The FranzParisi Criterion and Computational Tradeoffs in High Dimensional Statistics
Abstract
Many highdimensional statistical inference problems are believed to possess inherent computational hardness. Various frameworks have been proposed to give rigorous evidence for such hardness, including lower bounds against restricted models of computation (such as lowdegree functions), as well as methods rooted in statistical physics that are based on free energy landscapes. This paper aims to make a rigorous connection between the seemingly different lowdegree and freeenergy based approaches. We define a freeenergy based criterion for hardness and formally connect it to the wellestablished notion of lowdegree hardness for a broad class of statistical problems, namely all Gaussian additive models and certain models with a sparse planted signal. By leveraging these rigorous connections we are able to: establish that for Gaussian additive models the "algebraic" notion of lowdegree hardness implies failure of "geometric" local MCMC algorithms, and provide new lowdegree lower bounds for sparse linear regression which seem difficult to prove directly. These results provide both conceptual insights into the connections between different notions of hardness, as well as concrete technical tools such as new methods for proving lowdegree lower bounds.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 DOI:
 10.48550/arXiv.2205.09727
 arXiv:
 arXiv:2205.09727
 Bibcode:
 2022arXiv220509727B
 Keywords:

 Mathematics  Statistics Theory;
 Condensed Matter  Statistical Mechanics;
 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 Statistics  Machine Learning
 EPrint:
 52 pages, 1 figure