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Variational Bayesian inference for CP tensor completion with side information

Stanislav Budzinskiy, Nikolai Zamarashkin

TL;DR

The paper addresses CP tensor completion under limited observations by introducing a variational Bayesian framework that incorporates side information in the form of low-dimensional subspaces. It derives a closed-form, message-passing VB algorithm that jointly infers CP factors, hyperparameters, and an automatic effective rank through Gamma priors, extending the matrix case to tensors. Empirical results on synthetic data and real 3D Basel Face Model data show substantial reductions in the required number of observed entries when side information is available, along with insights into phase-transition behavior and rank determination. The work highlights the regularization benefits and practical impact of side information for tensor completion in data-scarce regimes.

Abstract

We propose a message passing algorithm, based on variational Bayesian inference, for low-rank tensor completion with automatic rank determination in the canonical polyadic format when additional side information (SI) is given. The SI comes in the form of low-dimensional subspaces the contain the fiber spans of the tensor (columns, rows, tubes, etc.). We validate the regularization properties induced by SI with extensive numerical experiments on synthetic and real-world data and present the results about tensor recovery and rank determination. The results show that the number of samples required for successful completion is significantly reduced in the presence of SI. We also discuss the origin of a bump in the phase transition curves that exists when the dimensionality of SI is comparable with that of the tensor.

Variational Bayesian inference for CP tensor completion with side information

TL;DR

The paper addresses CP tensor completion under limited observations by introducing a variational Bayesian framework that incorporates side information in the form of low-dimensional subspaces. It derives a closed-form, message-passing VB algorithm that jointly infers CP factors, hyperparameters, and an automatic effective rank through Gamma priors, extending the matrix case to tensors. Empirical results on synthetic data and real 3D Basel Face Model data show substantial reductions in the required number of observed entries when side information is available, along with insights into phase-transition behavior and rank determination. The work highlights the regularization benefits and practical impact of side information for tensor completion in data-scarce regimes.

Abstract

We propose a message passing algorithm, based on variational Bayesian inference, for low-rank tensor completion with automatic rank determination in the canonical polyadic format when additional side information (SI) is given. The SI comes in the form of low-dimensional subspaces the contain the fiber spans of the tensor (columns, rows, tubes, etc.). We validate the regularization properties induced by SI with extensive numerical experiments on synthetic and real-world data and present the results about tensor recovery and rank determination. The results show that the number of samples required for successful completion is significantly reduced in the presence of SI. We also discuss the origin of a bump in the phase transition curves that exists when the dimensionality of SI is comparable with that of the tensor.
Paper Structure (39 sections, 107 equations, 12 figures, 2 tables)

This paper contains 39 sections, 107 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Matrix completion with side information
  4. Probabilistic model: priors
  5. Variational Bayesian inference
  6. Optimal posterior distributions
  7. Factor matrices $U$ and $V$
  8. Precision matrix $\Lambda$
  9. Noise precision $\tau$
  10. Message passing updates
  11. Tensor completion with side information
  12. Probabilistic model: priors
  13. Optimal posterior distributions
  14. Canonical factors $U_l$
  15. Precision matrix $\Lambda$
  16. ...and 24 more sections

Figures (12)

  • Figure 1: Phase plots for noiseless matrix completion with rank $r = 3$ and perfect rank prediction $k = 3$: FBCP (a) and FBCP-SI with side information size $m = 30$ (b).
  • Figure 2: Phase plots for noiseless $3$-dimensional CP completion with rank $r = 3$ and perfect rank prediction $k = 3$: FBCP (a) and FBCP-SI with side information size $m = 30$ (b).
  • Figure 3: Phase plots for noiseless $3$-dimensional CP completion with rank $r = 3$, perfect rank prediction $k = 3$, and different sizes of side information: $m = 10$ (a), $m = 20$ (b), $m = 30$ (c), and $m = 40$ (d).
  • Figure 4: Phase plots for noiseless $4$-dimensional CP completion with rank $r = 3$, perfect rank prediction $k = 3$, and side information size $m = 30$ after different numbers of iterations: $N_{iter} = 20$ (a), $N_{iter} = 50$ (b), $N_{iter} = 100$ (c), and $N_{iter} = 200$ (d).
  • Figure 5: Determined rank for 3-dimensional CP completion with size $n = 100$, rank $r = 3$, side information size $m = 10$, rank prediction $k = 10$, varying levels of noise and different numbers of samples: $|\Omega| = 2.5 \cdot 10^2$ (a), $|\Omega| = 1 \cdot 10^3$ (b), $|\Omega| = 1.5 \cdot 10^3$ (c), and $|\Omega| = 2 \cdot 10^3$ (d). The curves show the averaged rank together with the 5th and 95th percentiles for threshold values $\varepsilon = 0.05$ and $\varepsilon = 0.01$.
  • ...and 7 more figures