Results 1  10
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16
Approximation of functions over redundant dictionaries using coherence
 Proc. of SODA
, 2003
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On the exponential convergence of matching pursuit in quasicoherent dictionaries
 IEEE TRANS. INFORMATION TH
, 2006
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Vector Greedy Algorithms
"... Our objective is to study nonlinear approximation with regard to redundant systems. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. Greedy type approximati ..."
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Cited by 51 (8 self)
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Our objective is to study nonlinear approximation with regard to redundant systems. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. Greedy type approximations proved to be convenient and efficient ways of constructing mterm approximants. We introduce and study vector greedy algorithms that are designed with aim of constructing mth greedy approximants simultaneously for a given finite number of elements. We prove convergence theorems and obtain some estimates for the rate of convergence of vector greedy algorithms when elements come from certain classes.
Algorithms for Subset Selection in Linear Regression
 STOC'08
, 2008
"... We study the problem of selecting a subset of k random variables to observe that will yield the best linear prediction of another variable of interest, given the pairwise correlations between the observation variables and the predictor variable. Under approximation preserving reductions, this proble ..."
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Cited by 30 (3 self)
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We study the problem of selecting a subset of k random variables to observe that will yield the best linear prediction of another variable of interest, given the pairwise correlations between the observation variables and the predictor variable. Under approximation preserving reductions, this problem is also equivalent to the“sparse approximation”problem of approximating signals concisely. We propose and analyze exact and approximation algorithms for several special cases of practical interest. We give an FPTAS when the covariance matrix has constant bandwidth, and exact algorithms when the associated covariance graph, consisting of edges for pairs of variables with nonzero correlation, forms a tree or has a large (known) independent set. Furthermore, we give an exact algorithm when the variables can be embedded into a line such that the covariance decreases exponentially in the distance, and a constantfactor approximation when the variables have no “conditional suppressor variables”. Much of our reasoning is based on perturbation results for the R 2 multiple correlation measure, frequently used as a measure for “goodnessoffit statistics”. It lies at the core of our FPTAS, and also allows us to extend exact algorithms to approximation algorithms when the matrix “nearly ” falls into one of the above classes. We also use perturbation analysis to prove approximation guarantees for the widely used “Forward Regression ” heuristic when the observation variables are nearly independent.
Simultaneous approximation by greedy algorithms
, 2003
"... Abstract. We study nonlinear mterm approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f ∈ H and any dictionary D an expansion into a series f = cj(f)ϕj(f), ϕj(f) ∈ ..."
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Cited by 17 (1 self)
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Abstract. We study nonlinear mterm approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f ∈ H and any dictionary D an expansion into a series f = cj(f)ϕj(f), ϕj(f) ∈ D, j = 1, 2,... j=1 with the Parseval property: ‖f ‖ 2 = ∑ j cj(f)  2. Following the paper of A. Lutoborski and the second author [21] we study analogs of the above expansions for a given finite number of functions f 1,..., f N with a requirement that the dictionary elements ϕj of these expansions are the same for all f i, i = 1,..., N. We study convergence and rate of convergence of such expansions which we call simultanious expansions. 1.
Optimised Orthogonal Matching Pursuit Approach
 IEEE Signal Processing Letters, Vol 9
, 2002
"... A recursive approach for shrinking coefficients of an atomic decomposition is proposed. The corresponding algorithm evolves so as to provide at each iteration a) the orthogonal projection of a signal onto a reduced subspace and b) the index of the coefficient to be disregarded in order to construct ..."
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Cited by 14 (8 self)
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A recursive approach for shrinking coefficients of an atomic decomposition is proposed. The corresponding algorithm evolves so as to provide at each iteration a) the orthogonal projection of a signal onto a reduced subspace and b) the index of the coefficient to be disregarded in order to construct a coarser approximation minimizing the norm of the residual error. EDICS Category: 1TFSR. 1
Sparse Representations are Most Likely to be the Sparsest Possible
 EURASIP Journal on Applied Signal Processing, Paper No. 96247
, 2004
"... and a full rank matrix D with N < L, we define the signal's overcomplete representations as all # satisfying S = D#. Among all the possible solutions, we have special interest in the sparsest one  the one minimizing 0 . Previous work has established that a representation is unique if i ..."
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Cited by 12 (2 self)
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and a full rank matrix D with N < L, we define the signal's overcomplete representations as all # satisfying S = D#. Among all the possible solutions, we have special interest in the sparsest one  the one minimizing 0 . Previous work has established that a representation is unique if it is sparse enough, requiring 0 < Spark(D)/2.
Greedy Basis Pursuit
, 2006
"... We introduce Greedy Basis Pursuit (GBP), a new algorithm for computing signal representations using overcomplete dictionaries. GBP is rooted in computational geometry and exploits an equivalence between minimizing the ℓ 1norm of the representation coefficients and determining the intersection of th ..."
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Cited by 7 (0 self)
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We introduce Greedy Basis Pursuit (GBP), a new algorithm for computing signal representations using overcomplete dictionaries. GBP is rooted in computational geometry and exploits an equivalence between minimizing the ℓ 1norm of the representation coefficients and determining the intersection of the signal with the convex hull of the dictionary. GBP unifies the different advantages of previous algorithms: like standard approaches to Basis Pursuit, GBP computes representations that have minimum ℓ 1norm; like greedy algorithms such as Matching Pursuit, GBP builds up representations, sequentially selecting atoms. We describe the algorithm, demonstrate its performance, and provide code. Experiments show that GBP can provide a fast alternative to standard linear programming approaches to Basis Pursuit.
On approximation of orientation distributions by means of spherical ridgelets
 In Proc. IEEE International Symposium on Biomedical Imaging: from Nano to Macro. IEEE
, 2008
"... Visualization and analysis of the microarchitecture of brain parenchyma by means of magnetic resonance imaging is nowadays believed to be one of the most powerful tools used for the assessment of various cerebral conditions as well as for understanding the intracerebral connectivity. Unfortunately, ..."
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Cited by 7 (0 self)
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Visualization and analysis of the microarchitecture of brain parenchyma by means of magnetic resonance imaging is nowadays believed to be one of the most powerful tools used for the assessment of various cerebral conditions as well as for understanding the intracerebral connectivity. Unfortunately, the conventional diffusion tensor imaging (DTI) used for estimating the local orientations of neural fibers, is incapable of performing reliably in the situations when a voxel of interest accommodates multiple fiber tracts. In this case, a much more accurate analysis is possible using the high angular resolution diffusion imaging (HARDI) that represents local diffusion by its apparent coefficients measured as a discrete function of spatial orientations. In this note, a novel approach to enhancing and modeling the HARDI signals using multiresolution bases of spherical ridgelets is presented. In addition to its desirable properties of being adaptive, sparsifying, and efficiently computable, the proposed modeling leads to analytical computation of the orientation distribution functions associated with the measured diffusion, thereby providing a fast and robust analytical solution for qball imaging.