A restricted Boltzmann machine (RBM) learns a probability distribution over its input samples and has numerous uses like dimensionality reduction, classification and generative modeling. Conventional RBMs accept vectorized data that dismisses potentially important structural information in the original tensor (multi-way) input. Matrix-variate and tensor-variate RBMs, named MvRBM and TvRBM, have been proposed but are all restrictive by model construction, which leads to a weak model expression power. This work presents the matrix product operator RBM (MPORBM) that utilizes a tensor network generalization of Mv TvRBM, preserves input formats in both the visible and hidden layers, and results in higher expressive power. A novel training algorithm integrating contrastive divergence and an alternating optimization procedure is also developed. Numerical experiments compare the MPORBM with the traditional RBM and MvRBM for data classification and image completion and denoising tasks. The expressive power of the MPORBM as a function of the MPO-rank is also investigated.
The Conditional Restricted Boltzmann Machine (CRBM) is a recently proposed model for time series that has a rich, distributed hidden state and permits simple, exact inference. We present a new model, based on the CRBM that preserves its most important computational properties and includes multiplicative three-way interactions that allow the effective interaction weight between two units to be modulated by the dynamic state of a third unit. We factor the three-way weight tensor implied by the multiplicative model, reducing the number of parameters from O(N3) to O(N2). The result is an efficient, compact model whose effectiveness we demonstrate by modeling human motion. Like the CRBM, our model can capture diverse styles of motion with a single set of parameters, and the three-way interactions greatly improve the model's ability to blend motion styles or to transition smoothly among them. Deep belief nets have been successful in modeling handwritten characters, but it has proved more difficult to apply them to real images. The problem lies in the restricted Boltzmann machine (RBM) which is used as a module for learning deep belief nets one layer at a time. The Gaussian-Binary RBMs that have been used to model real-valued data are not a good way to model the covariance structure of natural images. We propose a factored 3-way RBM that uses the states of its hidden units to represent abnormalities in the local covariance structure of an image. This provides a probabilistic framework for the widely used simple complex cell architecture. Our model learns binary features that work very well for object recognition on the “tiny images” data set. Even better features are obtained by then using standard binary RBM’s to learn a deeper model.