[Paper Review] Spectral computations for birth and death chains

Guan‐Yu Chen, Laurent Saloff‐Coste|arXiv (Cornell University)|May 2, 2013

Quantum chaos and dynamical systems9 references4 citations

TL;DR

This paper presents an iterative algorithm for computing the spectrum of birth and death chains on a path, with exponential convergence to the spectral gap. It introduces two schemes: one based on inverse iteration with monotonicity constraints, and another using a dichotomy method that guarantees convergence at a rate of log 2, enabling full spectral computation in O(n²) operations with high accuracy.

ABSTRACT

We consider the spectrum of birth and death chains on a $n$-path. An iterative scheme is proposed to compute any eigenvalue with exponential convergence rate independent of $n$. This allows one to determine the whole spectrum in order $n^2$ elementary operations. Using the same idea, we also provide a lower bound on the spectral gap, which is of the correct order on some classes of examples.

Motivation & Objective

• To develop a numerically efficient method for computing the full spectrum of birth and death chains on a finite path.
• To provide a fast and reliable algorithm for computing the spectral gap, a key quantity in the analysis of Markov chains.
• To establish theoretical convergence guarantees, including exponential rates independent of the path length n.
• To derive a lower bound on the spectral gap that matches the correct order of magnitude in known examples.

Proposed method

• An iterative scheme (A1) is proposed that uses inverse iteration with a monotonicity constraint derived from Miclo’s theorem on minimizers of the spectral gap.
• The algorithm initializes a function ψ₀ with a fixed value and iteratively updates it using a recurrence involving the Dirichlet form and eigenvalue estimate.
• A second scheme (A2) employs a bisection-like method on the eigenvalue parameter, updating bounds Lₖ and Uₖ based on the sign of the normalized mean of the iterated function.
• The method leverages the fact that all non-trivial eigenvectors of the generator matrix are orthogonal to the stationary measure π, ensuring mean-zero solutions.
• Theoretical convergence is established via analysis of the recurrence structure and the behavior of the Rayleigh quotient under iteration.
• The dichotomy method (A2) ensures that the error in the eigenvalue estimate halves at each step, achieving a convergence rate of log 2.

Experimental results

Research questions

• RQ1 Can the full spectrum of a birth and death chain on a path be computed efficiently, in O(n²) operations?
• RQ2 Does the iterative scheme converge to the true spectral gap with a rate independent of n?
• RQ3 Can a lower bound on the spectral gap be derived that matches the correct asymptotic order in known examples?
• RQ4 Is it possible to design a numerical scheme that guarantees exponential convergence with a known rate?
• RQ5 How do the spectral properties of birth and death chains behave under scaling limits, such as when the chain size grows?

Key findings

• The iterative scheme (A1) converges to the spectral gap with exponential rate, and for n ≥ 3, the sequence λₖ is strictly decreasing and bounded below by the true spectral gap.
• The dichotomy-based scheme (A2) achieves a provably exponential convergence rate of log 2, with error bounded by (U₀ - L₀)2⁻ᵏ for all k ≥ 0.
• For n = 2, the algorithm converges in one step to the exact spectral gap, regardless of initial parameters.
• The method allows full spectral computation in O(n²) elementary operations, making it practical for large n.
• The spectral gap is bounded below by a quantity matching the correct order of magnitude in examples such as the symmetric random walk on a path.
• In the limit of large n, the paper derives asymptotic expressions for the spectral gap under various scaling regimes, including cases where the birth/death rates scale with n² or slower.