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Efficient Trajectory Inference in Wasserstein Space Using Consecutive Averaging

Amartya Banerjee, Harlin Lee, Nir Sharon, Caroline Moosmüller

TL;DR

This work proposes methods for B-spline approximation and interpolation of point clouds through consecutive averaging that is intrinsic to the Wasserstein space and demonstrates linear convergence rates and rigorously evaluates the effectiveness of this method on cell data characterized by bifurcations, merges, and trajectory splitting scenarios like supercells.

Abstract

Capturing data from dynamic processes through cross-sectional measurements is seen in many fields, such as computational biology. Trajectory inference deals with the challenge of reconstructing continuous processes from such observations. In this work, we propose methods for B-spline approximation and interpolation of point clouds through consecutive averaging that is intrinsic to the Wasserstein space. Combining subdivision schemes with optimal transport-based geodesic, our methods carry out trajectory inference at a chosen level of precision and smoothness, and can automatically handle scenarios where particles undergo division over time. We prove linear convergence rates and rigorously evaluate our method on cell data characterized by bifurcations, merges, and trajectory splitting scenarios like $supercells$, comparing its performance against state-of-the-art trajectory inference and interpolation methods. The results not only underscore the effectiveness of our method in inferring trajectories but also highlight the benefit of performing interpolation and approximation that respect the inherent geometric properties of the data.

Efficient Trajectory Inference in Wasserstein Space Using Consecutive Averaging

TL;DR

This work proposes methods for B-spline approximation and interpolation of point clouds through consecutive averaging that is intrinsic to the Wasserstein space and demonstrates linear convergence rates and rigorously evaluates the effectiveness of this method on cell data characterized by bifurcations, merges, and trajectory splitting scenarios like supercells.

Abstract

Capturing data from dynamic processes through cross-sectional measurements is seen in many fields, such as computational biology. Trajectory inference deals with the challenge of reconstructing continuous processes from such observations. In this work, we propose methods for B-spline approximation and interpolation of point clouds through consecutive averaging that is intrinsic to the Wasserstein space. Combining subdivision schemes with optimal transport-based geodesic, our methods carry out trajectory inference at a chosen level of precision and smoothness, and can automatically handle scenarios where particles undergo division over time. We prove linear convergence rates and rigorously evaluate our method on cell data characterized by bifurcations, merges, and trajectory splitting scenarios like , comparing its performance against state-of-the-art trajectory inference and interpolation methods. The results not only underscore the effectiveness of our method in inferring trajectories but also highlight the benefit of performing interpolation and approximation that respect the inherent geometric properties of the data.
Paper Structure (38 sections, 1 theorem, 11 equations, 10 figures, 5 tables, 4 algorithms)

This paper contains 38 sections, 1 theorem, 11 equations, 10 figures, 5 tables, 4 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Problem formulation
  4. Discrete optimal transport and the Wasserstein space
  5. Splines through consecutive averaging
  6. WASSERSTEIN CURVE APPROXIMATION THROUGH CONSECUTIVE AVERAGING
  7. Trajectory inference
  8. Approximation and bending behavior
  9. Splitting trajectories
  10. Complexity analysis
  11. Convergence analysis
  12. EXTENSION OF WLR AND THE FOUR-POINT SCHEME
  13. EXPERIMENTS
  14. Splitting trajectories and non-uniform weights in supercells
  15. Constant number of points with uniform mass distribution
  16. ...and 23 more sections

Key Result

Theorem 1

Consider a sequence of measures $\mu_{t_0},\ldots,\mu_{t_T}$ with $t_0<\ldots < t_T$. Define the initial data by $\nu^{(0)} = [\nu^{(0)}_{t_j}]_{j=0}^T$ with $\nu^{(0)}_{t_j} = \mu_{t_j}$, and choose a smoothness degree $M$. Apply WLR (i.e. call alg:WLRm for the initial data, smoothness $M$, and ref where constant $C$ is a constant independent of $R$.

Figures (10)

  • Figure 1: Our proposed WLR successfully performs trajectory inference on Converging Gaussian dataset, which has different number of points per time step. While others maintain a fixed number of trajectories from initialization to termination, our method can split trajectories automatically. The fact that WLR can naturally deal with mass splitting phenomena is one of its major benefits. See Sec. \ref{['sec:experiments']} for details.
  • Figure 2: Illustration of the classical Lane-Riesenfeld (Alg. \ref{['alg:LRm']}) with $M=2$, $R=1$. This refinement step can be repeated to approximate a smooth cubic B-spline. (a) Doubling points; (b) First averaging; (c) Second averaging; (d) Refined points.
  • Figure 3: We provide an illustrative example in $\mathbb{R}^2$ to demonstrate how WLR infers trajectories and handles both uniform (Top) and non-uniform (Bottom) masses. The size of the point is proportional to $a_{{t_j}, i}$, the mass of point $i$ at time step $t_j$. In all cases, the trajectories respect the geometry of the Wasserstein space.
  • Figure 4: Illustrative example in $\mathbb{R}^2$ demonstrating exact interpolation via the Wasserstein four-point scheme on weighted circular Gaussians.
  • Figure 5: WLR produces smooth trajectories on CITE-seq supercells with non-uniformly distributed mass by automatically splitting trajectories. The size of the points is proportional to $a_{{t_j}, i}$, the mass of point $i$ at time step $t_j$. Only a subset $(n_t = 50)$ is visualized for clarity.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof