6 Search Results for "Srivastava, Nikhil"

Document
DOI: 10.4230/LIPIcs.ITCS.2023.42
Bit Complexity of Jordan Normal Form and Polynomial Spectral Factorization

Authors: Papri Dey, Ravi Kannan, Nick Ryder, and Nikhil Srivastava

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

Abstract
We study the bit complexity of two related fundamental computational problems in linear algebra and control theory. Our results are: (1) An Õ(n^{ω+3}a+n⁴a²+n^ωlog(1/ε)) time algorithm for finding an ε-approximation to the Jordan Normal form of an integer matrix with a-bit entries, where ω is the exponent of matrix multiplication. (2) An Õ(n⁶d⁶a+n⁴d⁴a²+n³d³log(1/ε)) time algorithm for ε-approximately computing the spectral factorization P(x) = Q^*(x)Q(x) of a given monic n× n rational matrix polynomial of degree 2d with rational a-bit coefficients having a-bit common denominators, which satisfies P(x)⪰0 for all real x. The first algorithm is used as a subroutine in the second one. Despite its being of central importance, polynomial complexity bounds were not previously known for spectral factorization, and for Jordan form the best previous best running time was an unspecified polynomial in n of degree at least twelve [Cai, 1994]. Our algorithms are simple and judiciously combine techniques from numerical and symbolic computation, yielding significant advantages over either approach by itself.
Cite as

Papri Dey, Ravi Kannan, Nick Ryder, and Nikhil Srivastava. Bit Complexity of Jordan Normal Form and Polynomial Spectral Factorization. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dey_et_al:LIPIcs.ITCS.2023.42,
  author =	{Dey, Papri and Kannan, Ravi and Ryder, Nick and Srivastava, Nikhil},
  title =	{{Bit Complexity of Jordan Normal Form and Polynomial Spectral Factorization}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{42:1--42:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.42},
  URN =		{urn:nbn:de:0030-drops-175450},
  doi =		{10.4230/LIPIcs.ITCS.2023.42},
  annote =	{Keywords: Symbolic algorithms, numerical algorithms, linear algebra}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.108
A Spectral Approach to Polytope Diameter

Authors: Hariharan Narayanan, Rikhav Shah, and Nikhil Srivastava

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

Abstract
We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for integer polytopes in terms of the length of the description of the polytope (in bits) and the minimum angle between facets of its polar. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a "giant component" of vertices, with measure 1-o(1) and polynomial diameter. Both bounds rely on spectral gaps - of a certain Schrödinger operator in the first case, and a certain continuous time Markov chain in the second - which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.
Cite as

Hariharan Narayanan, Rikhav Shah, and Nikhil Srivastava. A Spectral Approach to Polytope Diameter. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 108:1-108:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{narayanan_et_al:LIPIcs.ITCS.2022.108,
  author =	{Narayanan, Hariharan and Shah, Rikhav and Srivastava, Nikhil},
  title =	{{A Spectral Approach to Polytope Diameter}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{108:1--108:22},
  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.2022.108},
  URN =		{urn:nbn:de:0030-drops-157044},
  doi =		{10.4230/LIPIcs.ITCS.2022.108},
  annote =	{Keywords: Polytope diameter, Markov Chain}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.96
High-Girth Near-Ramanujan Graphs with Lossy Vertex Expansion

Authors: Theo McKenzie and Sidhanth Mohanty

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
Kahale proved that linear sized sets in d-regular Ramanujan graphs have vertex expansion at least d/2 and complemented this with construction of near-Ramanujan graphs with vertex expansion no better than d/2. However, the construction of Kahale encounters highly local obstructions to better vertex expansion. In particular, the poorly expanding sets are associated with short cycles in the graph. Thus, it is natural to ask whether the vertex expansion of high-girth Ramanujan graphs breaks past the d/2 bound. Our results are two-fold: 1) For every d = p+1 for prime p ≥ 3 and infinitely many n, we exhibit an n-vertex d-regular graph with girth Ω(log_{d-1} n) and vertex expansion of sublinear sized sets bounded by (d+1)/2 whose nontrivial eigenvalues are bounded in magnitude by 2√{d-1}+O(1/(log_{d-1} n)). 2) In any Ramanujan graph with girth Clog_{d-1} n, all sets of size bounded by n^{0.99C/4} have near-lossless vertex expansion (1-o_d(1))d. The tools in analyzing our construction include the nonbacktracking operator of an infinite graph, the Ihara-Bass formula, a trace moment method inspired by Bordenave’s proof of Friedman’s theorem [Bordenave, 2019], and a method of Kahale [Kahale, 1995] to study dispersion of eigenvalues of perturbed graphs.
Cite as

