Principal Curves In Metric Spaces And The Space Of Probability Measures

TL;DR

This work develops a theory of principal curves in general compact metric spaces and specializes to the 2-Wasserstein space to address trajectory inference from measure-valued data. It introduces a variational objective that trades off data-fit with curve length, proves existence and stability of minimizers, and provides discretization schemes with rigorous continuum limits. In the Wasserstein setting, the framework handles doubly empirical data and yields a consistency guarantee that the estimated curve recovers the true developmental path up to reparametrization, enabling seriation of time labels without explicit timing. The approach is validated on synthetic datasets, showing competitive seriation performance and offering extensions to nonlocal kernels and semi-supervised configurations, with broad implications for high-density, parallelized single-cell experiments.

Abstract

We introduce principal curves in Wasserstein space, and in general compact metric spaces. Our motivation for the Wasserstein case comes from optimal-transport-based trajectory inference, where a developing population of cells traces out a curve in Wasserstein space. Our framework enables new experimental procedures for collecting high-density time-courses of developing populations of cells: time-points can be processed in parallel (making it easier to collect more time-points). However, then the time of collection is unknown, and must be recovered by solving a seriation problem (or one-dimensional manifold learning problem). We propose an estimator based on Wasserstein principal curves, and prove it is consistent for recovering a curve of probability measures in Wasserstein space from empirical samples. This consistency theorem is obtained via a series of results regarding principal curves in compact metric spaces. In particular, we establish the validity of certain numerical discretization schemes for principal curves, which is a new result even in the Euclidean setting.

Figures (10)

• Figure 1: (a) An illustration showing ${\color{blue} \rho_t}$ (blue) at evenly-spaced time samples $t_1, \ldots, t_N$, with the samples comprising $\color{red} \hat{\rho}_{t_n}$ shown in red. (b) An illustration of a principal curve $\gamma_t$ (black curve) "fitting" the empirical measures $\color{red} \hat{\rho}_{t_n}$, each of which is represented by a single red dot. Note that the curve achieves low average projection distance, while the curve itself is not "too long". The straight lines connecting individual data points to their projections along the curve represent length-minimizing geodesics in the Wasserstein space.
• Figure 2: A visualization of one loop of the algorithm. (a) A local view of the situation in the discretized case. Here, the knots $\color{blue} \{\gamma_k\}_{k=1}^K$ are plotted with geodesic interpolations between adjacent points, ordered using a TSP solver. Also shown: The data points $\color{red} \{x_n\}_{n=1}^N$ and the Voronoi cells (dotted black lines). (b) An illustration of how the update step works. The position of each knot $\color{blue} \gamma_k$ affects the overall objective value via: (1) the average distance from $\color{blue} \gamma_k$ to the data points ${\color{red} x_n} \in I_k$, and (2) the distance from $\color{blue} \gamma_{k}$ to the adjacent knots $\color{blue} \gamma_{k-1}, \gamma_{k+1}$. In that sense, at the update step each $\color{blue} \gamma_k$ is "pulled" toward the points of $I_k$ (vectors drawn in gray) and $\{{\color{blue} \gamma_{k-1}, \gamma_{k+1}}\}$ (vectors draw in blue). (c) Moving the knot points according a weighted sum of the vectors in \ref{['fig:vc-vectors']}. In accordance with the weightings of the terms in the discretized functional in Step 5, each vector pointing to a ${\color{red} x_n} \in I_k$ is weighted by $1/N$, while the vectors pointing to $\color{blue} \gamma_{k-1}, \gamma_{k+1}$ are weighted by $\beta/(2K-2)$ (the factor of two arising because each $d^2({\color{blue} \gamma_k}, {\color{blue} \gamma_{k+1}})$ is split into one vector on $\color{blue} \gamma_k$ and one on $\color{blue} \gamma_{k+1}$). (d) The updated $\color{blue}\{ \gamma_k \}_{k=1}^K$ with the associated updated Voronoi cells.
• Figure 3: A simple curve of probability measures with 250 time points that undergoes a branching event (A) is fitted with a principle curve, using kernel bandwidth $h=0.037$ and length penalty $\beta=0.17$ (B). The color code bar to the right of panel (B) indicates the normalized time parameter for the underlying dataset (A) and the fitted principal curve (B) respectively. A performance comparison to other methods can be seen in (C), which indicates the seriation error for various choices of the number of time points, with a fixed budget of $10000$ total atoms measured. For spectral seriation, a kernel bandwidth of $\sigma = 0.5$ was used.
• Figure 4: A curve in the space of probability measures with a non-trivial change in direction, and which undergoes a branching event (A) is fitted with a principle curve with $h=0.01$, and $\beta=0.037$ (B). The curve contains 250 time points. The principle curve seriation method can be seen to obtain the best performance for a number of regimes with varying time points and 10000 total atoms (C). For spectral seriation, a kernel bandwidth $\sigma = 0.315$ was used.
• Figure 5: Illustration for the proof of Proposition \ref{['prop:r2-nonuniqueness']}. Take $\varphi_1, \varphi_2, \varphi_3$ to be the three distinct isometries indicated, applied to a curve with $\mu$ uniform on a triangle. In this case, $\varphi_3$ yields a distinct image.

Theorems & Definitions (57)

• Theorem 1.1: Consistency of Wasserstein principal curves
• Proposition 2.1: Existence of principal curves
• Proposition 2.2: Non-uniqueness of principal curves
• Proposition 2.3: Stability of principal curves with respect to data distribution
• Corollary 2.4
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.5: Discrete to continuum