Simon et al.’s contribution to the Discussion of ‘Statistical exploration of the Manifold Hypothesis’ by Whiteley et al.

原文摘要

We congratulate the authors on their impressive effort and commend their tour de force. Our comments follow below. Theoretical foundations. The authors contribution is driven by embedding the manifold hypothesis in a rigorous statistical framework. The Latent Manifold Model (LMM) plays a central role by integrating latent variables, random functions, and structured noise within a unified construction. Grounded in Mercer’s theorem, the approach links correlation kernels with manifold embeddings and explains why high-dimensional data often concentrates near low-dimensional subspaces. It also clarifies the empirical success of Principal Component Analysis (PCA) as an entry point. Nonetheless, the assumptions underlying homeomorphism and isometry results deserve scrutiny. Condition (A2), requiring functional distinguishability of latent points, may be too strong for biological or climate data, where near-collinear responses occur. Likewise, the assumption of isotropic kernels (radial symmetry) is limited, since real-world data often exhibit anisotropy and nonstationarity. While simple anisotropic cases can build upon the isotropic model, many situations require distinct treatments (Allard et al., 2016). Similar challenges arise in nonstationary settings, as discussed in Porcu et al. (2020) and Senoussi and Porcu (2022). Rather than viewing these as flaws, the framework’s flexibility invites extensions that relax its technical assumptions. PCA and statistical guarantees. Theorem 1 shows that PCA scores can recover the latent geometry, up to orthogonal transformation, as p/n→∞⁠. It extends to spiked covariance models, where Z is Euclidean and f is linear, aligning with two-to-infinity norm analyses (Cape et al., 2019). Unlike classical PCA results reliant on eigenvalue gaps, Whiteley et al. (2025) emphasize geometric fidelity: PCA preserves the manifold’s structure, not merely its dominant components. This justifies using PCA before nonlinear embeddings, ensuring that the data’s essential shape is already captured in lower dimensions. Future work could explore dependent features, infinite-rank kernels, and computational scalability. While the Wasserstein metric fits the model, it can be expensive for large n,p⁠; alternatives like Sinkhorn distances or GeomLoss (Cuturi, 2013; Feydy et al., 2019; Peyré & Cuturi, 2019) could provide efficient benchmarks alongside criteria such as profile likelihood and the ladle estimator (Luo & Li, 2016; Zhu & Ghodsi, 2006). Scientific insights. The empirical demonstrations show the framework’s versatility despite isotropic assumptions. Rotated face images reveal loop topology, single-cell RNA sequencing uncovers tree-like differentiation, and climate data reflect geographical constraints. These examples confirm that latent geometries align with meaningful scientific hypotheses. Importantly, deviations from isometry often carry scientific meaning rather than error: in climate data, path distortions correspond to mountains and seas (Porcu et al., 2025). Thus, manifold learning becomes a means of hypothesis generation—linking geometry with domain knowledge and reframing discrepancies as opportunities for discovery. No data have been used.

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