Spectral Graph Sparsification Research Articles

SPROUT: spectral sparsification helps restore the spatial structure at single-cell resolution.

Single-cell RNA sequencing thoroughly quantifies the individual cell transcriptomes but renounces the spatial structure. Conversely, recently emerged spatial transcriptomics technologies capture the cellular spatial structure but skimp cell or gene resolutions. Ligand-receptor interactions reveal the potential of cell proximity since they are spatially constrained. Cell–cell affinity values estimated by ligand–receptor interaction can partially represent the structure of cells but falsely include the pseudo affinities between distant or indirectly interacting cells. Here, we develop a software package, SPROUT, to reconstruct the single-cell resolution spatial structure from the transcriptomics data through diminished pseudo ligand–receptor affinities. For spatial data, SPROUT first curates the representative single-cell profiles for each spatial spot from a candidate library, then reduces the pseudo affinities in the intercellular affinity matrix by partial correlation, spectral graph sparsification, and spatial coordinates refinement. SPROUT embeds the estimated interactions into a low-dimensional space with the cross-entropy objective to restore the intercellular structures, which facilitates the discovery of dominant ligand–receptor pairs between neighboring cells at single-cell resolution. SPROUT reconstructed structures achieved shape Pearson correlations ranging from 0.91 to 0.97 on the mouse hippocampus and human organ tumor microenvironment datasets. Furthermore, SPROUT can solely de novo reconstruct the structures at single-cell resolution, i.e., reaching the cell-type proximity correlations of 0.68 and 0.89 between reconstructed and immunohistochemistry-informed spatial structures on a human developing heart dataset and a tumor microenvironment dataset, respectively.

Electrical flows over spanning trees

The network reconfiguration problem seeks to find a rooted tree T such that the energy of the (unique) feasible electrical flow over T is minimized. The tree requirement on the support of the flow is motivated by operational constraints in electricity distribution networks. The bulk of existing results on convex optimization over vertices of polytopes and on the structure of electrical flows do not easily give guarantees for this problem, while many heuristic methods have been developed in the power systems community as early as 1989. Our main contribution is to give the first provable approximation guarantees for the network reconfiguration problem. We provide novel lower bounds and corresponding approximation factors for various settings ranging from \(\min \{{\mathcal {O}}(m-n), {\mathcal {O}}(n)\}\) for general graphs, to \({\mathcal {O}}(\sqrt{n})\) over grids with uniform resistances on edges, and \({\mathcal {O}}(1)\) for grids with uniform edge resistances and demands. To obtain the result for general graphs, we propose a new method for (approximate) spectral graph sparsification, which may be of independent interest. Using insights from our theoretical results, we propose a general heuristic for the network reconfiguration problem that is orders of magnitude faster than existing methods in the literature, while obtaining comparable performance.

GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis

Spectral graph sparsification aims to find ultra-sparse subgraphs whose Laplacian matrix can well approximate the original Laplacian eigenvalues and eigenvectors. In recent years, spectral sparsification techniques 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 work 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 also 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 very large-scale integration computer-aided design, finite element analysis, as well as social and data networks frequently studied in many machine learning and data mining applications. For instance, a sparse SDD matrix with 40 million unknowns and 180 million nonzeros can be solved (1E-3 accuracy level) within 2 min using a single CPU core and about 6-GB memory.

Simple Parallel and Distributed Algorithms for Spectral Graph Sparsification

We describe simple algorithms for spectral graph sparsification, based on iterative computations of weighted spanners and sampling. Leveraging the algorithms of Baswana and Sen for computing spanners, we obtain the first distributed spectral sparsification algorithm in the CONGEST model. We also obtain a parallel algorithm with improved work and time guarantees, as well as other natural distributed implementations. Combining this algorithm with the parallel framework of Peng and Spielman for solving symmetric diagonally dominant linear systems, we get a parallel solver that is significantly more efficient in terms of the total work.

Faster Spectral Sparsification and Numerical Algorithms for SDD Matrices

We study algorithms for spectral graph sparsification. The input is a graph G with n vertices and m edges, and the output is a sparse graph G˜ that approximates G in an algebraic sense. Concretely, for all vectors x and any ϵ > 0, the graph G˜ satisfies (1-ϵ ) x T L G x ≤ x T L G˜ x ≤ (1+ϵ) x T L G x , where L G and G˜ are the Laplacians of G and G˜ respectively. The first contribution of this article applies to all existing sparsification algorithms that rely on solving solving linear systems on graph Laplacians. These algorithms are the fastest known to date. Specifically, we show that less precision is required in the solution of the linear systems, leading to speedups by an O (log n ) factor. We also present faster sparsification algorithms for slightly dense graphs: — An O ( m log n ) time algorithm that generates a sparsifier with O ( n log 3 n /ϵ 2 ) edges. — An O ( m log log n ) time algorithm for graphs with more than n log 5 n log log n edges. — An O ( m ) algorithm for graphs with more than n log 10 n edges. — An O ( m ) algorithm for unweighted graphs with more than n log 8 n edges. These bounds hold up to factors that are in O ( poly (log log n )) and are conjectured to be removable.