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Representational Transfer Learning for Matrix Completion

Yong He, Zeyu Li, Dong Liu, Kangxiang Qin, Jiahui Xie

TL;DR

The paper tackles matrix completion under noise by leveraging multiple related sources through representational transfer learning. It formalizes representational similarity as containment of left and right singular subspaces, enabling a two-stage strategy: (i) unsupervised subspace integration across sources to learn shared $U$ and $V$, and (ii) a low-dimensional regression in the reduced space to estimate the target $\Theta_0^*$. The authors provide oracle theory giving near-optimal convergence rates and post-transfer inference, and propose a non-oracle procedure with selective subspace integration and optional knowledge transfer to avoid negative transfer. Simulations and real-data experiments (e.g., COVID-19 CT images) demonstrate robustness and practical gains over baseline methods, highlighting the approach’s efficiency and scalability for streaming data and potential tensor extensions.

Abstract

We propose to transfer representational knowledge from multiple sources to a target noisy matrix completion task by aggregating singular subspaces information. Under our representational similarity framework, we first integrate linear representation information by solving a two-way principal component analysis problem based on a properly debiased matrix-valued dataset. After acquiring better column and row representation estimators from the sources, the original high-dimensional target matrix completion problem is then transformed into a low-dimensional linear regression, of which the statistical efficiency is guaranteed. A variety of extensional arguments, including post-transfer statistical inference and robustness against negative transfer, are also discussed alongside. Finally, extensive simulation results and a number of real data cases are reported to support our claims.

Representational Transfer Learning for Matrix Completion

TL;DR

The paper tackles matrix completion under noise by leveraging multiple related sources through representational transfer learning. It formalizes representational similarity as containment of left and right singular subspaces, enabling a two-stage strategy: (i) unsupervised subspace integration across sources to learn shared and , and (ii) a low-dimensional regression in the reduced space to estimate the target . The authors provide oracle theory giving near-optimal convergence rates and post-transfer inference, and propose a non-oracle procedure with selective subspace integration and optional knowledge transfer to avoid negative transfer. Simulations and real-data experiments (e.g., COVID-19 CT images) demonstrate robustness and practical gains over baseline methods, highlighting the approach’s efficiency and scalability for streaming data and potential tensor extensions.

Abstract

We propose to transfer representational knowledge from multiple sources to a target noisy matrix completion task by aggregating singular subspaces information. Under our representational similarity framework, we first integrate linear representation information by solving a two-way principal component analysis problem based on a properly debiased matrix-valued dataset. After acquiring better column and row representation estimators from the sources, the original high-dimensional target matrix completion problem is then transformed into a low-dimensional linear regression, of which the statistical efficiency is guaranteed. A variety of extensional arguments, including post-transfer statistical inference and robustness against negative transfer, are also discussed alongside. Finally, extensive simulation results and a number of real data cases are reported to support our claims.

Paper Structure

This paper contains 13 sections, 3 theorems, 18 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Similarity metrics
  3. Representational similarity
  4. Contributions and notations
  5. Oracle Transfer Learning Procedure
  6. Default implementation procedure
  7. Oracle Theory
  8. Non-Oracle Transfer Learning Procedure
  9. Selective subspace integration
  10. Optional knowledge transfer
  11. Numerical Simulation
  12. Real Data Experiment
  13. Discussion

Key Result

theorem 1

Under Assumptions assum:1 to assum:4, for $k\in \cI$, assume that $n_k\gtrsim p\log^{\tau} p$ for some $\tau\ge3$ and $K\lesssim (N/p\log p)^{1/2}$ as $n_0, n_k, p,q\rightarrow \infty$, we have

Figures (2)

  • Figure 1: Comparison of various methods under different scenarios, i.e., scenario A, B and C from left to right. The proposed non-oracle knowledge transfer algorithm demonstrates its advantage across the three given settings.
  • Figure 2: For each panel, from left to right are respectively: (a) the partially observed image; (b) the target completed image using the target dataset only; (c) the two-step completed image by debiasing the Euclidean mean of the sources; (d) the first slice of the tensor-Tucker completion estimator and (e) the representational transferred completed image by utilizing the subspace information from the sources.

Theorems & Definitions (3)

  • theorem 1: Representation transfer learning
  • corollary 1: Asymptotic normality
  • corollary 2: Local maximum