3 Search Results for "Valiant, Paul"


Document
Homogeneous Algebraic Complexity Theory and Algebraic Formulas

Authors: Pranjal Dutta, Fulvio Gesmundo, Christian Ikenmeyer, Gorav Jindal, and Vladimir Lysikov

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)


Abstract
We study algebraic complexity classes and their complete polynomials under homogeneous linear projections, not just under the usual affine linear projections that were originally introduced by Valiant in 1979. These reductions are weaker yet more natural from a geometric complexity theory (GCT) standpoint, because the corresponding orbit closure formulations do not require the padding of polynomials. We give the first complete polynomials for VF, the class of sequences of polynomials that admit small algebraic formulas, under homogeneous linear projections: The sum of the entries of the non-commutative elementary symmetric polynomial in 3 by 3 matrices of homogeneous linear forms. Even simpler variants of the elementary symmetric polynomial are hard for the topological closure of a large subclass of VF: the sum of the entries of the non-commutative elementary symmetric polynomial in 2 by 2 matrices of homogeneous linear forms, and homogeneous variants of the continuant polynomial (Bringmann, Ikenmeyer, Zuiddam, JACM '18). This requires a careful study of circuits with arity-3 product gates.

Cite as

Pranjal Dutta, Fulvio Gesmundo, Christian Ikenmeyer, Gorav Jindal, and Vladimir Lysikov. Homogeneous Algebraic Complexity Theory and Algebraic Formulas. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 43:1-43:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{dutta_et_al:LIPIcs.ITCS.2024.43,
  author =	{Dutta, Pranjal and Gesmundo, Fulvio and Ikenmeyer, Christian and Jindal, Gorav and Lysikov, Vladimir},
  title =	{{Homogeneous Algebraic Complexity Theory and Algebraic Formulas}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{43:1--43:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.43},
  URN =		{urn:nbn:de:0030-drops-195713},
  doi =		{10.4230/LIPIcs.ITCS.2024.43},
  annote =	{Keywords: Homogeneous polynomials, Waring rank, Arithmetic formulas, Border complexity, Geometric Complexity theory, Symmetric polynomials}
}
Document
Optimal Sub-Gaussian Mean Estimation in Very High Dimensions

Authors: Jasper C.H. Lee and Paul Valiant

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)


Abstract
We address the problem of mean estimation in very high dimensions, in the high probability regime parameterized by failure probability δ. For a distribution with covariance Σ, let its "effective dimension" be d_eff = {Tr(Σ)}/{λ_{max}(Σ)}. For the regime where d_eff = ω(log^2 (1/δ)), we show the first algorithm whose sample complexity is optimal to within 1+o(1) factor. The algorithm has a surprisingly simple structure: 1) re-center the samples using a known sub-Gaussian estimator, 2) carefully choose an easy-to-compute positive integer t and then remove the t samples farthest from the origin and 3) return the sample mean of the remaining samples. The core of the analysis relies on a novel vector Bernstein-type tail bound, showing that under general conditions, the sample mean of a bounded high-dimensional distribution is highly concentrated around a spherical shell.

Cite as

Jasper C.H. Lee and Paul Valiant. Optimal Sub-Gaussian Mean Estimation in Very High Dimensions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 98:1-98:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{lee_et_al:LIPIcs.ITCS.2022.98,
  author =	{Lee, Jasper C.H. and Valiant, Paul},
  title =	{{Optimal Sub-Gaussian Mean Estimation in Very High Dimensions}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{98:1--98:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.98},
  URN =		{urn:nbn:de:0030-drops-156942},
  doi =		{10.4230/LIPIcs.ITCS.2022.98},
  annote =	{Keywords: High-dimensional mean estimation}
}
Document
Block Rigidity: Strong Multiplayer Parallel Repetition Implies Super-Linear Lower Bounds for Turing Machines

Authors: Kunal Mittal and Ran Raz

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)


Abstract
We prove that a sufficiently strong parallel repetition theorem for a special case of multiplayer (multiprover) games implies super-linear lower bounds for multi-tape Turing machines with advice. To the best of our knowledge, this is the first connection between parallel repetition and lower bounds for time complexity and the first major potential implication of a parallel repetition theorem with more than two players. Along the way to proving this result, we define and initiate a study of block rigidity, a weakening of Valiant’s notion of rigidity [Valiant, 1977]. While rigidity was originally defined for matrices, or, equivalently, for (multi-output) linear functions, we extend and study both rigidity and block rigidity for general (multi-output) functions. Using techniques of Paul, Pippenger, Szemerédi and Trotter [Paul et al., 1983], we show that a block-rigid function cannot be computed by multi-tape Turing machines that run in linear (or slightly super-linear) time, even in the non-uniform setting, where the machine gets an arbitrary advice tape. We then describe a class of multiplayer games, such that, a sufficiently strong parallel repetition theorem for that class of games implies an explicit block-rigid function. The games in that class have the following property that may be of independent interest: for every random string for the verifier (which, in particular, determines the vector of queries to the players), there is a unique correct answer for each of the players, and the verifier accepts if and only if all answers are correct. We refer to such games as independent games. The theorem that we need is that parallel repetition reduces the value of games in this class from v to v^Ω(n), where n is the number of repetitions. As another application of block rigidity, we show conditional size-depth tradeoffs for boolean circuits, where the gates compute arbitrary functions over large sets.

Cite as

Kunal Mittal and Ran Raz. Block Rigidity: Strong Multiplayer Parallel Repetition Implies Super-Linear Lower Bounds for Turing Machines. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 71:1-71:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{mittal_et_al:LIPIcs.ITCS.2021.71,
  author =	{Mittal, Kunal and Raz, Ran},
  title =	{{Block Rigidity: Strong Multiplayer Parallel Repetition Implies Super-Linear Lower Bounds for Turing Machines}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{71:1--71:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.71},
  URN =		{urn:nbn:de:0030-drops-136101},
  doi =		{10.4230/LIPIcs.ITCS.2021.71},
  annote =	{Keywords: Block-rigidity, Matrix Rigidity, Parallel Repetition, Size-depth tradeoffs, Turing Machines}
}
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