Introduction to Clustering Large and High-Dimensional Data
Focusing on a few of the most important clustering algorithms, this book provides a detailed account of these major models in an information retrieval context.
TABLE OF CONTENTS Introduction to Clustering Large and High-Dimensional Data
Chapter 5: Linear Algebra Techniques
This chapter is devoted to a number of Linear Algebra techniques for clustering. In what follows the central place is occupied by the Principal Direction Divisive Partitioning algorithm (PDDP). This remarkable algorithm is especially efficient when applied to sparse high-dimensional data.
5.1. Two Approximation Problems
For a vector a and a line l in R n denote by P l( a) the orthogonal projection of a on l. For a set of vectors 
= { a 1 , , a m} and a line l in R n denote by P l( ) the set of projections {P l( a 1) , , P l( a m)}.
Problem 5.1.1. Let = { a 1 , , a m} be a set of vectors in R n . Find the lines l d and l v such that for each line l in R n one has:
and
Problem 5.1.2. Find out how the lines l d and l v are related.
5.2. Nearest Line
A line l in R n can be described by the parametric equation y + x t where y , x ? R n, x = 1, y T x = 0, and t is a scalar parameter. In what follows we fix a line l and compute the distance from a point a ? R n
Copyright Jacob Kogan 2007 under license agreement with Books24x7
