1. Hybrid Orthogonal Projection & Estimation (HOPE)

Each hidden layer in neural networks can be formulated as one HOPE model

In this paper, we propose a universal model for high-dimensional data, called the Hybrid Orthogonal Projection and Estimation (HOPE) model, which combines a linear orthogonal projection and a nite mixture model under a unied generative modelling framework. The HOPE model itself can be learned unsupervisedly from un-labelled data based on the maximum likelihood estimation as well as trained discriminatively from labelled data. More interestingly, we have shown the proposed HOPE models are closely related to neural networks (NNs) in a sense that each hidden layer can be reformulated as a HOPE model. As a result, the HOPE framework can be used as a novel tool to probe why and how NNs work, more importantly, it also provides several new learning algorithms to learn NNs either supervisedly or unsupervisedly. In this work, we have investigated the HOPE framework in learning NNs for several standard tasks, including image recognition on MNIST and speech recognition on TIMIT. Experimental results show that the HOPE framework yields signicant performance gains over the current stateof- the-art methods in various types of NN learning problems, including unsupervised feature learning, supervised or semi-supervised learning.