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Tensor Train Completion from Fiberwise Observations Along a Single Mode

Shakir Showkat Sofi, Lieven De Lathauwer

TL;DR

This work shows how to use only standard linear algebra operations to compute the tensor train decomposition of aspecific type of “fiber-wise” observed tensor, where some of the fibers of a tensor are either fully observed or entirely missing, unlike the usualentry-wise observations.

Abstract

Tensor completion is an extension of matrix completion aimed at recovering a multiway data tensor by leveraging a given subset of its entries (observations) and the pattern of observation. The low-rank assumption is key in establishing a relationship between the observed and unobserved entries of the tensor. The low-rank tensor completion problem is typically solved using numerical optimization techniques, where the rank information is used either implicitly (in the rank minimization approach) or explicitly (in the error minimization approach). Current theories concerning these techniques often study probabilistic recovery guarantees under conditions such as random uniform observations and incoherence requirements. However, if an observation pattern exhibits some low-rank structure that can be exploited, more efficient algorithms with deterministic recovery guarantees can be designed by leveraging this structure. This work shows how to use only standard linear algebra operations to compute the tensor train decomposition of a specific type of ``fiber-wise'' observed tensor, where some of the fibers of a tensor (along a single specific mode) are either fully observed or entirely missing, unlike the usual entry-wise observations. From an application viewpoint, this setting is relevant when it is easier to sample or collect a multiway data tensor along a specific mode (e.g., temporal). The proposed completion method is fast and is guaranteed to work under reasonable deterministic conditions on the observation pattern. Through numerical experiments, we showcase interesting applications and use cases that illustrate the effectiveness of the proposed approach.

Tensor Train Completion from Fiberwise Observations Along a Single Mode

TL;DR

This work shows how to use only standard linear algebra operations to compute the tensor train decomposition of aspecific type of “fiber-wise” observed tensor, where some of the fibers of a tensor are either fully observed or entirely missing, unlike the usualentry-wise observations.

Abstract

Tensor completion is an extension of matrix completion aimed at recovering a multiway data tensor by leveraging a given subset of its entries (observations) and the pattern of observation. The low-rank assumption is key in establishing a relationship between the observed and unobserved entries of the tensor. The low-rank tensor completion problem is typically solved using numerical optimization techniques, where the rank information is used either implicitly (in the rank minimization approach) or explicitly (in the error minimization approach). Current theories concerning these techniques often study probabilistic recovery guarantees under conditions such as random uniform observations and incoherence requirements. However, if an observation pattern exhibits some low-rank structure that can be exploited, more efficient algorithms with deterministic recovery guarantees can be designed by leveraging this structure. This work shows how to use only standard linear algebra operations to compute the tensor train decomposition of a specific type of ``fiber-wise'' observed tensor, where some of the fibers of a tensor (along a single specific mode) are either fully observed or entirely missing, unlike the usual entry-wise observations. From an application viewpoint, this setting is relevant when it is easier to sample or collect a multiway data tensor along a specific mode (e.g., temporal). The proposed completion method is fast and is guaranteed to work under reasonable deterministic conditions on the observation pattern. Through numerical experiments, we showcase interesting applications and use cases that illustrate the effectiveness of the proposed approach.

Paper Structure

This paper contains 25 sections, 2 theorems, 11 equations, 12 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries and Notation
  4. Organization
  5. TT Decomposition of Fully Observed Tensor
  6. TT Completion from Fiberwise Observations Along a Single Mode
  7. Piecewise Subspace Learning
  8. Subspace Constraint Approach
  9. Subspace Intersection Approach
  10. Algorithm
  11. Computing the Next-to-Last TT Core $\mathcal{G}^{(N\!-\!1)}$
  12. Computational Complexity
  13. Uniqueness Conditions
  14. Numerical Experiments
  15. Synthetic Data
  16. ...and 10 more sections

Key Result

Theorem 1

The TT cores $\{\mathcal{G}^{(n)}\in\mathbb{R}^{R_{n-1}\times I_n\times R_n}\}_{n=1}^N$ are uniquely determined (up to basis transformations) if the following hold:

Figures (12)

  • Figure 1: TT decomposition of $5$th-order tensor as a train of five core tensors, where $\mathcal{G}^{(1)} \in \mathbb{R}^{1 \times I_{1} \times R_{1}}$ and $\mathcal{G}^{(5)} \in \mathbb{R}^{R_{4} \times I_{5} \times 1}.$
  • Figure 2: The $n$th matrix unfolding of a tensor observed through fibers along a single mode is characterized by a structured observation pattern.
  • Figure 3: (a) Affine subspace $S_1$ corresponding to the first column $\mathbf{m}_{:1}$; (b) affine subspace $S_2$ corresponding to the second column $\mathbf{m}_{:2}$; (c) intersection of the affine subspaces $S_1 \cap S_2$.
  • Figure 4: (Left) The accuracy of our method is slightly lower than that of TT-WOPT and TMac-TT, which is expected due to its reliance on only standard NLA operations. (Right) However, it provides a significant computational speedup.
  • Figure 5: (Left) Accuracy improves as $I$ increases since relatively more data becomes available per parameter (with the missing rate and TT rank held constant). TT-WOPT achieves the highest accuracy. (Right) Meanwhile, our method outperforms all reference algorithms in terms of computation time.
  • ...and 7 more figures

Theorems & Definitions (7)

  • Definition 1: Isorank submatrix
  • Definition 2: Row overlap
  • Definition 3: Contraction
  • Example 1
  • Theorem 1: Algebraic conditions
  • Remark 1
  • Corollary 1: Generic conditions