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 nondecreasing 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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2018
- DOI:
- 10.48550/arXiv.1806.10648
- arXiv:
- arXiv:1806.10648
- Bibcode:
- 2018arXiv180610648R
- Keywords:
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- Mathematics - Statistics Theory;
- Statistics - Machine Learning;
- 62G08
- E-Print:
- To appear in Information and Inference: a Journal of the IMA