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

TL;DR

This paper introduces a randomized tensor Krylov subspace method using sketched Einstein products, significantly improving computational efficiency for large-scale image and video restoration tasks while maintaining convergence guarantees.

Contribution

It presents a novel randomized tensor global GMRES method based on sketched Einstein inner products, reducing computational costs in high-dimensional tensor problems.

Findings

Reduces orthogonalization cost in tensor Krylov methods

Maintains convergence properties with tensor subspace embeddings

Effective in large-scale image and video restoration applications

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.