Milan Stehlík and Martin Schlather’s contribution to the Discussion of ‘Statistical exploration of the manifold hypothesis’ by Whiteley et al.

原文摘要

We congratulate the authors of Whiteley et al. (2025) for opening up a challenging world of exploration of the manifold hypothesis. We fully agree that principal component analysis (PCA) is one the most important tool data analysis that the underlying spectral representation needs generalization. An alternative to a purely statistical modelling is to try to include as much knowledge as possible into the generalization. Schlather and Reinbott (2021) suggest an algebraic modelling based on a generalized notion of entropy (Schlather, 2026). As a consequence, the approach by Schlather and Reinbott (2021) allows also distributions without existing second moment entering the PCA. The approach preserves also the considered class of distributions (Reinbott & Janßen, 2025). Other important questions relate to Johnson–Lindenstrauss Flattening Lemma approaches. How much does your construction of the latent metric space relate to the real sparsity of the data and known results? The Johnson–Lindenstrauss lemma is a result concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. The lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly preserved. The map used for the embedding is at least Lipschitz and can even be taken to be an orthogonal projection. The essential algorithms for working with large datasets tend to become bogged down very quickly as dimension increases. It is therefore desirable to reduce the dimensionality of the data in a way that preserves its relevant structure. The Johnson–Lindenstrauss lemma is a classic result in this vein. Also, the lemma is tight up to a factor log(1/ε), i.e. there exists a set of points of size m that needs dimension Ω(log(m)ε2log(1/ε))⁠. One proof of the lemma takes f to be a suitable multiple of the orthogonal projection onto a random subspace of dimension n in RN and exploits the phenomenon of concentration of measure (Ledoux, 2001). For an extensive discussion, see Chapter 15 in Matoušek (2002). Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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