  In this paper, we study the problem of sampling from distributions of the
form p(x) \propto e^{-\beta f(x)} for some function f whose values and
gradients we can query. This mode of access to f is natural in the scenarios in
which such problems arise, for instance sampling from posteriors in parametric
Bayesian models. Classical results show that a natural random walk, Langevin
diffusion, mixes rapidly when f is convex. Unfortunately, even in simple
examples, the applications listed above will entail working with functions f
that are nonconvex -- for which sampling from p may in general require an
exponential number of queries.
  In this paper, we focus on an aspect of nonconvexity relevant for modern
machine learning applications: existence of invariances (symmetries) in the
function f, as a result of which the distribution p will have manifolds of
points with equal probability. First, we give a recipe for proving mixing time
bounds for Langevin diffusion as a function of the geometry of these manifolds.
Second, we specialize our arguments to classic matrix factorization-like
Bayesian inference problems where we get noisy measurements A(XX^T), X \in R^{d
\times k} of a low-rank matrix, i.e. f(X) = \|A(XX^T) - b\|^2_2, X \in R^{d
\times k}, and \beta the inverse of the variance of the noise. Such functions f
are invariant under orthogonal transformations, and include problems like
matrix factorization, sensing, completion. Beyond sampling, Langevin dynamics
is a popular toy model for studying stochastic gradient descent. Along these
lines, we believe that our work is an important first step towards
understanding how SGD behaves when there is a high degree of symmetry in the
space of parameters the produce the same output.
