Stay on the Manifold: Hierarchical Bayesian Trajectory-Family Transfer for Sparse-Target Reconstruction and Sampling

Where should reconstructions and new samples lie when a few observations reveal only fragments of a trajectory? We propose hierarchical Bayesian trajectory principal-manifold sampling (HB-TPM), which learns a family of smooth curves from ordered source tasks and adapts it to sparse targets. For unordered vectors, alternating progression assignments and coefficient updates estimate trajectory structure between and beyond the observed shots under the learned source family. Observation-aware extensions connect the trajectory prior to photon arrivals and image-encoded cardiac cycles. Our analysis characterizes the coefficient directions supplied by the prior and bounds sampling error conditional on curve recovery. Across 30 held-out vector tasks, HB-TPM reduces mean sliced-Wasserstein distance by 21.0–41.8% relative to a source-adapted VAE. Prior-strength and sampling ablations identify coefficient shrinkage as the main supported mechanism. On 100 simulated photon records, adaptation lowers mean relative trajectory RMSE by 50.2% versus target-only fitting and 23.6% versus source-mean prediction. Across three training seeds and five folds on 98 cardiac sequences, HB-TPM with latent-MSE training achieves the best mean reconstruction among the tested configurations. Together, the results demonstrate trajectory-family transfer for sparse-target reconstruction and trajectory-centered sampling when source and target tasks share comparable progression structure.