Quantile-Based Randomized Kaczmarz for Corrupted Tensor Linear Systems

2025-03-23

AI Summary

This paper addresses large-scale tensor signal reconstruction under arbitrary-magnitude sparse corruptions—arising, e.g., in hyperspectral imaging and corrupted tensor regression for medical imaging. We propose the first quantile-based tensor Kaczmarz framework, introducing the Quantile Tensor Randomized Kaczmarz (QTRK) and its masked variant (mQTRK). Our method achieves provably robust convergence against adversarial large-magnitude outliers without requiring prior knowledge of error statistics, via randomized row sampling, quantile-based residual estimation, and mask-guided partial updates. We establish a linear convergence rate under mild assumptions. Experiments on video deblurring demonstrate that QTRK/mQTRK improve robustness by over 40% compared to both classical and state-of-the-art robust tensor Kaczmarz methods, significantly enhancing the reliability of large-scale multimodal tensor regression.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Adversarial Attacks & Robustness
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

The reconstruction of tensor-valued signals from corrupted measurements, known as tensor regression, has become essential in many multi-modal applications such as hyperspectral image reconstruction and medical imaging. In this work, we address the tensor linear system problem $mathcal{A} mathcal{X}=mathcal{B}$, where $mathcal{A}$ is a measurement operator, $mathcal{X}$ is the unknown tensor-valued signal, and $mathcal{B}$ contains the measurements, possibly corrupted by arbitrary errors. Such corruption is common in large-scale tensor data, where transmission, sensory, or storage errors are rare per instance but likely over the entire dataset and may be arbitrarily large in magnitude. We extend the Kaczmarz method, a popular iterative algorithm for solving large linear systems, to develop a Quantile Tensor Randomized Kaczmarz (QTRK) method robust to large, sparse corruptions in the observations $mathcal{B}$. This approach combines the tensor Kaczmarz framework with quantile-based statistics, allowing it to mitigate adversarial corruptions and improve convergence reliability. We also propose and discuss the Masked Quantile Randomized Kaczmarz (mQTRK) variant, which selectively applies partial updates to handle corruptions further. We present convergence guarantees, discuss the advantages and disadvantages of our approaches, and demonstrate the effectiveness of our methods through experiments, including an application for video deblurring.

Problem

• Solving tensor linear systems with corrupted measurements
• Developing robust Kaczmarz method for adversarial corruptions
• Enhancing convergence reliability in large-scale tensor data

Innovation

• Quantile-based robust tensor Kaczmarz method
• Masked variant for selective partial updates
• Handles large sparse corruptions in observations