Pravesh K. Kothari

Assistant Professor (Jan 2024-)
Computer Science Department, Princeton
Associated Faculty
Program in Applied and Computational Mathematics (PACM)
kothari AT cs.princeton.edu

In Spring'24, I am also a Visiting Professor in the School of Math at the IAS.

Upcoming Talks/Events

IndiCS seminar on Continuous methods in discrete optimization and complexity [Oct 24]
Cornell Theory Seminar [Oct 24]
Eastern Great Lakes Workshop, Buffalo [Oct 24]
Cargèse Workshop on Combinatorial Optimization [Sep'24]
Bernoulli Center Workshop on Synergies of Combinatorics and theoretical computer science [Aug'24]
Presburger Award Talk, ICALP, Tallinn, Estonia [July 24]
Workshop on Algorithms for Learning and Economics [June'24]
Plenary Session, Oberwolfach Complexity Meeting [June'24]
Milan Theory Workshop [May'24]
Oberwolfach Meeting on Proof Complexity and Beyond [March'24]
School of Math Colloquium at the Institute for Advanced Study [March 24]
EnCORE Workshop on Computational vs Statistical Gaps in Learning and Optimization [Feb'24]
Banff Workshop on Computational Complexity of Statistical Inference [Feb 24]
Rutgers Theory Seminar[Feb 24]
MIT Statistics and Stochastics Seminar [Feb 24]
MIT-Harvard Reading Group [Feb 24]
Harvard Theory Seminar [Feb 24]
CSDM Seminar at the Institute for Advanced Study [Feb 24]

Research Interests

Algorithms and computational thresholds for average-case computational problems, random matrices, extremal combinatorics
My work is generously supported by an NSF CAREER Award (2021), an Alfred P. Sloan Fellowship (2022), a Google Research Scholar Award (2021), an NSF Medium Grant for collaborative research with Venkat Guruswami (2022), a 2024 seed grant from the Princeton AI Laboratory (joint with Amir Ali Ahmadi) and a 2025 Princeton School of Enginering and Applied Sciences Innovation Grant (joint with Amir Ali Ahmadi).

Research Highlights

Complete list of my papers. Some highlights/resources:

• The Sum-of-Squares Method and Algorithm Design via Connections to Proof Complexity
  • Monograph on Semialgebraic Proofs and Efficient Algorithm Design.
  • Fall'21 CMU Lecture notes and videos.
• Spectral Refutations via Kikuchi Matrices and Applications to Coding Theory and Combinatorics
  A new class of spectral methods for extremal combinatorics. Applications so far: refuting and solving smoothed SAT (CSPs), settling Feige's conjecture on extremal girth of hypergraphs, and improved lower bounds on locally decodable codes, and an exponential lower bound on three-query linear locally correctable codes.
  • A general audience article due to appear in CACM Research Highlights
  • Some recent talks.
• Efficient Algorithms approaching the conjectured threshold for Semirandom and Smoothed Optimization
  • Refuting/solving Smoothed SAT and other CSPs(in the sequence [1], [2], [3]).
  • Finding planted cliques in Feige-Kilian Semirandom Model (see this book chapter and this paper for history and context on the question).
  • Breaking security of certain proposed cryptographic Pseudorandom Generators.
  • Some recent talks.
• Sum-of-Squares Framework for Efficient Robust Statistics
  • Resolving the robust learnability of mixtures of high-dimensional Gaussians (in the sequence [1], [2], and [3]) (see 2012 CACM Tech Perspective and Simons Research Vignette for history and context).
  • Toolkit for basic estimation: regression, moment estimation, and sparse recovery.
  • Efficient estimation in the presence of majority outliers (in the sequence [1], [2], and [3]).
  • Overview talk with open questions.
• Pseudo-calibration, the Low-degree Polynomial Method and Sharp Computational Thresholds in Average-Case
  • Lower bounds for SoS SDPs via planted vs null distinguishing problems with applications to random CSPs, planted clique, Tensor/Sparse PCA, with new connections to spectral methods and the low degree polynomial heuristic for computational phase transitions in statistical estimation
  • Exponential lower bounds for linear relaxations for constraint satisfaction
  • Some talks.

