Sparsifying Sums of Positive Semidefinite Matrices

2025-08-11

AI Summary

This paper studies spectral sparsification of sums of positive semidefinite (PSD) matrices: constructing a sparse weighted sum that spectrally approximates the original sum in the operator norm—unifying graph spectral sparsification and Cayley graph generator sparsification. We introduce a novel parameter, the *connectivity threshold* $N^*(A)$, which breaks the classical $O(n)$ barrier and yields the first nontrivial upper bound for Cayley graph sparsification over general groups. Leveraging this parameter, we design a randomized polynomial-time algorithm based on importance sampling to achieve efficient spectral approximation. Our method constructs a spectral sparsifier using only $O(varepsilon^{-2} N^*(A) log n log r)$ matrices, and we prove that $N^*(A)$ is a tight lower bound. Applied to Cayley graphs, our result reduces the number of generators to $O(varepsilon^{-2} log^4 N)$, significantly improving prior bounds.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Computational social science

Abstract

In this paper, we revisit spectral sparsification for sums of arbitrary positive semidefinite (PSD) matrices. Concretely, for any collection of PSD matrices $mathcal{A} = {A_1, A_2, ldots, A_r} subset mathbb{R}^{n imes n}$, given any subset $T subseteq [r]$, our goal is to find sparse weights $μin mathbb{R}_{geq 0}^r$ such that $(1 - ε) sum_{i in T} A_i preceq sum_{i in T} μ_i A_i preceq (1 + ε) sum_{i in T} A_i.$ This generalizes spectral sparsification of graphs which corresponds to $mathcal{A}$ being the set of Laplacians of edges. It also captures sparsifying Cayley graphs by choosing a subset of generators. The former has been extensively studied with optimal sparsifiers known. The latter has received attention recently and was solved for a few special groups (e.g., $mathbb{F}_2^n$). Prior work shows any sum of PSD matrices can be sparsified down to $O(n)$ elements. This bound however turns out to be too coarse and in particular yields no non-trivial bound for building Cayley sparsifiers for Cayley graphs. In this work, we develop a new, instance-specific (i.e., specific to a given collection $mathcal{A}$) theory of PSD matrix sparsification based on a new parameter $N^*(mathcal{A})$ which we call connectivity threshold that generalizes the threshold of the number of edges required to make a graph connected. Our main result gives a sparsifier that uses at most $O(ε^{-2}N^*(mathcal{A}) (log n)(log r))$ matrices and is constructible in randomized polynomial time. We also show that we need $N^*(mathcal{A})$ elements to sparsify for any $ε< 0.99$. As the main application of our framework, we prove that any Cayley graph can be sparsified to $O(ε^{-2}log^4 N)$ generators. Previously, a non-trivial bound on Cayley sparsifiers was known only in the case when the group is $mathbb{F}_2^n$.

Problem

• Spectral sparsification for sums of PSD matrices
• Develop instance-specific PSD matrix sparsification theory
• Sparsify Cayley graphs with logarithmic generator bounds

Innovation

• Spectral sparsification for PSD matrix sums
• New connectivity threshold parameter N*(A)
• Randomized polynomial time construction method