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On minimax density estimation via measure transport

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arxiv 2207.10231 v2 pith:4SZAHROE submitted 2022-07-20 math.ST math.PRstat.MLstat.TH

Sven Wang , Youssef Marzouk This is my paper · ORCID
classification math.STmath.PRstat.MLstat.TH
keywords estimatorsmeasuretransportdensitypenalizedchosenconvergenceestablish
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abstract

We study the convergence properties, in Hellinger and related distances, of nonparametric density estimators based on measure transport. These estimators represent the measure of interest as the pushforward of a chosen reference distribution under a transport map, where the map is chosen via a maximum likelihood objective (equivalently, minimizing an empirical Kullback-Leibler loss) or a penalized version thereof. We establish concentration inequalities for a general class of penalized measure transport estimators, by combining techniques from M-estimation with analytical properties of the transport-based density representation. We then demonstrate the implications of our theory for the case of triangular Knothe-Rosenblatt (KR) transports on the $d$-dimensional unit cube, and show that both penalized and unpenalized versions of such estimators achieve minimax optimal convergence rates over H\"older classes of densities. Specifically, we establish optimal rates for unpenalized nonparametric maximum likelihood estimation over bounded H\"older-type balls, and then for certain Sobolev-penalized estimators and sieved wavelet estimators.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    stat.ML 2026-02 conditional novelty 7.0 of 10

    For Gaussian distributions, slice-matching to an isotropic target with decaying step sizes converges at rate O(k^{-(2α-1)}) in expectation.

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    Conditional Brenier-map estimators provably converge to true conditionals, with mean map error decaying like N^{-1/4} (slow) or sqrt(log N / N) (fast), and these rates carry over to an idealized optimal-transport filter.

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    A new multi-fidelity Bayesian transport map method learns non-Gaussian joint distributions across spatial scales and outperforms existing emulators in downscaling climate fields from small training sets.

  4. Robust Learnability of Sample-Compressible Distributions under Noisy or Adversarial Perturbations

    stat.ML 2025-06 reject novelty 6.0 of 10

    Sample-compressible distribution families are shown to be PAC-learnable under additive noise and adversarial corruption, given new stability and low-frequency assumptions.

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