[Paper Review] Fast and Smooth Interpolation on Wasserstein Space

Sinho Chewi, Julien Clancy+4 more|arXiv (Cornell University)|Oct 22, 2020

Advanced Numerical Analysis Techniques37 references4 citations

TL;DR

This paper proposes a novel method for smooth, particle-flow-based interpolation of probability measures on Wasserstein space by reducing the problem to Euclidean spline interpolation via Monge maps. It achieves the first approximation guarantees on Wasserstein space and enables efficient, interpretable trajectories for applications like single-cell genomics, with a key contribution being the use of thin-plate splines for measure-valued surfaces.

ABSTRACT

We propose a new method for smoothly interpolating probability measures using the geometry of optimal transport. To that end, we reduce this problem to the classical Euclidean setting, allowing us to directly leverage the extensive toolbox of spline interpolation. Unlike previous approaches to measure-valued splines, our interpolated curves (i) have a clear interpretation as governing particle flows, which is natural for applications, and (ii) come with the first approximation guarantees on Wasserstein space. Finally, we demonstrate the broad applicability of our interpolation methodology by fitting surfaces of measures using thin-plate splines.

Motivation & Objective

• To develop a method for smooth, trajectory-aware interpolation of probability measures on Wasserstein space that preserves particle flow interpretation.
• To overcome limitations of prior P-spline methods, which lack unique particle trajectories and have high computational cost.
• To provide the first approximation guarantees for measure-valued interpolation on Wasserstein space.
• To enable efficient and interpretable reconstruction of individual cell trajectories in single-cell genomics.
• To extend the method to surface fitting using thin-plate splines for multidimensional measure interpolation.

Proposed method

• The method reduces interpolation on Wasserstein space to standard Euclidean cubic spline fitting by constructing Monge maps between consecutive measures.
• It uses the composition of cumulative distribution functions (CDFs) and their inverses to define transport maps between measures, ensuring monotonicity and Monge solutions.
• For Gaussian measures, the interpolated quantiles are computed via thin-plate splines applied to means and standard deviations separately.
• The particle trajectories are derived from the spline interpolation of the transport maps, ensuring smooth, deterministic paths from initial to interpolated positions.
• The approach leverages existing Euclidean spline toolboxes, enabling fast computation with only N Monge map evaluations and standard spline fitting.
• For non-Gaussian measures, the method relies on empirical CDFs and inverse CDFs to define the transport maps, assuming absolute continuity.

Experimental results

Research questions

• RQ1 Can we construct smooth, deterministic particle trajectories that interpolate given probability measures on Wasserstein space?
• RQ2 How can we ensure the interpolated curves reflect the geometry of optimal transport while maintaining computational efficiency?
• RQ3 What are the approximation guarantees for measure-valued interpolation in the Wasserstein space?
• RQ4 Can we extend the method to interpolate surfaces of measures using thin-plate splines?
• RQ5 Why do existing P-spline methods fail to produce unique particle trajectories, and how can this be overcome?

Key findings

• The proposed method produces Monge solutions, ensuring that each initial particle has a unique, smooth trajectory through the interpolation process.
• The method achieves the first known approximation guarantees for interpolation on Wasserstein space, providing theoretical bounds on convergence.
• For Gaussian measures, the interpolated distribution at any point is fully determined by thin-plate splines applied to the means and standard deviations, with quantiles given by $ m_x + \Phi^{-1}(\alpha) \cdot |s_x| $.
• The computational cost is significantly reduced compared to prior P-spline methods, requiring only N evaluations of Monge maps and standard spline fitting.
• Empirical results on single-cell data and weather station temperature distributions show smooth, interpretable interpolation with clear particle flow interpretation.
• The method successfully reconstructs trajectories in a way that is both mathematically sound and practically deployable, outperforming non-smooth or non-unique alternatives.