Online Coresets for Conditional Flow Matching: Learnt Support-Size and Transport Transfer

Flow matching learns a velocity field that transports a source distribution to a data law. In the conditional setting, endpoint memory determines both interpolation inputs and supervised velocity targets. Retaining full history can preserve obsolete regimes under drift, while a fixed small buffer can erase conditional structure. We formulate endpoint memory as a condition-wise model-order problem and introduce Online Memory Compression and Identification with Maximum Mean Discrepancy (OMCI–MMD). The method approximates each conditional kernel mean with a sparse weighted subset of observed endpoints and selects the first support budget whose fully corrective approximation reaches a predeclared MMD sampling radius. Under stationary conditional independence and bounded kernels, we derive support-size brackets that account for approximation error along the computed path. A recent-window analysis separates sampling error, stale-law bias, candidate availability, and numerical error, with current-law guarantees requiring a valid staleness bound. Under explicit reproducing-kernel Hilbert-space witness assumptions, we establish when endpoint MMD controls Wasserstein, Sinkhorn, and fixed-path flow-matching losses. The selected cardinality is frozen before downstream losses are inspected, permitting subsequent fixed-size refinement. In a controlled two-condition, ten-atom neural-flow study, selected support remains at six atoms for the stationary condition and grows to eight after a programmed shift. The selected endpoint laws improve terminal conditional MMD-squared over undersized and stale-history alternatives, while the full-current reference achieves lower held-out velocity error and terminal MMD-squared. These results connect statistically calibrated endpoint selection to conditional generation and quantify its approximation tradeoff.