Rigorous theoretical analysis of Rotational Variational Inference (RoVI)

Develop a rigorous theoretical analysis of rotational variational inference (RoVI), which jointly optimizes over rotations O ∈ O(d) and product measures μ ∈ P_{2,ac}(R)^⊗d to minimize KL(O_# μ || π), by establishing algorithmic convergence guarantees and general approximation bounds under explicit assumptions on the target distribution π.

Background

The paper proposes RoVI to mitigate MFVI mode collapse by augmenting the mean-field family with rotations and introduces a gradient-based algorithm involving alternating updates of transport parameters and the rotation. Empirical studies and a bound for Gaussian mixtures illustrate promise, but formal guarantees are lacking.

The authors explicitly note the absence of a rigorous theory for RoVI, pointing to open directions such as convergence analysis (iteration complexity) and approximation guarantees across classes of non-log-concave targets.