Response to Reviewer JGBa
We thank the reviewer for the encouraging remarks and insightful comments. We are glad to note your positive evaluation of the paper's quality of writing, and we appreciate your favorable comments regarding its robustness, clarity, and meaningful contributions.
We agree with the reviewer that the proof techniques presented in this paper are largely built upon the works done in the recent literature. However, as already noted by the reviewer, applying the proof technique, particularly in the MMD context requires additional intricacies and has to be carefully dealt with. Indeed as already pointed out on page 6, A1 is a strong assumption. Certainly, in the context of GANs, a less stringent assumption can be employed since the learning task becomes considerably simpler. As observed by the reviewer, a parallel challenge arose in the analysis of autoencoders, as elucidated by Liu et al. (2023). They addressed this issue by exploring chart-autoencoders, which incorporate additional components in the network architecture compared to conventional autoencoders. We thank the reviewer for the suggestions. Indeed compact $\tilde{d}$-dimensional differentiable manifolds have a Minkowski dimension of at most $\tilde{d}$. Thus, the main result, i.e Theorem 8 and the subsequent hold with $d_\mu$ replaced with $\tilde{d}$, when the data-support is a $\tilde{d}$-dimensional differentiable manifold. A similar result holds for other examples such as affine convex sets, self-similar sets, etc. as highlighted in Proposition 9 of Weed and Bach (2019). We thank the reviewer for stimulating discussion and will include the following discussion in the revised manuscript.
We recall that we call a set $\mathcal{M}$ is $\tilde{d}$-regular w.r.t. the $\tilde{d}$-dimensional Hausdorff measure $\mathbb{H}^{\tilde{d}}$ if $$\mathbb{H}(B_\varrho(x, r)) \asymp r^{\tilde{d}},$$
for all $x \in \mathcal{M}$ (see Definition 6 of Weed and Bach (2019)). It is known (Mattila, 1999) that if $\mathcal{M}$ is $\tilde{d}$-regular, then the Minkowski dimension of $\mathcal{M}$ is $\tilde{d}$. Thus, when $\text{supp}(\mu)$ is $\tilde{d}$-regular, $d_\mu = \tilde{d}$. Since compact $\tilde{d}$-dimensional differentiable manifolds are $\tilde{d}$-regular (Proposition 9 of Weed and Bach (2019)), this implies that for when $\operatorname{supp}(\mu)$ is a compact differentiable $\tilde{d}$-dimensional manifold, the error rates for the sample estimates scale as in Theorem 8, with $d_\mu$ replaced with $\tilde{d}$. A similar result holds when $\text{supp}(\mu)$ is a nonempty, compact convex set spanned by an affine space of dimension $\tilde{d}$; the relative boundary of a nonempty, compact convex set of dimension $\tilde{d} + 1$; or a self-similar set with similarity dimension $\tilde{d}$.