# Similarity-Aware Spectral Sparsification by Edge Filtering

arXiv: https://arxiv.org/abs/1711.05135 · HTML: https://mlsanity.com/paper/1711.05135/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Nov 2017.

## Built on by (most cited first, 3 of 3)

- [HyperEF: Spectral Hypergraph Coarsening by Effective-Resistance Clustering](https://mlsanity.com/paper/2210.14813/) (2022)
- [inGRASS: Incremental Graph Spectral Sparsification via Low-Resistance-Diameter Decomposition](https://mlsanity.com/paper/2402.16990/) (2024)
- [SHyPar: A Spectral Coarsening Approach to Hypergraph Partitioning](https://mlsanity.com/paper/2410.10875/) (2024)

## Abstract (arXiv)

In recent years, spectral graph sparsification techniques that can compute ultra-sparse graph proxies have been extensively studied for accelerating various numerical and graph-related applications. Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from the original graph to form the backbone of the sparsifier, and then recover small portions of spectrally-critical off-tree edges to the spanning tree to significantly improve the approximation quality. However, it is not clear how many off-tree edges should be recovered for achieving a desired spectral similarity level within the sparsifier. Motivated by recent graph signal processing techniques, this paper proposes a similarity-aware spectral graph sparsification framework that leverages efficient spectral off-tree edge embedding and filtering schemes to construct spectral sparsifiers with guaranteed spectral similarity (relative condition number) level. An iterative graph densification scheme is introduced to facilitate efficient and effective filtering of off-tree edges for highly ill-conditioned problems. The proposed method has been validated using various kinds of graphs obtained from public domain sparse matrix collections relevant to VLSI CAD, finite element analysis, as well as social and data networks frequently studied in many machine learning and data mining applications.