arXiv subjects

Zhuo Feng

Publications and source records attributed to Zhuo Feng.

28 records · Page 2Linked to original sources

arXiv2020

SF-GRASS: Solver-Free Graph Spectral Sparsification

Recent spectral graph sparsification techniques have shown promising performance in accelerating many numerical and graph algorithms, such as iterative methods for solving large sparse matrices, spectral partitioning of undirected graphs, vectorless verification of power/thermal grids, representation learning of large graphs, etc. However, prior spectral graph sparsification methods rely on fast Laplacian matrix solvers that are usually challenging to implement in practice. This work, for the first time, introduces a solver-free approach (SF-GRASS) for spectral graph sparsification by leveraging emerging spectral graph coarsening and graph signal processing (GSP) techniques. We introduce a local spectral embedding scheme for efficiently identifying spectrally-critical edges that are key to preserving graph spectral properties, such as the first few Laplacian eigenvalues and eigenvectors. Since the key kernel functions in SF-GRASS can be efficiently implemented using sparse-matrix-vector-multiplications (SpMVs), the proposed spectral approach is simple to implement and inherently parallel friendly. Our extensive experimental results show that the proposed method can produce a hierarchy of high-quality spectral sparsifiers in nearly-linear time for a variety of real-world, large-scale graphs and circuit networks when compared with the prior state-of-the-art spectral method.

cs.DS↗

arXiv2020

GRASPEL: Graph Spectral Learning at Scale

Learning meaningful graphs from data plays important roles in many data mining and machine learning tasks, such as data representation and analysis, dimension reduction, data clustering, and visualization, etc. In this work, for the first time, we present a highly-scalable spectral approach (GRASPEL) for learning large graphs from data. By limiting the precision matrix to be a graph Laplacian, our approach aims to estimate ultra-sparse (tree-like) weighted undirected graphs and shows a clear connection with the prior graphical Lasso method. By interleaving the latest high-performance nearly-linear time spectral methods for graph sparsification, coarsening and embedding, ultra-sparse yet spectrally-robust graphs can be learned by identifying and including the most spectrally-critical edges into the graph. Compared with prior state-of-the-art graph learning approaches, GRASPEL is more scalable and allows substantially improving computing efficiency and solution quality of a variety of data mining and machine learning applications, such as spectral clustering (SC), and t-Distributed Stochastic Neighbor Embedding (t-SNE). {For example, when comparing with graphs constructed using existing methods, GRASPEL achieved the best spectral clustering efficiency and accuracy.

cs.LG↗

arXiv2020

A Unified Approach to Scalable Spectral Sparsification of Directed Graphs

Recent spectral graph sparsification research allows constructing nearly-linear-sized subgraphs that can well preserve the spectral (structural) properties of the original graph, such as the first few eigenvalues and eigenvectors of the graph Laplacian, leading to the development of a variety of nearly-linear time numerical and graph algorithms. However, there is not a unified approach that allows for truly-scalable spectral sparsification of both directed and undirected graphs. In this work, we prove the existence of linear-sized spectral sparsifiers for general directed graphs and introduce a practically-efficient and unified spectral graph sparsification approach that allows sparsifying real-world, large-scale directed and undirected graphs with guaranteed preservation of the original graph spectra. By exploiting a highly-scalable (nearly-linear complexity) spectral matrix perturbation analysis framework for constructing nearly-linear sized (directed) subgraphs, it enables us to well preserve the key eigenvalues and eigenvectors of the original (directed) graph Laplacians. The proposed method has been validated using various kinds of directed graphs obtained from public domain sparse matrix collections, showing promising results for solving directed graph Laplacians, spectral embedding, and partitioning of general directed graphs, as well as approximately computing (personalized) PageRank vectors.

cs.DS↗

arXiv2020

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 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 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 VLSI CAD, 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 two minutes using a single CPU core and about 6GB memory.

cs.DS↗

arXiv2020

GraphZoom: A multi-level spectral approach for accurate and scalable graph embedding

Graph embedding techniques have been increasingly deployed in a multitude of different applications that involve learning on non-Euclidean data. However, existing graph embedding models either fail to incorporate node attribute information during training or suffer from node attribute noise, which compromises the accuracy. Moreover, very few of them scale to large graphs due to their high computational complexity and memory usage. In this paper we propose GraphZoom, a multi-level framework for improving both accuracy and scalability of unsupervised graph embedding algorithms. GraphZoom first performs graph fusion to generate a new graph that effectively encodes the topology of the original graph and the node attribute information. This fused graph is then repeatedly coarsened into much smaller graphs by merging nodes with high spectral similarities. GraphZoom allows any existing embedding methods to be applied to the coarsened graph, before it progressively refine the embeddings obtained at the coarsest level to increasingly finer graphs. We have evaluated our approach on a number of popular graph datasets for both transductive and inductive tasks. Our experiments show that GraphZoom can substantially increase the classification accuracy and significantly accelerate the entire graph embedding process by up to 40.8x, when compared to the state-of-the-art unsupervised embedding methods.

