Sublinear-time quadratic minimization via spectral decomposition of matrices

Amit Levi; Yuichi Yoshida

doi:10.4230/LIPIcs.APPROX-RANDOM.2018.17

Sublinear-time quadratic minimization via spectral decomposition of matrices

Amit Levi, Yuichi Yoshida

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix with a polylogarithmic number of blocks, such that in each block all the entries are the same, and the pseudorandom part has a small spectral norm, achieving better error bound than the existing decomposition theorem of Frieze and Kannan (FOCS'96). As an additional application of the decomposition theorem, we give a sublinear-time approximation algorithm for computing the top singular values of a matrix.

Original language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018
Editors	Eric Blais, Jose D. P. Rolim, David Steurer, Klaus Jansen
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print)	9783959770859
DOIs	https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.17
State	Published - 1 Aug 2018
Externally published	Yes
Event	21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 - Princeton, United States Duration: 20 Aug 2018 → 22 Aug 2018

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	116
ISSN (Print)	1868-8969

Conference

Conference	21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018
Country/Territory	United States
City	Princeton
Period	20/08/18 → 22/08/18

Bibliographical note

Publisher Copyright:
© 2018 Aditya Bhaskara and Srivatsan Kumar.

Keywords

Approximation Algorithms
Graph limits
Matrix spectral decomposition
Qudratic function minimization

ASJC Scopus subject areas

Software

Access to Document

10.4230/LIPIcs.APPROX-RANDOM.2018.17

Cite this

Levi, A., & Yoshida, Y. (2018). Sublinear-time quadratic minimization via spectral decomposition of matrices. In E. Blais, J. D. P. Rolim, D. Steurer, & K. Jansen (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018 Article 17 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 116). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.17

Levi, Amit ; Yoshida, Yuichi. / Sublinear-time quadratic minimization via spectral decomposition of matrices. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018. editor / Eric Blais ; Jose D. P. Rolim ; David Steurer ; Klaus Jansen. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018. (Leibniz International Proceedings in Informatics, LIPIcs).

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abstract = "We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix with a polylogarithmic number of blocks, such that in each block all the entries are the same, and the pseudorandom part has a small spectral norm, achieving better error bound than the existing decomposition theorem of Frieze and Kannan (FOCS'96). As an additional application of the decomposition theorem, we give a sublinear-time approximation algorithm for computing the top singular values of a matrix.",

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author = "Amit Levi and Yuichi Yoshida",

note = "Publisher Copyright: {\textcopyright} 2018 Aditya Bhaskara and Srivatsan Kumar.; 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 ; Conference date: 20-08-2018 Through 22-08-2018",

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Levi, A & Yoshida, Y 2018, Sublinear-time quadratic minimization via spectral decomposition of matrices. in E Blais, JDP Rolim, D Steurer & K Jansen (eds), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018., 17, Leibniz International Proceedings in Informatics, LIPIcs, vol. 116, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018, Princeton, United States, 20/08/18. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.17

Sublinear-time quadratic minimization via spectral decomposition of matrices. / Levi, Amit; Yoshida, Yuichi.
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018. ed. / Eric Blais; Jose D. P. Rolim; David Steurer; Klaus Jansen. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018. 17 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 116).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Sublinear-time quadratic minimization via spectral decomposition of matrices

AU - Levi, Amit

AU - Yoshida, Yuichi

PY - 2018/8/1

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N2 - We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix with a polylogarithmic number of blocks, such that in each block all the entries are the same, and the pseudorandom part has a small spectral norm, achieving better error bound than the existing decomposition theorem of Frieze and Kannan (FOCS'96). As an additional application of the decomposition theorem, we give a sublinear-time approximation algorithm for computing the top singular values of a matrix.

AB - We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix with a polylogarithmic number of blocks, such that in each block all the entries are the same, and the pseudorandom part has a small spectral norm, achieving better error bound than the existing decomposition theorem of Frieze and Kannan (FOCS'96). As an additional application of the decomposition theorem, we give a sublinear-time approximation algorithm for computing the top singular values of a matrix.

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SN - 9783959770859

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BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018

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Levi A, Yoshida Y. Sublinear-time quadratic minimization via spectral decomposition of matrices. In Blais E, Rolim JDP, Steurer D, Jansen K, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2018. 17. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.APPROX-RANDOM.2018.17

Sublinear-time quadratic minimization via spectral decomposition of matrices

Abstract

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Conference

Bibliographical note

Keywords

ASJC Scopus subject areas

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