2 Search Results for "Jiang, Shunhua"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.10
The Bit Complexity of Dynamic Algebraic Formulas and Their Determinants

Authors: Emile Anand, Jan van den Brand, Mehrdad Ghadiri, and Daniel J. Zhang

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Many iterative algorithms in computer science require repeated computation of some algebraic expression whose input varies slightly from one iteration to the next. Although efficient data structures have been proposed for maintaining the solution of such algebraic expressions under low-rank updates, most of these results are only analyzed under exact arithmetic (real-RAM model and finite fields) which may not accurately reflect the more limited complexity guarantees of real computers. In this paper, we analyze the stability and bit complexity of such data structures for expressions that involve the inversion, multiplication, addition, and subtraction of matrices under the word-RAM model. We show that the bit complexity only increases linearly in the number of matrix operations in the expression. In addition, we consider the bit complexity of maintaining the determinant of a matrix expression. We show that the required bit complexity depends on the logarithm of the condition number of matrices instead of the logarithm of their determinant. Finally, we discuss rank maintenance and its connections to determinant maintenance. Our results have wide applications ranging from computational geometry (e.g., computing the volume of a polytope) to optimization (e.g., solving linear programs using the simplex algorithm).
Cite as

Emile Anand, Jan van den Brand, Mehrdad Ghadiri, and Daniel J. Zhang. The Bit Complexity of Dynamic Algebraic Formulas and Their Determinants. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{anand_et_al:LIPIcs.ICALP.2024.10,
  author =	{Anand, Emile and van den Brand, Jan and Ghadiri, Mehrdad and Zhang, Daniel J.},
  title =	{{The Bit Complexity of Dynamic Algebraic Formulas and Their Determinants}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.10},
  URN =		{urn:nbn:de:0030-drops-201538},
  doi =		{10.4230/LIPIcs.ICALP.2024.10},
  annote =	{Keywords: Data Structures, Online Algorithms, Bit Complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.79
A Faster Interior-Point Method for Sum-Of-Squares Optimization

Authors: Shunhua Jiang, Bento Natura, and Omri Weinstein

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
We present a faster interior-point method for optimizing sum-of-squares (SOS) polynomials, which are a central tool in polynomial optimization and capture convex programming in the Lasserre hierarchy. Let p = ∑_i q²_i be an n-variate SOS polynomial of degree 2d. Denoting by L : = binom(n+d,d) and U : = binom(n+2d,2d) the dimensions of the vector spaces in which q_i’s and p live respectively, our algorithm runs in time Õ(LU^{1.87}). This is polynomially faster than state-of-art SOS and semidefinite programming solvers [Jiang et al., 2020; Huang et al., 2021; Papp and Yildiz, 2019], which achieve runtime Õ(L^{0.5} min{U^{2.37}, L^{4.24}}). The centerpiece of our algorithm is a dynamic data structure for maintaining the inverse of the Hessian of the SOS barrier function under the polynomial interpolant basis [Papp and Yildiz, 2019], which efficiently extends to multivariate SOS optimization, and requires maintaining spectral approximations to low-rank perturbations of elementwise (Hadamard) products. This is the main challenge and departure from recent IPM breakthroughs using inverse-maintenance, where low-rank updates to the slack matrix readily imply the same for the Hessian matrix.
Cite as

Shunhua Jiang, Bento Natura, and Omri Weinstein. A Faster Interior-Point Method for Sum-Of-Squares Optimization. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 79:1-79:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{jiang_et_al:LIPIcs.ICALP.2022.79,
  author =	{Jiang, Shunhua and Natura, Bento and Weinstein, Omri},
  title =	{{A Faster Interior-Point Method for Sum-Of-Squares Optimization}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{79:1--79:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.79},
  URN =		{urn:nbn:de:0030-drops-164205},
  doi =		{10.4230/LIPIcs.ICALP.2022.79},
  annote =	{Keywords: Interior Point Methods, Sum-of-squares Optimization, Dynamic Matrix Inverse}
}
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