Impact of flexibility when modeling latent dynamics versus data-space dynamics

Determine the impact of increasing flexibility in the parameterization of the posterior reparameterization function F(ε, t, X) and the diffusion term g(z_t, t) within SDE Matching on modeling dynamical processes in a latent space compared to modeling dynamics directly in the original data space, including its effects on the quality of learned trajectories and predictive performance.

Background

SDE Matching enables simulation-free training of latent stochastic differential equations by directly parameterizing posterior marginals via an invertible reparameterization function F(ε, t, X). To preserve finite variational bounds, the posterior process shares the diffusion term g(z_t, t) with the prior process. This parameterization introduces trade-offs: F must be smooth, invertible in ε, and allow efficient computation of the conditional score ∇_{z_t} log q(z_t|X), which constrains posterior flexibility.

The authors highlight that while SDE Matching does not constrain the generative prior process, developing methods that allow more flexible parameterizations of F and g is a promising direction. A key unresolved issue is understanding how such flexibility affects modeling when learning dynamics in a latent space versus learning dynamics directly in the original data space, motivating a focused investigation of the comparative consequences and benefits of these choices.