Convergence Analysis of Quantile Randomized Kaczmarz Method for Linear Systems with Time-Varying Noise and Corruption

Core Concepts

The Quantile Randomized Kaczmarz (QRK) method converges at least linearly in expectation up to a convergence horizon even when the linear system is perturbed by time-varying noise and corruption. The rate of convergence depends only on the corruption rate, while the convergence horizon depends on both the corruption rate and the time-varying noise.

Abstract

The paper considers solving highly overdetermined linear systems Ax = b that are perturbed by both time-varying noise n(k) and corruption c(k) in the measurement vector b(k) = b + n(k) + c(k). The authors analyze the convergence of the Quantile Randomized Kaczmarz (QRK) method in this setting.
Key highlights:

QRK is a variant of the Randomized Kaczmarz method that avoids updates corresponding to highly corrupted data by only updating if the sampled residual magnitude is less than a sufficient fraction of the residual magnitudes.
The authors prove that QRK converges at least linearly in expectation up to a convergence horizon, even in the presence of time-varying noise and corruption.
The rate of convergence depends only on parameters determined by the time-varying corruption, in particular the corruption rate.
The convergence horizon depends on both the corruption rate and the time-varying noise in the system.
The authors also provide a lower bound on the probability that the indices of the corrupted equations can be identified by examining the largest entries of the residual.
Numerical experiments are presented to illustrate the theoretical results.

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Key Insights Distilled From

On Quantile Randomized Kaczmarz for Linear Systems with Time-Varying Noise and Corruption

by Nestor Coria... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2403.19874.pdf

Deeper Inquiries

How would the convergence of QRK be affected if the corruption rate β is allowed to vary over time

In the context of the convergence analysis of QRK, allowing the corruption rate β to vary over time would introduce additional complexity to the method. The convergence of QRK could be affected in several ways:

Changing Convergence Rates: As the corruption rate β varies over time, the convergence rate of QRK may fluctuate. Higher corruption rates could lead to slower convergence, while lower corruption rates may allow for faster convergence.
Impact on Convergence Horizon: The convergence horizon of QRK, which determines how quickly the method converges to a solution, may also be influenced by the varying corruption rate. Higher corruption rates could extend the convergence horizon, requiring more iterations to reach a satisfactory solution.
Adaptation Challenges: Adapting the QRK algorithm to handle time-varying corruption rates would require dynamic adjustments to the quantile thresholds and sampling strategies to effectively identify and mitigate the impact of corrupted data points.

Can the QRK method be extended to handle more general types of time-varying perturbations beyond just noise and corruption

The QRK method can potentially be extended to handle more general types of time-varying perturbations beyond noise and corruption by incorporating adaptive mechanisms and robust estimation techniques. Some possible extensions could include:

Dynamic Quantile Thresholds: Adapting the quantile thresholds based on the characteristics of the time-varying perturbations to effectively identify and filter out corrupted data points.
Robust Estimation: Integrating robust estimation methods into QRK to enhance the method's resilience to various types of perturbations, such as outliers, drifts, or systematic errors in the data.
Online Learning: Implementing online learning strategies to continuously update the QRK algorithm based on real-time feedback and evolving data patterns, enabling it to adapt to changing perturbations over time.

What are the implications of the QRK method for practical applications involving large-scale linear systems with unreliable data, such as in machine learning, medical imaging, or sensor networks

The implications of the QRK method for practical applications involving large-scale linear systems with unreliable data are significant:

Robustness to Corruption: QRK's ability to converge even in the presence of time-varying noise and corruption makes it well-suited for real-world scenarios where data quality may fluctuate.
Improved Data Quality: By effectively filtering out corrupted data points during the iterative process, QRK can enhance the overall quality and reliability of the solutions obtained from large-scale linear systems.
Efficient Computation: The low-memory footprint and attractive theoretical guarantees of QRK make it a computationally efficient choice for handling large-scale data problems with unreliable measurements, such as those encountered in machine learning, medical imaging, and sensor networks.