The Experts below are selected from a list of 99807 Experts worldwide ranked by ideXlab platform
Thomas Strohmer - One of the best experts on this subject based on the ideXlab platform.
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a Performance Guarantee for spectral clustering
SIAM Journal on Mathematics of Data Science, 2021Co-Authors: March T. Boedihardjo, Shaofeng Deng, Thomas StrohmerAbstract:The two-step spectral clustering method, which consists of the Laplacian eigenmap and a rounding step, is a widely used method for graph partitioning. It can be seen as a natural relaxation to the ...
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A Performance Guarantee for Spectral Clustering.
arXiv: Machine Learning, 2020Co-Authors: March T. Boedihardjo, Shaofeng Deng, Thomas StrohmerAbstract:The two-step spectral clustering method, which consists of the Laplacian eigenmap and a rounding step, is a widely used method for graph partitioning. It can be seen as a natural relaxation to the NP-hard minimum ratio cut problem. In this paper we study the central question: when is spectral clustering able to find the global solution to the minimum ratio cut problem? First we provide a condition that naturally depends on the intra- and inter-cluster connectivities of a given partition under which we may certify that this partition is the solution to the minimum ratio cut problem. Then we develop a deterministic two-to-infinity norm perturbation bound for the the invariant subspace of the graph Laplacian that corresponds to the $k$ smallest eigenvalues. Finally by combining these two results we give a condition under which spectral clustering is Guaranteed to output the global solution to the minimum ratio cut problem, which serves as a Performance Guarantee for spectral clustering.
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Certifying Global Optimality of Graph Cuts via Semidefinite Relaxation: A Performance Guarantee for Spectral Clustering
Foundations of Computational Mathematics, 2019Co-Authors: Shuyang Ling, Thomas StrohmerAbstract:Spectral clustering has become one of the most widely used clustering techniques when the structure of the individual clusters is non-convex or highly anisotropic. Yet, despite its immense popularity, there exists fairly little theory about Performance Guarantees for spectral clustering. This issue is partly due to the fact that spectral clustering typically involves two steps which complicated its theoretical analysis: First, the eigenvectors of the associated graph Laplacian are used to embed the dataset, and second, k-means clustering algorithm is applied to the embedded dataset to get the labels. This paper is devoted to the theoretical foundations of spectral clustering and graph cuts. We consider a convex relaxation of graph cuts, namely ratio cuts and normalized cuts, that makes the usual two-step approach of spectral clustering obsolete and at the same time gives rise to a rigorous theoretical analysis of graph cuts and spectral clustering. We derive deterministic bounds for successful spectral clustering via a spectral proximity condition that naturally depends on the algebraic connectivity of each cluster and the inter-cluster connectivity. Moreover, we demonstrate by means of some popular examples that our bounds can achieve near optimality. Our findings are also fundamental to the theoretical understanding of kernel k-means. Numerical simulations confirm and complement our analysis.
Jwoyuh Wu - One of the best experts on this subject based on the ideXlab platform.
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an improved rip based Performance Guarantee for sparse signal recovery via orthogonal matching pursuit
IEEE Transactions on Information Theory, 2014Co-Authors: Linghua Chang, Jwoyuh WuAbstract:A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit (OMP) in K iterations (when there is no noise) is that the restricted isometry constant (RIC) of the sensing matrix satisfies δ K+1 <; (1/√(K) + 1). In the noisy case, this RIC upper bound along with a requirement on the minimal signal entry magnitude is known to Guarantee exact support identification. In this paper, we show that, in the presence of noise, a relaxed RIC upper bound δ K+1 <; (√(4K + 1) - 1/2K) together with a relaxed requirement on the minimal signal entry magnitude suffices to achieve perfect support identification using OMP. In the noiseless case, our result asserts that such a relaxed RIC upper bound can ensure exact support recovery in K iterations: this narrows the gap between the so far best known bound δ K+1 <; (1/√(K( + 1)) and the ultimate Performance Guarantee δ K+1 = (1/(K)). Our approach relies on a newly established near orthogonality condition, characterized via the achievable angles between two orthogonal sparse vectors upon compression, and, thus, better exploits the knowledge about the geometry of the compressed space. The proposed near orthogonality condition can be also exploited to derive less restricted sufficient conditions for signal reconstruction in two other compressive sensing problems, namely, compressive domain interference cancellation and support identification via the subspace pursuit algorithm.
