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In this study, we prove that an intrinsic low dimensionality of covariates isthe main factor that determines the performance of deep neural networks (DNNs).DNNs generally provide outstanding empirical performance. Hence, numerousstudies have actively investigated the theoretical properties of DNNs tounderstand their underlying mechanisms. In particular, the behavior of DNNs interms of high-dimensional data is one of the most critical questions. However,this issue has not been sufficiently investigated from the aspect ofcovariates, although high-dimensional data have practically low intrinsicdimensionality. In this study, we derive bounds for an approximation error anda generalization error regarding DNNs with intrinsically low dimensionalcovariates. We apply the notion of the Minkowski dimension and develop a novelproof technique. Consequently, we show that convergence rates of the errors byDNNs do not depend on the nominal high dimensionality of data, but on its lowerintrinsic dimension. We further prove that the rate is optimal in the minimaxsense. We identify an advantage of DNNs by showing that DNNs can handle abroader class of intrinsic low dimensional data than other adaptive estimators.Finally, we conduct a numerical simulation to validate the theoretical results.

Adaptive Approximation and Generalization of Deep Neural Network with Intrinsic Dimensionality