Synchronization of Unbalanced Dynamical Optimal Transport across Multiple Spaces

TL;DR

It is proved that UnSyncOT can be reduced to a single-space problem: the Monge model becomes a Benamou-Brenier problem with a metric-modified kinetic energy, and the Kantorovich model yields a nonlocal action induced by the synchronization operator, both of which fit within a dissipation-distance formulation.

Abstract

Many biological systems are observed through heterogeneous modalities, requiring transport models that couple dynamics across spaces while allowing mass variation. To address this challenge, we introduce Unbalanced Synchronized Optimal Transport (UnSyncOT), a novel dynamical framework that synchronizes transport-reaction flows between spaces via either geometric embeddings (Monge type) or Markov kernels (Kantorovich type). For both cases we prove that UnSyncOT can be reduced to a single-space problem: the Monge model becomes a Benamou-Brenier problem with a metric-modified kinetic energy, and the Kantorovich model yields a nonlocal action induced by the synchronization operator, both of which fit within a dissipation-distance formulation. We also analyze the pure transport (Wasserstein) and pure reaction (Fisher-Rao) limits and derive structural properties. For the Kantorovich case we propose an approximate UnSyncOT by introducing a Hellinger-Kantorovich based trapezoidal time discretization of the secondary action for efficient computation. Finally we present staggered-grid discretizations and primal-dual solvers, validate the convergence, stability, and efficiency, and demonstrate coherent dynamics reconstructions across spaces.

Figures (5)

• Figure 3.1: Schematic of unbalanced synchronized optimal transport. There is a given mapping from the primary space $X^{(1)}$ to each secondary space $X^{(i)}$, for $i=2,\cdots, d$.
• Figure 7.1: a) Mass curve comparison when $\bar{\rho}_0=N(\cdot;0.3, 0.05)$ and $\bar{\rho}_1=N(\cdot;0.7, 0.05)$. b) Mass curve comparison when $\bar{\rho}_0=N(\cdot;0.3, 0.05)$ and $\bar{\rho}_1=1.2N(\cdot;0.7, 0.05)$. $M = 64$ and $Q=32$ for both cases. Recall that for $X=[0,1]^2$, the grid sizes are $\Delta_x = 1/M, \Delta_y = 1/N, \Delta_t=1/Q$. If $X=[0,1]$, the grid is only determined by $M$ and $Q$.
• Figure 7.2: a) The evolving density $\rho$ and growth $H=\rho g$ in the primary space at different time points when $\mathbf T(x,y)=(x,y,\exp{(-\frac{(x-0.5)^2+(y-0.5)^2}{2\times 0.15^2})})$ and $c_2 = 0.01$ or $0.05$. b) The evolving density $\rho$ and growth $H=\rho g$ in the primary space at different time points when $\mathbf T(x,y)=(x,y,\sin(2\pi x)\sin(2\pi y))$ and $c_2=0.01$ or $0.1$. The map $\mathbf T$ is visualized as contours. In both examples, $\alpha=\beta=1$.
• Figure 7.3: a) The evolving density $\rho$ in the primary space at different time points given by WFR or Kantorovich UnSyncOT for the first Monge UnSyncOT example. Here $Y=\mathbf T(x)$ where $\mathbf T(x,y)=(x,y,\exp{(-\frac{(x-0.5)^2+(y-0.5)^2}{2\times 0.15^2})})$,$\alpha=\beta=1$, and $c_2/c_1=20$. b) The background color example. $\alpha=\beta=1$ and $c_2=0.5$. Top: the evolving density $\xi = \mathcal{T}_\mathrm{K}(\rho)$ in the primary space at different time points. The RGB color is shown in the background. Bottom: The evolving density in the secondary color space. Note that $\xi_0$ and $\xi_1$ are supported on the same region in the color space, and Kantorovich UnSyncOT is able to take that fact into account.
• Figure 7.4: a) The map $\mathbf T(x,y)=(x,y,10z(x,y))$. The data manifold $X_{\text{data}}$ occupies the black region. b) The evolving density $\rho$ in the primary space at different time points when $c_2 = 0.0$ or $0.9$ for the PHATE example. $X_{\text{data}}$ is visualized as the region inside the red contour. When $c_2=0.9$, most part of $\rho_t$ is confined inside $X_{\text{data}}$. c) The normalized simulated scRNA-seq and scATAC-seq data. d) The evolving density $\rho$ in the primary space at different time points when $c_2 = 0.0$ or $0.5$ for the multimodal example. In both examples, $\alpha=\beta=1$. When $c_2=0.5$, a branching apparently occurs.

Theorems & Definitions (44)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Corollary 3.3