Construction of neural networks for realization of localized deep learning

TL;DR

This paper introduces a deep neural network approach for localized manifold learning, with layered tasks for dimensionality, bias, and variance reduction, providing theoretical approximation bounds based on sample size.

Contribution

It proposes a novel deep-net architecture with layered learning tasks tailored for manifold data, advancing theoretical understanding of deep learning approximation capabilities.

Findings

Achieves an approximation order of O(m^{-2s/(2s+d)}) for regression functions.

Replaces ambient space dimension with manifold dimension in theoretical bounds.

Provides a framework for layered deep learning targeting specific learning tasks.

Abstract

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 also designed to eliminate 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.