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183
Greed is Good: Algorithmic Results for Sparse Approximation
, 2004
"... This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho’s basis pursuit (BP) paradigm can recover the optimal representa ..."
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Cited by 520 (6 self)
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This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho’s basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasiincoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function is introduced to quantify the level of incoherence. This analysis unifies all the recent results on BP and extends them to OMP. Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP is an approximation algorithm for the sparse problem over a quasiincoherent dictionary. That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms.
Just Relax: Convex Programming Methods for Identifying Sparse Signals in Noise
, 2006
"... This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that ..."
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Cited by 293 (1 self)
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This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis.
Just relax: Convex programming methods for subset selection and sparse approximation
, 2004
"... Abstract. Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electric ..."
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Cited by 90 (4 self)
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Abstract. Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical engineering, applied mathematics and statistics, but small theoretical progress has been made over the last fifty years. Subset selection and sparse approximation both admit natural convex relaxations, but the literature contains few results on the behavior of these relaxations for general input signals. This report demonstrates that the solution of the convex program frequently coincides with the solution of the original approximation problem. The proofs depend essentially on geometric properties of the ensemble of elementary signals. The results are powerful because sparse approximation problems are combinatorial, while convex programs can be solved in polynomial time with standard software. Comparable new results for a greedy algorithm, Orthogonal Matching Pursuit, are also stated. This report should have a major practical impact because the theory applies immediately to many realworld signal processing problems. 1.
TENSOR RANK AND THE ILLPOSEDNESS OF THE BEST LOWRANK APPROXIMATION PROBLEM
"... There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, te ..."
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Cited by 69 (10 self)
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There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rankr approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders and ranks, regardless of the choice of norm (or even Brègman divergence). Moreover, we show that in many instances these counterexamples have positive volume: they cannot be regarded as isolated phenomena. In one extreme case, we exhibit a tensor space in which no rank3 tensor has an optimal rank2 approximation. The notable exceptions to this misbehavior are rank1 tensors and order2 tensors (i.e. matrices). In a more positive spirit, we propose a natural way of overcoming the illposedness of the lowrank approximation problem, by using weak solutions when true solutions do not exist. For this to work, it is necessary to characterize the set of weak solutions, and we do this in the case of rank 2, order 3 (in arbitrary dimensions). In our work we emphasize the importance of closely studying concrete lowdimensional examples as a first step towards more general results. To this end, we present a detailed analysis of equivalence classes of 2 × 2 × 2 tensors, and we develop methods for extending results upwards to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory be given a semialgebraic description; in other words, can be determined by a system of polynomial inequalities. We study some of these polynomials in cases of interest to us; in particular we make extensive use of the hyperdeterminant ∆ on R 2×2×2.
Learning Eigenfunctions Links Spectral Embedding And Kernel PCA
 NEURAL COMPUTATION
, 2004
"... In this paper, we show a direct relation between spectral embedding methods and kernel PCA, and how both are special cases of a more general learning problem, that of learning the principal eigenfunctions of an operator defined from a kernel and the unknown data generating density. Whereas ..."
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Cited by 65 (6 self)
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In this paper, we show a direct relation between spectral embedding methods and kernel PCA, and how both are special cases of a more general learning problem, that of learning the principal eigenfunctions of an operator defined from a kernel and the unknown data generating density. Whereas
KPCA plus LDA: a complete kernel Fisher discriminant framework for feature extraction and recognition
 IEEE Transactions on Pattern Analysis and Machine Intelligence
"... Abstract—This paper examines the theory of kernel Fisher discriminant analysis (KFD) in a Hilbert space and develops a twophase KFD framework, i.e., kernel principal component analysis (KPCA) plus Fisher linear discriminant analysis (LDA). This framework provides novel insights into the nature of K ..."
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Cited by 54 (4 self)
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Abstract—This paper examines the theory of kernel Fisher discriminant analysis (KFD) in a Hilbert space and develops a twophase KFD framework, i.e., kernel principal component analysis (KPCA) plus Fisher linear discriminant analysis (LDA). This framework provides novel insights into the nature of KFD. Based on this framework, the authors propose a complete kernel Fisher discriminant analysis (CKFD) algorithm. CKFD can be used to carry out discriminant analysis in “double discriminant subspaces. ” The fact that, it can make full use of two kinds of discriminant information, regular and irregular, makes CKFD a more powerful discriminator. The proposed algorithm was tested and evaluated using the FERET face database and the CENPARMI handwritten numeral database. The experimental results show that CKFD outperforms other KFD algorithms. Index Terms—Kernelbased methods, subspace methods, principal component analysis (PCA), Fisher linear discriminant analysis (LDA or FLD), feature extraction, machine learning, face recognition, handwritten digit recognition. æ 1
LocalityPreserving Hashing in Multidimensional Spaces
 In Proceedings of the 29th ACM Symposium on Theory of Computing
, 1997
"... this paper was published in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages 618625, 1997 ..."
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Cited by 50 (4 self)
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this paper was published in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages 618625, 1997
Matrix Radiance Transfer
, 2003
"... Precomputed Radiance Transfer allows interactive rendering of objects illuminated by lowfrequency environment maps, including selfshadowing and interreflections. The expensive integration of incident lighting is partially precomputed and stored as matrices.Incorporating anisotropic, glossy BRDFs i ..."
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Cited by 39 (4 self)
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Precomputed Radiance Transfer allows interactive rendering of objects illuminated by lowfrequency environment maps, including selfshadowing and interreflections. The expensive integration of incident lighting is partially precomputed and stored as matrices.Incorporating anisotropic, glossy BRDFs into precomputed radiance transfer has been previously shown to be possible, but none of the previous methods offer realtime performance. We propose a new method, matrix radiance transfer, which significantly speeds up exit radiance computation and allows anisotropic BRDFs. We generalize the previous radiance transfer methods to work with a matrix representation of the BRDF and optimize exit radiance computation by expressing the exit radiance in a new, directionally locally supported basis set instead of the spherical harmonics. To determine exit radiance, our method performs four dot products per vertex in contrast to previous methods, where a full matrixvector multiply is required. Image quality can be controlled by adapting the number of basis functions. We compress our radiance transfer matrices through principal component analysis (PCA). We show that it is possible to render directly from the PCA representation, which also enables the user to trade interactively between quality and speed.
Bounds for the entries of matrix functions with applications to preconditioning
 BIT
, 1999
"... Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a wellknown result of Demko, Moss and Smith on the decay of the inverse we show that when A is banded, the entries of f(A)are bounded in an exponentially decaying manner away ..."
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Cited by 33 (14 self)
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Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a wellknown result of Demko, Moss and Smith on the decay of the inverse we show that when A is banded, the entries of f(A)are bounded in an exponentially decaying manner away from the main diagonal. Bounds obtained by representing the entries of f(A) in terms of Riemann–Stieltjes integrals and by approximating such integrals by Gaussian quadrature rules are also considered. Applications of these bounds to preconditioning are suggested and illustrated by a few numerical examples.