Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing

Zeyuan Allen-Zhu; Ankit Garg; Yuanzhi Li; Rafael Oliveira; Avi Wigderson

doi:10.1145/3188745.3188942

Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing

Zeyuan Allen-Zhu, Ankit Garg, Yuanzhi Li, Rafael Oliveira, Avi Wigderson

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

49 Scopus citations

Abstract

We propose a new second-order method for geodesically convex optimization on the natural hyperbolic metric over positive definite matrices. We apply it to solve the operator scaling problem in time polynomial in the input size and logarithmic in the error. This is an exponential improvement over previous algorithms which were analyzed in the usual Euclidean, “commutative” metric (for which the above problem is not convex). Our method is general and applicable to other settings. As a consequence, we solve the equivalence problem for the left-right group action underlying the operator scaling problem. This yields a deterministic polynomial-time algorithm for a new class of Polynomial Identity Testing (PIT) problems, which was the original motivation for studying operator scaling.

Original language	English
Title of host publication	STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
Editors	Monika Henzinger, David Kempe, Ilias Diakonikolas
Publisher	Association for Computing Machinery
Pages	44-50
Number of pages	7
ISBN (Electronic)	9781450355599
DOIs	https://doi.org/10.1145/3188745.3188942
State	Published - 20 Jun 2018
Externally published	Yes
Event	50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States Duration: 25 Jun 2018 → 29 Jun 2018

Publication series

Name	Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)	0737-8017

Conference

Conference	50th Annual ACM Symposium on Theory of Computing, STOC 2018
Country/Territory	United States
City	Los Angeles
Period	25/06/18 → 29/06/18

Bibliographical note

Publisher Copyright:
© 2018 Copyright held by the owner/author(s).

Keywords

Capacity
Geodesic convex
Operator scaling
Orbit-closure intersection
Positive definite

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10.1145/3188745.3188942

Cite this

Allen-Zhu, Z., Garg, A., Li, Y., Oliveira, R., & Wigderson, A. (2018). Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing. In M. Henzinger, D. Kempe, & I. Diakonikolas (Eds.), STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (pp. 44-50). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3188745.3188942

Allen-Zhu, Zeyuan ; Garg, Ankit ; Li, Yuanzhi et al. / Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing. STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. editor / Monika Henzinger ; David Kempe ; Ilias Diakonikolas. Association for Computing Machinery, 2018. pp. 44-50 (Proceedings of the Annual ACM Symposium on Theory of Computing).

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abstract = "We propose a new second-order method for geodesically convex optimization on the natural hyperbolic metric over positive definite matrices. We apply it to solve the operator scaling problem in time polynomial in the input size and logarithmic in the error. This is an exponential improvement over previous algorithms which were analyzed in the usual Euclidean, “commutative” metric (for which the above problem is not convex). Our method is general and applicable to other settings. As a consequence, we solve the equivalence problem for the left-right group action underlying the operator scaling problem. This yields a deterministic polynomial-time algorithm for a new class of Polynomial Identity Testing (PIT) problems, which was the original motivation for studying operator scaling.",

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Allen-Zhu, Z, Garg, A, Li, Y, Oliveira, R & Wigderson, A 2018, Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing. in M Henzinger, D Kempe & I Diakonikolas (eds), STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. Proceedings of the Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, pp. 44-50, 50th Annual ACM Symposium on Theory of Computing, STOC 2018, Los Angeles, United States, 25/06/18. https://doi.org/10.1145/3188745.3188942

Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing. / Allen-Zhu, Zeyuan; Garg, Ankit; Li, Yuanzhi et al.
STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. ed. / Monika Henzinger; David Kempe; Ilias Diakonikolas. Association for Computing Machinery, 2018. p. 44-50 (Proceedings of the Annual ACM Symposium on Theory of Computing).

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

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Allen-Zhu Z, Garg A, Li Y, Oliveira R, Wigderson A. Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing. In Henzinger M, Kempe D, Diakonikolas I, editors, STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. Association for Computing Machinery. 2018. p. 44-50. (Proceedings of the Annual ACM Symposium on Theory of Computing). doi: 10.1145/3188745.3188942

Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing

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Keywords

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