Tensor decomposition is an effective approach to compress over-parameterized neural networks and to enable their deployment on resource-constrained hardware platforms. However, directly applying tensor compression in the training process is a challenging task due to the difficulty of choosing a proper tensor rank. In order to achieve this goal, this paper proposes a Bayesian tensorized neural network. Our Bayesian method performs automatic model compression via an adaptive tensor rank determination. We also present approaches for posterior density calculation and maximum a posteriori (MAP) estimation for the end-to-end training of our tensorized neural network. We provide experimental validation on a fully connected neural network, a CNN and a residual neural network where our work produces @math to @math more compact neural networks directly from the training.
Tensor Rank Determination. Two main approaches are used for rank determination in tensor completion: low-rank optimization and Bayesian inference. Optimization methods mainly rely on some generalization of the matrix nuclear norm @cite_28 to tensors. The most popular approaches place a low-rank objective on tensor unfoldings @cite_11 @cite_27 @cite_9 , but the computation is expensive for high-order tensors. The Frobenius norms of CP factors are used as a regularization for 3-way tensor completion @cite_15 , but this does not generalize to high-order tensors either. Bayesian methods can directly infer the tensor rank in CP or Tucker tensor completion through low-rank priors @cite_25 @cite_2 @cite_23 @cite_15 @cite_13 . In the CP and Tucker formats, the ranks of different tensor factors do not couple with each other. This is not true in the tensor-train format (see ). Furthermore, in tensor completion the observed data is a linear mapping of a tensor, whereas the mapping is nonlinear in neural networks. Therefore, previous work on tensor rank determination cannot be directly applied to tensorized neural networks.
@cite_28 @cite_9 @cite_27 @cite_23 @cite_2 @cite_15 @cite_13 @cite_25 @cite_11
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CANDECOMP PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank . In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. To learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance. Low rank tensor representation underpins much of recent progress in tensor completion. In real applications, however, this approach is confronted with two challenging problems, namely (1) tensor rank determination; (2) handling real tensor data which only approximately fulfils the low-rank requirement. To address these two issues, we develop a data-adaptive tensor completion model which explicitly represents both the low-rank and non-low-rank structures in a latent tensor. Representing the non-low-rank structure separately from the low-rank one allows priors which capture the important distinctions between the two, thus enabling more accurate modelling, and ultimately, completion. Through defining a new tensor rank, we develop a sparsity induced prior for the low-rank structure, with which the tensor rank can be automatically determined. The prior for the non-low-rank structure is established based on a mixture of Gaussians which is shown to be flexible enough, and powerful enough, to inform the completion process for a variety of real tensor data. With these two priors, we develop a Bayesian minimum mean squared error estimate (MMSE) framework for inference which provides the posterior mean of missing entries as well as their uncertainty. Compared with the state-of-the-art methods in various applications, the proposed model produces more accurate completion results. A novel regularizer of the PARAFAC decomposition factors capturing the tensor's rank is proposed in this paper, as the key enabler for completion of three-way data arrays with missing entries. Set in a Bayesian framework, the tensor completion method incorporates prior information to enhance its smoothing and prediction capabilities. This probabilistic approach can naturally accommodate general models for the data distribution, lending itself to various fitting criteria that yield optimum estimates in the maximum-a-posteriori sense. In particular, two algorithms are devised for Gaussian- and Poisson-distributed data, that minimize the rank-regularized least-squares error and Kullback-Leibler divergence, respectively. The proposed technique is able to recover the “ground-truth” tensor rank when tested on synthetic data, and to complete brain imaging and yeast gene expression datasets with 50 and 15 of missing entries respectively, resulting in recovery errors at -11 dB and -15 dB. Higher-order low-rank tensors naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data reconstruction, and so on. We propose a new model to recover a low-rank tensor by simultaneously performing low-rank matrix factorizations to the all-mode matricizations of the underlying tensor. An alternating minimization algorithm is applied to solve the model, along with two adaptive rank-adjusting strategies when the exact rank is not known. &nbsp Phase transition plots reveal that our algorithm can recover a variety of synthetic low-rank tensors from significantly fewer samples than the compared methods, which include a matrix completion method applied to tensor recovery and two state-of-the-art tensor completion methods. Further tests on real-world data show similar advantages. Although our model is non-convex, our algorithm performs consistently throughout the tests and gives better results than the compared methods, some of which are based on convex models. In addition, subsequence convergence of our algorithm can be established in the sense that any limit point of the iterates satisfies the KKT condtions.