Notes on computational-to-statistical gaps: predictions using statistical physics

TL;DR

This paper discusses heuristics derived from statistical physics to predict regimes where statistical problems are theoretically solvable but lack efficient algorithms, highlighting computational-to-statistical gaps.

Contribution

It introduces non-rigorous physics-based heuristics to identify computational-to-statistical gaps in statistical problems.

Findings

Heuristics can predict computational barriers in statistical problems.

The methods are based on mature statistical physics tools.

The approach helps understand regimes where problems are solvable but computationally hard.

Abstract

In these notes we describe heuristics to predict computational-to-statistical gaps in certain statistical problems. These are regimes in which the underlying statistical problem is information-theoretically possible although no efficient algorithm exists, rendering the problem essentially unsolvable for large instances. The methods we describe here are based on mature, albeit non-rigorous, tools from statistical physics. These notes are based on a lecture series given by the authors at the Courant Institute of Mathematical Sciences in New York City, on May 16th, 2017.