Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination (1401.6497v2)

Abstract: 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.

• The paper presents a novel hierarchical Bayesian model for CP factorization that uses sparsity-inducing priors to automatically determine the tensor rank.
• It develops an efficient deterministic Bayesian inference algorithm that scales linearly with data size, reducing the need for manual rank specification.
• Empirical results show the method effectively predicts missing tensor entries, demonstrating robust performance in noisy and real-world applications.

Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination

The paper "Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination" presents a novel approach to addressing challenges in tensor factorization, particularly issues of incomplete data and automatic rank determination in CP (CANDECOMP/PARAFAC) decomposition. The authors propose a Bayesian methodology to tackle the complexities associated with the determination of the tensor rank, which is a notable difficulty in CP factorization.

Key Contributions

The paper formulates CP factorization through a hierarchical probabilistic model, incorporating a fully Bayesian treatment with sparsity-inducing priors over the latent factors. This framework enables automatic rank determination, providing a significant advantage by eliminating the need for manual rank specification, which is often intractable and computationally intensive.

1. Model Framework: The proposed method employs a hierarchical prior structure that facilitates automatic rank determination by leveraging a sparsity-inducing prior across multiple latent factors. This component is crucial in constraining the number of components in the factor matrices to a minimum, capturing the multimodal interactions comprehensively.
2. Inference Methodology: The authors develop an efficient deterministic Bayesian inference algorithm that scales linearly with data size. This approach contrasts with traditional methods, which can suffer from overfitting and require heuristic or cross-validation-based rank determination, thus demonstrating practical advantages in computational efficiency.
3. Predictive Performance: The method provides predictive distributions over missing entries. This attribute is particularly beneficial in real-world applications where data is incomplete or noisy. The Bayesian nature of the model allows it to handle uncertainty effectively, offering predictions complete with distribution information rather than point estimates.
4. Empirical Validation: Extensive simulations on synthetic data illustrate the capacity of the proposed method to recover the true CP rank while avoiding overfitting, even with substantial data missing. Real-world applications, such as image inpainting and facial image synthesis, demonstrate the method’s effectiveness compared to existing state-of-the-art tensor factorization and completion approaches.

Implications and Future Directions

The development of a fully Bayesian CP factorization framework has both theoretical and practical implications. Theoretically, the approach provides an elegant solution to the NP-complete problem of tensor rank determination within the CP framework. In practice, the method's application to fields such as image processing, face recognition, and brain signal processing highlights its versatility and robustness in high-dimensional and noisy data environments.

The automatic rank determination and the scalable inference mechanism set a precedent for future research in tensor analysis. Potential developments may involve extending the Bayesian framework to other tensor decomposition methods, exploring non-Gaussian noise models, or integrating auxiliary information to further enhance predictive capabilities.

In conclusion, the paper presents a comprehensive and rigorously developed method for CP factorization and completion, offering significant improvements in precision, efficiency, and applicability to real-world problems. This work sets the stage for further innovations in the field of tensor analysis, paving the way for more robust and scalable solutions in handling complex multidimensional data.