Construction of Neural Networks for Realization of Localized Deep Learning

Frontiers in Applied Mathematics and Statistics

Chui, Charles K.¹  Zhou, Ding-Xuan²  Lin, Shao-Bo³ 

[1] Department of Mathematics, Hong Kong Baptist University, Hong Kong;Department of Mathematics, Wenzhou University, China;Department of Statistics, Stanford University, United States

关键词: Deep nets;    Learning Theory;    deep learning;    Manifold Learning;    Learning Rate;   

DOI  :  10.3389/fams.2018.00014

学科分类：数学（综合）

来源: Frontiers

【 摘 要 】

The subject of deep learning has recently attracted users of machine learning from various disciplines, including: medical diagnosis and bioinformatics, financial market analysis and online advertisement, speech and handwriting recognition, computer vision and natural language processing, time series forecasting, and search engines. However, theoretical development of deep learning is still at its infancy. The objective of this paper is to introduce a deep neural network (also called deep-net) approach to localized manifold learning, with each hidden layer endowed with a specific learning task. For the purpose of illustrations, we only focus on deep-nets with three hidden layers, with the first layer for dimensionality reduction, the second layer for bias reduction, and the third layer for variance reduction. A feedback component is also designed to deal with outliers. The main theoretical result in this paper is the order $\mathcal O\left(m^{-2s/(2s+d)}\right)$ of approximation of the regression function with regularity $s$, in terms of the number $m$ of sample points, where the (unknown) manifold dimension $d$ replaces the dimension $D$ of the sampling (Euclidean) space for shallow nets.

【 授权许可】

CC BY