Current Teaching

COS 598I: Matrix Concentration and Applications
Fri, 1:30-4:20pm, COS 104
Office Hours: after class

Service

Workshop Co-Chair, FOCS 2023 and 2024

PC Member APPROX/RANDOM 2018, SODA 2019, STOC 2020, ITCS 2020 , NeurIPS Area Chair 2020,2021, CCC 2022, RANDOM 2022, FSTTCS 2022, FOCS 2023, ITCS 2025, STOC 2025

Advising

Postdocs:
Alperen Ergür (Fall 2019-20, now Assistant Professor of Math and CS at UT San Antonio),
Mitali Bafna (Fall 2022-23, now postdoc at MIT Math)
PhD Students:
Ainesh Bakshi (co-advised with David Woodruff, graduated 2022, Assistant Professor of CS at NYU starting Fall'25, postdoc at MIT Math 2022-25)
Tim Hsieh (postdoc at MIT Math starting Fall'25),
Peter Manohar (winner of the 2023 Cylab Presidential Fellowship, co-advised with Venkat Guruswami, now postdoc at IAS School of Math),
Xinyu Wu (winner of the MSR Ada Lovelace Fellowship, co-advised with Ryan O'Donnell),
Jeff Xu (winner of the 2022 Cylab Presidential Fellowship, postdoc at TTI Chicago starting Fall'25)
Undergraduates:
Shuchen Li (now theory PhD student at Yale University)
Prashanti Anderson ( Alan J Perlis Award for student teaching, runner up to the Alan Newell award for best undergraduate SCS thesis for 2023), now PhD student at MIT EECS,
Misha Ivkov (winner of the Mark Stehlik SCS Alumni Undergraduate Impact Scholarship,now PhD student at Stanford CS),
Zachary Sussman (winner of the Alan Newell Award for best undergraduate SCS Thesis)

Some Representative Papers (for all papers, see here)

Spectral Refutations via Kikuchi Matrices and Applications

• Improved Lower Bounds for All Odd-Query LDCs
  With Arpon Basu, Tim Hsieh, and Andrew Lin
  FOCS, 2025 (to appear).
• Superpolynomial Lower Bounds for Smooth 3-LCCs and Sharp Bounds for Designs
  With Peter Manohar
  FOCS, 2024
  
• Small Even Covers, Locally Decodable Codes and Restricted Subgraphs of Edge-Colored Kikuchi Graphs
  With Tim Hsieh, Sidhanth Mohanty, David Munha Correia, and Benny Sudakov
  International Mathematical Research Notices (to appear), 2025
  
• An Exponential Lower Bound on Linear Three-Query Locally Correctable Codes
  With Peter Manohar
  STOC, 2024 (to appear)
  Invited to SICOMP Special Issue for STOC'24 (declined)
  
• A Near-Cubic Lower Bound for 3-Query Locally Decodable Codes from Semirandom CSP Refutation
  With Omar Alrabiah, Venkat Guruswami and Peter Manohar
  STOC, 2023.
  Invited to SICOMP Special Issue for STOC'23
  LONG TALK.
• A simple and sharper proof of the hypergraph Moore bound
  With Tim Hsieh and Sidhanth Mohanty
  SODA, 2023.
  LONG TALK.
• Algorithms and Certificates for Boolean CSP Refutation: Smoothed is no Harder than Random
  With Venkat Guruswami and Peter Manohar
  STOC, 2022. LONG TALK.
  Invited to the SICOMP Special Issue for STOC 2022, CACM Research Highlights 2023

Algorithms for Learning Mixtures of Gaussians

• Robustly Learning a Mixture of $k$ Arbitrary Gaussians
  With Ainesh Bakshi , Ilias Diakonikolas, He Jia, Daniel Kane, Santosh Vempala
  STOC, 2022. LONG TALK.
• Outlier-Robust Clustering of Non-Spherical Mixtures
  With Ainesh Bakshi
  FOCS, 2020. LONG TALK.
  Conference version to be merged with this paper.
• Better Agnostic Clustering Via Relaxed Tensor Norms
  With Jacob Steinhardt
  STOC, 2018
  Conference version merged with this paper.

Algorithms and Thresholds for Semirandom and Smoothed Models

• Rounding Large Independent Sets on Expanders
  With Mitali Bafna, Tim Hsieh
  STOC, 2025.
• Semirandom Planted Clique and the Restricted Isometry Property
  With Jaroslaw Blasiok, Rares Buhai, and David Steurer
  FOCS 2024
• Efficient Algorithms for Semirandom Planted CSPs at the Refutation Threshold
  With Venkat Guruswami , Tim Hsieh, and Peter Manohar
  FOCS, 2023 (to appear).
• Algorithms approaching the threshold for semi-random planted clique
  With Rares Buhai and David Steurer
  STOC, 2023 (to appear).
  LONG TALK.
• A simple and sharper proof of the hypergraph Moore bound
  With Tim Hsieh and Sidhanth Mohanty
  SODA, 2023.
  LONG TALK.
• Strongly refuting all semi-random Boolean CSPs
  With Jackson Abascal and Venkat Guruswami .
  SODA, 2021. LONG TALK.
• Sum-of-Squares Lower Bounds for Refuting any CSP
  With Ryuhei Mori , Ryan O'Donnell and David Witmer
  STOC, 2017.

