lect9

lect9 - Dimensionality reduction Outline From distances to...

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Dimensionality reduction
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Outline From distances to points : MultiDimensional Scaling (MDS) FastMap Dimensionality Reductions or data projections Random projections Principal Component Analysis (PCA)
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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 ?
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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?
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How can we do that? (Algorithm)
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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
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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
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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?
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Problem (revisited) Given distance matrix D between
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This document was uploaded on 10/05/2010.

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lect9 - Dimensionality reduction Outline From distances to...

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