2 Search Results for "Franks, W. Cole"

Document

DOI: 10.4230/LIPIcs.CCC.2024.14

Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture

Authors: Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract

When a group acts on a set, it naturally partitions it into orbits, giving rise to orbit problems. These are natural algorithmic problems, as symmetries are central in numerous questions and structures in physics, mathematics, computer science, optimization, and more. Accordingly, it is of high interest to understand their computational complexity. Recently, Bürgisser et al. (2021) gave the first polynomial-time algorithms for orbit problems of torus actions, that is, actions of commutative continuous groups on Euclidean space. In this work, motivated by theoretical and practical applications, we study the computational complexity of robust generalizations of these orbit problems, which amount to approximating the distance of orbits in ℂⁿ up to a factor γ ≥ 1. In particular, this allows deciding whether two inputs are approximately in the same orbit or far from being so. On the one hand, we prove the NP-hardness of this problem for γ = n^Ω(1/log log n) by reducing the closest vector problem for lattices to it. On the other hand, we describe algorithms for solving this problem for an approximation factor γ = exp(poly(n)). Our algorithms combine tools from invariant theory and algorithmic lattice theory, and they also provide group elements witnessing the proximity of the given orbits (in contrast to the algebraic algorithms of prior work). We prove that they run in polynomial time if and only if a version of the famous number-theoretic abc-conjecture holds - establishing a new and surprising connection between computational complexity and number theory.

Cite as

Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson. Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 14:1-14:48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{burgisser_et_al:LIPIcs.CCC.2024.14,
  author =	{B\"{u}rgisser, Peter and Do\u{g}an, Mahmut Levent and Makam, Visu and Walter, Michael and Wigderson, Avi},
  title =	{{Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{14:1--14:48},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.14},
  URN =		{urn:nbn:de:0030-drops-204100},
  doi =		{10.4230/LIPIcs.CCC.2024.14},
  annote =	{Keywords: computational invariant theory, geometric complexity theory, orbit problems, abc-conjecture, closest vector problem}
}

Document

DOI: 10.4230/LIPIcs.CCC.2021.13

Barriers for Recent Methods in Geodesic Optimization

Authors: W. Cole Franks and Philipp Reichenbach

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract

We study a class of optimization problems including matrix scaling, matrix balancing, multidimensional array scaling, operator scaling, and tensor scaling that arise frequently in theory and in practice. Some of these problems, such as matrix and array scaling, are convex in the Euclidean sense, but others such as operator scaling and tensor scaling are geodesically convex on a different Riemannian manifold. Trust region methods, which include box-constrained Newton’s method, are known to produce high precision solutions very quickly for matrix scaling and matrix balancing (Cohen et. al., FOCS 2017, Allen-Zhu et. al. FOCS 2017), and result in polynomial time algorithms for some geodesically convex problems like operator scaling (Garg et. al. STOC 2018, Bürgisser et. al. FOCS 2019). One is led to ask whether these guarantees also hold for multidimensional array scaling and tensor scaling. We show that this is not the case by exhibiting instances with exponential diameter bound: we construct polynomial-size instances of 3-dimensional array scaling and 3-tensor scaling whose approximate solutions all have doubly exponential condition number. Moreover, we study convex-geometric notions of complexity known as margin and gap, which are used to bound the running times of all existing optimization algorithms for such problems. We show that margin and gap are exponentially small for several problems including array scaling, tensor scaling and polynomial scaling. Our results suggest that it is impossible to prove polynomial running time bounds for tensor scaling based on diameter bounds alone. Therefore, our work motivates the search for analogues of more sophisticated algorithms, such as interior point methods, for geodesically convex optimization that do not rely on polynomial diameter bounds.

Cite as

W. Cole Franks and Philipp Reichenbach. Barriers for Recent Methods in Geodesic Optimization. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 13:1-13:54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{franks_et_al:LIPIcs.CCC.2021.13,
  author =	{Franks, W. Cole and Reichenbach, Philipp},
  title =	{{Barriers for Recent Methods in Geodesic Optimization}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{13:1--13:54},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.13},
  URN =		{urn:nbn:de:0030-drops-142879},
  doi =		{10.4230/LIPIcs.CCC.2021.13},
  annote =	{Keywords: Geodesically Convex Optimization, Weight Margin, Moment Polytope, Diameter Bounds, Tensor Scaling, Matrix Scaling}
}

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1 Bürgisser, Peter
1 Doğan, Mahmut Levent
1 Franks, W. Cole
1 Makam, Visu
1 Reichenbach, Philipp
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2 Theory of computation → Algebraic complexity theory
1 Computing methodologies → Algebraic algorithms
1 Computing methodologies → Combinatorial algorithms
1 Mathematics of computing → Combinatorics
1 Theory of computation → Continuous optimization

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1 Diameter Bounds
1 Geodesically Convex Optimization
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