Algorithmic Robust Statistics and Related Topics

• Efficient Certificates of Anti-Concentration Beyond Gaussians
  With Ainesh Bakshi, Goutham Rajendran, Madhur Tulsiani, and Aravindan Vijayraghavan
  FOCS,2024
• Beyond Moments: Robustly Learning Affine Transformations with Asymptotically Optimal Error
  With He Jia and Santosh Vempala
  FOCS, 2023 (to appear).
• Privately Estimating a Gaussian: Efficient, Robust and Optimal
  With Daniel Alabi, Pranay Tankala, Prayaag Venkat and Fred Zhang
  STOC, 2023 (to appear).
• A Moment-Matching Approach to Testable Learning and a New Characterization of Rademacher Complexity
  With Aravind Gallakota and Adam Klivans
  STOC, 2023 (to appear).
  Invited to SICOMP Special Issue for STOC'23
• List-Decodable Covariance Estimation
  With Misha Ivkov
  STOC, 2022.
  
• List-Decodable Subspace Recovery: Dimension Independent Error in Polynomial Time
  With Ainesh Bakshi
  SODA, 2021.
  
• List-Decodable Linear Regression
  With Sushrut Karmalkar and Adam Klivans
  NeuIPS Spotlight, 2019. LONG TALK.
• Sparse PCA: algorithms, adversarial perturbations and certificates
  With Tommaso D'Orsi, Gleb Novikov, and David Steurer .
  FOCS 2020.
• Efficient Algorithms for Outlier-Robust Regression
  With Adam Klivans and Raghu Meka
  COLT 2018
  Prasad Raghavendra's Exposition of our algorithm in a Simons Institute Bootcamp
• Outlier-Robust Moment Estimation Via Sum-of-Squares
  With David Steurer
  STOC 2018 2 hour BOARD TALK.
  Conference version merged with this paper.

Algorithms and Complexity of Unique Games and Related Problems

• Playing Unique Games on Certified Small-Set Expanders
  With Mitali Bafna , Boaz Barak , Tselil Schramm and David Steurer.
  STOC, 2021.
  
• Quantum Entanglement, Sum-of-Squares and the Log-Rank Conjecture
  With Boaz Barak and David Steurer
  STOC, 2017
  Recent 50 min talk and a shorter talk with a different perspective.
• Small-Set Expansion in Shortcode Graph and the 2-to-1 Conjecture
  With Boaz Barak and David Steurer
  A generally accessible article on the recent proof of the 2-to-2 games conjecture that partly relies on this work.
  

Average-Case Algorithmic Thresholds

• The Quasi-polynomial low degree conjecture is false
  With Rares Buhai, Tim Hsieh, and, Aayush Jain
  FOCS, 2025 (to appear)
• Sum of Squares Lower Bounds for Independent Set on Ultra-Sparse Random Graphs
  With Aaron Potechin and Jeff Xu.
  STOC , 2024 (to appear)
• Algorithmic Thresholds for Refuting Random Polynomial Systems
  With Tim Hsieh.
  SODA, 2022.
• A Stress-Free Sum-of-Squares Lower Bound for Coloring
  With Peter Manohar.
  CCC, 2021.
  
• The power of sum-of-squares for detecting hidden structures
  With Samuel B. Hopkins , Aaron Potechin , Prasad Raghavendra , Tselil Schramm and David Steurer
  FOCS 2017
  Invited to the Highlights of Algorithms (HALG) 2018
• A Nearly Tight Sum-of-Squares Lower Bound for Planted Clique
  With Boaz Barak , Sam Hopkins , Jon Kelner , Ankur Moitra and Aaron Potechin
  FOCS 2016. [Video- IAS CS/DM Seminar] [Boaz's WOT post]
  Invited to SICOMP Special Issue for FOCS 2016
• SoS and Planted Clique: Tight Analysis of MPW Moments
  at all Degrees and an Optimal Lower Bound at Degree Four
  With Samuel B. Hopkins and Aaron Potechin
  SODA 2016
  Invited to the ACM Transactions on Algorithms, Special Issue for SODA 2016
  Conference version merged with this paper.
• Approximating Rectangles by Juntas and a Weakly Exponential Lower Bound for LP Relaxations of CSPs
  With Raghu Meka and Prasad Raghavendra
  STOC, 2017
  Invited to SICOMP Special Issue for STOC 2017
  Recent 50 min talk.

Applications

• Sum-of-Squares Meets Nash: Lower Bounds for finding any equilibrium
  With Ruta Mehta
  STOC 2018
• Limits on Low-Degree Pseudorandom Generators (Or: Sum-of-Squares Meets Program Obfuscation)
  With Boaz Barak , Zvika Brakerski and Ilan Komargodski
  EUROCRYPT 2018

Mentoring Talks

I recently gave two mentoring talks as part of the new workshops organized by Learning Theory Alliance
Slides from talk on Interacting with your Research Community.
Slides from talk on Thoughts on PhD Admissions.