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Existing theories on deep nonparametric regression have shown that when theinput data lie on a low-dimensional manifold, deep neural networks can adapt tothe intrinsic data structures. In real world applications, such an assumptionof data lying exactly on a low dimensional manifold is stringent. This paperintroduces a relaxed assumption that the input data are concentrated around asubset of $\mathbb{R}^d$ denoted by $\mathcal{S}$, and the intrinsic dimensionof $\mathcal{S}$ can be characterized by a new complexity notation -- effectiveMinkowski dimension. We prove that, the sample complexity of deep nonparametricregression only depends on the effective Minkowski dimension of $\mathcal{S}$denoted by $p$. We further illustrate our theoretical findings by consideringnonparametric regression with an anisotropic Gaussian random design$N(0,\Sigma)$, where $\Sigma$ is full rank. When the eigenvalues of $\Sigma$have an exponential or polynomial decay, the effective Minkowski dimension ofsuch an Gaussian random design is $p=\mathcal{O}(\sqrt{\log n})$ or$p=\mathcal{O}(n^\gamma)$, respectively, where $n$ is the sample size and$\gamma\in(0,1)$ is a small constant depending on the polynomial decay rate.Our theory shows that, when the manifold assumption does not hold, deep neuralnetworks can still adapt to the effective Minkowski dimension of the data, andcircumvent the curse of the ambient dimensionality for moderate sample sizes.

Effective Minkowski Dimension of Deep Nonparametric Regression: Function Approximation and Statistical Theories