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an improved rip based Performance Guarantee for sparse signal recovery via orthogonal matching pursuit
International Symposium on Communications Control and Signal Processing, 2014Co-Authors: Linghua Chang, Jwoyuh WuAbstract:A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit in K iterations is that the restricted isometry constant of the sensing matrix satisfies δ K+1 <; 1/(√K + 1). By exploiting a “near orthogonality” condition specified in terms of the achievable angles between two orthogonal sparse vectors upon compression, this paper shows that the requirement on δ K+1 can be further relaxed to δ k+1 <; √4k+1 - 1\2K. This result thus narrows the gap between the so far best known bound and the ultimate Performance Guarantee δ K+1 <; 1/√K that is conjectured by Dai and Milenkovic in 2009.
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an improved rip based Performance Guarantee for sparse signal recovery via orthogonal matching pursuit
arXiv: Information Theory, 2014Co-Authors: Linghua Chang, Jwoyuh WuAbstract:A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit (OMP) in K iterations is that the restricted isometry constant of the sensing matrix satisfies delta_K+1<1/(sqrt(delta_K+1)+1). By exploiting an approximate orthogonality condition characterized via the achievable angles between two orthogonal sparse vectors upon compression, this paper shows that the upper bound on delta can be further relaxed to delta_K+1<(sqrt(1+4*delta_K+1)-1)/(2K).This result thus narrows the gap between the so far best known bound and the ultimate Performance Guarantee delta_K+1<1/(sqrt(delta_K+1)) that is conjectured by Dai and Milenkovic in 2009. The proposed approximate orthogonality condition is also exploited to derive less restricted sufficient conditions for signal reconstruction in several compressive sensing problems, including signal recovery via OMP in a noisy environment, compressive domain interference cancellation, and support identification via the subspace pursuit algorithm.
Linghua Chang - One of the best experts on this subject based on the ideXlab platform.
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an improved rip based Performance Guarantee for sparse signal recovery via orthogonal matching pursuit
IEEE Transactions on Information Theory, 2014Co-Authors: Linghua Chang, Jwoyuh WuAbstract:A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit (OMP) in K iterations (when there is no noise) is that the restricted isometry constant (RIC) of the sensing matrix satisfies δ K+1 <; (1/√(K) + 1). In the noisy case, this RIC upper bound along with a requirement on the minimal signal entry magnitude is known to Guarantee exact support identification. In this paper, we show that, in the presence of noise, a relaxed RIC upper bound δ K+1 <; (√(4K + 1) - 1/2K) together with a relaxed requirement on the minimal signal entry magnitude suffices to achieve perfect support identification using OMP. In the noiseless case, our result asserts that such a relaxed RIC upper bound can ensure exact support recovery in K iterations: this narrows the gap between the so far best known bound δ K+1 <; (1/√(K( + 1)) and the ultimate Performance Guarantee δ K+1 = (1/(K)). Our approach relies on a newly established near orthogonality condition, characterized via the achievable angles between two orthogonal sparse vectors upon compression, and, thus, better exploits the knowledge about the geometry of the compressed space. The proposed near orthogonality condition can be also exploited to derive less restricted sufficient conditions for signal reconstruction in two other compressive sensing problems, namely, compressive domain interference cancellation and support identification via the subspace pursuit algorithm.
