On Symmetric Lanczos Quadrature for Stochastic Trace Estimation

A common approach to approximating quadratic forms of matrix functions is to use a quadrature rule derived from the Lanczos process, known as a Lanczos quadrature. Although symmetric quadrature rules are computationally favorable, it has remained unclear whether a symmetric Lanczos quadrature is practically feasible. In this work, we resolve this ambiguity by establishing necessary and sufficient conditions for the existence of symmetric Lanczos quadratures. We show that the sufficient condition can be met for a class of Jordan-Wielandt matrices by carefully constructing initial vectors with specific distributions for the Lanczos algorithm. Applying such a symmetric Lanczos quadrature to compute the Estrada index of bipartite or directed graphs ensures that the resulting stochastic trace estimators are unbiased. Furthermore, we observe that the variance of the quadratic form estimator based on the symmetric Lanczos quadrature is lower than that of the standard estimator.

💡 Research Summary

This paper addresses a long‑standing question in numerical linear algebra: under what circumstances can a Lanczos‑derived quadrature rule be made symmetric, and can such a symmetric Lanczos quadrature be employed effectively in stochastic trace estimation? The authors first formalize the notion of a symmetric Lanczos quadrature as one in which, for every iteration level m, the quadrature nodes (the Ritz values) appear in ±‑pairs around zero and the corresponding weights are identical for each pair. While symmetric quadrature rules are known to reduce error bounds for analytic integrands, prior work had not clarified when the Lanczos process itself would generate such symmetry, especially when the starting vector is drawn at random.

The theoretical contribution begins with a necessary condition (Theorem 3.1). It states that for a symmetric matrix A to produce a symmetric quadrature at all possible Lanczos steps, the spectrum of A must be symmetric about zero (i.e., eigenvalues occur as ±λ pairs) and the initial vector v⁽¹⁾ must have a coordinate representation μ = Qᵀv⁽¹⁾ that is an “r‑partial absolute palindrome”: the absolute values of the components associated with non‑zero eigenvalues must be symmetric. This condition is derived in exact arithmetic and highlights that without such a structure the Lanczos tridiagonal will inevitably break symmetry due to unequal contributions from asymmetric components of μ.

The authors then provide a sufficient condition (Theorem 3.3 and Theorem 3.4) that is constructive and does not require prior knowledge of the matrix rank. They focus on Jordan‑Wielandt matrices of the form
A =