Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination

By Qibin Zhao, Liqing Zhang, Andrzej S. Cichocki. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2014

595 citations recorded by the source corpus.

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.

Subfields: Computational Mathematics, Computational Mechanics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/c83c9b03a171c96c18fe74dae8e7000c279c7401

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Citations in the corpus

Efficient Tensor Completion for Color Image and Video Recovery: Low-Rank Tensor Train (2016, methodology): “in determining the optimal TT rank and weights via automatic [46] and/or adaptive methods [21].”
Tensors for Data Mining and Data Fusion (2016, methodology): “In addition to CORCONDIA, various other methods have been proposed, ranging from Minimum Description Length (MDL) [Araujo et al. 2014; Metzler and Miettinen 2015] to Bayesian [Mørup and Hansen 2009; Zhao et al. 2015].”
Infrared Dim and Small Target Detection via Multiple Subspace Learning and Spatial-Temporal Patch-Tensor Model (2021, background): “The target detection problem is transformed into recovering low-rank component and sparse component [37]– [39], which is also widely used in background subtraction and image denoising [40]–[42].”
Hyperspectral Image Super-Resolution via Non-local Sparse Tensor Factorization (2017, methodology): “In the past years, tensor factorization has been successfully applied into multiframe data denoising [10, 21], completion [41, 17, 40], compressive sensing [36] and classification [35].”
Bayesian Temporal Factorization for Multidimensional Time Series Prediction (2019, background): “For example, a Bayesian tensor factorization is proposed in [31], which can automatically determine the CP rank.”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, methodology): “Low rank structure of tensor data has been successfully utilized in applications including missing data imputation (Liu et al., 2013; Zhao et al., 2015), multilinear robust principal component analysis (Inoue et al.”
Tensor Ring Decomposition (2016, background): “These two models have been widely investigated and applied in various real-world applications, e.g., [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22].”