AI 中文总结
该研究针对流形值时间序列提出黎曼因子模型（RFM），推导了其在高维渐近下的收敛速率，通过模拟数据和美国股票协方差序列验证了模型的可解释性与预测性能。
AI 中文摘要
我们提出了一种黎曼因子模型（RFM），这是一个用于分析在黎曼流形上观测到的潜在高维时间序列数据的新框架。这类时间序列出现在多个应用领域，包括经济学、金融学、医学成像以及基因组学和微生物组研究中。所提出的模型具有几何感知能力，可考虑数据中固有的非线性。在高维渐近 regime 下，即允许流形维度随样本量 $n$ 发散时，我们为估计的载荷空间建立了收敛速率。特别地，在短记忆和强因子条件下，我们得到了与维度无关的 $n^{-1/2}$ 速率，该速率与高维线性因子模型的收敛速率相匹配。我们通过在 Bures--Wasserstein 流形和球面乘积上的模拟时间序列，以及对选定美国股票收益率的月度实现协方差的应用，展示了所提出的 RFM 的有限样本性能——这些实现协方差被建模为 Bures--Wasserstein 流形中的时间序列，其中 RFM 提供了明显可解释的因子并产生了有竞争力的预测性能。
英文摘要
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.
