Spatial Data: Dimensionality Reduction CSC444 Techniques In this - PowerPoint PPT Presentation

Spatial Data: Dimensionality Reduction CSC444 Techniques

In this subfield, we think of a data point as a vector in R^n (what could possibly go wrong?)

“Linear” dimensionality reduction: Reduction is achieved by multiplying a point by a single matrix for every point.

Regular Scatterplots • Every data point is a vector:   v 0 v 1     v 2   v 3 • Every scatterplot is produced by a very simple matrix:  1 � 0 0 0 0 1 0 0  1 � 0 0 0 0 0 1 0

What about other matrices?

Grand Tour (Asimov, 1985) http://cscheid.github.io/lux/demos/tour/tour.html

Is there a best matrix? How do we think about that?

Linear Algebra review • Vectors • Inner Products • Lengths • Angles • Bases • Linear Transformations and Eigenvectors

Principal Component Analysis 0.2 0.1 Species Petal.Length setosa Petal.Width PC2 0.0 versicolor virginica Sepal.Length − 0.1 Sepal.Width − 0.2 − 0.10 − 0.05 0.00 0.05 0.10 0.15 PC1

Principal Component Analysis • Given data set as matrix X in R^(d x n), ~ 1 1 T ) = XH • Center matrix: ˜ ~ X = X ( I − n X T ˜ • is a matrix of inner products of centered rows of X ˜ X X T ˜ • Compute eigendecomposition of ˜ X X T ˜ ˜ X = U Σ U T • The principal components are the first few rows of U Σ 1 / 2

What if we don’t have coordinates, but distances? “Classical” Multidimensional Scaling

http://www.math.pku.edu.cn/teachers/yaoy/Fall2011/ lecture11.pdf

What if we don’t distances, but similarities? Classical Multidimensional Scaling works, too!

Borg and Groenen, Modern Multidimensional Scaling

Borg and Groenen, Modern Multidimensional Scaling

Classical Multidimensional Scaling • Algorithm: B = − 1 • Given , create D ij = | X i − X j | 2 2 HDH T • PCA of B is equal to the PCA of X • (Huh?!)

“Nonlinear” dimensionality reduction (ie: projection is not a matrix operation)

Data might have “high- order” structure

http://isomap.stanford.edu/Supplemental_Fig.pdf

We might want to minimize something else besides “di ff erence between squared distances” t-SNE: di ff erence between neighbor ordering Why not distances?

The curse of Dimensionality • High dimensional space looks nothing like low- dimensional space • Most distances become meaningless