arXiv subjects

Yiming Zang

Publications and source records attributed to Yiming Zang.

6 recordsLinked to original sources

arXiv2026

High-dimensional Semi-supervised Classification via the Fermat Distance

Semi-supervised classification, where unlabeled data are massive but labeled data are limited, often arises in machine learning applications. We address this challenge under high-dimensional data by leveraging the manifold and cluster assumptions. Based on the Fermat distance, a density-sensitive metric that naturally encodes the cluster assumption, we propose the weighted $k$-nearest neighbors (NN) classifier and multidimensional scaling (MDS)-induced classifiers. The use of MDS with a large target dimension allows the effective application of linear classifiers to complex manifold data. Theoretically, we derive a sharp lower bound for the expected excess risk within clusters and prove that the weighted $k$-NN classifier utilizing the true Fermat distance is minimax optimal. Furthermore, we explicitly quantify the utility of unlabeled data by showing that the error arising from estimating the Fermat distance decays exponentially with the pooled sample size. Such a rate is much faster than the related rates in the literature. Extensive experiments on synthetic and real datasets demonstrate competitive or superior performance of our approaches compared to state-of-the-art graph-based semi-supervised classifiers.

stat.ML↗

arXiv2026

Semi-supervised Classification for Noisy Functional Data with Application to Astronomical Spectra

Despite its extensive development for multivariate data, semi-supervised learning remains underdeveloped for functional data, especially under discrete and noisy observations. We develop a density-sensitive semi-supervised framework for functional data supported on a low-dimensional manifold by adapting the Fermat distance to reconstructed trajectories. The resulting pairwise distances are used to construct a weighted $k$-nearest-neighbor classifier and multidimensional-scaling-based classifiers. To accommodate massive datasets commonly seen in semi-supervised applications, we design a computationally efficient estimation procedure tailored for discrete and noisy functional observations. Theoretically, we establish exponentially decaying convergence rates of the $k$-NN classifier and the consistency of the estimated Fermat distance. Crucially, our results reveal that incorporating unlabeled data may not lead to improved classification accuracy without a sufficiently fast-growing individual sampling rate, precisely due to discrete and noisy observations. In most simulation settings satisfying the manifold and cluster assumptions, the proposed classifiers outperform the supervised benchmarks considered; in the Gaia spectra analysis, they attain higher agreement with high-confidence proxy labels.

stat.ME↗

arXiv2025

Supervised Manifold Learning for Functional Data

Classification is a core topic in functional data analysis. A large number of functional classifiers have been proposed in the literature, most of which are based on functional principal component analysis or functional regression. In contrast, we investigate this topic from the perspective of manifold learning. It is assumed that functional data lie on an unknown low-dimensional manifold, and we expect that superior classifiers can be developed based on the manifold structure. To this end, we propose a novel proximity measure that takes the label information into account to learn the low-dimensional representations, also known as the supervised manifold learning outcomes. When the outcomes are coupled with multivariate classifiers, the procedure induces a new family of functional classifiers. In theory, we prove that our functional classifier induced by the $k$-NN classifier is asymptotically optimal. In practice, we show that our method, coupled with several classical multivariate classifiers, achieves highly competitive classification performance compared to existing functional classifiers across both synthetic and real data examples. Supplementary materials are available online.

stat.ME↗

arXiv2023

Generalized Ricci surfaces

We consider smooth Riemannian surfaces whose curvature $K$ satisfies the relation $Δ\log|K-c|=aK+b$ away from points where $K=c$ for some $(a,b,c)\in\mathbb{R}^3$, which we call generalized Ricci surfaces. We prove some isometric immersion theorems allowing points where $K=c$ using properties of log-harmonic functions. For instance, we obtain a characterization of Riemannian surfaces that locally admit minimal isometric immersions, possibly with umbilical points, into a $3$-dimensional Riemannian manifold of constant sectional curvature. We also give an application to convex affine spheres. Finally, we study compact generalized Ricci surfaces: we obtain topological obstructions and construct examples.

math.DG↗

arXiv2022

Constructions of helicoidal minimal surfaces and minimal annuli in $\widetilde{E(2)}$

In this article, we construct two one-parameter families of properly embedded minimal surfaces in a three-dimensional Lie group $\widetilde{E(2)}$, which is the universal covering of the group of rigid motions of Euclidean plane endowed with a left-invariant Riemannian metric. The first one can be seen as a family of helicoids, while the second one is a family of catenoidal minimal surfaces. The main tool that we use for the construction of these surfaces is a Weierstrass-type representation introduced by Meeks, Mira, Pérez and Ros for minimal surfaces in Lie groups of dimension three. In the end, we study the limit of the catenoidal minimal surfaces. As an application of this limit case, we get a new proof of a half-space theorem for minimal surfaces in $\widetilde{E(2)}$.

math.DG↗

arXiv2021

Non-positively curved Ricci surfaces with catenoidal ends

A Ricci surface is defined as a Riemannian surface $(M,g_M)$ whose Gauss curvature satisfies the differential equation $KΔK + g_M(dK,dK) + 4K^3=0$. Andrei Moroianu and Sergiu Moroianu proved that a Ricci surface with non-positive Gauss curvature admits locally a minimal immersion into $\mathbb{R}^3$. In this paper, we are interested in studying non-compact orientable Ricci surfaces with catenoidal ends. We use an analogue of the Weierstrass data to obtain some classification results for such Ricci surfaces. We also give an existence result for positive genus Ricci surfaces with catenoidal ends.

math.DG↗