2022/09/29 by Qiang Liu, Liu, Qiang · 64 citations
Computer Science · #Advanced Neural Network Applications #Adversarial Robustness in Machine Learning #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2209.14577
openalex publication_date 2022/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a flow-based approach to the optimal transport (OT) problem between two continuous distributions π0,π1 on ℝd, of minimizing a transport cost 𝔼[c(X1-X0)] in the set of couplings (X0,X1) whose marginal distributions on X0,X1 equals π0,π1, 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.