We improve the best known upper bounds on the density of corner-free sets over quasirandom groups from inverse poly-logarithmic to quasi-polynomial. We make similarly substantial improvements to the best known lower bounds on the communication complexity of a large class of permutation functions in the 3-player Number-on-Forehead model. Underpinning both results is a general combinatorial theorem that extends the recent work of Kelley, Lovett, and Meka (STOC'24), itself a development of ideas from the breakthrough result of Kelley and Meka on three-term arithmetic progressions (FOCS'23).

In this talk, we will address areas of recent work centered around the themes of fairness and foundations in machine learning as well as highlight the challenges in this area. We will discuss recent results involving linear algebraic tools for learning, such as methods in non-negative matrix factorization that include tailored approaches for fairness. We will showcase our derived theoretical guarantees as well as practical applications of those approaches.  Then, we will discuss new foundational results that theoretically justify phenomena like benign overfitting in neural networks.  Throughout the talk, we will include example applications from collaborations with community partners, using machine learning to help organizations with fairness and justice goals. 

Robust Statistics is a classical topic that dates back to the seminal work of Huber in the 1980s. In essence, the main goal of the field is to account for the effect of outliers when performing estimation tasks, such as mean estimation. A recent line of research, inspired by the seminal work of Catoni, has revisited some classical problems in robust statistics from a non-asymptotic perspective. The goal of this short seminar series is to introduce the key ideas related to robust estimation and discuss various notions of robustness, including heavy-tailed distributions and adversarial contamination. The primary example will be the mean estimation problem, and if time permits, I will also cover covariance estimation

In 2004 J. Friedman wrote a ~100 page paper proving a conjecture of Alon which stated that random d-regular graphs are nearly optimal expanders. Since then, Friedman's result has been refined and generalized in several directions, perhaps most notably by Bordenave and Collins who in 2019 established strong convergence of independent permutation matrices (a massive generalization of Friedman's theorem), a result that led to groundbreaking results in spectral theory and geometry.  

In this talk I will present joint work with C. Chen, J. Tropp and R. van Handel, where we introduce a new proof technique that allows one to convert qualitative results in random matrix theory into quantitative ones. This technique yields a fundamentally new approach to the study of strong convergence which is more flexible and significantly simpler than the existing techniques. Concretely, we're able to obtain  (1) a remarkably short of Friedman's theorem (2) a quantitative version of the result of Bordenave and Collins (3) a proof of strong convergence for arbitrary stable representations of the symmetric group, which constitutes a substantial generalization of the result of Bordenave and Collins. 

Gaussian elimination with partial pivoting (GEPP) remains the most used dense linear solver. For a nxn matrix A, GEPP results in the factorization PA = LU where L and U are lower and upper triangular matrices and P is a permutation matrix. If A is a random matrix, then the associated permutation from the P factor is random. When is this a uniform permutation? How many disjoint cycles are in its cycle decomposition (which equivalently answers how many GEPP pivot movements are needed on A)? What is the longest increasing subsequence of this permutation? We will provide some statistical answers to these questions for select random matrix ensembles and transformations. For particular butterfly permutations, we will present full distributional descriptions for these particular statistics. Moreover, we introduce a random butterfly matrix ensemble that induces the Haar measure on the full 2-Sylow subgroup of the symmetric group on a set of size 2ⁿ.