Title: Bootstrap AMG for spectral clustering

Abstract

Graph Laplacian is a popular tool for analyzing graphs, particularly in graph partitioning and clustering. Given a notion of similarity (via an adjacency matrix), graph clustering refers to identifying different groups such that vertices in the same group are more similar compared to vertices across different groups. Data clustering can be reformulated in terms of a graph clustering problem when the given set of data is represented as a graph, also known as similarity graph. In this context, eigenvectors of the graph Laplacian are often used to obtain a new geometric representation of the original data set that generally enhances cluster properties and improves cluster detection. Here, we apply a bootstrap algebraic multigrid (AMG) method that constructs a set of vectors associated with the graph Laplacian. These vectors, referred to as algebraically smooth ones, span a low-dimensional Euclidean space, which we use to represent the data, enabling cluster detection both in synthetic and in realistic well-clustered graphs. We show that, in the case of a good quality bootstrap AMG, the computed smooth vectors employed in the construction of the final AMG operator, which by construction is spectrally equivalent to the originally given graph Laplacian, accurately approximate the space in themore » lower portion of the spectrum of the preconditioned operator. Thus, our approach can be viewed as a spectral clustering technique associated with the generalized spectral problem (Laplace operator versus the final AMG operator), and hence, it can be seen as an extension of the classical spectral clustering that employs a standard eigenvalue problem.« less

Authors:

D'Ambra, Pasqua  ^[1];

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Cutillo, Luisa ^[2]; Vassilevski, Panayot S. ^[3]

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+ Show Author Affiliations

1. National Research Council (CNR), Naples (Italy). Inst. for Applied Computing
2. Univ. of Leeds (United Kingdom). School of Mathematics
3. Portland State Univ., OR (United States). Dept. of Mathematics and Statistics; Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing\

Publication Date:

Wed Apr 03 00:00:00 EDT 2019

Research Org.:

Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Org.:

USDOE National Nuclear Security Administration (NNSA); European Union (EU); National Science Foundation (NSF)

OSTI Identifier:

1669240

Alternate Identifier(s):

OSTI ID: 1504994

Report Number(s):

LLNL-JRNL-765277
Journal ID: ISSN 2577-7408; 955345

Grant/Contract Number:  

AC52-07NA27344; 676629; DMS-1619640

Resource Type:

Accepted Manuscript

Journal Name:

Computational and Mathematical Methods

Additional Journal Information:

Journal Volume: 1; Journal Issue: 2; Journal ID: ISSN 2577-7408

Publisher:

Wiley

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; algebraically smooth vectors; bootstrap AMG; graph Laplacian; spectral clustering