  Many high-dimensional 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 low-degree 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 low-degree
and free-energy based approaches. We define a free-energy based criterion for
hardness and formally connect it to the well-established notion of low-degree
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 low-degree hardness implies failure of
"geometric" local MCMC algorithms, and provide new low-degree 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 low-degree lower bounds.
