A principled minimax training procedure over input convex neural networks learns the optimal quadratic-cost transport map as the gradient of a convex potential.
Uncoupled isotonic regression via minimum Wasserstein deconvolution
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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.
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cs.LG 1years
2019 1verdicts
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Optimal transport mapping via input convex neural networks
A principled minimax training procedure over input convex neural networks learns the optimal quadratic-cost transport map as the gradient of a convex potential.