3 Search Results for "Derksen, Harm"

Document
DOI: 10.4230/LIPIcs.ITCS.2023.6
On Identity Testing and Noncommutative Rank Computation over the Free Skew Field

Authors: V. Arvind, Abhranil Chatterjee, Utsab Ghosal, Partha Mukhopadhyay, and C. Ramya

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
The identity testing of rational formulas (RIT) in the free skew field efficiently reduces to computing the rank of a matrix whose entries are linear polynomials in noncommuting variables [Hrubeš and Wigderson, 2015]. This rank computation problem has deterministic polynomial-time white-box algorithms [Ankit Garg et al., 2016; Ivanyos et al., 2018] and a randomized polynomial-time algorithm in the black-box setting [Harm Derksen and Visu Makam, 2017]. In this paper, we propose a new approach for efficient derandomization of black-box RIT. Additionally, we obtain results for matrix rank computation over the free skew field and construct efficient linear pencil representations for a new class of rational expressions. More precisely, we show: - Under the hardness assumption that the ABP (algebraic branching program) complexity of every polynomial identity for the k×k matrix algebra is 2^Ω(k) [Andrej Bogdanov and Hoeteck Wee, 2005], we obtain a subexponential-time black-box RIT algorithm for rational formulas of inversion height almost logarithmic in the size of the formula. This can be seen as the first "hardness implies derandomization" type theorem for rational formulas. - We show that the noncommutative rank of any matrix over the free skew field whose entries have small linear pencil representations can be computed in deterministic polynomial time. While an efficient rank computation was known for matrices with noncommutative formulas as entries [Ankit Garg et al., 2020], we obtain the first deterministic polynomial-time algorithms for rank computation of matrices whose entries are noncommutative ABPs or rational formulas. - Motivated by the definition given by Bergman [George M Bergman, 1976], we define a new class of rational functions where a rational function of inversion height at most h is defined as a composition of a noncommutative r-skewed circuit (equivalently an ABP) with inverses of rational functions of this class of inversion height at most h-1 which are also disjoint. We obtain a polynomial-size linear pencil representation for this class which gives a white-box deterministic polynomial-time identity testing algorithm for the class.
Cite as

V. Arvind, Abhranil Chatterjee, Utsab Ghosal, Partha Mukhopadhyay, and C. Ramya. On Identity Testing and Noncommutative Rank Computation over the Free Skew Field. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 6:1-6:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arvind_et_al:LIPIcs.ITCS.2023.6,
  author =	{Arvind, V. and Chatterjee, Abhranil and Ghosal, Utsab and Mukhopadhyay, Partha and Ramya, C.},
  title =	{{On Identity Testing and Noncommutative Rank Computation over the Free Skew Field}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{6:1--6:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.6},
  URN =		{urn:nbn:de:0030-drops-175093},
  doi =		{10.4230/LIPIcs.ITCS.2023.6},
  annote =	{Keywords: Algebraic Complexity, Identity Testing, Non-commutative rank}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.9
Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors

Authors: Harm Derksen, Visu Makam, and Jeroen Zuiddam

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract
Since the seminal works of Strassen and Valiant it has been a central theme in algebraic complexity theory to understand the relative complexity of algebraic problems, that is, to understand which algebraic problems (be it bilinear maps like matrix multiplication in Strassen’s work, or the determinant and permanent polynomials in Valiant’s) can be reduced to each other (under the appropriate notion of reduction). In this paper we work in the setting of bilinear maps and with the usual notion of reduction that allows applying linear maps to the inputs and output of a bilinear map in order to compute another bilinear map. As our main result we determine precisely how many independent scalar multiplications can be reduced to a given bilinear map (this number is called the subrank, and extends the concept of matrix diagonalization to tensors), for essentially all (i.e. generic) bilinear maps. Namely, we prove for a generic bilinear map T : V × V → V where dim(V) = n that θ(√n) independent scalar multiplications can be reduced to T. Our result significantly improves on the previous upper bound from the work of Strassen (1991) and Bürgisser (1990) which was n^{2/3 + o(1)}. Our result is very precise and tight up to an additive constant. Our full result is much more general and applies not only to bilinear maps and 3-tensors but also to k-tensors, for which we find that the generic subrank is θ(n^{1/(k-1)}). Moreover, as an application we prove that the subrank is not additive under the direct sum. The subrank plays a central role in several areas of complexity theory (matrix multiplication algorithms, barrier results) and combinatorics (e.g., the cap set problem and sunflower problem). As a consequence of our result we obtain several large separations between the subrank and tensor methods that have received much interest recently, notably the slice rank (Tao, 2016), analytic rank (Gowers-Wolf, 2011; Lovett, 2018; Bhrushundi-Harsha-Hatami-Kopparty-Kumar, 2020), geometric rank (Kopparty-Moshkovitz-Zuiddam, 2020), and G-stable rank (Derksen, 2020). Our proofs of the lower bounds rely on a new technical result about an optimal decomposition of tensor space into structured subspaces, which we think may be of independent interest.
Cite as

Harm Derksen, Visu Makam, and Jeroen Zuiddam. Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 9:1-9:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{derksen_et_al:LIPIcs.CCC.2022.9,
  author =	{Derksen, Harm and Makam, Visu and Zuiddam, Jeroen},
  title =	{{Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{9:1--9:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.9},
  URN =		{urn:nbn:de:0030-drops-165716},
  doi =		{10.4230/LIPIcs.CCC.2022.9},
  annote =	{Keywords: tensors, bilinear maps, complexity, subrank, diagonalization, generic tensors, random tensors, reduction, slice rank}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.57
Efficient Black-Box Identity Testing for Free Group Algebras

Authors: V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
Hrubeš and Wigderson [Pavel Hrubeš and Avi Wigderson, 2014] initiated the study of noncommutative arithmetic circuits with division computing a noncommutative rational function in the free skew field, and raised the question of rational identity testing. For noncommutative formulas with inverses the problem can be solved in deterministic polynomial time in the white-box model [Ankit Garg et al., 2016; Ivanyos et al., 2018]. It can be solved in randomized polynomial time in the black-box model [Harm Derksen and Visu Makam, 2017], where the running time is polynomial in the size of the formula. The complexity of identity testing of noncommutative rational functions, in general, remains open for noncommutative circuits with inverses. We solve the problem for a natural special case. We consider expressions in the free group algebra F(X,X^{-1}) where X={x_1, x_2, ..., x_n}. Our main results are the following. 1) Given a degree d expression f in F(X,X^{-1}) as a black-box, we obtain a randomized poly(n,d) algorithm to check whether f is an identically zero expression or not. The technical contribution is an Amitsur-Levitzki type theorem [A. S. Amitsur and J. Levitzki, 1950] for F(X, X^{-1}). This also yields a deterministic identity testing algorithm (and even an expression reconstruction algorithm) that is polynomial time in the sparsity of the input expression. 2) Given an expression f in F(X,X^{-1}) of degree D and sparsity s, as black-box, we can check whether f is identically zero or not in randomized poly(n,log s, log D) time. This yields a randomized polynomial-time algorithm when D and s are exponential in n.
Cite as

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Efficient Black-Box Identity Testing for Free Group Algebras. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 57:1-57:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{arvind_et_al:LIPIcs.APPROX-RANDOM.2019.57,
  author =	{Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha},
  title =	{{Efficient Black-Box Identity Testing for Free Group Algebras}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{57:1--57:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.57},
  URN =		{urn:nbn:de:0030-drops-112723},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.57},
  annote =	{Keywords: Rational identity testing, Free group algebra, Noncommutative computation, Randomized algorithms}
}
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