70 research outputs found
Large Scale Spectral Clustering Using Approximate Commute Time Embedding
Spectral clustering is a novel clustering method which can detect complex
shapes of data clusters. However, it requires the eigen decomposition of the
graph Laplacian matrix, which is proportion to and thus is not
suitable for large scale systems. Recently, many methods have been proposed to
accelerate the computational time of spectral clustering. These approximate
methods usually involve sampling techniques by which a lot information of the
original data may be lost. In this work, we propose a fast and accurate
spectral clustering approach using an approximate commute time embedding, which
is similar to the spectral embedding. The method does not require using any
sampling technique and computing any eigenvector at all. Instead it uses random
projection and a linear time solver to find the approximate embedding. The
experiments in several synthetic and real datasets show that the proposed
approach has better clustering quality and is faster than the state-of-the-art
approximate spectral clustering methods
Quantum Speedup for Graph Sparsification, Cut Approximation and Laplacian Solving
Graph sparsification underlies a large number of algorithms, ranging from
approximation algorithms for cut problems to solvers for linear systems in the
graph Laplacian. In its strongest form, "spectral sparsification" reduces the
number of edges to near-linear in the number of nodes, while approximately
preserving the cut and spectral structure of the graph. In this work we
demonstrate a polynomial quantum speedup for spectral sparsification and many
of its applications. In particular, we give a quantum algorithm that, given a
weighted graph with nodes and edges, outputs a classical description of
an -spectral sparsifier in sublinear time
. This contrasts with the optimal classical
complexity . We also prove that our quantum algorithm is optimal
up to polylog-factors. The algorithm builds on a string of existing results on
sparsification, graph spanners, quantum algorithms for shortest paths, and
efficient constructions for -wise independent random strings. Our algorithm
implies a quantum speedup for solving Laplacian systems and for approximating a
range of cut problems such as min cut and sparsest cut.Comment: v2: several small improvements to the text. An extended abstract will
appear in FOCS'20; v3: corrected a minor mistake in Appendix
HIGH-PERFORMANCE SPECTRAL METHODS FOR COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS
Recent research shows that by leveraging the key spectral properties of eigenvalues and eigenvectors of graph Laplacians, more efficient algorithms can be developed for tackling many graph-related computing tasks. In this dissertation, spectral methods are utilized for achieving faster algorithms in the applications of very-large-scale integration (VLSI) computer-aided design (CAD)
First, a scalable algorithmic framework is proposed for effective-resistance preserving 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 three key components: a spectrum-preserving node aggregation and reduction scheme, a spectral graph sparsification framework with iterative edge weight scaling, as well as effective-resistance preserving post-scaling and iterative solution refinement schemes. We show that the resultant spectrally-reduced graphs can robustly preserve the first few nontrivial eigenvalues and eigenvectors of the original graph Laplacian and thus allow for developing highly-scalable spectral graph partitioning and circuit simulation algorithms.
Based on the framework of the spectral graph reduction, a Sparsified graph-theoretic Algebraic Multigrid (SAMG) is proposed for solving large Symmetric Diagonally Dominant (SDD) matrices. The proposed SAMG framework allows efficient construction of nearly-linear sized graph Laplacians for coarse-level problems while maintaining good spectral approximation during the AMG setup phase by leveraging a scalable spectral graph sparsification engine. Our experimental results show that the proposed method can offer more scalable performance than existing graph-theoretic AMG solvers for solving large SDD matrices in integrated circuit (IC) simulations, 3D-IC thermal analysis, image processing, finite element analysis as well as data mining and machine learning applications.
Finally, the spectral methods are applied to power grid and thermal integrity verification applications. This dissertation introduces a vectorless power grid and thermal integrity verification framework that allows computing worst-case voltage drop or thermal profiles across the entire chip under a set of local and global workload (power density) constraints. To address the computational challenges introduced by the large 3D mesh-structured thermal grids, we apply the spectral graph reduction approach for highly-scalable vectorless thermal (or power grids) verification of large chip designs. The effectiveness and efficiency of our approach have been demonstrated through extensive experiments
A Parallel Solver for Graph Laplacians
Problems from graph drawing, spectral clustering, network flow and graph
partitioning can all be expressed in terms of graph Laplacian matrices. There
are a variety of practical approaches to solving these problems in serial.
However, as problem sizes increase and single core speeds stagnate, parallelism
is essential to solve such problems quickly. We present an unsmoothed
aggregation multigrid method for solving graph Laplacians in a distributed
memory setting. We introduce new parallel aggregation and low degree
elimination algorithms targeted specifically at irregular degree graphs. These
algorithms are expressed in terms of sparse matrix-vector products using
generalized sum and product operations. This formulation is amenable to linear
algebra using arbitrary distributions and allows us to operate on a 2D sparse
matrix distribution, which is necessary for parallel scalability. Our solver
outperforms the natural parallel extension of the current state of the art in
an algorithmic comparison. We demonstrate scalability to 576 processes and
graphs with up to 1.7 billion edges.Comment: PASC '18, Code: https://github.com/ligmg/ligm
A simple, combinatorial algorithm for solving SDD systems in nearly-linear time
Original manuscript January 28, 2013In this paper, we present a simple combinatorial algorithm that solves symmetric diagonally dominant (SDD) linear systems in nearly-linear time. It uses little of the machinery that previously appeared to be necessary for a such an algorithm. It does not require recursive preconditioning, spectral sparsification, or even the Chebyshev Method or Conjugate Gradient. After constructing a "nice" spanning tree of a graph associated with the linear system, the entire algorithm consists of the repeated application of a simple update rule, which it implements using a lightweight data structure. The algorithm is numerically stable and can be implemented without the increased bit-precision required by previous solvers. As such, the algorithm has the fastest known running time under the standard unit-cost RAM model. We hope the simplicity of the algorithm and the insights yielded by its analysis will be useful in both theory and practice.National Science Foundation (U.S.) (Award 0843915)National Science Foundation (U.S.) (Award 1111109)Alfred P. Sloan Foundation (Research Fellowship)National Science Foundation (U.S.). Graduate Research Fellowship Program (Grant 1122374
Quantum speedup for graph sparsification, cut approximation and Laplacian solving
Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, "spectral sparsification" reduces the number of edges to near-linear in the number of nodes, while approximately preserving the cut and spectral structure of the graph. The breakthrough work by Benczúr and Karger (STOC'96) and Spielman and Teng (STOC'04) showed that sparsification can be done optimally in time near-linear in the number of edges of the original graph.
In this work we show that quantum algorithms allow to speed up spectral sparsification, and thereby many of the derived algorithms. Given adjacency-list access to a weighted graph with
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