Data Mapping and Finite Difference Learning

TL;DR

This paper introduces a data mapping framework for RBMs that allows real-valued data, interprets contrastive divergence as finite difference approximation, and supports various activation functions, enhancing flexibility in dimensionality reduction and feature extraction.

Contribution

It presents a novel data mapping for RBMs, reinterprets contrastive divergence as finite difference learning, and enables the use of non-sigmoid activation functions.

Findings

Nodes can handle real-valued matrix data without probabilistic interpretation.

CD1 is shown to be a finite difference approximation of gradient descent.

Flexible activation functions improve low-dimensional data reduction and feature extraction.

Abstract

Restricted Boltzmann machine (RBM) is a two-layer neural network constructed as a probabilistic model and its training is to maximize a product of probabilities by the contrastive divergence (CD) scheme. In this paper a data mapping is proposed to describe the relationship between the visible and hidden layers and the training is to minimize a squared error on the visible layer by a finite difference learning. This paper presents three new properties in using the RBM: 1) nodes on the visible and hidden layers can take real-valued matrix data without a probabilistic interpretation; 2) the famous CD1 is a finite difference approximation of the gradient descent; 3) the activation can take non-sigmoid functions such as identity, relu and softsign. The data mapping provides a unified framework on the dimensionality reduction, the feature extraction and the data representation pioneered and developed by Hinton and his colleagues. As an approximation of the gradient descent, the finite difference learning is applicable to both directed and undirected graphs. Numerical experiments are performed to verify these new properties on the very low dimensionality reduction, the collinearity of timer series data and the use of flexible activations.