Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inference

Predicting how a dynamical unit evolves over time - how an individual ages, an epidemic spreads, or a physical system degrades - typically requires dense longitudinal tracking. When only extremely sparse or entirely cross-sectional data is available, inferring individualized, continuous-time trajectories is fundamentally ill-posed. Existing methods force a strict compromise: sequence models (e.g. latent ODEs) require dense longitudinal data, while cross-sectional methods (e.g. optimal transport, flow matching-based) map aggregate populations, losing individual dynamics. In this paper, we demonstrate that this dichotomy can be broken. We introduce CADENCE, a principled probabilistic framework that recovers continuous individual trajectories from isolated snapshots by anchoring latent dynamics to static, individual-level contexts. We provide novel identifiability guarantees for single-timepoint trajectory inference. By combining a score-based spatial encoder (bijective Probability Flow ODE) to eliminate diffeomorphic ambiguities with a Soft Mixture-of-Experts (SMoE) router, we show that individual dynamical parameters and routing function are jointly identifiable. Across a suite of benchmarks spanning physical systems to real-world biological data, CADENCE, trained strictly on extremely sparse snapshots with context structure, matches or exceeds the performance of state-of-the-art sequential models trained on dense, full-trajectory data.

Single-cell trajectory inference from destructive time-course snapshots is fundamentally ill-posed: neither cross-time cell correspondences nor continuous trajectories are observed, so the snapshot distributions alone do not uniquely determine the underlying dynamics. Existing optimal transport and flow-based methods typically couple cells by Euclidean proximity at observed clock times, which can misalign trajectories when development is asynchronous and cells sampled at the same experimental time occupy different latent pseudotime stages. We propose PACE, a trajectory inference framework that recovers geometry-consistent continuous transport dynamics from destructive time-course snapshots through three coupled components. First, PACE constructs a state- and time-dependent anisotropic Riemannian metric that assigns low transport cost along locally supported tangent directions while penalizing normal velocity components. Second, it alternates between refining cross-time couplings under the induced path-action cost and fitting endpoint-preserving neural bridges between adjacent snapshots. Third, it distills the learned bridge dynamics into a global continuous-time velocity field over cellular states. Across seven controlled and biological datasets covering nine held-out reconstruction experiments, PACE achieves the strongest overall reconstruction performance, reducing MMD, Wasserstein-1 distance, and Wasserstein-2 distance by 23.7% on average relative to the strongest competing baseline. PACE also improves RNA-velocity alignment by 15.4% on an embryoid body differentiation benchmark, without requiring explicit cell pairing, lineage tracing, or RNA-velocity supervision during training. Code is available at https://github.com/AI4Science-WestlakeU/PACE.

Suppose we observe the first $n$ points of a sequence of random variables having length $n+1$, and wish to estimate a functional of the unobserved final point and the empirical measure of the $n$ observed training points. Such next-token functionals include the probability that the next token is novel (also known as the surprise probability), the tail probability of the minimum distance between the next token and training points, and the test error of a classifier trained on the observed points. All of these quantities are classically estimated by the leave-one-out method, which is inconsistent under temporal dependence. We propose a leave-a-window-out estimator, which deletes a window of length $τ$ after each index before forming the empirical measure and reduces to leave-one-out at $τ= 1$. Under natural assumptions, we show that the error of our estimator decays at a parametric rate for any stationary $β$-mixing process that also admits a Marton coupling. Our results thus cover several natural functionals on a large class of stochastic processes. We complement these upper bounds with a sharp minimax lower bound for estimating the surprise probability on mixing Markov chains. Simulations on Markov chains, moving-average processes, and autoregressive processes show that our estimator succeeds in many scenarios where leave-one-out and add-constant baselines fail.