Georgia Institute of Technology College of Sciences

High-dimensional inference problems such as sparse PCA and planted clique often exhibit statistical-vs-computational tradeoffs whereby there is no known polynomial-time algorithm matching the performance of the optimal estimator. I will discuss an emerging framework -- based on the so-called low-degree likelihood ratio -- for precisely predicting these tradeoffs and giving rigorous evidence for computational hardness in the conjectured hard regime. This method was originally proposed in a sequence of works on the sum-of-squares hierarchy, and the key idea is to study whether or not there exists a low-degree polynomial that succeeds at a given statistical task.

In the second part of the talk, I will give an application to the algorithmic problem of finding an approximate ground state of the SK (Sherrington-Kirkpatrick) spin glass model. I will explain two variants of this problem: "optimization" and "certification." While optimization can be solved in polynomial time [Montanari'18], we give rigorous evidence (in the low-degree framework) that certification cannot be. This result reveals a fundamental discrepancy between two classes of algorithms: local search succeeds while convex relaxations fail.

Based on joint work with Afonso Bandeira and Tim Kunisky (https://arxiv.org/abs/1902.07324 and https://arxiv.org/abs/1907.11636).

 A powerful method for analyzing graphs is to first apply regularity lemmas, which roughly state that one can partition the graph into a few parts so that it looks mostly random between the parts, and then apply probabilistic tools from there.  The drawback of this approach is that it only works in general when the input graph is very dense: standard regularity lemmas are trivial already for n-node graphs on "only" <= n^{1.99} edges.

In this work we prove extensions of several standard regularity lemmas to sparse graphs, which are nontrivial so long as the graph spectrum is not too far from that of a random graph.  We then apply our notion of "spectral pseudorandomness" to port several notable regularity-based results in combinatorics and theoretical computer science down to sparser graphs.

Joint work with Santosh Vempala.

Let X be an elliptic surface with a section defined over a number field. Specialization theorems by Néron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly large than the latter. In this talk, I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre. This is joint work with Dan Loughran.