2 Search Results for "Chen, Sitan"


Document
Symmetric Sparse Boolean Matrix Factorization and Applications

Authors: Sitan Chen, Zhao Song, Runzhou Tao, and Ruizhe Zhang

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


Abstract
In this work, we study a variant of nonnegative matrix factorization where we wish to find a symmetric factorization of a given input matrix into a sparse, Boolean matrix. Formally speaking, given {𝐌} ∈ {ℤ}^{m× m}, we want to find {𝐖} ∈ {0,1}^{m× r} such that ‖ {𝐌} - {𝐖} {𝐖}^⊤ ‖₀ is minimized among all {𝐖} for which each row is k-sparse. This question turns out to be closely related to a number of questions like recovering a hypergraph from its line graph, as well as reconstruction attacks for private neural network training. As this problem is hard in the worst-case, we study a natural average-case variant that arises in the context of these reconstruction attacks: {𝐌} = {𝐖} {𝐖}^{⊤} for {𝐖} a random Boolean matrix with k-sparse rows, and the goal is to recover {𝐖} up to column permutation. Equivalently, this can be thought of as recovering a uniformly random k-uniform hypergraph from its line graph. Our main result is a polynomial-time algorithm for this problem based on bootstrapping higher-order information about {𝐖} and then decomposing an appropriate tensor. The key ingredient in our analysis, which may be of independent interest, is to show that such a matrix {𝐖} has full column rank with high probability as soon as m = Ω̃(r), which we do using tools from Littlewood-Offord theory and estimates for binary Krawtchouk polynomials.

Cite as

Sitan Chen, Zhao Song, Runzhou Tao, and Ruizhe Zhang. Symmetric Sparse Boolean Matrix Factorization and Applications. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 46:1-46:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{chen_et_al:LIPIcs.ITCS.2022.46,
  author =	{Chen, Sitan and Song, Zhao and Tao, Runzhou and Zhang, Ruizhe},
  title =	{{Symmetric Sparse Boolean Matrix Factorization and Applications}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{46:1--46:25},
  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.2022.46},
  URN =		{urn:nbn:de:0030-drops-156422},
  doi =		{10.4230/LIPIcs.ITCS.2022.46},
  annote =	{Keywords: Matrix factorization, tensors, random matrices, average-case complexity}
}
Document
RANDOM
The Product of Gaussian Matrices Is Close to Gaussian

Authors: Yi Li and David P. Woodruff

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)


Abstract
We study the distribution of the matrix product G₁ G₂ ⋯ G_r of r independent Gaussian matrices of various sizes, where G_i is d_{i-1} × d_i, and we denote p = d₀, q = d_r, and require d₁ = d_{r-1}. Here the entries in each G_i are standard normal random variables with mean 0 and variance 1. Such products arise in the study of wireless communication, dynamical systems, and quantum transport, among other places. We show that, provided each d_i, i = 1, …, r, satisfies d_i ≥ C p ⋅ q, where C ≥ C₀ for a constant C₀ > 0 depending on r, then the matrix product G₁ G₂ ⋯ G_r has variation distance at most δ to a p × q matrix G of i.i.d. standard normal random variables with mean 0 and variance ∏_{i = 1}^{r-1} d_i. Here δ → 0 as C → ∞. Moreover, we show a converse for constant r that if d_i < C' max{p,q}^{1/2}min{p,q}^{3/2} for some i, then this total variation distance is at least δ', for an absolute constant δ' > 0 depending on C' and r. This converse is best possible when p = Θ(q).

Cite as

Yi Li and David P. Woodruff. The Product of Gaussian Matrices Is Close to Gaussian. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 35:1-35:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{li_et_al:LIPIcs.APPROX/RANDOM.2021.35,
  author =	{Li, Yi and Woodruff, David P.},
  title =	{{The Product of Gaussian Matrices Is Close to Gaussian}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{35:1--35:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.35},
  URN =		{urn:nbn:de:0030-drops-147281},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.35},
  annote =	{Keywords: random matrix theory, total variation distance, matrix product}
}
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