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[2510.17886] Graphical model for factorization and completion of relatively high rank tensors by sparse sampling
Summary
This paper presents a graphical model for factorization and completion of high-rank tensors using sparse sampling, addressing challenges in data completion for recommendation systems.
Why It Matters
As data becomes increasingly sparse in various applications, particularly in recommendation systems, this research provides theoretical insights and practical algorithms that can enhance data completion methods. The findings could improve the performance of machine learning models in real-world scenarios where data is often incomplete.
Key Takeaways
- Introduces a new graphical model for tensor factorization under sparse sampling conditions.
- Demonstrates the applicability of message-passing algorithms in high-dimensional settings.
- Develops a replica theory to analyze statistical inference performance, avoiding common pitfalls of Gaussian assumptions.
Statistics > Machine Learning arXiv:2510.17886 (stat) [Submitted on 18 Oct 2025 (v1), last revised 17 Feb 2026 (this version, v2)] Title:Graphical model for factorization and completion of relatively high rank tensors by sparse sampling Authors:Angelo Giorgio Cavaliere, Riki Nagasawa, Shuta Yokoi, Tomoyuki Obuchi, Hajime Yoshino View a PDF of the paper titled Graphical model for factorization and completion of relatively high rank tensors by sparse sampling, by Angelo Giorgio Cavaliere and 4 other authors View PDF Abstract:We consider tensor factorizations based on sparse measurements of the components of relatively high rank tensors. The measurements are designed in a way that the underlying graph of interactions is a random graph. The setup will be useful in cases where a substantial amount of data is missing, as in completion of relatively high rank matrices for recommendation systems heavily used in social network services. In order to obtain theoretical insights on the setup, we consider statistical inference of the tensor factorization in a high dimensional limit, which we call as dense limit, where the graphs are large and dense but not fully connected. We build message-passing algorithms and test them in a Bayes optimal teacher-student setting in some specific cases. We also develop a replica theory to examine the performance of statistical inference in the dense limit based on a cumulant expansion. The latter approach allows one to avoid blind usage of Gaussian an...
