# When Bayesian Tensor Completion Meets Multioutput Gaussian Processes: Functional Universality and Rank Learning

arXiv: https://arxiv.org/abs/2512.21486 · HTML: https://mlsanity.com/paper/2512.21486/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 63 references that are in the map.
- First posted on arXiv 25 Dec 2025.

## Builds on (most cited first, 26 of 26)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2212.00262/) (2022)
- [Characteristic and Universal Tensor Product Kernels](https://mlsanity.com/paper/1708.08157/) (2017)
- [Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion](https://mlsanity.com/paper/1909.06982/) (2019)
- [Scaling Gaussian Process Regression with Derivatives](https://mlsanity.com/paper/1810.12283/) (2018)
- [Scalable Gaussian Process Variational Autoencoders](https://mlsanity.com/paper/2010.13472/) (2020)
- [Integrated Sensing and Communication with Massive MIMO: A Unified Tensor Approach for Channel and Target Parameter Estimation](https://mlsanity.com/paper/2401.01738/) (2024)
- [Multivariate Gaussian and Student-t Process Regression for Multi-output Prediction](https://mlsanity.com/paper/1703.04455/) (2017)
- [Deep Spectrum Cartography: Completing Radio Map Tensors Using Learned Neural Models](https://mlsanity.com/paper/2105.00177/) (2021)
- [Rethinking Bayesian Learning for Data Analysis: The Art of Prior and Inference in Sparsity-Aware Modeling](https://mlsanity.com/paper/2205.14283/) (2022)
- [Functional Bayesian Tucker Decomposition for Continuous-indexed Tensor Data](https://mlsanity.com/paper/2311.04829/) (2023)
- [Linear Multiple Low-Rank Kernel Based Stationary Gaussian Processes Regression for Time Series](https://mlsanity.com/paper/1904.09559/) (2019)
- [Towards Flexible Sparsity-Aware Modeling: Automatic Tensor Rank Learning Using The Generalized Hyperbolic Prior](https://mlsanity.com/paper/2009.02472/) (2020)
- [Channel Estimation for Intelligent Reflecting Surface Assisted MIMO Systems: A Tensor Modeling Approach](https://mlsanity.com/paper/2008.04766/) (2020)
- [Tensor Train Factorization and Completion under Noisy Data with Prior Analysis and Rank Estimation](https://mlsanity.com/paper/2010.06564/) (2020)
- [Conditional Expectation Propagation](https://mlsanity.com/paper/1910.12360/) (2019)
- [Bayesian Low-rank Matrix Completion with Dual-graph Embedding: Prior Analysis and Tuning-free Inference](https://mlsanity.com/paper/2203.10044/) (2022)
- [A Bayesian Tensor Factorization Model via Variational Inference for Link Prediction](https://mlsanity.com/paper/1409.8276/) (2014)
- [Striking The Right Balance: Three-Dimensional Ocean Sound Speed Field Reconstruction Using Tensor Neural Networks](https://mlsanity.com/paper/2308.04930/) (2023)

## Built on by (most cited first, 1 of 1)

- [Kernel Regression with Tensor Trains and Hadamard Overparameterization](https://mlsanity.com/paper/2607.17390/) (2026)

## Abstract (arXiv)

Functional tensor decomposition can analyze multi-dimensional data with real-valued indices, paving the path for applications in machine learning and signal processing. A limitation of existing approaches is the assumption that the tensor rank-a critical parameter governing model complexity-is known. However, determining the optimal rank is a non-deterministic polynomial-time hard (NP-hard) task and there is a limited understanding regarding the expressive power of functional low-rank tensor models for continuous signals. We propose a rank-revealing functional Bayesian tensor completion (RR-FBTC) method. Modeling the latent functions through carefully designed multioutput Gaussian processes, RR-FBTC handles tensors with real-valued indices while enabling automatic tensor rank determination during the inference process. We establish the universal approximation property of the model for continuous multi-dimensional signals, demonstrating its expressive power in a concise format. To learn this model, we employ the variational inference framework and derive an efficient algorithm with closed-form updates. Experiments on both synthetic and real-world datasets demonstrate the effectiveness and superiority of the RR-FBTC over state-of-the-art approaches. The code is available at https://github.com/OceanSTARLab/RR-FBTC.