker model. Thus, the problem is reduced to solve a largescale underdetermined linear system of equations possessing a Kronecker structure, for which we have developed a greedy algorithm called KroneckerOMP as a generalization of the classical Orthogonal Matching Pursuit (OMP) algorithm for vectors. We also introduce the concept of multiway blocksparse representation of Nway arrays and develop a new greedy algorithm that exploits not only the Kronecker structure but also blocksparsity. This allows us to derive a very fast and memory efficient algorithm called NBOMP (Nway Block OMP).