Charles K. Chui

Yong Zheng Ong, Charles K. Chui, Haizhao Yang

This paper introduces a cross adversarial source separation (CASS) framework via autoencoder, a new model that aims at separating an input signal consisting of a mixture of multiple components into individual components defined via adversarial learning and autoencoder fitting. CASS unifies popular generative networks like auto-encoders (AEs) and generative adversarial networks (GANs) in a single framework. The basic building block that filters the input signal and reconstructs the $i$-th target component is a pair of deep neural networks $\mathcal{EN}_i$ and $\mathcal{DE}_i$ as an encoder for dimension reduction and a decoder for component reconstruction, respectively. The decoder $\mathcal{DE}_i$ as a generator is enhanced by a discriminator network $\mathcal{D}_i$ that favors signal structures of the $i$-th component in the $i$-th given dataset as guidance through adversarial learning. In contrast with existing practices in AEs which trains each Auto-Encoder independently, or in GANs that share the same generator, we introduce cross adversarial training that emphasizes adversarial relation between any arbitrary network pairs $(\mathcal{DE}_i,\mathcal{D}_j)$, achieving state-of-the-art performance especially when target components share similar data structures.

Charles K. Chui, Shao-Bo Lin, Ding-Xuan Zhou

Based on the tree architecture, the objective of this paper is to design deep neural networks with two or more hidden layers (called deep nets) for realization of radial functions so as to enable rotational invariance for near-optimal function approximation in an arbitrarily high dimensional Euclidian space. It is shown that deep nets have much better performance than shallow nets (with only one hidden layer) in terms of approximation accuracy and learning capabilities. In particular, for learning radial functions, it is shown that near-optimal rate can be achieved by deep nets but not by shallow nets. Our results illustrate the necessity of depth in neural network design for realization of rotation-invariance target functions.

Charles K. Chui, Shao-Bo Lin, Ding-Xuan Zhou

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.