AI 中文总结
研究双噪声扰动模型下张量随机化Kaczmarz方法，先分析标准TRK，再引入谱阻尼张量随机化Kaczmarz方法（SD - TRK），证明其误差递推式，给出基于FFT的实现，通过数值实验展示SD - TRK在噪声和病态设置下相对于标准TRK的稳定行为。
AI 中文摘要
张量随机化Kaczmarz（TRK）方法是t积框架下张量线性系统的有效行作用求解器。本文研究其在双噪声扰动模型下的行为，该模型中系统张量和右侧张量均被破坏。首先分析标准TRK并推导含两项的预期误差递推式，解释了观测张量系统不一致时可能出现的噪声限制和半收敛行为。接着引入谱阻尼张量随机化Kaczmarz方法（SD - TRK），证明了其将误差传播与噪声注入分离的预期误差递推式，明确了速度 - 稳健性权衡。还给出基于FFT的实现，允许在实践中使用频率相关的阻尼参数。数值实验表明SD - TRK在噪声和病态设置下相对于标准TRK的稳定行为，还进行了相同噪声重建管道下的双程图像重建比较。
英文摘要
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.
Comments36 pages, 7 figures
