[Paper Review] Average polynomial time for eigenvector computations

Diego Armentano, Carlos Beltrán+1 more|arXiv (Cornell University)|Oct 8, 2014

Matrix Theory and Algorithms12 references3 citations

TL;DR

This paper presents two algorithms—deterministic and randomized—for computing eigenvalues and eigenvectors of complex Gaussian matrices, using homotopy continuation with Newton-type steps. Both algorithms terminate with probability 1 and run in average polynomial time, specifically O(n⁹), with a key contribution being the analysis of the expected squared condition number for the eigenproblem, which underpins the complexity bounds.

ABSTRACT

We describe two algorithms for the eigenvalue, eigenvector problem which, on input a Gaussian matrix with complex entries, finish with probability 1 and in average polynomial time.

Motivation & Objective

• To design algorithms that compute eigenpairs of complex matrices with guaranteed termination and average polynomial-time complexity.
• To address the challenge of eigenvector computation under high condition numbers, especially near multiple eigenvalues.
• To establish theoretical bounds on the expected squared condition number for the eigenvalue-eigenvector problem.
• To demonstrate that homotopy-based methods can achieve average polynomial cost, unlike standard numerical algorithms such as QR with shifts.
• To provide a framework for analyzing average-case complexity in eigenvalue problems using probabilistic and geometric techniques.

Proposed method

• Uses a homotopy continuation method based on Newton’s method, tracking paths from a known solution to the target matrix.
• Employs a randomized algorithm that computes one eigenpair at a time, with a step size controlled by the condition number μ(A,λ,v).
• Applies a variant of Newton’s method with adaptive precision control, ensuring robustness to round-off and input errors.
• Introduces a geometric framework using the projective space P(C^n) and the metric d_P^2 to measure distance between eigenpairs.
• Uses the condition number μ(A,λ,v) = ||((I - vv*)(λI - A))†|| as a central complexity parameter, bounding path length and step size.
• Applies concentration and perturbation bounds (e.g., Lemma 33 and Proposition 34) to control the variation of μ along paths, ensuring stability.

Experimental results

Research questions

• RQ1 Can a deterministic algorithm compute all n eigenpairs of a complex matrix in average polynomial time with probability 1 of termination?
• RQ2 Can a randomized algorithm compute a single eigenpair in average polynomial time while maintaining robustness to numerical errors?
• RQ3 What is the expected value of the squared condition number μ² for the eigenvalue-eigenvector problem under the complex Gaussian distribution?
• RQ4 How does the condition number μ(A,λ,v) vary along homotopy paths, and can this variation be bounded to ensure polynomial complexity?
• RQ5 Can the average-case complexity of homotopy-based eigensolvers be rigorously analyzed and bounded, especially compared to standard numerical linear algebra methods?

Key findings

• The average running time of the deterministic algorithm is O(n⁹), and the same bound holds for the randomized algorithm, both terminating with probability 1.
• The expected value of the squared condition number μ² for the eigenproblem is bounded by a universal constant independent of n, which is crucial for the average polynomial cost.
• The randomized algorithm’s complexity could be improved to O(n⁶) if a certain open question about the growth of μ along paths is resolved.
• The algorithms are robust to input and round-off errors, with precision requirements that scale near degeneracy but do not destroy the average polynomial nature of the cost.
• The variation of μ along homotopy paths is controlled via metric bounds in projective space, ensuring that step sizes remain stable and the number of steps remains polynomial on average.
• The analysis confirms that standard numerical algorithms like unshifted QR or Rayleigh quotient shifted QR do not satisfy the same termination and complexity guarantees, especially under average-case analysis.