lect9

# lect9 - Dimensionality reduction Outline From distances to...

This preview shows pages 1–10. Sign up to view the full content.

Dimensionality reduction

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Outline From distances to points : MultiDimensional Scaling (MDS) FastMap Dimensionality Reductions or data projections Random projections Principal Component Analysis (PCA)
Multi-Dimensional Scaling (MDS) So far we assumed that we know both data points X and distance matrix D between these points What if the original points X are not known but only distance matrix D is known? Can we reconstruct X or some approximation of X ?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Problem Given distance matrix D between n points Find a k -dimensional representation of every x i point i So that d(x i ,x j ) is as close as possible to D(i,j) Why do we want to do that?
How can we do that? (Algorithm)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
High-level view of the MDS algorithm Randomly initialize the positions of n points in a k -dimensional space Compute pairwise distances D’ for this placement Compare D’ to D Move points to better adjust their pairwise distances (make D’ closer to D ) Repeat until D’ is close to D
The MDS algorithm Input: n x n distance matrix D Random n points in the k -dimensional space (x 1 , …,x n ) stop = false while not stop totalerror = 0.0 For every i,j compute D’(i,j)=d(x i ,x j ) error = (D(i,j)-D’(i,j))/D(i,j) totalerror +=error For every dimension m : x im = (x im -x jm )/D’(i,j)*error If totalerror small enough, stop = true

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Questions about MDS Running time of the MDS algorithm O(n 2 I), where I is the number of iterations of the algorithm MDS does not guarantee that metric property is maintained in d’ Faster? Guarantee of metric property?
Problem (revisited) Given distance matrix D between

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This document was uploaded on 10/05/2010.

### Page1 / 27

lect9 - Dimensionality reduction Outline From distances to...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online