Global optimality under amenable symmetry constraints
Peter Orbanz
We ask whether there exists a function or measure that (1) minimizes a given
convex functional or risk and (2) satisfies a symmetry property specified by an
amenable group of transformations. Examples of such symmetry properties are
invariance, equivariance, or quasi-invariance. Our results draw on old ideas of
Stein and Le Cam and on approximate group averages that appear in ergodic
theorems for amenable groups. A class of convex sets known as orbitopes in
convex analysis emerges as crucial, and we establish properties of such
orbitopes in nonparametric settings. We also show how a simple device called a
cocycle can be used to reduce different forms of symmetry to a single problem.
As applications, we obtain results on invariant kernel mean embeddings and a
Monge-Kantorovich theorem on optimality of transport plans under symmetry
constraints. We also explain connections to the Hunt-Stein theorem on invariant
tests.