Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Faster Spectral Density Estimation and Sparsification in the Nuclear Norm

Yujia Jin, Ishani Karmarkar, Christopher Musco, Aaron Sidford, Apoorv Vikram Singh

TL;DR

This work introduces additive nuclear sparsification, a weaker yet powerful form of graph sparsification that preserves spectral information of the normalized adjacency matrix via the nuclear norm. It shows how to deterministically compute $\varepsilon$-additive nuclear sparsifiers with $O(n\varepsilon^{-2})$ nonzeros in sublinear time, enabling both faster randomized and first sublinear-time deterministic spectral density estimation (SDE). By leveraging a two-stage approach—sparsify then apply moment/matrix-trace methods to recover the spectrum—the authors achieve an $O(n\varepsilon^{-3})$ randomized SDE in the adjacency-query model and a deterministic variant with running time near-linear in $n$. They establish near-optimal sparsity and query lower bounds, and they separate nucleus-based sparsification from additive spectral sparsification, including results in a weaker one-step random-walk query model. The framework yields the first deterministic sublinear-time SDE and broadens sublinear spectral techniques to weighted graphs and graphical sparsifiers, with implications for scalable spectral analysis of large networks.

Abstract

We consider the problem of estimating the spectral density of the normalized adjacency matrix of an $n$-node undirected graph. We provide a randomized algorithm that, with $O(nε^{-2})$ queries to a degree and neighbor oracle and in $O(nε^{-3})$ time, estimates the spectrum up to $ε$ accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an $O(nε^{-7})$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $ε$, a $2^{O(ε^{-1})}$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call nuclear sparsification. We provide an $O(nε^{-2})$-query and $O(nε^{-2})$-time algorithm for computing $O(nε^{-2})$-sparse nuclear sparsifiers. We show that this bound is optimal in both its sparsity and query complexity, and we separate our results from the related notion of additive spectral sparsification. Of independent interest, we show that our sparsification method also yields the first deterministic algorithm for spectral density estimation that scales linearly with $n$ (sublinear in the representation size of the graph).

Faster Spectral Density Estimation and Sparsification in the Nuclear Norm

TL;DR

This work introduces additive nuclear sparsification, a weaker yet powerful form of graph sparsification that preserves spectral information of the normalized adjacency matrix via the nuclear norm. It shows how to deterministically compute -additive nuclear sparsifiers with nonzeros in sublinear time, enabling both faster randomized and first sublinear-time deterministic spectral density estimation (SDE). By leveraging a two-stage approach—sparsify then apply moment/matrix-trace methods to recover the spectrum—the authors achieve an randomized SDE in the adjacency-query model and a deterministic variant with running time near-linear in . They establish near-optimal sparsity and query lower bounds, and they separate nucleus-based sparsification from additive spectral sparsification, including results in a weaker one-step random-walk query model. The framework yields the first deterministic sublinear-time SDE and broadens sublinear spectral techniques to weighted graphs and graphical sparsifiers, with implications for scalable spectral analysis of large networks.

Abstract

We consider the problem of estimating the spectral density of the normalized adjacency matrix of an -node undirected graph. We provide a randomized algorithm that, with queries to a degree and neighbor oracle and in time, estimates the spectrum up to accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small , a time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call nuclear sparsification. We provide an -query and -time algorithm for computing -sparse nuclear sparsifiers. We show that this bound is optimal in both its sparsity and query complexity, and we separate our results from the related notion of additive spectral sparsification. Of independent interest, we show that our sparsification method also yields the first deterministic algorithm for spectral density estimation that scales linearly with (sublinear in the representation size of the graph).
Paper Structure (25 sections, 28 theorems, 52 equations, 1 figure)

This paper contains 25 sections, 28 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Sparsification Approach to Spectral Density Estimation
  3. Comparison to Prior Work on Sparsification
  4. Applications to Sublinear Time Spectral Density Estimation
  5. Lower Bounds for Nuclear Sparsification
  6. Random Walk Query Model
  7. Paper Organization and Preliminaries
  8. Paper Organization.
  9. Graph Notation.
  10. Vector and Matrix Notation.
  11. Other Notation.
  12. Preliminary Facts.
  13. Additive Nuclear Sparsifiers
  14. Applications to Spectral Density Estimation
  15. Lower Bounds for Nuclear Sparsification
  16. ...and 10 more sections

Key Result

Theorem 1.4

There is a deterministic method (Algorithm alg:nuclear) that, for any $\varepsilon\in(0,1)$, returns a $2n \varepsilon^{-2}$-sparse $\varepsilon$-additive nuclear sparsifier for any undirected weighted graph $G$, and runs in $O(n \varepsilon^{-2})$ time in the adjacency query model (def:query_model)

Figures (1)

  • Figure 1: Illustration of construction of $\tilde{E}, G_1$ and $G_2$ in the proof of Theorem \ref{['lemma:spectral_deterministic_lb']}.

Theorems & Definitions (53)

  • Definition 1.2: Additive Nuclear Approximation and Sparsification
  • Definition 1.3: Adjacency query model
  • Theorem 1.4: Sublinear Time Nuclear Sparsification
  • Definition 1.5: Spectral Sparsifier, spielman2011spectral
  • Definition 1.6: Additive Spectral Sparsifier
  • Theorem 1.7
  • Theorem 1.8: Randomized Sublinear Time SDE
  • Theorem 1.9: Deterministic Sublinear Time SDE
  • Theorem 1.10: Sparsity Lower Bound
  • Theorem 1.11: Query Lower Bound
  • ...and 43 more