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an improved rip based Performance Guarantee for sparse signal recovery via orthogonal matching pursuit
International Symposium on Communications Control and Signal Processing, 2014Co-Authors: Linghua Chang, Jwoyuh WuAbstract:A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit in K iterations is that the restricted isometry constant of the sensing matrix satisfies δ K+1 <; 1/(√K + 1). By exploiting a “near orthogonality” condition specified in terms of the achievable angles between two orthogonal sparse vectors upon compression, this paper shows that the requirement on δ K+1 can be further relaxed to δ k+1 <; √4k+1 - 1\2K. This result thus narrows the gap between the so far best known bound and the ultimate Performance Guarantee δ K+1 <; 1/√K that is conjectured by Dai and Milenkovic in 2009.
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an improved rip based Performance Guarantee for sparse signal recovery via orthogonal matching pursuit
arXiv: Information Theory, 2014Co-Authors: Linghua Chang, Jwoyuh WuAbstract:A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit (OMP) in K iterations is that the restricted isometry constant of the sensing matrix satisfies delta_K+1<1/(sqrt(delta_K+1)+1). By exploiting an approximate orthogonality condition characterized via the achievable angles between two orthogonal sparse vectors upon compression, this paper shows that the upper bound on delta can be further relaxed to delta_K+1<(sqrt(1+4*delta_K+1)-1)/(2K).This result thus narrows the gap between the so far best known bound and the ultimate Performance Guarantee delta_K+1<1/(sqrt(delta_K+1)) that is conjectured by Dai and Milenkovic in 2009. The proposed approximate orthogonality condition is also exploited to derive less restricted sufficient conditions for signal reconstruction in several compressive sensing problems, including signal recovery via OMP in a noisy environment, compressive domain interference cancellation, and support identification via the subspace pursuit algorithm.
Shlomi Rubinstein - One of the best experts on this subject based on the ideXlab platform.
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an approximation algorithm for maximum triangle packing
Discrete Applied Mathematics, 2006Co-Authors: Refael Hassin, Shlomi RubinsteinAbstract:We present a randomized (89/169 - e)-approximation algorithm for the weighted maximum triangle packing problem, for any given e > 0. This is the first algorithm for this problem whose Performance Guarantee is better than ½. The algorithm also improves the best-known approximation bound for the maximum 2-edge path packing problem.
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an approximation algorithm for maximum triangle packing
European Symposium on Algorithms, 2004Co-Authors: Refael Hassin, Shlomi RubinsteinAbstract:We present a randomized \(\left({89\over169}-\epsilon\right)\)-approximation algorithm for the weighted maximum triangle packing problem, for any given e> 0. This is the first algorithm for this problem whose Performance Guarantee is better than \({1\over2}\). The algorithm also improves the best known approximation bound for the maximum 2-edge path packing problem.
Shuyang Ling - One of the best experts on this subject based on the ideXlab platform.
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Certifying Global Optimality of Graph Cuts via Semidefinite Relaxation: A Performance Guarantee for Spectral Clustering
Foundations of Computational Mathematics, 2019Co-Authors: Shuyang Ling, Thomas StrohmerAbstract:Spectral clustering has become one of the most widely used clustering techniques when the structure of the individual clusters is non-convex or highly anisotropic. Yet, despite its immense popularity, there exists fairly little theory about Performance Guarantees for spectral clustering. This issue is partly due to the fact that spectral clustering typically involves two steps which complicated its theoretical analysis: First, the eigenvectors of the associated graph Laplacian are used to embed the dataset, and second, k-means clustering algorithm is applied to the embedded dataset to get the labels. This paper is devoted to the theoretical foundations of spectral clustering and graph cuts. We consider a convex relaxation of graph cuts, namely ratio cuts and normalized cuts, that makes the usual two-step approach of spectral clustering obsolete and at the same time gives rise to a rigorous theoretical analysis of graph cuts and spectral clustering. We derive deterministic bounds for successful spectral clustering via a spectral proximity condition that naturally depends on the algebraic connectivity of each cluster and the inter-cluster connectivity. Moreover, we demonstrate by means of some popular examples that our bounds can achieve near optimality. Our findings are also fundamental to the theoretical understanding of kernel k-means. Numerical simulations confirm and complement our analysis.
