Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linear-Sized Spectral Sparsifiers and the Kadison-Singer Problem

Phevos Paschalidis, Ashley Zhuang

TL;DR

The paper formalizes how the Kadison-Singer result proved by Marcus, Spielman, and Srivastava implies the Batson–Spielman–Srivastava theorem on linear-sized spectral sparsifiers for graphs. It develops a two-tier strategy: first, for graphs with bounded leverage scores, it derives linear-sized sparsifiers by recursively applying an MSS corollary to partitions of the edge set; second, for general graphs, it introduces a recursive edge-splitting scheme that bounds leverage scores and reweights edges to obtain a sparsifier with $|E|=O(n/\epsilon^2)$ while preserving a spectral approximation within $\epsilon$, i.e., $e^{-\epsilon}L_G \preceq L_H \preceq e^{\epsilon}L_G$. This work both solidifies the intuition that MSS strengthens BSS and provides an accessible construction for linear-sized spectral sparsifiers, albeit with constants that differ from the optimal twice-Ramanujan bound. It also suggests possible extensions to stronger notions of spectral approximation and other graph-operator settings, highlighting the broader impact of MSS on practical sparsification techniques.

Abstract

The Kadison-Singer Conjecture, as proved by Marcus, Spielman, and Srivastava (MSS) [Ann. Math. 182, 327-350 (2015)], has been informally thought of as a strengthening of Batson, Spielman, and Srivastava's theorem that every undirected graph has a linear-sized spectral sparsifier [SICOMP 41, 1704-1721 (2012)]. We formalize this intuition by using a corollary of the MSS result to derive the existence of spectral sparsifiers with a number of edges linear in their number of vertices for all undirected, weighted graphs. The proof consists of two steps. First, following a suggestion of Srivastava [Asia Pac. Math. Newsl. 3, 15-20 (2013)], we show the result in the special case of graphs with bounded leverage scores by repeatedly applying the MSS corollary to partition the graph, while maintaining an appropriate bound on the leverage scores of each subgraph. Then, we extend to the general case by constructing a recursive algorithm that repeatedly (i) divides edges with high leverage scores into multiple parallel edges and (ii) uses the bounded leverage score case to sparsify the resulting graph.

Linear-Sized Spectral Sparsifiers and the Kadison-Singer Problem

TL;DR

The paper formalizes how the Kadison-Singer result proved by Marcus, Spielman, and Srivastava implies the Batson–Spielman–Srivastava theorem on linear-sized spectral sparsifiers for graphs. It develops a two-tier strategy: first, for graphs with bounded leverage scores, it derives linear-sized sparsifiers by recursively applying an MSS corollary to partitions of the edge set; second, for general graphs, it introduces a recursive edge-splitting scheme that bounds leverage scores and reweights edges to obtain a sparsifier with while preserving a spectral approximation within , i.e., . This work both solidifies the intuition that MSS strengthens BSS and provides an accessible construction for linear-sized spectral sparsifiers, albeit with constants that differ from the optimal twice-Ramanujan bound. It also suggests possible extensions to stronger notions of spectral approximation and other graph-operator settings, highlighting the broader impact of MSS on practical sparsification techniques.

Abstract

The Kadison-Singer Conjecture, as proved by Marcus, Spielman, and Srivastava (MSS) [Ann. Math. 182, 327-350 (2015)], has been informally thought of as a strengthening of Batson, Spielman, and Srivastava's theorem that every undirected graph has a linear-sized spectral sparsifier [SICOMP 41, 1704-1721 (2012)]. We formalize this intuition by using a corollary of the MSS result to derive the existence of spectral sparsifiers with a number of edges linear in their number of vertices for all undirected, weighted graphs. The proof consists of two steps. First, following a suggestion of Srivastava [Asia Pac. Math. Newsl. 3, 15-20 (2013)], we show the result in the special case of graphs with bounded leverage scores by repeatedly applying the MSS corollary to partition the graph, while maintaining an appropriate bound on the leverage scores of each subgraph. Then, we extend to the general case by constructing a recursive algorithm that repeatedly (i) divides edges with high leverage scores into multiple parallel edges and (ii) uses the bounded leverage score case to sparsify the resulting graph.
Paper Structure (9 sections, 12 theorems, 7 equations, 1 algorithm)

This paper contains 9 sections, 12 theorems, 7 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Spectral Sparsification.
  3. Connections to the Kadison-Singer Problem.
  4. Our Contribution.
  5. Related Work.
  6. Linear-sized sparsifiers for bounded leverage scores
  7. Extension to general case
  8. Conclusions & Further Directions
  9. Acknowledgements

Key Result

Theorem 1.1

For every $0 < \epsilon < 1$, every undirected, weighted graph $G$ with $n$ vertices contains a reweighted subgraph $H$ with $O(n/\epsilon^2)$ edges such that $H$ is an $\epsilon$-spectral approximation of $G$.

Theorems & Definitions (17)

  • Definition 1.1: The Laplacian
  • Definition 1.2: Spectral approximation
  • Theorem 1.1: Weaker version of Theorem 1.1, batson2012twice
  • Theorem 1.2: Theorem 2, srivastava2013discrepancy. Implied by Corollary 1.5, marcus2015interlacing
  • Definition 1.3: Leverage Score
  • Theorem 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.1
  • Remark 2.1
  • ...and 7 more