Theo McKenzie and Sidhanth Mohanty. High-Girth Near-Ramanujan Graphs with Lossy Vertex Expansion. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 96:1-96:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{mckenzie_et_al:LIPIcs.ICALP.2021.96,
  author =	{McKenzie, Theo and Mohanty, Sidhanth},
  title =	{{High-Girth Near-Ramanujan Graphs with Lossy Vertex Expansion}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{96:1--96:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.96},
  URN =		{urn:nbn:de:0030-drops-141655},
  doi =		{10.4230/LIPIcs.ICALP.2021.96},
  annote =	{Keywords: expander graphs, Ramanujan graphs, vertex expansion, girth}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.12
High-Dimensional Expanders from Expanders

Authors: Siqi Liu, Sidhanth Mohanty, and Elizabeth Yang

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
We present an elementary way to transform an expander graph into a simplicial complex where all high order random walks have a constant spectral gap, i.e., they converge rapidly to the stationary distribution. As an upshot, we obtain new constructions, as well as a natural probabilistic model to sample constant degree high-dimensional expanders. In particular, we show that given an expander graph G, adding self loops to G and taking the tensor product of the modified graph with a high-dimensional expander produces a new high-dimensional expander. Our proof of rapid mixing of high order random walks is based on the decomposable Markov chains framework introduced by [Jerrum et al., 2004].
Cite as

Siqi Liu, Sidhanth Mohanty, and Elizabeth Yang. High-Dimensional Expanders from Expanders. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 12:1-12:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{liu_et_al:LIPIcs.ITCS.2020.12,
  author =	{Liu, Siqi and Mohanty, Sidhanth and Yang, Elizabeth},
  title =	{{High-Dimensional Expanders from Expanders}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{12:1--12:32},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.12},
  URN =		{urn:nbn:de:0030-drops-116974},
  doi =		{10.4230/LIPIcs.ITCS.2020.12},
  annote =	{Keywords: High-Dimensional Expanders, Markov Chains}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.85
Combinatorial Lower Bounds for 3-Query LDCs

Authors: Arnab Bhattacharyya, L. Sunil Chandran, and Suprovat Ghoshal

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
A code is called a q-query locally decodable code (LDC) if there is a randomized decoding algorithm that, given an index i and a received word w close to an encoding of a message x, outputs x_i by querying only at most q coordinates of w. Understanding the tradeoffs between the dimension, length and query complexity of LDCs is a fascinating and unresolved research challenge. In particular, for 3-query binary LDC’s of dimension k and length n, the best known bounds are: 2^{k^o(1)} ≥ n ≥ Ω ̃(k²). In this work, we take a second look at binary 3-query LDCs. We investigate a class of 3-uniform hypergraphs that are equivalent to strong binary 3-query LDCs. We prove an upper bound on the number of edges in these hypergraphs, reproducing the known lower bound of Ω ̃(k²) for the length of strong 3-query LDCs. In contrast to previous work, our techniques are purely combinatorial and do not rely on a direct reduction to 2-query LDCs, opening up a potentially different approach to analyzing 3-query LDCs.
Cite as

Arnab Bhattacharyya, L. Sunil Chandran, and Suprovat Ghoshal. Combinatorial Lower Bounds for 3-Query LDCs. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 85:1-85:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bhattacharyya_et_al:LIPIcs.ITCS.2020.85,
  author =	{Bhattacharyya, Arnab and Chandran, L. Sunil and Ghoshal, Suprovat},
  title =	{{Combinatorial Lower Bounds for 3-Query LDCs}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{85:1--85:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.85},
  URN =		{urn:nbn:de:0030-drops-117704},
  doi =		{10.4230/LIPIcs.ITCS.2020.85},
  annote =	{Keywords: Coding theory, Graph theory, Hypergraphs}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.5
Real Stability Testing

Authors: Prasad Raghavendra, Nick Ryder, and Nikhil Srivastava

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
We give a strongly polynomial time algorithm which determines whether or not a bivariate polynomial is real stable. As a corollary, this implies an algorithm for testing whether a given linear transformation on univariate polynomials preserves real-rootedness. The proof exploits properties of hyperbolic polynomials to reduce real stability testing to testing nonnegativity of a finite number of polynomials on an interval.
Cite as

Prasad Raghavendra, Nick Ryder, and Nikhil Srivastava. Real Stability Testing. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{raghavendra_et_al:LIPIcs.ITCS.2017.5,
  author =	{Raghavendra, Prasad and Ryder, Nick and Srivastava, Nikhil},
  title =	{{Real Stability Testing}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{5:1--5:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.5},
  URN =		{urn:nbn:de:0030-drops-81965},
  doi =		{10.4230/LIPIcs.ITCS.2017.5},
  annote =	{Keywords: real stable polynomials, hyperbolic polynomials, real rootedness, moment matrix, sturm sequence}
}
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