
# Pravesh K. Kothari

Assistant Professor
Theory Group
Computer Science Department, CMU
Gates 7105
praveshk AT cs.cmu.edu

## Research

I am broadly interested in theoretical computer science. My work clusters around the meta-question:
What structure in inputs makes efficient computation possible?
Can we formulate a picture of efficient computation and demarcate its boundaries based on the presence and absence of such structures?

One playground for such investigations is average-case complexity -- the study of the complexity of random instances of computational problems. So that forms a large part of my work -- both in finding algorithms based on new structures and in proving that there are no better algorithms (often under additional conjectures/restricted models). But frequently, it also turns out that studying average-case computation reveals structures that one can then find analogs of in much more general semi-random instances.

As an example, see this recent broadly accessible talk for an overview of my work that uses the sum-of-squares method to develop a theory of high dimensional robust statistics -- the science of statistical estimation from data that deviates in an unknown way from the chosen model. A prominent example of this research is our recent solution (recent talk) to the problem of robustly learning a mixture of Gaussians.

My research is generously supported by an NSF Career Award (2021-2026) on The Nature of Average-Case Computation and an Award from the Google Research Scholar Program for Efficient Algorithms for Robust Machine Learning.

## Recent/Upcoming Events/Talks

Simons Workshop on Rigorous Evidence for Information-Computation Trade-offs [Sep 21]
TCS Plus Seminar: Double Feature with Ankur Moitra on Robust Learning of Mixture of $k$ Arbitrary Gaussians [June 21]
POEMA Workshop [Feb 21]
Simons Workshop on High Dimensional Learning and Testing [Dec 20]
Online CSP Seminar [Nov 20]
Simons Workshop on Computational Phase Transitions [Sep 20]
TAMU High Dimensional Probability Seminar [Aug 20]
ICERM Workshop on Symmetry, Randomness, and Computations in Real Algebraic Geometry[Aug 20]
Polynomials as an Algorithmic Paradigm, PolyAlg Seminar [June 20]
Workshop on Extension Complexity and Lifting Theorems, FSTTCS [Dec 2019]
International Conference on Continuous Optimization, Berlin [Aug 2019]
SIAM Conference in Applied Algebraic Geometry, Bern [Jul 2019]
BIRS Workshop on Algebraic Techniques in Computational Complexity, Banff [July 2019]

## Service

Postdoc: Alperen Ergür (Now Assistant Professor of Math and CS at UT San Antonio)
Students: Ainesh Bakshi (co-advised with David Woodruff), Tim Hsieh, Xinyu Wu (co-advised with Ryan O'Donnell), Jeff Xu.

## 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.

## Current Teaching

15-854B: Advanced Approximation Algorithms (with Anupam Gupta)
MWF, 10:10-11:30 am. First meeting Aug 30

## Selected Recent Papers (for all papers, see here)

Strongly refuting all semi-random Boolean CSPs
With Jackson Abascal and Venkat Guruswami .
SODA, 2021 (to appear). LONG TALK.

Outlier-Robust Clustering of Non-Spherical Mixtures
With Ainesh Bakshi
FOCS, 2020. LONG TALK.
Conference version to be merged with this paper.

List-Decodable Linear Regression
With Sushrut Karmalkar and Adam Klivans
NeuIPS Spotlight, 2019. LONG TALK.

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.

Outlier-Robust Moment Estimation Via Sum-of-Squares
With David Steurer
STOC 2018 (conference version to be merged with the paper below) 2 hour BOARD TALK.

Better Agnostic Clustering Via Relaxed Tensor Norms
With Jacob Steinhardt STOC 2018 BOARD TALK with a simpler, complete proof!.
(conference version to be merged with the paper above)

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

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

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.

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 .

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 to be merged with Tight Lower Bounds for Planted Clique in the Degree-4 SOS Program by Tselil Schramm and Prasad Raghavendra .)

Provable Submodular Minimization Using Wolfe's Algorithm
With Deeparnab Chakrabarty and Prateek Jain
NIPS 2014 (Oral Presentation)