Fukang Zhu and Xiangyu Guo’s contribution to the Discussion of ‘Statistical exploration of the Manifold Hypothesis’ by Whiteley et al.
本文是对一篇关于流形假设统计探索论文的讨论，提出了三个需要进一步阐述的问题：潜在度量空间的定义、主成分分析后单位向量的网络自回归模型构建、以及平稳性的不同形式及其几何性质。
We congratulate the authors on their introducing theoretical explanations and applications of the manifold hypothesis in statistics based on the latent metric space model. Using the continuity of the mean correlation kernel in the latent metric space model, the authors employ a feature map to connect the metric space of latent variables to a Euclidean space. This links the geometric and statistical properties of the model. Furthermore, the smoothness of the mean correlation kernel in the latent metric space model enables the model to concentrate on a low-dimensional subspace, providing the theoretical basis for the manifold hypothesis. When dealing with real data, the authors proposed applying principal component analysis to approximate the low-dimensional subspace in which the data concentrate. This provides a statistical foundation for a variety of tools that have been developed to analyse data based on the manifold hypothesis. We hold the view that certain parts of the paper require further elaboration. First, we focus on the definition of the latent metric space in Section 2. We observe that the latent metric space may exhibit certain similarities to the latent position described in Raftery (2017) and Kaur et al. (2024). In Figure 7, the authors mentioned that ‘Estimated kernel as a function of latent positions.’ We believe it is worthwhile elaborating on the relationship between them. In addition, it is interesting to construct latent metric space models similar to latent position network models in Kaur et al. (2024) and Kaur and Rastelli (2024). Second, Section 4.4 states that projecting the vectors obtained from principal component analysis onto the unit hypersphere results in a set of unit vectors, which can form a nearest neighbour graph. This implies that the set of unit vectors contains embedded network structures. Could this network structure be used to construct a network autoregressive model among these unit vectors, similar to the approaches adopted by X. Zhu et al. (2017) and Armillotta and Fokianos (2023)? We would like to know more about the feasibility of constructing a network autoregressive model or related models. It should be noted that Martin et al. (2025) proposed a network informed restricted vector autoregressive model by drawing inspiration from the workflow rationalized in this article: linear dimension reduction via principal component analysis, embedding, clustering, and graph construction. Third, we observe that stationarity is mentioned in Section 3.3. However, only the weak stationarity of the random functions corresponding to each component is introduced here. We believe that further research is needed to provide additional explanations of stationarity to enhance comprehensiveness. For example, local stationarity and stationarity on Riemannian manifolds introduced in J. Zhu et al. (2024) can be considered, as they address the gap that an absence of a formal definition of stationarity that is tailored to manifold time series, which overcomes the limitations of weak stationarity. It would also be interesting to discuss whether different types of stationarity give rise to distinct geometric properties. This work is supported by Science and Technology Development Plan Project of Jilin Province of China (No. 20250102003JC) and National Natural Science Foundation of China (No. 12271206). None declared. The author replied later in writing as follows: