On Symmetric Lanczos Quadrature for Trace Estimation

TL;DR

The paper addresses when Lanczos-based Gauss quadrature yields symmetric nodes and weights for trace estimation, establishing necessary and sufficient conditions linked to spectral symmetry and initial-vector structure. It provides explicit constructions of starting vectors for Jordan-Wielandt matrices to guarantee symmetric Ritz values, enabling unbiased trace estimators for functions such as exp, which underpin Estrada index calculations on bipartite or directed graphs. The work also introduces partial-Rademacher strategies to reduce variance while preserving unbiasedness, and demonstrates practical computational gains through extensive numerical experiments on synthetic and real-world networks. Overall, the results offer a principled path to efficient and reliable trace estimation in large-scale symmetric and Jordan-Wielandt-type settings with direct applications to network analysis and beyond.

Abstract

The Golub-Welsch algorithm computes Gauss quadrature rules with the nodes and weights generated from the symmetric tridiagonal matrix in the Lanczos process. While symmetric Lanczos quadrature (in exact arithmetic) theoretically reduces computational costs, its practical feasibility for trace estimation remains uncertain. This paper resolves this ambiguity by establishing sufficient and necessary conditions for the symmetry of Lanczos quadratrure. For matrices of Jordan-Wielandt type, we provide guidance on selecting initial vectors for the Lanczos algorithm that guarantees symmetric quadrature nodes and weights. More importantly, regarding Estrada index computations in bipartite graphs or directed ones, our method would not only save computational costs, but also ensure the unbiasedness of trace estimators.

Figures (4)

• Figure 1: Plots of discrete measure functions \ref{['eq:measure_f']} and the locations of Ritz values in four cases
• Figure 2: 100 trials of stochastic Lanczos quadrature estimators of $\mathrm{tr}(e^{\beta \textbf{A}})$ with $\beta = 1$, $m=100$, synthetic bipartite matrix $\textbf{A}$ and different initial vectors. $\tilde{\textbf{z}}$ denote upper and lower partial-Rademacher vectors, while all elements of $\textbf{z}$ are Rademacher distributed
• Figure 3: 100 trials of stochastic Lanczos quadrature estimators of $\mathrm{tr}(e^{\beta \textbf{A}})$ with $\beta = 0.5/\lambda_{\max}$, $m=100$, bipartite matrix based on email-Eu-core-temporal data set PBL17DH11 and different initial vectors. $\tilde{\textbf{z}}$ denote upper and lower partial-Rademacher vectors, while all elements of $\textbf{z}$ are Rademacher distributed
• Figure 4: 100 trials of stochastic Lanczos quadrature estimators of $\mathrm{tr}(e^{\beta \textbf{A}})$ with $\beta = 1/\lambda_{\max}$, $m=100$, bipartite matrix based on Notre Dame networks data set PBL17DH11 and different initial vectors. $\tilde{\textbf{z}}$ denote upper (of size $392400$) and lower (of size $127823$) partial-Rademacher vectors, while all elements of $\textbf{z}$ are Rademacher distributed

Theorems & Definitions (16)

• Theorem 2.1
• Theorem 2.2
• Definition 3.1
• Remark 3.2
• Theorem 3.3
• proof
• Lemma 3.4
• proof
• Remark 3.5
• Theorem 3.6