变指数Besov空间上回归问题深度学习的估计误差分析
Estimation error analysis of deep learning on the regression problem on the variable exponent Besov space
摘要：
深度学习在各个领域都取得了显著的成功，包括图像和语音识别。深度学习成功的因素之一是它的高特征提取能力。在这项研究中，我们专注于深度学习的自适应性;因此，我们处理可变指数Besov空间，该空间根据输入位置$x$具有不同的平滑度。换句话说，估计的难度在域内是不均匀的。我们分析了变指数Besov空间的一般逼近误差以及深度学习的逼近和估计误差。我们注意到，当目标函数光滑度较低的区域较小且维数较大时，基于自适应性的改进是显着的。此外，线性估计的优越性示出的估计误差的收敛速度。
英文摘要：
Deep learning has achieved notable success in various fields, including image and speech recognition. One of the factors in the successful performance of deep learning is its high feature extraction ability. In this study, we focus on the adaptivity of deep learning; consequently, we treat the variable exponent Besov space, which has a different smoothness depending on the input location $x$. In other words, the difficulty of the estimation is not uniform within the domain. We analyze the general approximation error of the variable exponent Besov space and the approximation and estimation errors of deep learning. We note that the improvement based on adaptivity is remarkable when the region upon which the target function has less smoothness is small and the dimension is large. Moreover, the superiority to linear estimators is shown with respect to the convergence rate of the estimation error.
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