Martin Schlather and Milan Stehlík’s contribution to the Discussion of ‘Statistical exploration of the manifold hypothesis’ by N. Whiteley et al.
本文是对一篇关于流形假设的论文的讨论，指出其中一些数学断言不够精确，例如命题1的证明、弱依赖的定义以及某些收敛结论的不足。
Many thanks for your enlightening paper. We feel that some of the assertions should be more precise from a mathematical point of view to fully appreciate the paper. For instance, Proposition 1 does not need any generalization of the Karhunen–Loève expansion for its proof, since it is an application of the expansion, at best: rewriting Wjk as ck⟨Xj,ukf⟩, we see that the linearity in the representation of Yi only needs the assumption that the ukf build an orthonormal basis, so that any vector can be represented as a weighted sum of the basis vectors, where the weights are the scalar products, i.e. Xj=∑k⟨Xj,ukf⟩ukf=∑kWjk(ck−1ukf). Given the ukf, the requirement that ∑jWjk2=1 determines the ck uniquely up to sign. Only for the last assertion, ∑jWjkWjℓ=0, k≠j, the Karhunen–Loève expansion is convenient by offering an adequate orthonormal basis, but, by far, it is not the only possible choice, in general. Trivially, the formula for ck simplifies when the ukf stem for the Karhunen–Loève expansion. Continuing with Section 3.1, the notion of weak dependence is not defined. The clarification is important as simple conditions, such as Xj(z) and Xi(z′) having a correlation less than ε for all z,z′∈Z, i≠j, do not entail small correlations among the Wj,k. Further, since the property, that an−bn→0 as n→∞ for bounded sequences (an)n and (bn)n, is not enough to conclude the existence of limn→∞an or limn→∞bn, the requirement of a large number of summands is probably not enough to conclude proximity to a limit. Section 3.2 ends with the claim that a certain difference of norms converges to 0, which is surely not true in general. In the proof of Proposition 3, it is stated that Assumption 8 holds, i.e. that the given positive definite kernel f has an extension from a manifold Z to its enclosing ball Z~. It is unclear how the authors conclude this from the form of f given in Proposition 3. There are no new data associated with this article.