Manifold-consistent velocity

中文導讀

這頁講一個幾何約束:細胞落在一個 low-dimensional manifold M 上,真正的 velocity 必須落在該點的 tangent space T_pM 裡。多數方法估出來的 velocity 不滿足這條件(有 normal component = noise)。GraphVelo 用 tangent space projection(TSP)把 velocity 投回 T_pM,同時保留方向跟 magnitude(cosine kernel 只留方向)。延伸出 MacK genes(velocity sign 跟 manifold 一致的高信心 基因)跟 cross-modality velocity transformation。注意:這是 geometric consistency,不是 physical-time grounding——它讓 velocity 更乾淨、可保留 relative speed,但不 anchor absolute time。

Definition

A velocity field is manifold-consistent if each cell’s velocity lies in the tangent space T_pM of the low-dimensional data manifold M on which cells reside. Velocities inferred by splicing-based methods generally violate this — they carry a normal component (noise) off the manifold. Enforcing the tangent constraint cleans the field while, done correctly, preserving magnitude (unlike the direction-only cosine kernel).

Tangent-space projection (TSP)

The mechanism in GraphVelo: build a kNN graph; the neighbor displacements δ_ij = x_j − x_i span a local approximation of T_pM; express the measured velocity as v_i = Σ φ_ij δ_ij and solve for φ by minimizing a loss that (a) matches the original velocity magnitude and (b) matches the cosine-kernel direction (asymptotically correct per La Manno / Li et al.), with L2 regularization. Result: a smooth field tangent to M that keeps both direction and speed.

Two extensions

• MacK genes (Manifold-consistent Kinetics): genes whose splicing-velocity sign agrees with the manifold direction; a high-confidence subset from which whole-genome velocity is inferred — robust to MURK / complex-kinetics genes.
• Velocity transformation: via local linear embedding, velocity moves between representations and modalities (RNA↔chromatin↔spatial), justified by the Whitney embedding theorem. This is what lets velocity reach multi-omics data.

A fuller cousin: ddHodge

ddHodge takes the same geometry-preserving stance but goes further: instead of only projecting velocity onto the tangent space, it patches local PCA tangent spaces with a sheaf (connection) Laplacian and performs a full hodge-decomposition (gradient / curl / harmonic) plus acceleration. Both reject UMAP velocity-embedding for destroying manifold geometry; GraphVelo cleans+transports the field, ddHodge dissects it.

Tangency by construction: VeloCycle

GraphVelo projects an existing velocity onto the tangent space post hoc; ddHodge dissects the field; VeloCycle (velocycle-2024) builds the constraint in — it jointly estimates the manifold and a velocity field parameterized as an autonomous function V(x) of the manifold coordinates, tangent by construction, in one Bayesian generative model. There is no off-manifold component to remove because none is ever produced. On the cell cycle (a 1D periodic manifold) this is what makes the velocity dynamically consistent and, crucially, what lets a single angular speed ω(φ) be defined and scaled to real time.

Relation to physical time

Geometric consistency is usually orthogonal to physical-time-grounding: TSP makes the field cleaner and retains relative speed (a step methods that normalize discard), but does not anchor absolute time — magnitude is inherited from the input method’s scale. The exception is VeloCycle, where the manifold constraint is not just cosmetic: by reducing the dynamics to a single known-geometry coordinate (the cycle phase), it makes the velocity a scalar angular speed that a measured half-life can convert to validated metric time. So manifold-constraint + known geometry + a measured rate can combine to ground physical time — the lesson the general (grn-informed-velocity) and absolute-rate (metabolic-labeling) axes feed into.

Related

GraphVelo · ddHodge · VeloCycle · velocycle-2024 · hodge-decomposition · RNA velocity · splicing-kinetics-ode · physical-time-grounding · dynamo · metabolic-labeling · FlowVelo · optimal-transport