Entrywise Approximation for Matrix Inversion and Linear Systems

TL;DR

This work develops and analyzes entrywise, multiplicative (ε) approximations to matrix inverses for diagonally dominant classes (RDDL and SDDM), motivated by probabilities and hitting times in random walks. It introduces two algorithmic frameworks—Threshold Decay (for fixed-point inputs) and Shortcutter (for subcubic inversions)—and provides concrete bit-operation bounds for both solving linear systems and inverting matrices in floating-point and fixed-point regimes. For floating-point inputs, it proves a cubic-time algorithm for RDDL inversion and shows APSP-based hardness implying no subcubic- time improvement under the APSP conjecture; for fixed-point inputs, it delivers near-linear to subcubic bit-complexities (e.g., $ ilde{O}(m n^{0.5})$ to invert SDDM, $ ilde{O}(m n^{1+o(1)})$ for RDDL, and variants). The paper also connects these algebraic procedures to graph-theoretic quantities such as all-pairs escape probabilities and hitting times, and establishes reductions from APSP that highlight the optimality of the proposed cubic-time approach in the floating-point setting. Overall, the results offer new entrywise approximation tools with strong theoretical guarantees and practical impact for graph-based computations where tiny probabilities or entries must be recovered accurately.

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 $n^{ω+1}$, respectively, where $m$ is the number of nonzero entries in the matrix, $n$ is its size, and $ω$ is the matrix multiplication exponent. We present algorithms that compute entrywise $\exp(ε)$-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 $\tilde{O}(m\sqrt{n})$ bit operations, and another for computing an entrywise approximate inverse using $\tilde{O}(mn)$ bit operations. (2) For RDDL matrices, we present an algorithm for solving a linear system using $\tilde{O}(mn^{1+o(1)})$ bit operations, and two algorithms for computing an entrywise approximate inverse: one using $\tilde{O}(n^{ω+0.5})$ bit operations, and the other using $\tilde{O}(mn^{1.5+o(1)})$ bit operations.

Figures (9)

• Figure 1: Exponentially small escape probability.
• Figure 2: Exponentially large hitting time.
• Figure 3: The Threshold Decay framework
• Figure 4: The entrywise approximate linear system solver for RDDL matrices
• Figure 5: The entrywise approximate linear system solver for SDDM matrices given a prediciton

Theorems & Definitions (97)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Example 1.1
• Definition 2.1: RDDL and SDDM Matrices
• Definition 2.2: Associated Graph of an RDDL Matrix