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Title: Permuted and unlinked monotone regression in $\mathbb{R}^d$: An approach based on mixture modeling and optimal transport Authors:  Martin Slawski - George Mason Univ (United States) [presenting]
Bodhisattva Sen - Columbia University (United States)
Abstract: Suppose the aim is to learn a map between $d$-dimensional inputs and $d$-dimensional noisy outputs, without observing (input, output)-pairs, but only separate unordered lists of inputs and outputs. We show that the notion of cyclical monotonicity of the underlying map is sufficient for identification and estimation in the unordered setting. We study restoration of the correct correspondence of (input,output)-pairs (``permutation recovery'') and develop a computationally efficient and easy-to-use algorithm for denoising based on the Kiefer-Wolfowitz nonparametric maximum likelihood estimator and techniques from the theory of optimal transport. We provide explicit upper bounds on the associated mean squared denoising error under Gaussian noise. Numerical studies corroborate our theoretical analysis.