Abstract:  We construct approximations to high-dimensional probability measures as low-dimensional updates of a reference measure. Specifically, we replace the high-dimensional likelihood function, i.e., the ratio of the densities, with a composition of (i) a feature map identifying the leading principal feature components of the target measure, and (ii) a low-dimensional profile function. When the reference measure satisfies certain functional inequalities from Markov semigroup theory, we show that our algorithm constructs approximations with certifiable error guarantees with respect to the Amari alpha-divergences. Notably, we demonstrate that there exists particular linear feature maps which universally control the "worst-case" error for a wide range of alpha-divergences. Furthermore, by leveraging recent improvements to the functional inequalities we consider, we are able to improve our error guarantees by an order of magnitude. These refinements also enable us to provide guarantees on the "best-case" error of our approximations. Preliminary numerical experiments suggest that our certificates of optimality are essentially tight.