论文标题
深度神经网络的优势用于估算高空图上奇异性的功能
Advantage of Deep Neural Networks for Estimating Functions with Singularity on Hypersurfaces
论文作者
论文摘要
我们开发了最小值分析，以描述深神经网络（DNN）的性能优于其他标准方法的原因。对于非参数回归问题，众所周知，许多标准方法都达到平滑函数的最小估计误差的最佳速率，因此，识别DNN的理论优势并不直接。这项研究试图通过考虑一类非平滑函数在高度曲面上具有奇异性的估计来填补这一空白。我们的发现如下：（i）我们得出DNN估计量的概括误差，并证明其收敛速率几乎是最佳的。 （ii）我们阐明了估计问题的相图，该图描述了DNNS优于一类估计器的情况，包括内核方法，高斯过程方法等。我们还表明，DNNS优于基于谐波分析的估计器。 DNN的这种优势来自以下事实：奇异性的形状可以通过其多层结构成功处理。
We develop a minimax rate analysis to describe the reason that deep neural networks (DNNs) perform better than other standard methods. For nonparametric regression problems, it is well known that many standard methods attain the minimax optimal rate of estimation errors for smooth functions, and thus, it is not straightforward to identify the theoretical advantages of DNNs. This study tries to fill this gap by considering the estimation for a class of non-smooth functions that have singularities on hypersurfaces. Our findings are as follows: (i) We derive the generalization error of a DNN estimator and prove that its convergence rate is almost optimal. (ii) We elucidate a phase diagram of estimation problems, which describes the situations where the DNNs outperform a general class of estimators, including kernel methods, Gaussian process methods, and others. We additionally show that DNNs outperform harmonic analysis based estimators. This advantage of DNNs comes from the fact that a shape of singularity can be successfully handled by their multi-layered structure.
