Vishesh Jain



Department of Statistics, Stanford University
Office:
Email:   visheshj at stanford dot edu
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I am a Stein Fellow in the Department of Statistics at Stanford University. Starting Fall 2022, I will be an Assistant Professor in the department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago. In May, 2020, I received a Ph.D. in mathematics from MIT, where I was fortunate to have been advised by Elchanan Mossel.

I am broadly interested in probability, combinatorics, and theoretical computer science.

Publications and Preprints

Invertibility of random matrices

  1. Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial, joint with Asaf Ferber, Ashwin Sah, and Mehtaab Sawhney. To appear in Mathematical Proceedings of the Cambridge Philosophical Society. [arXiv:2106.04049]
  2. Rank deficiency of random matrices, joint with Ashwin Sah and Mehtaab Sawhney. To appear in Electronic Communications in Probability. [arXiv:2103.02467]
  3. On the smallest singular value of symmetric random matrices, joint with Ashwin Sah and Mehtaab Sawhney. To appear in Combinatorics, Probability and Computing. [arXiv:2011.02344]
  4. Singularity of discrete random matrices, joint with Ashwin Sah and Mehtaab Sawhney. To appear in Geometric and Functional Analysis (GAFA). [arXiv:2010.06554]
  5. Sharp invertibility of random Bernoulli matrices, joint with Ashwin Sah and Mehtaab Sawhney. Merged with above for publication. [arXiv:2010.06553]
  6. The smallest singular value of dense random regular digraphs, joint with Ashwin Sah and Mehtaab Sawhney. To appear in International Mathematics Research Notices (IMRN). [arXiv:2008.04755]
  7. Quantitative invertibility of random matrices: a combinatorial perspective. Discrete Analysis, 2021:10. [Journal][arXiv:1908.11255]
  8. Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices. Israel Journal of Mathematics, 242, pp. 461--500 (2021). [Journal][arXiv:1904.10592]
  9. On the counting problem in inverse Littlewood-Offord theory, joint with Asaf Ferber, Kyle Luh, and Wojciech Samotij. Journal of the London Mathematical Society, vol. 103, issue 4, pp. 1333--1362 (2021). [Journal][arXiv:1904.10425]
  10. Singularity of random symmetric matrices -- a combinatorial approach to improved bounds, joint with Asaf Ferber. Forum of Mathematics, Sigma, vol. 7, e22, 29 pages (2019). [Journal][arXiv:1809.04718]

Counting, sampling, and Markov chains

  1. Dimension reduction for maximum matchings and the Fastest Mixing Markov Chain, joint with Huy Tuan Pham and Thuy Duong Vuong. Submitted. [arXiv:2203.03858]
  2. Entropic Independence II: Optimal Sampling and Concentration via Restricted Modified Log-Sobolev Inequalities, joint with Nima Anari, Frederic Koehler, Huy Tuan Pham, and Thuy Duong Vuong. To appear in the 54th ACM Symposium on the Theory of Computing (STOC 2022). [arXiv:2111.03247]
  3. Approximate counting and sampling via local central limit theorems, joint with Will Perkins, Ashwin Sah and Mehtaab Sawhney. To appear in the 54th ACM Symposium on the Theory of Computing (STOC 2022). [arXiv:2108.01161]
  4. Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models, joint with Nima Anari, Frederic Koehler, Huy Tuan Pham, and Thuy Duong Vuong. To appear in the 54th ACM Symposium on the Theory of Computing (STOC 2022). [arXiv:2106.04105]
  5. Spectral independence, coupling, and the spectral gap of the Glauber dynamics, joint with Huy Tuan Pham and Thuy Duong Vuong. To appear in Information Processing Letters. [arXiv:2105.01201]
  6. On the sampling Lovász Local Lemma for atomic constraint satisfaction problems, joint with Huy Tuan Pham and Thuy Duong Vuong. Submitted. [arXiv:2102.08342]
  7. Towards the sampling Lovász Local Lemma, joint with Huy Tuan Pham and Thuy Duong Vuong. To appear in 62nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2021). [arXiv:2011.12196]
  8. Perfectly sampling \(k \geq (8/3 + o(1))\Delta\)-colorings in graphs, joint with Ashwin Sah and Mehtaab Sawhney. 53rd ACM Symposium on the Theory of Computing (STOC 2021). [Conference][arXiv:2007.06360]
  9. Fast and memory-optimal dimension reduction using Kac's walk, joint with Natesh Pillai, Ashwin Sah, Mehtaab Sawhney, and Aaron Smith. To appear in Annals of Applied Probability. [arXiv:2003.10069]

