Reformulation of RBM to Unify Linear and Nonlinear Dimensionality Reduction

A restricted Boltzmann machine (RBM) is a two-layer neural network with shared weights and has been extensively studied for dimensionality reduction, data representation and recommendation systems in the literature. The traditional RBM requires a probabilistic interpretation of the values on both layers and a Markov chain Monte Carlo (MCMC) procedure to generate samples during the training. The contrastive divergence (CD) is efficient to train the RBM but its convergence has not been proved mathematically. In this paper, using a maximum a posteriori (MAP) estimate and the expectation maximization (EM) algorithm, we show that the CD algorithm without MCMC is convergent for the conditional likelihood object function. Another key contribution in this paper is the reformulation of the RBM into a deterministic model. Within the reformulated RBM, the CD algorithm without MCMC approximates the gradient descent (GD) method. This reformulated RBM can take the continuous scalar and vector variables on the nodes with flexibility in choosing the activation functions. Numerical experiments show its capability in both linear and nonlinear dimensionality reduction, and, for the nonlinear dimensionality reduction, the reformulated RBM can outperform principal component analysis (PCA) by choosing the proper activation functions. Finally, we demonstrate its application to vector-valued nodes for the CIFAR-10 dataset (color images) and the multivariate sequence data, which cannot be configured naturally with the traditional RBM. This work not only provides theoretical insights regarding the traditional RBM but also unifies the linear and nonlinear dimensionality reduction for scalar and vector variables.