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.
Real-life data are extensively multiway. Researchers have been motivated to develop corresponding multiway RBMs. For example, @cite_9 proposed a factored conditional RBM for modeling motion style. In their model, both historical and current motion vectors are considered as inputs so that the pairwise association between them is captured. However, since the visible layer is in vector form, the spatial information in the multiway data is not retained. @cite_1 a three-way factored conditional RBM was proposed where a three-way weight tensor is employed to capture the correlations between the input, output, and hidden variables. However, their training data still requires vectorization.
@cite_9 @cite_1
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Many real-world problems, such as web image analysis, document categorization and product recommendation, often exhibit dual-heterogeneity: heterogeneous features obtained in multiple views, and multiple tasks might be related to each other through one or more shared views. To address these Multi-Task Multi-View (MTMV) problems, we propose a tensor-based framework for learning the predictive multilinear structure from the full-order feature interactions within the heterogeneous data. The usage of tensor structure is to strengthen and capture the complex relationships between multiple tasks with multiple views. We further develop efficient multilinear factorization machines (MFMs) that can learn the task-specific feature map and the task-view shared multilinear structures, without physically building the tensor. In the proposed method, a joint factorization is applied to the full-order interactions such that the consensus representation can be learned. In this manner, it can deal with the partially incomplete data without difficulty as the learning procedure does not simply rely on any particular view. Furthermore, the complexity of MFMs is linear in the number of parameters, which makes MFMs suitable to large-scale real-world problems. Extensive experiments on four real-world datasets demonstrate that the proposed method significantly outperforms several state-of-the-art methods in a wide variety of MTMV problems. Given a visual history, multiple future outcomes for a video scene are equally probable, in other words, the distribution of future outcomes has multiple modes. Multimodality is notoriously hard to handle by standard regressors or classifiers: the former regress to the mean and the latter discretize a continuous high dimensional output space. In this work, we present stochastic neural network architectures that handle such multimodality through stochasticity: future trajectories of objects, body joints or frames are represented as deep, non-linear transformations of random (as opposed to deterministic) variables. Such random variables are sampled from simple Gaussian distributions whose means and variances are parametrized by the output of convolutional encoders over the visual history. We introduce novel convolutional architectures for predicting future body joint trajectories that outperform fully connected alternatives DBLP:journals corr WalkerDGH16 . We introduce stochastic spatial transformers through optical flow warping for predicting future frames, which outperform their deterministic equivalents DBLP:journals corr PatrauceanHC15 . Training stochastic networks involves an intractable marginalization over stochastic variables. We compare various training schemes that handle such marginalization through a) straightforward sampling from the prior, b) conditional variational autoencoders NIPS2015_5775,DBLP:journals corr WalkerDGH16 , and, c) a proposed K-best-sample loss that penalizes the best prediction under a fixed "prediction budget". We show experimental results on object trajectory prediction, human body joint trajectory prediction and video prediction under varying future uncertainty, validating quantitatively and qualitatively our architectural choices and training schemes. In view of the advantages of deep networks in producing useful representation, the generated features of different modality data (such as image, audio) can be jointly learned using Multimodal Restricted Boltzmann Machines (MRB-M). Recently, audiovisual speech recognition based the M-RBM has attracted much attention, and the MRBM shows its effectiveness in learning the joint representation across audiovisual modalities. However, the built networks have weakness in modeling the multimodal sequence which is the natural property of speech signal. In this paper, we will introduce a novel temporal multimodal deep learning architecture, named as Recurrent Temporal Multimodal RB-M (RTMRBM), that models multimodal sequences by transforming the sequence of connected MRBMs into a probabilistic series model. Compared with existing multimodal networks, it's simple and efficient in learning temporal joint representation. We evaluate our model on audiovisual speech datasets, two public (AVLetters and AVLetters2) and one self-build. The experimental results demonstrate that our approach can obviously improve the accuracy of recognition compared with standard MRBM and the temporal model based on conditional RBM. In addition, RTMRBM still outperforms non-temporal multimodal deep networks in the presence of the weakness of long-term dependencies. We review the method of Parallel Factor Analysis, which simultaneously fits multiple two-way arrays or ‘slices’ of a three-way array in terms of a common set of factors with differing relative weights in each ‘slice’. Mathematically, it is a straightforward generalization of the bilinear model of factor (or component) analysis (xij = ΣRr = 1airbjr) to a trilinear model (xijk = ΣRr = 1airbjrckr). Despite this simplicity, it has an important property not possessed by the two-way model: if the latent factors show adequately distinct patterns of three-way variation, the model is fully identified; the orientation of factors is uniquely determined by minimizing residual error, eliminating the need for a separate ‘rotation’ phase of analysis. The model can be used several ways. It can be directly fit to a three-way array of observations with (possibly incomplete) factorial structure, or it can be indirectly fit to the original observations by fitting a set of covariance matrices computed from the observations, with each matrix corresponding to a two-way subset of the data. Even more generally, one can simultaneously analyze covariance matrices computed from different samples, perhaps corresponding to different treatment groups, different kinds of cases, data from different studies, etc. To demonstrate the method we analyze data from an experiment on right vs. left cerebral hemispheric control of the hands during various tasks. The factors found appear to correspond to the causal influences manipulated in the experiment, revealing their patterns of influence in all three ways of the data. Several generalizations of the parallel factor analysis model are currently under development, including ones that combine parallel factors with Tucker-like factor ‘interactions’. Of key importance is the need to increase the method's robustness against nonstationary factor structures and qualitative (nonproportional) factor change.