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Dimensional reduction, PCAPowerPoint Presentation

Dimensional reduction, PCA

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Dimensional reduction, PCA. Curse of dimensionality. The higher the dimension, the more data is needed to draw any conclusion Probability density estimation: Continuous: histograms Discrete: k-factorial designs Decision rules: Nearest-neighbor and K-nearest neighbor.

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Curse of dimensionality

- The higher the dimension, the more data is needed to draw any conclusion
- Probability density estimation:
- Continuous: histograms
- Discrete: k-factorial designs

- Decision rules:
- Nearest-neighbor and K-nearest neighbor

How to reduce dimension?

- Assume we know something about the distribution
- Parametric approach: assume data follow distributions within a family H

- Example: counting histograms for 10-D data needs lots of bins, but knowing it’s normal allows to summarize the data in terms of sufficient statistics
- (Number of bins)10 v.s. (10 + 10*11/2)

Linear dimension reduction

- Normality assumption is crucial for linear methods
- Examples:
- Principle Components Analysis (also Latent Semantic Indexing)
- Factor Analysis
- Linear discriminant analysis

Covariance structure of multivariate Gaussian

- 2-dimensional example
- No correlations --> diagonal covariance matrix, e.g.
- Special case: = I
- - log likelihood Euclidean distance to the center

Variance in each dimension

Correlation between dimensions

Covariance structure of multivariate Gaussian

- Non-zero correlations --> full covariance matrix, COV(X1,X2) 0
- E.g. =

- Nice property of Gaussians: closed under linear transformation
- This means we can remove correlation by rotation

Covariance structure of multivariate Gaussian

- Rotation matrix: R = (w1, w2), where w1, w2 are two unit vectors perpendicular to each other
- Rotation by 90 degree
- Rotation by 45 degree

w1

w2

w1 w2

w1

w2

Covariance structure of multivariate Gaussian

- Matrix diagonalization: any 2X2 covariance matrix A can be written as:
- Interpretation: we can always find a rotation to make the covariance look “nice” -- no correlation between dimensions
- This IS PCA when applied to N dimensions

Rotation!

w3

3-D: 3 coordinates

w1

w2

Computation of PCA- The new coordinates uniquely identify the rotation
- In computation, it’s easier to identify one coordinate at a time.
- Step 1: centering the data
- X <-- X - mean(X)
- Want to rotate around the center

Computation of PCA

- Step 2: finding a direction of projection that has the maximal variance
- Linear projection of X onto vector w:
- Projw(X) = XNXd * wdX1 (X centered)

- Now measure the stretch
- This is sample variance = Var(X*w)

X

x

w

Computation of PCA

- Step 3: formulate this as a constrained optimization problem
- Objective of optimization: Var(X*w)
- Need constraint on w: (otherwise can explode), only consider the direction, not the scaling

- So formally:argmax||w||=1 Var(X*w)

Computation of PCA

- Recall single variable case:Var(a*X) = a2 Var(X)
- Apply to multivariate case using matrix notation:Var(X*w) = wT XT X w = wTCov(X) w
- Cov(X) is a dXd matrixSymmetric (easy)
- For any y, yTCov(X) y > 0

Computation of PCA

- Going back to the optimization problem:= max||w||=1 Var(X*w)= max||w||=1 wTCOV(X) w
- The answer is the largest eigenvalue for COV(X)

w1

The first

Principle Component!

(see demo)

More principle components

- We keep looking among all the projections perpendicular to w1
- Formally:max||w2||=1,w2w1 wTCov(X) w
- This turns out to be another eigenvector corresponding to the 2nd largest eigenvalue(see demo)

w2

New coordinates!

Rotation

- Can keep going until we find all projections/coordinates w1,w2,…,wd
- Putting them together, we have a big matrix W=(w1,w2,…,wd)
- W is called an orthogonal matrix
- This corresponds to a rotation (sometimes plus reflection) of the pancake
- This pancake has no correlation between dimensions (see demo)

When does dimension reduction occur?

- Decomposition of covariance matrix
- If only the first few ones are significant, we can ignore the rest, e.g.

2-D coordinates of X

An application of PCA

- Latent Semantic Indexing in document retrieval
- Documents as vectors of word counts
- Try to extract some “features” by linear combination of word counts
- The underlying geometry unclear (mean? Distance?)
- The meaning of principle components unclear (rotation?)

#market

#stock

#bonds

Summary of PCA:

- PCA looks for:
- A sequence of linear, orthogonal projections that reveal interesting structure in data (rotation)

- Defining “interesting”:
- Maximal variance under each projection
- Uncorrelated structure after projection

Departure from PCA

- 3 directions of divergence
- Other definitions of “interesting”?
- Linear Discriminant Analysis
- Independent Component Analysis

- Other methods of projection?
- Linear but not orthogonal: sparse coding
- Implicit, non-linear mapping

- Turning PCA into a generative model
- Factor Analysis

- Other definitions of “interesting”?

Re-thinking “interestingness”

- It all depends on what you want
- Linear Disciminant Analysis (LDA): supervised learning
- Example: separating 2 classes

Maximal separation

Maximal variance

Re-thinking “interestingness”

- Most high-dimensional data look like Gaussian under linear projections
- Maybe non-Gaussian is more interesting
- Independent Component Analysis
- Projection pursuits

- Example: ICA projection of 2-class data

Most unlike Gaussian (e.g. maximize kurtosis)

w2

w3

w1

w4

The “efficient coding” perspective- Sparse coding:
- Projections do not have to be orthogonal
- There can be more basis vectors than the dimension of the space
- Representation using over-complete basis

Basis expansion

p << d; compact coding (PCA)

p > d; sparse coding

“Interesting” can be expensive

- Often faces difficult optimization problems
- Need many constraints
- Lots of parameter sharing
- Expensive to compute, no longer an eigenvalue problem

PCA’s relatives: Factor Analysis

- PCA is not a generative model: reconstruction error is not likelihood
- Sensitive to outliers
- Hard to build into bigger models

- Factor Analysis: adding a measurement noise to account for variability

observation

Measurement noise

N(0,R), R diagonal

Loading matrix (scaled PC’s)

Factors: spherical Gaussian N(0,I)

PCA’s relatives: Factor Analysis

- Generative view: sphere --> stretch and rotate --> add noise
- Learning: a version of EM algorithm

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