We present a flow-based approach to the optimal transport (OT) problem between two continuous distributions $\pi_0,\pi_1$ on $\mathbb{R}^d$, of minimizing a transport cost $\mathbb{E}[c(X_1-X_0)]$ in the set of couplings $(X_0,X_1)$ whose marginal distributions on $X_0,X_1$ equals $\pi_0,\pi_1$, respectively, where $c$ is a cost function. Our method iteratively constructs a sequence of neural ordinary differentiable equations (ODE), each learned by solving a simple unconstrained regression problem, which monotonically reduce the transport cost while automatically preserving the marginal constraints. This yields a monotonic interior approach that traverses inside the set of valid couplings to decrease the transport cost, which distinguishes itself from most existing approaches that enforce the coupling constraints from the outside. The main idea of the method draws from rectified flow, a recent approach that simultaneously decreases the whole family of transport costs induced by convex functions $c$ (and is hence multi-objective in nature), but is not tailored to minimize a specific transport cost. Our method is a single-object variant of rectified flow that guarantees to solve the OT problem for a fixed, user-specified convex cost function $c$.
翻译:我们对最佳运输(OT)问题提出了一种基于流动的办法,即两个连续分配问题($$\pi_0,\pi_1美元,$$mathbb{R ⁇ d$),两个连续分配问题(OT),一个是最大限度地减少运输费用($mathbb{E})[c(X_1-X_0)],一个是一组连带费用(X_0,X_1美元),其边际分配在X_0,X_1美元等于美元/pi_0,\pi_1美元,其中C美元是一个成本函数。我们的方法迭代地构建了一个神经普通可变异方程式(ODE)的序列,每个序列都是通过解决一个简单的不受限制的回归问题(ODE)来学习的,它单调降低运输费用,同时自动保留边际限制。这产生了一种单调的内置方法,在一套有效的连带组合的组合中,它与大多数从外部实施连带制约的方法有区别。这种方法的主要想法是从纠正流中得出,最近一种同时减少整个运输费用组合成本(On-comx函数所引引出的),一个固定的固定的固定的固定成本(也是一种固定的固定的固定的固定的固定的路径。