machine learning for signal...

Machine Learning for Signal Processing

Eigenfaces and Eigenrepresentations

Class 6. 16 Sep 2014

Instructor: Bhiksha Raj

16 Sep 2014 11-755/18-797 1

Administrivia

• Project teams?

• Project proposals? – Please send proposals to TA, and cc me

– Rahul Raj is no longer a TA

• Reminder: Assignment 1 due in ?? days

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Recall: Representing images

• The most common element in the image: background

– Or rather large regions of relatively featureless shading

– Uniform sequences of numbers

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Adding more bases

• Checkerboards with different variations

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BWPROJECTION

ageBpinvW

ageBW

Im)(

Im

] [

.

.

...Im

3213

2

1

332211

BBBBw

w

w

W

BwBwBwage

B1 B2 B3 B4 B5 B6

Getting closer at 625 bases!

“Bases”

• “Bases” are the “standard” units such that all instances can be expressed a weighted combinations of these units

• Ideal requirements: Bases must be orthogonal

• Checkerboards are one choice of bases – Orthogonal

– But not “smooth”

• Other choices of bases: Complex exponentials, Wavelets, etc..

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B1 B2 B3 B4 B5 B6

...332211 BwBwBwimage

Data specific bases?

• Issue: All the bases we have considered so far are data agnostic

– Checkerboards, Complex exponentials, Wavelets..

– We use the same bases regardless of the data we analyze • Image of face vs. Image of a forest

• Segment of speech vs. Seismic rumble

• How about data specific bases

– Bases that consider the underlying data • E.g. is there something better than checkerboards to describe

faces

• Something better than complex exponentials to describe music?

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The Energy Compaction Property

• Define “better”?

• The description

• The ideal:

– If the description is terminated at any point, we should

still get most of the information about the data

• Error should be small

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NN BwBwBwBwX ...332211

iii BwBwBwX ...ˆ2211

2ˆ

ii XXError

1 ii ErrorError

• A collection of images

– All normalized to 100x100 pixels

• What is common among all of them?

– Do we have a common descriptor?

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Data-specific description of faces

A typical face

• Assumption: There is a “typical” face that captures most of

what is common to all faces

– Every face can be represented by a scaled version of a typical face

– We will denote this face as V

• Approximate every face f as f = wf V

• Estimate V to minimize the squared error

– How? What is V?

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The typical face

A collection of least squares typical faces

• Assumption: There are a set of K “typical” faces that captures most of all faces

• Approximate every face f as f = wf,1 V1+ wf,2 V2 + wf,3 V3 +.. + wf,k Vk

– V2 is used to “correct” errors resulting from using only V1. So on average

– V3 corrects errors remaining after correction with V2

– And so on..

– V = [V1 V2 V3]

• Estimate V to minimize the squared error

– How? What is V?

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2

1,1,

2

2,2,1,1, )( ffffff VwfVwVwf

2

2,2,1,1,

2

3,3,2,2,1,1, )( )( ffffffffff VwVwfVwVwVwf

A recollection

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M =

N =

S=Pinv(N) M

? U =

• Finding the best explanation of music M in terms of notes N

• Also finds the score S of M in terms of N

MNpinvS

MNSU

)(

How about the other way?

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N = M Pinv(S)

M =

N = ? ?

S =

U =

• Finding the notes N given music M and score S • Also finds best explanation of M in terms of S

)( SpinvMN

MNSU

Finding Everything

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Find the four notes and their score that generate the closest approximation to M

M =

N = ? ?

S =

U =

?

MNSU

The same problem

• Here V, W and U are ALL unknown and must be determined

– Such that the squared error between U and F is minimum

• For each face

– f = wf,1 V1+ wf,2 V2 + wf,3 V3 +.. + wf,K VK

• For the collection of faces: F ≈ V W

– V is D x K and W is K x N

• D is the no. of pixels, N, is the no. of faces in the set

– Find V and W such that ||F – VW||2 is minimized 14

F =

U = Approximation

W

V

Typical faces

Abstracting the problem: Finding the FIRST typical face

• Each “point” represents a face in “pixel space”

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Pixel 1

Pix

el 2



• Any “typical face” V is a vector in this space

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Pixel 1

Pix

el 2

V



• The “typical face” V is a vector in this space

• The approximation wf, V for any face f is the projection of f onto V

• The distance between f and its projection wfV is the projection error for f

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Pixel 1

Pix

el 2

V

f

error


• Every face in our data will suffer error when approximated by its projection on V

• The total squared length of all error lines is the total squared projection error

16 Sep 2014 11-755/18-797 18

Pixel 1

Pix

el 2

V


• The problem of finding the first typical face V1: Find the V for which the total projection error is minimum!