cs.LG↗

arXiv2019

Observation of Complete Photonic Bandgap in Low Refractive Index Contrast Inverse Rod-Connected Diamond Structured Chalcogenides

Three-dimensional complete photonic bandgap materials or photonic crystals block light propagation in all directions. The rod-connected diamond structure exhibits the largest photonic bandgap known to date and supports a complete bandgap for the lowest refractive index contrast ratio down to $n_{high}/n_{low} \sim 1.9$. We confirm this threshold by measuring a complete photonic bandgap in the infrared region in Sn--S--O $(n\sim1.9)$ and Ge--Sb--S--O $(n\sim2)$ inverse rod-connected diamond structures. The structures were fabricated using a low-temperature chemical vapor deposition process via a single-inversion technique. This provides a reliable fabrication technique of complete photonic bandgap materials and expands the library of backfilling materials, leading to a wide range of future photonic applications.

physics.optics↗

arXiv2018

Nearly-Linear Time Spectral Graph Reduction for Scalable Graph Partitioning and Data Visualization

This paper proposes a scalable algorithmic framework for spectral reduction of large undirected graphs. The proposed method allows computing much smaller graphs while preserving the key spectral (structural) properties of the original graph. Our framework is built upon the following two key components: a spectrum-preserving node aggregation (reduction) scheme, as well as a spectral graph sparsification framework with iterative edge weight scaling. We show that the resulting spectrally-reduced graphs can robustly preserve the first few nontrivial eigenvalues and eigenvectors of the original graph Laplacian. In addition, the spectral graph reduction method has been leveraged to develop much faster algorithms for multilevel spectral graph partitioning as well as t-distributed Stochastic Neighbor Embedding (t-SNE) of large data sets. We conducted extensive experiments using a variety of large graphs and data sets, and obtained very promising results. For instance, we are able to reduce the "coPapersCiteseer" graph with 0.43 million nodes and 16 million edges to a much smaller graph with only 13K (32X fewer) nodes and 17K (950X fewer) edges in about 16 seconds; the spectrally-reduced graphs also allow us to achieve up to 1100X speedup for spectral graph partitioning and up to 60X speedup for t-SNE visualization of large data sets.

cs.DS↗

arXiv2018

Towards Scalable Spectral Clustering via Spectrum-Preserving Sparsification

The eigendeomposition of nearest-neighbor (NN) graph Laplacian matrices is the main computational bottleneck in spectral clustering. In this work, we introduce a highly-scalable, spectrum-preserving graph sparsification algorithm that enables to build ultra-sparse NN (u-NN) graphs with guaranteed preservation of the original graph spectrums, such as the first few eigenvectors of the original graph Laplacian. Our approach can immediately lead to scalable spectral clustering of large data networks without sacrificing solution quality. The proposed method starts from constructing low-stretch spanning trees (LSSTs) from the original graphs, which is followed by iteratively recovering small portions of "spectrally critical" off-tree edges to the LSSTs by leveraging a spectral off-tree embedding scheme. To determine the suitable amount of off-tree edges to be recovered to the LSSTs, an eigenvalue stability checking scheme is proposed, which enables to robustly preserve the first few Laplacian eigenvectors within the sparsified graph. Additionally, an incremental graph densification scheme is proposed for identifying extra edges that have been missing in the original NN graphs but can still play important roles in spectral clustering tasks. Our experimental results for a variety of well-known data sets show that the proposed method can dramatically reduce the complexity of NN graphs, leading to significant speedups in spectral clustering.

cs.LG↗

arXiv2018

Similarity-Aware Spectral Sparsification by Edge Filtering

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.

cs.DS↗

arXiv2015

Unconventional superconductivity in the layered iron germanide YFe$_2$Ge$_2$

Since the discovery of superconductivity in LaFePO in 2006, numerous iron-based superconductors have been identified within diverse structure families, all of which combine iron with a group-V (pnictogen) or group-VI (chalco- gen) element. Unconventional superconductivity is extremely rare among transition metal compounds outside these layered iron systems and the cuprates, and it is almost universally associated with highly anisotropic electronic properties and nearly 2D Fermi surface geometries. The iron-based intermetallic YFe$_2$Ge$_2$ features a 3D Fermi surface and a strongly enhanced low temperature heat capacity, which signals strong electronic correlations. We present data from a new generation of high quality samples of YFe$_2$Ge$_2$, which show superconducting transition anomalies below 1.8 K in thermodynamic as well as transport measurements, establishing that superconductivity is intrinsic in this layered iron compound outside the known superconducting iron pnictide or chalcogenide families. The Fermi surface geometry of YFe$_2$Ge$_2$ resembles that of KFe$_2$As$_2$ in the high pressure collapsed tetragonal phase, in which superconductivity at temperatures as high as 10 K has recently been reported, suggesting an underlying connection between the two systems.

cond-mat.str-el↗