A Spectrally Damped Tensor Randomized Kaczmarz Method for Doubly Noisy Tensor Systems

Tensor randomized Kaczmarz (TRK) methods are efficient row-action solvers for tensor linear systems under the t-product framework.

We study their behavior under a doubly noisy perturbation model.

In this model, both the system tensor and the right-hand side tensor are corrupted.

We first analyze standard TRK and derive an expected error recursion with two terms.

One term is contractive, and the other is a persistent perturbation term.

This explains the noise-limited and semi-convergent behavior that can occur when the observed tensor system is inconsistent.

We then introduce a spectrally damped tensor randomized Kaczmarz method (SD-TRK).

We prove an expected error recursion for SD-TRK that separates error propagation from noise injection.

The bound makes explicit a speed-robustness trade-off.

We also give an FFT-based implementation that applies the damped update slice-wise in the Fourier domain.

This implementation allows frequency-dependent damping parameters in practice.

Numerical experiments on synthetic tensor systems illustrate the stabilization behavior of SD-TRK relative to standard TRK in noisy and ill-conditioned settings.

We also include a two-pass image reconstruction comparison under the same noisy reconstruction pipeline.