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Randomized Tensor Krylov Subspace Methods via Sketched Einstein Product with Applications to Image and Video Restoration

Achraf Badahmane

TL;DR

This work introduces a sketched Einstein inner product constructed via mode-wise random projections and develops a randomized tensor global Arnoldi process, which significantly reduces orthogonalization cost while preserving convergence properties under tensor subspace embedding assumptions.

Abstract

We develop a randomized extension of tensor Krylov subspace methods based on the Einstein product for solving large-scale multilinear systems arising in image and video restoration. The classical tensor global GMRES method relies on Frobenius inner products and full tensor orthogonalization, which become computationally expensive for high-dimensional problems. We introduce a sketched Einstein inner product constructed via mode-wise random projections and develop a randomized tensor global Arnoldi process. The resulting Randomized Tensor Global GMRES (RTG-GMRES) method significantly reduces orthogonalization cost while preserving convergence properties under tensor subspace embedding assumptions. Residual bounds, perturbation analysis and projected Tikhonov regularization are derived. The proposed method provides an efficient framework for solving ill-posed multidimensional problems arising in color image and video restoration.

Randomized Tensor Krylov Subspace Methods via Sketched Einstein Product with Applications to Image and Video Restoration

TL;DR

This work introduces a sketched Einstein inner product constructed via mode-wise random projections and develops a randomized tensor global Arnoldi process, which significantly reduces orthogonalization cost while preserving convergence properties under tensor subspace embedding assumptions.

Abstract

We develop a randomized extension of tensor Krylov subspace methods based on the Einstein product for solving large-scale multilinear systems arising in image and video restoration. The classical tensor global GMRES method relies on Frobenius inner products and full tensor orthogonalization, which become computationally expensive for high-dimensional problems. We introduce a sketched Einstein inner product constructed via mode-wise random projections and develop a randomized tensor global Arnoldi process. The resulting Randomized Tensor Global GMRES (RTG-GMRES) method significantly reduces orthogonalization cost while preserving convergence properties under tensor subspace embedding assumptions. Residual bounds, perturbation analysis and projected Tikhonov regularization are derived. The proposed method provides an efficient framework for solving ill-posed multidimensional problems arising in color image and video restoration.
Paper Structure (22 sections, 1 theorem, 99 equations, 1 table, 2 algorithms)

This paper contains 22 sections, 1 theorem, 99 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Einstein Product
  4. Frobenius Inner Product
  5. Sketched Einstein Inner Product
  6. Compressed Tensor
  7. Tensor Subspace Embedding
  8. Equivalent Vectorized Form
  9. Tensor Matrix Subspace Embedding
  10. Tensor Singular-Value Bounds
  11. Block Krylov Subspace Methods via Sketched Einstein Product
  12. Block GMRES via Sketched Einstein Product
  13. Block Golub--Kahan via Sketched Einstein Product
  14. Numerical Results
  15. Tensor Model for RGB Image Deblurring
  16. ...and 7 more sections

Key Result

Proposition 1

If $\mathcal{S}$ is an $\varepsilon$-subspace embedding for $\mathcal{U}$, then for every unfolding $V_{(k)}$,

Theorems & Definitions (5)

  • Definition 3.1
  • Definition 3.2: Tensor $\varepsilon$-Subspace Embedding
  • Definition 3.3: Tensor Subspace Embedding – Norm Form
  • Proposition 1: Tensor Singular-Value Bounds
  • proof