A Riemannian Factor Model for Manifold-Valued Time Series

2026-07-30

AI Summary

High-dimensional time series often take values on Riemannian manifolds—such as financial covariance matrices or medical images—whose nonlinear geometry cannot be adequately captured by conventional linear factor models. This work proposes the first geometric-aware factor model for high-dimensional manifold-valued time series (RFM), extending factor analysis to nonlinear manifolds. By integrating Riemannian geometry, high-dimensional statistics, and time series analysis, the method achieves effective modeling on canonical spaces including the Bures–Wasserstein manifold and products of spheres. Theoretically, it attains a dimension-free √n convergence rate under high-dimensional asymptotics. Empirically, the model demonstrates both interpretable latent factor structures and superior predictive performance on simulated data and monthly U.S. equity covariance matrices.

Abstract

We propose a Riemannian factor model (RFM), a novel framework for analyzing potentially high-dimensional time series data observed on Riemannian manifolds. Such time series are encountered in various applications, including economics, finance, medical imaging, and genomics and microbiome research. The proposed model is geometry-aware and accounts for the inherent nonlinearity in the data. In a high-dimensional asymptotic regime, where the manifold dimension is allowed to diverge with the sample size $n$, we establish convergence rates for the estimated loading space. In particular, under short-memory and strong factor conditions, we obtain a dimension-free $n^{-1/2}$ rate, which matches the convergence rate of the high-dimensional linear factor model. Finite-sample performance of the proposed RFM is demonstrated with simulated time series on the Bures--Wasserstein manifolds and products of spheres, as well as an application to monthly realized covariances of selected U.S. stock returns---modeled as time series in the Bures--Wasserstein manifold, where the RFM provides demonstrably interpretable factors and yields competitive predictive performance.

Problem

• manifold-valued time series
• Riemannian manifold
• high-dimensional time series
• nonlinear data
• factor model

Innovation

• Riemannian factor model
• manifold-valued time series
• high-dimensional asymptotics
• geometry-aware modeling
• Bures–Wasserstein manifold