# Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination

arXiv: https://arxiv.org/abs/1401.6497 · HTML: https://mlsanity.com/paper/1401.6497/

## Key facts

- Cited by 65 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jan 2014.

## Built on by (most cited first, 30 of 65)

- [Tensor Methods in Computer Vision and Deep Learning](https://mlsanity.com/paper/2107.03436/) (2021)
- [Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2212.00262/) (2022)
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- [DeepTensor: Low-Rank Tensor Decomposition with Deep Network Priors](https://mlsanity.com/paper/2204.03145/) (2022)
- [Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion](https://mlsanity.com/paper/2110.08774/) (2021)
- [Noisy Tensor Completion via Low-rank Tensor Ring](https://mlsanity.com/paper/2203.08857/) (2022)
- [Permutation Search of Tensor Network Structures via Local Sampling](https://mlsanity.com/paper/2206.06597/) (2022)
- [Rethinking Bayesian Learning for Data Analysis: The Art of Prior and Inference in Sparsity-Aware Modeling](https://mlsanity.com/paper/2205.14283/) (2022)
- [Alternating Local Enumeration (TnALE): Solving Tensor Network Structure Search with Fewer Evaluations](https://mlsanity.com/paper/2304.12875/) (2023)
- [SVDinsTN: A Tensor Network Paradigm for Efficient Structure Search from Regularized Modeling Perspective](https://mlsanity.com/paper/2305.14912/) (2023)
- [Effective Tensor Completion via Element-wise Weighted Low-rank Tensor Train with Overlapping Ket Augmentation](https://mlsanity.com/paper/2109.05736/) (2021)
- [Tensor Recovery Based on A Novel Non-convex Function Minimax Logarithmic Concave Penalty Function](https://mlsanity.com/paper/2206.13506/) (2022)
- [Nonparametric Sparse Tensor Factorization with Hierarchical Gamma Processes](https://mlsanity.com/paper/2110.10082/) (2021)
- [Nonparametric Embeddings of Sparse High-Order Interaction Events](https://mlsanity.com/paper/2207.03639/) (2022)
- [tnGPS: Discovering Unknown Tensor Network Structure Search Algorithms via Large Language Models (LLMs)](https://mlsanity.com/paper/2402.02456/) (2024)
- [Block-Term Tensor Decomposition Model Selection and Computation: The Bayesian Way](https://mlsanity.com/paper/2101.02931/) (2021)
- [Bayesian Low-rank Matrix Completion with Dual-graph Embedding: Prior Analysis and Tuning-free Inference](https://mlsanity.com/paper/2203.10044/) (2022)
- [Scalable Spatiotemporally Varying Coefficient Modelling with Bayesian Kernelized Tensor Regression](https://mlsanity.com/paper/2109.00046/) (2021)
- [Streaming data preprocessing via online tensor recovery for large environmental sensor networks](https://mlsanity.com/paper/2109.00596/) (2021)
- [To Fold or Not to Fold: Graph Regularized Tensor Train for Visual Data Completion](https://mlsanity.com/paper/2306.11123/) (2023)
- [Efficient Nonparametric Tensor Decomposition for Binary and Count Data](https://mlsanity.com/paper/2401.07711/) (2024)
- [Interpretable Bayesian Tensor Network Kernel Machines with Automatic Rank and Feature Selection](https://mlsanity.com/paper/2507.11136/) (2025)
- [Multi-Tensor Network Representation for High-Order Tensor Completion](https://mlsanity.com/paper/2109.04022/) (2021)
- [FRAPPE: underlineFast underlineRank underlineApproximation with underlineExplainable Features for Tensors](https://mlsanity.com/paper/2206.09316/) (2022)
- [Bayesian Complementary Kernelized Learning for Multidimensional Spatiotemporal Data](https://mlsanity.com/paper/2208.09978/) (2022)
- [A Lightweight Method for Modeling Confidence in Recommendations with Learned Beta Distributions](https://mlsanity.com/paper/2308.03186/) (2023)
- [Scalable Bayesian Tensor Ring Factorization for Multiway Data Analysis](https://mlsanity.com/paper/2412.03321/) (2024)
- [Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCA](https://mlsanity.com/paper/2501.04565/) (2025)
- [Functional Complexity-adaptive Temporal Tensor Decomposition](https://mlsanity.com/paper/2502.06164/) (2025)

## Abstract (arXiv)

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.