A Matrix Hyperbolic Cosine Algorithm and Applications

Zouzias, Anastasios

doi:10.1007/978-3-642-31594-7_71

Cited by 36 publications

(33 citation statements)

References 43 publications

(78 reference statements)

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“…Similar results exist in [1]. We should also mention the work in [39], which corresponds to a derandomization of the randomized sampling algorithm in [13].…”

Section: Related Worksupporting

confidence: 66%

Provable deterministic leverage score sampling

Papailiopoulos¹,

Kyrillidis

Boutsidis

2014

Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining

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We explain theoretically a curious empirical phenomenon: "Approximating a matrix by deterministically selecting a subset of its columns with the corresponding largest leverage scores results in a good low-rank matrix surrogate". In this work, we provide a novel theoretical analysis of deterministic leverage score sampling. We show that such sampling can be provably as accurate as its randomized counterparts, if the leverage scores follow a moderately steep power-law decay. We support this power-law assumption by providing empirical evidence that such decay laws are abundant in real-world data sets. We then demonstrate empirically the performance of deterministic leverage score sampling, which many times matches or outperforms the state-of-the-art techniques.

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“…Similar results exist in [1]. We should also mention the work in [39], which corresponds to a derandomization of the randomized sampling algorithm in [13].…”

Section: Related Worksupporting

confidence: 66%

Provable deterministic leverage score sampling

Papailiopoulos¹,

Kyrillidis

Boutsidis

2014

Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining

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“…Their algorithm constructs a (1 ± )-spectral sparsifier of size O(n · poly(log n, −1 )) in nearly linear time. This result has seen several improvements in recent years [SS11,bHS16,Zou12,ZLO15]. The state of the art in the sequential model is an algorithm by Lee and Sun [LS15] that computes a (1 ± )-spectral sparsifier of size O(n −2 ) in nearly linear time.…”

Section: Dynamic Spectral Sparsifiermentioning

confidence: 99%

On Fully Dynamic Graph Sparsifiers

Abraham

Durfee

Koutis

et al. 2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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We initiate the study of fast dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after each update in the graph. Our three main results are as follows. First, we give a fully dynamic algorithm for maintaining a (1 ± )-spectral sparsifier with amortized update time poly(log n, −1 ). Second, we give a fully dynamic algorithm for maintaining a (1 ± )-cut sparsifier with worst-case update time poly(log n, −1 ). Both sparsifiers have size n · poly(log n, −1 ). Third, we apply our dynamic sparsifier algorithm to obtain a fully dynamic algorithm for maintaining a (1 + )-approximation to the value of the maximum flow in an unweighted, undirected, bipartite graph with amortized update time poly(log n, −1 ).

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“…Recently, Zouzias [26] made progress in improving the running time of the spectral sparsification result of [1]; can we get a similar improvement for the 2-set algorithms presented here? Or perhaps, can we trade off the running time with randomization in those algorithms?…”

Section: Frobenius Norm Approximation Note That a Lower Bound For Thmentioning

confidence: 99%

Near-Optimal Column-Based Matrix Reconstruction

Boutsidis¹,

Drineas²,

2014

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Abstract. We consider low-rank reconstruction of a matrix using a subset of its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are: (i) the use of fast approximate SVD-like decompositions for column-based matrix reconstruction, and (ii) two deterministic algorithms for selecting rows from matrices with orthonormal columns, building upon the sparse representation theorem for decompositions of the identity that appeared in [1].

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A Matrix Hyperbolic Cosine Algorithm and Applications

Cited by 36 publications

References 43 publications

Provable deterministic leverage score sampling

Provable deterministic leverage score sampling

On Fully Dynamic Graph Sparsifiers

Near-Optimal Column-Based Matrix Reconstruction

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