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

• URL: http://arxiv.org/abs/2512.21486v1
• Date: Thu, 25 Dec 2025 03:15:52 GMT
• Title: When Bayesian Tensor Completion Meets Multioutput Gaussian Processes: Functional Universality and Rank Learning
• Authors: Siyuan Li, Shikai Fang, Lei Cheng, Feng Yin, Yik-Chung Wu, Peter Gerstoft, Sergios Theodoridis,
• Abstract summary: Functional tensor decomposition can analyze multi-dimensional data with real-valued indices.<n>We propose a rank-revealing functional low-rank tensor completion (RR-F) method.<n>We establish the universal approximation property of the model for continuous multi-dimensional signals.
• Score: 53.17227599983122
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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