Numerical analysis

  1. Optimal and algorithmic norm regularization of random matrices, joint with Ashwin Sah and Mehtaab Sawhney. To appear in Proceedings of the American Mathematical Society. [arXiv:2012.00175]
  2. On the smoothed analysis of the smallest singular value with discrete noise, joint with Ashwin Sah and Mehtaab Sawhney. To appear in Bulletin of the London Mathematical Society. [arXiv:2009.01699]
  3. On the real Davies' conjecture, joint with Ashwin Sah and Mehtaab Sawhney. To appear in Annals of Probability. [arXiv:2005.08908]

Graph decompositions

  1. Towards the linear arboricity conjecture, joint with Asaf Ferber and Jacob Fox. Journal of Combinatorial Theory, Series B, Volume 142, May 2020, Pages 56--79. [Journal][arXiv:1809.04716]
  2. Number of 1-factorizations of regular high-degree graphs, joint with Asaf Ferber and Benny Sudakov. Combinatorica, 2020. [Journal][arXiv:1803.10360]
  3. On the k-planar local crossing number, joint with John Asplund, Thao Do, and Arran Hamm. Discrete Mathematics, vol. 342, issue 4, pp. 927--933 (2019). [Journal][arXiv:1804.02117]
  4. 1-factorizations of pseudorandom graphs, joint with Asaf Ferber. 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2018). Random Structures and Algorithms. [Conference][Journal][arXiv:1803.10361]
  5. The probability of selecting k edge-disjoint Hamilton cycles in the complete graph, joint with Asaf Ferber and Kaarel Haenni. Not intended for publication. [arXiv:2001.01149]
  6. Uniformity-independent minimum degree conditions for perfect matchings in hypergraphs, joint with Asaf Ferber. Not intended for publication. [arXiv:1903.12207]

Mean-field approximation

  1. Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective, joint with Frederic Koehler and Andrej Risteski. 51st ACM Symposium on the Theory of Computing (STOC 2019). [Conference][arXiv:1808.07226]
  2. The Vertex Sample Complexity of Free Energy is Polynomial, joint with Frederic Koehler and Elchanan Mossel. 31st Annual Conference on Learning Theory (COLT 2018). [Conference][arXiv:1802.06129]
  3. The Mean-Field Approximation: Information Inequalities, Algorithms, and Complexity, joint with Frederic Koehler and Elchanan Mossel. 31st Annual Conference on Learning Theory (COLT 2018). [Conference][arXiv:1802.06126]

Anticoncentration

  1. Anticoncentration versus the number of subset sums, joint with Ashwin Sah and Mehtaab Sawhney. Advances in Combinatorics, 2021:6. [Journal][arXiv:2101.07726]
  2. On the number of Hadamard matrices via anti-concentration, joint with Asaf Ferber and Yufei Zhao. To appear in Combinatorics, Probability and Computing. [arXiv:1808.07222]

Universality of ESDs

  1. Circular law for random block band matrices with genuinely sublinear bandwidth, joint with Indrajit Jana, Kyle Luh, and Sean O'Rourke. To appear in Journal of Mathematical Physics. [arXiv:2008.03850]
  2. Universality and least singular values of random product matrices: a simplified approach, joint with Rohit Chaudhuri and Natesh Pillai. Bernoulli, vol. 27, no. 4, pages 2519-2531 (2021). [Journal][arXiv:2007.03595]
  3. A note on the universality of ESDs of inhomogeneous random matrices, joint with Sandeep Silwal. Latin American Journal of Probability and Mathematical Statistics (ALEA), vol. 18, 1047–1059 (2021). [Journal][arXiv:2006.05418]
  4. The strong circular law: a combinatorial view. Random Matrices: Theory and Applications, vol. 10, No. 03, 2150031 (2021). [Journal][arXiv:1904.11108]

Miscellaneous

  1. Optimal minimization of the covariance loss, joint with Ashwin Sah and Mehtaab Sawhney. Submitted. [arXiv:2205.01773]
  2. Accuracy-Memory Tradeoffs and Phase Transitions in Belief Propagation, joint with Frederic Koehler, Jingbo Liu, and Elchanan Mossel. 32nd Annual Conference on Learning Theory (COLT 2019). [Conference][arXiv:1905.10031]
  3. On discontinuity of planar optimal transport maps, joint with Otis Chodosh, Michael Lindsey, Lyuboslav Panchev, and Yanir Rubinstein. Journal of Topology and Analysis, vol. 7, no. 2, pp. 239--260 (2015). [Journal][arXiv:1312.2929]
  4. A Counterexample to the "Majority is Least Stable" Conjecture. Not intended for publication. [arXiv:1703.07657]

Expository

  1. An expanded version of a MathOverflow comment by Nazarov, giving a simple proof of an inequality due to Feige.
  2. Unedited and informal notes from a three-hour reading group talk on the Balogh-Morris-Samotij proof of the hypergraph containers theorem .
  3. Unedited and informal notes from a two-hour reading group talk on Peter Keevash's Counting Designs.

Upcoming talks

Teaching, Mentoring, and TAing