Junhyung Chang and Xiaoyu Lei’s contribution to the Discussion of ‘Statistical exploration of the Manifold Hypothesis’ by Whiteley et al.

We congratulate the authors on an insightful paper that spans both a general theory of manifold emergence, and methodology for exploring hidden manifold structure in high-dimensional data. In summary, assuming the result of Theorem 1 holds, the effectiveness of PCA in elucidating the geometry of the latent domain from high-dimensional data is closely related to the statistical properties of the random functions (Xj)1≤j≤p such as distinguishability and stationarity. To further explore this relationship, we conduct a simulation study of how the choice of random functions affects the performance of downstream spectral clustering. Let the latent variables (Zi)1≤i≤500 be i.i.d. random elements of a unit circle that form two clusters as in Figure 1a. Let the random functions (Xj)1≤j≤p be i.i.d. real-valued zero-mean Gaussian processes with covariance function exp(−‖x−x′‖22l2)⁠, where the length-scale parameter l controls how fast the correlation between two points decays as they grow further apart in Euclidean distance. Further, let Y~i=(Xj(Zi))1≤j≤p denote the p-dimensional embedding of Zi via the random functions, and Ei=(Eij)1≤j≤p the noise vector, where Eij∼i.i.d.N(0,σ2)⁠, so that the observed data is Yi=Y~i+Ei⁠. We investigate how well spherical spectral clustering based on (spherical k-means clustering on the spherically projected PCA embeddings of Yi⁠, namely ζi⁠) recovers the cluster structure in the latent variables across different values of σ and l. (a) Latent variables (Zi)1≤i≤500 placed on the unit circle. They are sampled from a two-peak distribution by combining two von Mises distributions (Mardia & Jupp, 2009, Section 3.5.4), Zi∼i.i.d.0.6⋅VM(0,4)+0.4⋅VM(π,4)⁠. The latent variables form two clusters that align with the two peaks of the sampling distribution and are distinguished by colour; (b) histogram of latent variables according to their radians; (c) miscluster rate for various pairs of p and σ, fixed parameters: l=0.5,n=500,s=30⁠; (d) miscluster rate for various pairs of p and l, fixed parameters: σ=1,n=500,s=30⁠. We observe that increasing σ slows down the convergence of (ζi)1≤i≤500 to (ϕ(Zi))1≤i≤500⁠, masking the effect of (Y~i)1≤i≤500 and subsequently degrading the spectral clustering performance (Figure 1c). Furthermore, a smaller l implies weaker correlations among (Y~i)1≤i≤500⁠, making Y~i behave more like pure noise, while a larger l implies stronger correlations among (Y~i)1≤i≤500⁠, causing them to concentrate together and become less distinguishable. As the spectral clustering performance in Figure 1d demonstrates, the latent cluster structure is best preserved for values of l where the covariance function correctly assigns discernibly higher correlation for within-cluster pairs compared to between-cluster pairs. From a methodological perspective, the consistency result of Theorem 1 can be complemented by distributional results in certain cases, allowing for hypothesis testing of the latent domain. As a concrete example, assume Y is the sum of a matrix Y~⁠, which is deterministic conditioned on (Zi)1≤i≤n⁠, and an i.i.d. zero-mean noise matrix E⁠. Namely, this is a special case of the Latent Metric Model (LMM) where the functions (Xj)1≤j≤p are deterministic. Then, a distributional theory similar to that in Chang and Cape (2025) can be derived to test a baseline null hypothesis such as H0⁠: Zis a singleton set, via the distribution of ‖sgn(uYT1n)uY−1n1n‖2,∞⁠, where (s12,uY) is the leading eigenpair of YYT⁠, and 1n is the length-n, all-ones vector associated with the leading left singular vector of Y~ under the proposed null where Y~ has n repeated rows. Extending such singular vector distributional results to the more general LMM is nontrivial, as there are multiple sources of randomness, and ϕ is generally not computable, nor intended to be explicitly computed. Hence, identifying and testing null hypotheses via principal components for general manifolds is a potential direction for future research. The code for simulations are available upon request from the author, Xiaoyu Lei.