Identifiability of ODK drift and diffusion objectives

Establish whether the $L_q$ and $L_p$ objectives for Ordered Diffusion Kernels are identifiable from the assumed observations, or determine why identifiability is impossible, so that drift and state-dependent diffusion can be interpreted reliably after optimization.

Background

The paper introduces the $L_q$ and $L_p$ maximum-likelihood objectives to infer ordering or drift information together with constant or state-dependent diffusion from independently sampled observations, with the $L_p$ objective additionally using sampling times. The authors demonstrate empirically that these objectives can recover the dynamics of Ornstein–Uhlenbeck and nonlinear stochastic differential equations, but they also observe that the optimization landscape may be flat and that deterministic drift and stochastic diffusion can compensate for one another.

This creates a fundamental identifiability question: the same observed evolution might potentially be represented through different combinations of ordering gradients and spatially varying diffusion. The paper reports only weak empirical evidence for identifiability and explicitly leaves a theoretical result—or an explanation of non-identifiability—unresolved.