Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral Sparsification by Deterministic Discrepancy Walk

Lap Chi Lau, Robert Wang, Hong Zhou

TL;DR

This work presents a deterministic matrix discrepancy walk that unifies spectral sparsification and discrepancy minimization. By introducing a regularized potential $\Phi(x)$ and carefully constrained updates in large subspaces, the authors obtain a deterministic matrix partial coloring theorem and translate it into a suite of linear-sized, deterministic sparsification constructions, including unit-circle and singular-value notions, as well as graphical spectral sketches and effective-resistance sparsifiers. The approach avoids probabilistic tools and convex-geometry arguments of prior work, offering a streamlined, self-contained analysis that handles linear-subspace constraints such as degree preservation. The resulting framework both simplifies existing results and yields new, near-optimal deterministic constructions across directed and undirected graphs with broad applicability to spectral queries and Laplacian-type computations.

Abstract

Spectral sparsification and discrepancy minimization are two well-studied areas that are closely related. Building on recent connections between these two areas, we generalize the "deterministic discrepancy walk" framework by Pesenti and Vladu [SODA~23] for vector discrepancy to matrix discrepancy, and use it to give a simpler proof of the matrix partial coloring theorem of Reis and Rothvoss [SODA~20]. Moreover, we show that this matrix discrepancy framework provides a unified approach for various spectral sparsification problems, from stronger notions including unit-circle approximation and singular-value approximation to weaker notions including graphical spectral sketching and effective resistance sparsification. In all of these applications, our framework produces improved results with a simpler and deterministic analysis.

Spectral Sparsification by Deterministic Discrepancy Walk

TL;DR

This work presents a deterministic matrix discrepancy walk that unifies spectral sparsification and discrepancy minimization. By introducing a regularized potential and carefully constrained updates in large subspaces, the authors obtain a deterministic matrix partial coloring theorem and translate it into a suite of linear-sized, deterministic sparsification constructions, including unit-circle and singular-value notions, as well as graphical spectral sketches and effective-resistance sparsifiers. The approach avoids probabilistic tools and convex-geometry arguments of prior work, offering a streamlined, self-contained analysis that handles linear-subspace constraints such as degree preservation. The resulting framework both simplifies existing results and yields new, near-optimal deterministic constructions across directed and undirected graphs with broad applicability to spectral queries and Laplacian-type computations.

Abstract

Spectral sparsification and discrepancy minimization are two well-studied areas that are closely related. Building on recent connections between these two areas, we generalize the "deterministic discrepancy walk" framework by Pesenti and Vladu [SODA~23] for vector discrepancy to matrix discrepancy, and use it to give a simpler proof of the matrix partial coloring theorem of Reis and Rothvoss [SODA~20]. Moreover, we show that this matrix discrepancy framework provides a unified approach for various spectral sparsification problems, from stronger notions including unit-circle approximation and singular-value approximation to weaker notions including graphical spectral sketching and effective resistance sparsification. In all of these applications, our framework produces improved results with a simpler and deterministic analysis.
Paper Structure (20 sections, 23 theorems, 62 equations)

This paper contains 20 sections, 23 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. Technical Overview
  4. Background
  5. Linear Algebra
  6. Spectral Graph Theory
  7. Spectral Sparsification for Directed Graphs
  8. Deterministic Matrix Partial Coloring
  9. The Deterministic Discrepancy Walk Framework
  10. Potential Function and Maximum Eigenvalue Bound
  11. Restricted Subspaces
  12. Spectral Argument
  13. From Matrix Partial Coloring to Spectral Sparsification
  14. Spectral Sparsification with Linear Subspace Constraints
  15. Unit-Circle Sparsification for Undirected Graphs
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $A_1, \dots, A_m \in \mathbb{R}^{n \times n}$ be symmetric matrices such that $\sum_{i=1}^m |A_i| \preccurlyeq I_d$. Let $\mathcal{C} \subseteq \mathbb{R}^m$ be a set of good partial fractional colorings defined as In addition, let $\mathcal{H} \subseteq \mathbb{R}^m$ be a linear subspace of dimension $c \cdot m$ for some constant $c \geq \frac{4}{5}$. There is a deterministic polynomial time

Theorems & Definitions (36)

  • Theorem 1.1: Deterministic Matrix Partial Coloring
  • Theorem 1.2: Deterministic Matrix Sparsification
  • Theorem 1.3: Linear-Sized UC-Sparsifiers of Undirected Graphs
  • Theorem 1.4: Improved SV-Sparsifiers of Directed Graphs
  • Theorem 1.5: Improved Graphical Spectral Sketching
  • Theorem 1.6: Improved Effective Resistance Sparsification
  • Lemma 2.1
  • Theorem 2.2: Cauchy Interlacing Theorem
  • Lemma 2.3: RR20
  • Lemma 2.4: RR20
  • ...and 26 more