Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2311.11934

Estimation of entropy-regularized optimal transport maps between non-compactly supported measures

Matthew Werenski·James M. Murphy·Shuchin Aeron
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper addresses the problem of estimating entropy-regularized optimal transport (EOT) maps with squared-Euclidean cost between source and target measures that are subGaussian. In the case that the target measure is compactly supported or strongly log-concave, we show that for a recently proposed in-sample estimator, the expected squared $L^2$-error decays at least as fast as $O(n^{-1/3})$ where $n$ is the sample size. For the general subGaussian case we show that the expected $L^1$-error decays at least as fast as $O(n^{-1/6})$, and in both cases we have polynomial dependence on the regularization parameter. While these results are suboptimal compared to known results in the case of compactness of both the source and target measures (squared $L^2$-error converging at a rate $O(n^{-1})$) and for when the source is subGaussian while the target is compactly supported (squared $L^2$-error converging at a rate $O(n^{-1/2})$), their importance lie in eliminating the compact support requirements. The proof technique makes use of a bias-variance decomposition where the variance is controlled using standard concentration of measure results and the bias is handled by T1-transport inequalities along with sample complexity results in estimation of EOT cost under subGaussian assumptions. Our experimental results point to a looseness in controlling the variance terms and we conclude by posing several open problems.

Explore related subjects

stat.MLcs.ITcs.LGmath.ITmath.STstat.TH

Keep this discovery

BibTeXRIS

Matthew Werenski, James M. Murphy, Shuchin Aeron. 2023-11-20. Estimation of entropy-regularized optimal transport maps between non-compactly supported measures. https://arxiv.org/abs/2311.11934

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Hilbert-Valued Functional Decomposition Framework for Explaining Time-Dependent Outputs

Feature-based explanations quantify features' influence on model predictions, but are primarily designed for scalar outputs. In many applications, however, outputs are functional or multivariate, such as time-dependent trajectories in demand forecasting. Consequently, existing approaches typically explain each output location independently, ignoring dependencies across the output components. We address this limitation by developing a unified framework for feature-based explanations of time-dependent outputs. Specifically, we generalize functional decomposition to Hilbert-valued prediction functions and extend an existing feature-based explanation framework to this setting. Our framework introduces kernel-based output representations that enable time-dependency-aware explanations at multiple levels of temporal granularity, including time-specific, time-resolved, and time-aggregated, while providing a unified view in which existing methods arise as special cases. We validate our framework on synthetic and real-world data, including intraday financial market volatility prediction and energy demand forecasting.

stat.ML↗
arXiv2026

Risk-Averse Decision Making with Multi-Level Reliability Guarantees

Many applications in engineering, including wireless broadcasting, require designs that provide performance certificates at different target outage levels. This paper studies the problem of maximizing the weighted average of such certificates in the presence of uncertainty about the true system state. The problem is shown to be equivalent to an optimization over nested prediction sets, connecting to the literature on conformal prediction and extending prior art on single-level risk-averse decision making. Furthermore, we derive a dual formulation that decouples optimization across input values. Numerical experiments on a diversity-based wireless transmission system illustrate the cost of enforcing multi-level certificates with a single shared policy and trace the Pareto trade-off between multiple reliability levels.

stat.ML↗
arXiv2026

A distribution-free certification framework for trustworthy crash-severity prediction

Crash-severity models inform screening, dispatch and site prioritization, yet are deployed without a finite-sample statement of what one prediction means. Off-the-shelf guarantees fail here, because the features that make crash severity distinctive defeat them: the KABCO outcome is ordinal, the recorded label is a field assessment agreeing with medical severity about half the time, erring in a structured way, and deployment crosses jurisdictions and years calibration never saw. We develop a certification layer that wraps any severity model unmodified, with distribution-free guarantees using this structure: contiguous ordinal sets that read as "B or worse"; per-class validity for any pre-declared partition, with an oracle efficiency characterization; transfer of coverage to unobserved true severity through a declared reporting band, with a worst-case sharpness result; a one-sided certificate under deployment shift; and severity-weighted risk control. The guarantees compose with an attributable slack budget. The same analysis bounds what certification can achieve. A certified set's informativeness is governed by a functional of the true law that no base model can evade and that cannot be lower-bounded distribution-free; given a declared misreporting channel identified from record-linkage data, a nonvacuous lower bound on that floor becomes computable. On 5.2 million Texas records across seven base models spanning four decades, the layer attaches identical validity and certifies, on the vulnerable road users, a model-independent floor on set width that no base model beats, separating it from a remainder that stays bounded but distribution-free unidentifiable. The framework is released as an open-source package with theorem-level tests.

stat.ML↗