arXiv subjects

Yujing Huang

Publications and source records attributed to Yujing Huang.

3 recordsLinked to original sources

arXiv2026

Intrinsic Riemannian Cross-covariance for Manifold-valued Random Objects

Covariance estimation yields a fundamental second-order statistic underlying representation learning, dimension reduction, and dependence modeling. While covariance has been well understood in Euclidean spaces, it is ill-defined for random objects residing on nonlinear Riemannian manifolds, which increasingly arise in modern machine learning applications involving shapes, symmetric positive definite (SPD) matrices, etc. This paper introduces an intrinsic Riemannian cross-covariance for manifold-valued random objects. Our approach defines covariance and correlation by transporting local variations to a common tangent space via parallel transport, yielding a second-order descriptor that is independent of arbitrary coordinate choices. We establish that the proposed covariance inherits desirable properties of its Euclidean counterparts and characterize its asymptotic behavior. Numerical studies on spheres and SPD manifolds, together with real-data experiments on heart valve shapes in Kendall's shape space, demonstrate the effectiveness of our estimators and verify the stated properties. Our results position the Riemannian covariance as a fundamental tool for second-order learning and analysis in non-Euclidean representation spaces.

math.ST↗

arXiv2026

Beyond Euclidean Summaries: Online Change Point Detection for Distribution-Valued Data

Existing online change-point detection (CPD) methods rely on fixed-dimensional Euclidean summaries, implicitly assuming that distributional changes are well captured by moment-based or feature-based representations. They can obscure important changes in distributional shape or geometry. We propose an intrinsic distribution-valued CPD framework that treats streaming batch data as a stochastic process on the 2-Wasserstein space. Our method detects changes in the law of this process by mapping each empirical distribution to a tangent space relative to a pre-change Fr\'echet barycenter, yielding a reference-centered local linearization of 2-Wasserstein space. This representation enables sequential detectors by adapting classical multivariate monitoring statistics to tangent fields. We provide theoretical guarantees and demonstrate, via synthetic and real-world experiments, that our approach detects complex distributional shifts with reduced detection delay at matched $\mathrm{ARL}_0$ compared with moments-based and model-free baselines. The code is available at https://github.com/yyzeng43/IDD-icml .

stat.ME↗

arXiv2024

Distribution-in-distribution-out Regression

Regression analysis with probability measures as input predictors and output response has recently drawn great attention. However, it is challenging to handle multiple input probability measures due to the non-flat Riemannian geometry of the Wasserstein space, hindering the definition of arithmetic operations, hence additive linear structure is not well-defined. In this work, a distribution-in-distribution-out regression model is proposed by introducing parallel transport to achieve provable commutativity and additivity of newly defined arithmetic operations in Wasserstein space. The appealing properties of the DIDO regression model can serve a foundation for model estimation, prediction, and inference. Specifically, the Fréchet least squares estimator is employed to obtain the best linear unbiased estimate, supported by the newly established Fréchet Gauss-Markov Theorem. Furthermore, we investigate a special case when predictors and response are all univariate Gaussian measures, leading to a simple close-form solution of linear model coefficients and $R^2$ metric. A simulation study and real case study in intraoperative cardiac output prediction are performed to evaluate the performance of the proposed method.

stat.ME↗