We study the properties of differentiable neural networks activated by rectified power unit (RePU) functions. We show that the partial derivatives of RePU neural networks can be represented by RePUs mixed-activated networks and derive upper bounds for the complexity of the function class of derivatives of RePUs networks. We establish error bounds for simultaneously approximating $C^s$ smooth functions and their derivatives using RePU-activated deep neural networks. Furthermore, we derive improved approximation error bounds when data has an approximate low-dimensional support, demonstrating the ability of RePU networks to mitigate the curse of dimensionality. To illustrate the usefulness of our results, we consider a deep score matching estimator (DSME) and propose a penalized deep isotonic regression (PDIR) using RePU networks. We establish non-asymptotic excess risk bounds for DSME and PDIR under the assumption that the target functions belong to a class of $C^s$ smooth functions. We also show that PDIR has a robustness property in the sense it is consistent with vanishing penalty parameters even when the monotonicity assumption is not satisfied. Furthermore, if the data distribution is supported on an approximate low-dimensional manifold, we show that DSME and PDIR can mitigate the curse of dimensionality.

翻译：我们研究了由整流幂单元（RePU）函数激活的可微分神经网络的性质。我们证明RePU神经网络的偏导数可通过RePU混合激活网络表示，并推导了RePU网络导数函数类复杂度的上界。建立了使用RePU激活深度神经网络同时逼近$C^s$光滑函数及其导数的误差界。此外，当数据具有近似低维支撑时，我们推导了改进的逼近误差界，展示了RePU网络缓解维数灾难的能力。为阐明结果的有效性，我们考虑深度得分匹配估计器（DSME），并提出基于RePU网络的惩罚深度保序回归（PDIR）。在目标函数属于$C^s$光滑函数类的假设下，建立了DSME和PDIR的非渐近超额风险界。同时证明即使单调性假设不满足，PDIR仍具有惩罚参数趋于零时一致性估计的鲁棒性质。进一步，若数据分布支撑于近似低维流形，我们表明DSME和PDIR可缓解维数灾难。