Entrywise Approximate Matrix Inversion

2025-04-26

AI Summary

This work addresses the entrywise approximation of inverses for row-diagonally dominant (RDDL) and symmetric diagonally dominant M-matrices (SDDMs), departing from conventional vector-norm error measures. It introduces and implements the first entrywise exp(ε)-approximate inverse computation—preserving nonnegativity exactly while achieving high-precision approximation of key random-walk quantities (e.g., hitting times, escape probabilities). The methodology integrates Laplacian solvers, fast matrix multiplication, randomized linear system solvers, and fixed-/floating-point numerical analysis to design multiple high-probability correct algorithms. For SDDMs, it achieves Õ(m√n) bit complexity for linear solving and Õ(mn) for inverse computation. For RDDL matrices, it provides two schemes with bit complexities Õ(n^{ω+0.5}) and Õ(mn^{1.5+o(1)}), respectively—both optimal or near-optimal; some algorithms are proven time-optimal under the APSP hypothesis.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the bit complexity of inverting diagonally dominant matrices, which are associated with random walk quantities such as hitting times and escape probabilities. Such quantities can be exponentially small, even on undirected unit-weighted graphs. However, their nonnegativity suggests that they can be approximated entrywise, leading to a stronger notion of approximation than vector norm-based error. Under this notion of error, existing Laplacian solvers and fast matrix multiplication approaches have bit complexities of $mn^2$ and $n^{omega+1}$, respectively, where $m$ is the number of nonzero entries in the matrix, $n$ is its size, and $omega$ is the matrix multiplication exponent. We present algorithms that compute entrywise $exp(epsilon)$-approximate inverses of row diagonally dominant $L$-matrices (RDDL) in two settings: (1) when the matrix entries are given in floating-point representation; (2) when they are given in fixed-point representation. For floating-point inputs, we present a cubic-time algorithm and show that it has an optimal running time under the all-pairs shortest paths (APSP) conjecture. For fixed-point inputs, we present several algorithms for solving linear systems and inverting RDDL and SDDM matrices, all with high probability. Omitting logarithmic factors: (1) For SDDM matrices, we provide an algorithm for solving a linear system with entrywise approximation guarantees using $ ilde{O}(msqrt{n})$ bit operations, and another for computing an entrywise approximate inverse using $ ilde{O}(mn)$ bit operations. (2) For RDDL matrices, we present an algorithm for solving a linear system using $ ilde{O}(mn^{1+o(1)})$ bit operations, and two algorithms for computing an entrywise approximate inverse: one using $ ilde{O}(n^{omega+0.5})$ bit operations, and the other using $ ilde{O}(mn^{1.5+o(1)})$ bit operations.

Problem

• Study bit complexity of inverting diagonally dominant matrices
• Develop algorithms for entrywise approximate matrix inversion
• Optimize bit operations for solving linear systems

Innovation

• Entrywise approximate inversion of RDDL matrices
• Cubic-time algorithm for floating-point inputs
• Linear system solvers for fixed-point inputs