Optimal transport

中文導讀

Optimal transport (OT) 在 single-cell 裡常用來把不同時間點的 population 對起來 (Waddington-OT 那類),或當 flow-matching 的 target coupling。它的 dynamical 版本( Schrödinger bridge)在已知 sampling time 的 time-series 上能給帶時間的 transport。對 physical-time-grounding 來說關鍵差別是:OT 有沒有吃 real sampling time——有(time-series) 才可能 metric,只有 snapshot 一樣只給 relative geometry。FlowVelo 的 transport 框架要面對 同一個問題。

Definition

The problem of moving one probability distribution to another at minimal cost; in single-cell, used to couple cell populations across timepoints (e.g. Waddington-OT) or as the target coupling for flow-matching. The dynamical form (Schrödinger bridge) yields a time-indexed stochastic transport between distributions.

Relevance to physical time

OT/Schrödinger-bridge methods that ingest a real-time series with known sampling times can carry metric time (the sampling times are the anchor). Applied to a single snapshot, OT recovers relative geometry only — same limitation as the splicing-kinetics-ode scale degeneracy (see physical-time-grounding). This is the design tension FlowVelo must resolve explicitly.

Related

flow-matching · transport-operators · FlowVelo · physical-time-grounding · RNA velocity