Uncoupled isotonic regression via minimum Wasserstein deconvolution
Abstract
Isotonic regression is a standard problem in shape-constrained estimation where the goal is to estimate an unknown non-decreasing regression function $f$ from independent pairs $(x_i, y_i)$ where ${\mathbb{E}}[y_i]=f(x_i), i=1, \ldots n$. While this problem is well understood both statistically and computationally, much less is known about its uncoupled counterpart, where one is given only the unordered sets $\{x_1, \ldots , x_n\}$ and $\{y_1, \ldots , y_n\}$. In this work, we leverage tools from optimal transport theory to derive minimax rates under weak moments conditions on $y_i$ and to give an efficient algorithm achieving optimal rates. Both upper and lower bounds employ moment-matching arguments that are also pertinent to learning mixtures of distributions and deconvolution.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Apr 02, 2019
- Source ID
- 10.1093/imaiai/iaz006
Entities
People
- Jonathan Weed
- Philippe Rigollet
Organizations
- Josephine De Karman Fellowship Trust
- Massachusetts Institute of Technology
- National Science Foundation
- Office of Naval Research
- Silicon Valley Community Foundation