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Randomized Kaczmarz solver for noisy linear systems
Deanna Needell
TL;DR
The paper addresses the limited convergence theory for Kaczmarz methods when linear systems are noisy. It analyzes randomized row projections and proves that the noisy method converges exponentially to a matrix-dependent error threshold, while examples establish sharpness in the general setting.
Problem
The paper addresses the lack of readily obtainable, comparable theoretical convergence rates for Kaczmarz methods, especially when systems are corrupted by noise.
Method
The paper analyzes a randomized Kaczmarz method that iteratively projects onto equation hyperplanes while selecting rows with probabilities proportional to their Euclidean norms.
Results
O(n) iterations yield exponential convergence, and the noisy method reaches a matrix-dependent error threshold at the same rate as in the error-free case.
Takeaways & Limitations
The method is most relevant to very large systems, where its O(n^2) runtime can improve on Gaussian elimination's O(mn^2) runtime.
Takeaways & Limitations
The row-selection strategy is not optimal, and the general noisy-case error bound is shown to be sharp.
Abstract
from arXiv · showhide
The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax=b. Theoretical convergence rates for this algorithm were largely unknown until recently when work was done on a randomized version of the algorithm. It was proved that for overdetermined systems, the randomized Kaczmarz method converges with expected exponential rate, independent of the number of equations in the system. Here we analyze the case where the system Ax=b is corrupted by noise, so we consider the system where Ax is approximately b + r where r is an arbitrary error vector. We prove that in this noisy version, the randomized method reaches an error threshold dependent on the matrix A with the same rate as in the error-free case. We provide examples showing our results are sharp in the general context.
1. Introduction
The paper introduces randomized Kaczmarz as an iterative projection method with provable expected exponential convergence, then motivates its noisy extension and summarizes its computational trade-offs and sharpness.
- Kaczmarz method: The Kaczmarz method solves overdetermined linear systems through alternating orthogonal projections onto the equations' solution hyperplanes.Each iteration projects the current estimate onto one equation's solution space.
- Randomized Kaczmarz: Randomized Kaczmarz selects rows randomly with probabilities proportional to their Euclidean norms, replacing sequential cycling.The row-norm probabilities are computationally efficient but require knowledge of the row norms.
- Convergence theory: The randomized method has an expected exponential convergence rate independent of the number m of equations, under the stated full-column-rank assumption.The analysis uses a mixed condition quantity R associated with the matrix and the algorithm's sampling probabilities.
- Computational implications: O(n) iterations and O(n^2) total runtime follow from the exponential convergence bound, making the method attractive for very large systems.For comparison, Gaussian elimination is stated to require O(mn^2), while randomized Kaczmarz can access only randomly selected rows.
- Limitations and trade-offs: The selected row-sampling strategy is not optimal, although its proven convergence rates are optimal under that strategy and it often performs well computationally.The paper frames this as a balance between computational efficiency and provable guarantees.
- Noisy systems: The paper studies noise because prior exponential-rate results addressed the error-free setting; it reports convergence to a specified noise-dependent error bound.It also provides a simple example showing that the bound is sharp in the general setting.
2. Main Results
The paper analyzes randomized Kaczmarz for noisy systems, proving exponential convergence to an error threshold and showing that this bound is sharp in a general setting.
- Problem setting: The noisy method applies randomized Kaczmarz to systems Ax ≈ b + r with an arbitrary error vector r.The analysis starts from a consistent system Ax = b and adds noise to the right-hand side.
- Sharpness: The identity-matrix example with r = (1, 1, . . . , 1) shows that the general noisy error bound is sharp when γ = 1.The noisy solution is x = r, whereas the unperturbed solution is x = 0.
- Error threshold: The limiting error between the iterates and the original solution is governed by the theorem’s noise-dependent bound.The paper compares this threshold with perturbation-theory bounds and notes that the bounds are similar in spirit.
- Main theorem: Theorem 2.1 establishes an expected error bound for the kth iterate of noisy randomized Kaczmarz under full column rank and consistency of Ax = b.The expectation is taken over the algorithm’s random row choices.
- Geometric interpretation: Noise translates each original solution hyperplane in the direction of its row normal, producing parallel noisy hyperplanes.The shifted noisy spaces are described as affine subspaces generated from the unperturbed spaces by row-normal translations.
- Scope: The geometric lemma does not require the perturbed system to be consistent, because each noisy solution space may be nonempty without their total intersection being required.This distinguishes individual shifted hyperplanes from consistency of the full noisy system.
3. Numerical Examples
Numerical studies compare randomized Kaczmarz errors with theoretically predicted thresholds across Gaussian, partial Fourier, and Bernoulli matrices. The observed errors converge exponentially and remain close to the predicted thresholds.
- 100 trials on 2000 × 100 Gaussian matrices with independent Gaussian noise of norm 0.02 showed exponential convergence toward the predicted threshold.The systems were homogeneous, with x = 0 and b = 0.
- The experiments also used partial Fourier matrices with m = 700 and n = 101, and matrices with Bernoulli 0/1 entries.The partial Fourier construction used nonuniformly sampled values relevant to signal-processing applications.
- Across all matrix types, practical errors were quite close to the theoretical threshold predicted for each trial.Figure 2 compares actual error as a thin line with predicted threshold as a thick line.
- The mean values of R were 163.2, 428.6, and 162.4 in the displayed experiments.