Lucas Pesenti

We resolve a conjecture of Feige (2008) concerning the optimal tradeoff between density and girth in hypergraphs, generalizing the classical Moore bound for graphs. Our main technical ingredient is a new application of the polynomial method in combinatorics.

Inspired by a generic decomposition result of Talagrand (2021), we determine the sparsity level at which Erdős-Rényi hypergraphs exhibit a spectral gap. Our proof overcomes the polylogarithmic losses in the dimension that arise in standard approaches to bounding the injective norm of sparse random tensors.

We study the dynamics of first-order methods through the limiting traffic distribution of the input matrix. This yields the first analysis of Approximate Message Passing on deterministic matrices, partially resolving conjectures of Marinari, Parisi, and Ritort (1994), and a new general AMP iteration whose limiting dynamics are conditionally Gaussian, answering questions of Wang, Zhong, and Fan (2022).

We establish a tighter connection between how well the indicator function of a set can be approximated by low-degree polynomials in Gaussian space and the Gaussian surface area of that set. As a consequence, we obtain improved agnostic learning algorithms for several classes of concepts.

We introduce a new approach for analyzing nonlinear iterative algorithms on random matrices using Fourier analysis. As the dimension of the input matrix goes to infinity, the Fourier basis simplifies, allowing us to implement heuristic reasoning based on the cavity method into rigorous arguments.

We present rounding schemes for SDP relaxations of the problem of maximizing cubic polynomials over the hypercube or the sphere, complementing the well-developed theory for quadratic polynomials. This answers a question posed by Trevisan (2021) on the existence of certificates for the optimum value of cubic polynomials matching the guarantees of a search algorithm of Khot and Naor (2007).

We analyze a new algorithmic framework for discrepancy minimization based on regularization. We demonstrate how varying the regularizer allows us to reinterpret several classical results in discrepancy theory, ranging from Spencer's theorem (1985) to Banaszczyk's bounds (1998).

The union bound is a classical tool in the probabilistic method for proving the existence of objects with extremal features. In my Ph.D. thesis, I explore an alternative approach: proving existence through the design and analysis of algorithms from continuous optimization that explicitly construct these objects.