ACPL: A Coordinate-System Prior for Topology-Aware Directed Trajectory Inference on Cyclic Biological Manifolds

Abstract

Directed trajectory inference from single-cell RNA sequencing data requires graph construction algorithms that respect the geometric structure of cyclical biological processes. Minimum spanning tree approaches systematically produce backward edges across cyclical manifolds through Euclidean shortcuts at the loop closure, inverting the inferred direction of division at its most biologically consequential transition. We introduce ACPL (Arc+LOESS Polar-Linearization), a coordinate-system prior that transforms two-dimensional UMAP embeddings into polar coordinates aligned with the manifold's angular structure and admits only forward edges along LOESS-smoothed cumulative arc length. We prove that the resulting directed graph is acyclic by construction at O(N log N) cost and that LOESS smoothing is mathematically necessary to avoid a trivial accuracy artefact. Empirical evaluation across five biological datasets is reported in full: ACPL achieves 97.0% structural accuracy on Spellman yeast and 75.9% on Nestorowa haematopoietic stem cells, with all pairwise bootstrap confidence intervals crossing zero. The method is robust to UMAP hyperparameter choice (SD = 0.77 pp across 25 configurations) and survives spatial autocorrelation correction by hundreds of orders of magnitude. An R implementation is available at https://github.com/shreyosecret/ACPL under the MIT License.