arXiv subjects

Houren Hong

Publications and source records attributed to Houren Hong.

3 recordsLinked to original sources

arXiv2026

Uniform Convergence of Generalized Conditional Fr\'echet Means with Applications to Weighted Fr\'echet Aggregation and Exceedance Set Estimation

The statistical analysis of object oriented data in non-Euclidean spaces heavily relies on generalized conditional Fr\'echet means, notably in the context of Fr\'echet regression. However, establishing the uniform convergence of these estimators presents several theoretical challenges. The difficulties are caused primarily by the absence of linear structures in general metric spaces, rendering standard techniques for verifying the asymptotic uniform equicontinuity of the estimator largely intractable. To overcome this limitation, this paper introduces an alternative theoretical framework for establishing uniform convergence that bypasses the need to verify uniform equicontinuity, under a novel structural condition on the empirical cost function of the generalized conditional Fr\'echet means. We demonstrate that this analytical condition is satisfied by various prominent Fr\'echet regression models across broad classes of metric spaces. Leveraging these foundational uniform convergence guarantees, we subsequently extend two widely used frameworks from Euclidean to non-Euclidean spaces: (i) a weighted Fr\'echet aggregation framework that facilitates both distributed regression and robust median-of-means regression; and (ii) an exceedance set estimation framework to identify critical covariate regions where the conditional generalized Fr\'echet mean surpasses a prescribed threshold, alongside a metric to quantify the aggregate magnitude of the exceedance. The theoretical properties of these proposed methods are empirically validated through Monte Carlo simulations and an application to dynamic transportation networks in New York City.

stat.ME↗

arXiv2026

Robust Estimation of Location in Matrix Manifolds Using the Projected Frobenius Median

We propose a robust method for location estimation in various matrix manifolds based on the projected Frobenius median, which is closely related to the spatial median. This method applies broadly to matrix manifolds, including Stiefel and Grassmann manifolds, Kendall shape spaces as well as to projective Stiefel manifolds, a type of quotient space of a Stiefel manifold. Our approach involves computation of the Frobenius median in an ambient Euclidean space followed by projection onto the relevant matrix manifold. Our estimation method is computationally attractive, has a unique solution provided the sample data are not colinear in the ambient Euclidean space, has desirable robustness features and has appropriate equivariance properties under natural groups of transformations. We establish asymptotic normality under mild conditions and derive the influence function for matrix manifolds of interest. Simulation studies on the rank-1 complex Grassmann manifold and the projective Stiefel manifold further show the applicability and robustness of our method. We also apply our method to a real-world earthquake moment tensor dataset.

stat.ME↗

arXiv2025

A Robust Extrinsic Single-index Model for Spherical Data

Regression with a spherical response is challenging due to the absence of linear structure, making standard regression models inadequate. Existing methods, mainly parametric, lack the flexibility to capture the complex relationship induced by spherical curvature, while methods based on techniques from Riemannian geometry often suffer from computational difficulties. The non-Euclidean structure further complicates robust estimation, with very limited work addressing this issue, despite the common presence of outliers in directional data. This article introduces a new semi-parametric approach, the extrinsic single-index model (ESIM) and its robust estimation, to address these limitations. We establish large-sample properties of the proposed estimator with a wide range of loss functions and assess their robustness using the influence function and standardized influence function. Specifically, we focus on the robustness of the exponential squared loss (ESL), demonstrating comparable efficiency and superior robustness over least squares loss under high concentration. We also examine how the tuning parameter for the ESL balances efficiency and robustness, providing guidance on its optimal choice. The computational efficiency and robustness of our methods are further illustrated via simulations and applications to geochemical compositional data.

stat.ME↗