Robust Estimation under the Wasserstein Distance

Code

• github.com/sbnietert/robust-ot
  OfficialIn paperpytorch★ 8

Abstract

We study the problem of robust distribution estimation under the Wasserstein distance, a popular discrepancy measure between probability distributions rooted in optimal transport (OT) theory. Given n samples from an unknown distribution , of which n are adversarially corrupted, we seek an estimate for with minimal Wasserstein error. To address this task, we draw upon two frameworks from OT and robust statistics: partial OT (POT) and minimum distance estimation (MDE). We prove new structural properties for POT and use them to show that MDE under a partial Wasserstein distance achieves the minimax-optimal robust estimation risk in many settings. Along the way, we derive a novel dual form for POT that adds a sup-norm penalty to the classic Kantorovich dual for standard OT. Since the popular Wasserstein generative adversarial network (WGAN) framework implements Wasserstein MDE via Kantorovich duality, our penalized dual enables large-scale generative modeling with contaminated datasets via an elementary modification to WGAN. Numerical experiments demonstrating the efficacy of our approach in mitigating the impact of adversarial corruptions are provided.

Tasks

Reproductions