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Title: A Matrix Computation View of the FastMap and RobustMap Dimension Reduction Algorithms

Abstract

Given a set of pairwise object distances and a dimension $$k$$, FastMap and RobustMap algorithms compute a set of $$k$$-dimensional coordinates for the objects. These metric space embedding methods implicitly assume a higher-dimensional coordinate representation and are a sequence of translations and orthogonal projections based on a sequence of object pair selections (called pivot pairs). We develop a matrix computation viewpoint of these algorithms that operates on the coordinate representation explicitly using Householder reflections. The resulting Coordinate Mapping Algorithm (CMA) is a fast approximate alternative to truncated principal component analysis (PCA) and it brings the FastMap and RobustMap algorithms into the mainstream of numerical computation where standard BLAS building blocks are used. Motivated by the geometric nature of the embedding methods, we further show that truncated PCA can be computed with CMA by specific pivot pair selections. Describing FastMap, RobustMap, and PCA as CMA computations with different pivot pair choices unifies the methods along a pivot pair selection spectrum. We also sketch connections to the semi-discrete decomposition and the QLP decomposition.

Authors:
 [1]
  1. ORNL
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). National Center for Computational Sciences (NCCS)
Sponsoring Org.:
USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE Office of Science (SC)
OSTI Identifier:
978769
DOE Contract Number:  
DE-AC05-00OR22725
Resource Type:
Journal Article
Journal Name:
SIAM Journal on Matrix Analysis and Applications
Additional Journal Information:
Journal Volume: 31; Journal Issue: 3
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ALGORITHMS; DIMENSIONS; METRICS; Householder reflection; metric space embedding; principal components analysis; singular value decomposition; BLAS.

Citation Formats

Ostrouchov, George. A Matrix Computation View of the FastMap and RobustMap Dimension Reduction Algorithms. United States: N. p., 2009. Web.
Ostrouchov, George. A Matrix Computation View of the FastMap and RobustMap Dimension Reduction Algorithms. United States.
Ostrouchov, George. 2009. "A Matrix Computation View of the FastMap and RobustMap Dimension Reduction Algorithms". United States.
@article{osti_978769,
title = {A Matrix Computation View of the FastMap and RobustMap Dimension Reduction Algorithms},
author = {Ostrouchov, George},
abstractNote = {Given a set of pairwise object distances and a dimension $k$, FastMap and RobustMap algorithms compute a set of $k$-dimensional coordinates for the objects. These metric space embedding methods implicitly assume a higher-dimensional coordinate representation and are a sequence of translations and orthogonal projections based on a sequence of object pair selections (called pivot pairs). We develop a matrix computation viewpoint of these algorithms that operates on the coordinate representation explicitly using Householder reflections. The resulting Coordinate Mapping Algorithm (CMA) is a fast approximate alternative to truncated principal component analysis (PCA) and it brings the FastMap and RobustMap algorithms into the mainstream of numerical computation where standard BLAS building blocks are used. Motivated by the geometric nature of the embedding methods, we further show that truncated PCA can be computed with CMA by specific pivot pair selections. Describing FastMap, RobustMap, and PCA as CMA computations with different pivot pair choices unifies the methods along a pivot pair selection spectrum. We also sketch connections to the semi-discrete decomposition and the QLP decomposition.},
doi = {},
url = {https://www.osti.gov/biblio/978769}, journal = {SIAM Journal on Matrix Analysis and Applications},
number = 3,
volume = 31,
place = {United States},
year = {Thu Jan 01 00:00:00 EST 2009},
month = {Thu Jan 01 00:00:00 EST 2009}
}