On Solving Asymmetric Diagonally Dominant Linear Systems in Sublinear Time

2025-09-17

AI Summary

This work studies sublinear-time estimation of the inner product $t^ op x^*$, where $x^*$ solves the asymmetric diagonally dominant linear system $Mx = b$. For row- or column-diagonally dominant matrices, we introduce the novel notion of *maximum $p$-norm gap*, the first unified convergence criterion for the asymmetric setting, enabling a common theoretical framework for both forward and backward push methods. Methodologically, we integrate Neumann series expansion, random-walk sampling, local pushing, and bidirectional composition to achieve efficient local approximation. Under bounded $p$-norm gap, we obtain sublinear-time algorithms across multiple regimes, substantially improving prior complexity bounds. Moreover, we establish the first rigorous lower bound for this inner-product estimation problem, closing a fundamental gap in the literature.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We initiate a study of solving a row/column diagonally dominant (RDD/CDD) linear system $Mx=b$ in sublinear time, with the goal of estimating $t^{ op}x^*$ for a given vector $tin R^n$ and a specific solution $x^*$. This setting naturally generalizes the study of sublinear-time solvers for symmetric diagonally dominant (SDD) systems [AKP19] to the asymmetric case. Our first contributions are characterizations of the problem's mathematical structure. We express a solution $x^*$ via a Neumann series, prove its convergence, and upper bound the truncation error on this series through a novel quantity of $M$, termed the maximum $p$-norm gap. This quantity generalizes the spectral gap of symmetric matrices and captures how the structure of $M$ governs the problem's computational difficulty. For systems with bounded maximum $p$-norm gap, we develop a collection of algorithmic results for locally approximating $t^{ op}x^*$ under various scenarios and error measures. We derive these results by adapting the techniques of random-walk sampling, local push, and their bidirectional combination, which have proved powerful for special cases of solving RDD/CDD systems, particularly estimating PageRank and effective resistance on graphs. Our general framework yields deeper insights, extended results, and improved complexity bounds for these problems. Notably, our perspective provides a unified understanding of Forward Push and Backward Push, two fundamental approaches for estimating random-walk probabilities on graphs. Our framework also inherits the hardness results for sublinear-time SDD solvers and local PageRank computation, establishing lower bounds on the maximum $p$-norm gap or the accuracy parameter. We hope that our work opens the door for further study into sublinear solvers, local graph algorithms, and directed spectral graph theory.

Problem

• Solving asymmetric diagonally dominant linear systems sublinearly
• Estimating linear functionals of specific solutions efficiently
• Generalizing symmetric solvers to asymmetric cases with new framework

Innovation

• Neumann series solution with truncation error bounds
• Random-walk sampling and local push techniques adaptation
• Bidirectional combination framework unifying Forward and Backward Push