论文标题
整流流：最佳运输的边际保存方法
Rectified Flow: A Marginal Preserving Approach to Optimal Transport
论文作者
论文摘要
我们提出了一种基于流量的最佳传输方法（OT）问题的方法，在两个连续分布之间$π_0，π_1$上的$ \ Mathbb {r}^d $，以最小化运输成本$ \ MATHBB {e} [c（x_1-x_0）$ in of of couplings $（x_1 $ equalter）$（x_1-x_0）$ narg（x_1-x_0）$ equplings $（x_0，x_1）$ x $（x_1）$ x $π_0，π_1$，其中$ c $是成本函数。我们的方法迭代地构建了一系列神经普通可区分方程（ode），每种方法都通过解决简单的无约束回归问题来学习，从而单调地降低了运输成本，同时自动保留边缘约束。这产生了一种单调的内部方法，该方法在有效耦合的集合中穿越以降低运输成本，这将自身与大多数现有方法区分开来，从而强制执行耦合约束与外部。该方法的主要思想是从整流流程中获取的，这是一种最近的方法，同时降低了由凸函数$ c $（因此本质上是多目标）引起的整个运输成本的家族，但并非量身定制以最大程度地减少特定的运输成本。我们的方法是整流流的单对象变体，可确保解决固定的，用户指定的凸成本函数$ c $的ot问题。
We present a flow-based approach to the optimal transport (OT) problem between two continuous distributions $π_0,π_1$ 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 $π_0,π_1$, 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 $c$.