• This “minimum squared error” V is our “best” first typical face

• It is also the first Eigen face

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Pixel 1

Pix

el 2

V





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Pixel 1

Pix

el 2

V





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Pixel 1

Pix

el 2

V





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Pixel 1

Pix

el 2

V





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Pixel 1

Pix

el 2

V1

Formalizing the Problem: Error from approximating a single vector

• Consider: approximating x = wv

– E.g x is a face, and “v” is the “typical face”

• Finding an approximation wv which is closest to x

– In a Euclidean sense

– Basically projecting x onto v

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x

y

v

x

Approximating: x = wv

Formalizing the Problem: Error from approximating a single vector

• Projection of a vector x on to a vector v

• Assuming v is of unit length:

16 Sep 2014 11-755/18-797 25

x

y

v

x

v

xvvx

T

ˆ

xvvxTˆ

xvvxxxTerror ˆ

vvTx

x-vvTx

2

error squared xvvxT

Approximating: x = wv

Error from approximating a single vector

• Minimum squared approximation error from approximating x as it as wv

• Optimal value of w: w = vTx

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x

y

v

x

vvTx

x-vvTx

2

)( xvvxxTe


• Error from projecting a vector x on to a vector onto a unit vector v

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x

y

v

x

vvTx

x-vvTx

2

)( xvvxxTe

xvvxvvxxxvvxxvvxxTTTTTTTe )(

xvvvvxxvvxxvvxxxTTTTTTTT



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x

y

v

x

vvTx

x-vvTx

xvvvvxxvvxxvvxxxTTTTTTTT

=1


2

)( xvvxxTe



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x

y

v

x

vvTx

x-vvTx

xvvxxvvxxvvxxxTTTTTTT

xvvxxxxTTTe )(


2

)( xvvxxTe


16 Sep 2014 11-755/18-797 30

x

y

v

x

vvTx

x-vvTx

This is the very familiar pythogoras’ theorem!!

e(x) = xTx – xTv.vTx

Length of projection

Error for many vectors

• Error for one vector:

• Error for many vectors

• Goal: Estimate v to minimize this error!

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xvvxxxxTTTe )(

x

y

v

i

i

TT

ii

T

i

i

ieE xvvxxxx )( i i

i

TT

ii

T

i xvvxxx

Error for many vectors

• Total error:

• Add constraint: vTv = 1

• Constrained objective to minimize:

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x

y

v

1 vvxvvxxxT

i i

i

TT

ii

T

iE

i i

i

TT

ii

T

iE xvvxxx

Two Matrix Identities

• Derivative w.r.t v

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vxxv

vxxv

v

xvvx TTTTT

d

d

d

d2

vv

vv2

d

d T

Minimizing error

• Differentiating w.r.t v and equating to 0

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x

y

v

i

T

ii 022 vvxx

1 vvxvvxxxT

i i

i

TT

ii

T

iE

vvxx

i

T

ii

The correlation matrix

• The encircled term is the correlation matrix

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vvxx

i

T

ii

X = Data Matrix

XT =

Tra

nsposed

Data

Matr

ix

Correlation =

NxxxX ... 21RXXxx T

i

T

ii

The best “basis”

• The minimum-error basis is found by solving

• v is an Eigen vector of the correlation matrix R

– is the corresponding Eigen value

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x

y

v

vRv

What about the total error?

• xTv = vTx (inner product)

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i i

i

TT

ii

T

iE xvvxxx

i i

T

ii

T

i

T

iE vxxvxx

i

T

i

T

i

i

T

i

T

i

i

T

i

T

iE vvxxvvxxRvvxx

i

i

T

iE xx

i i

T

ii

T

i

T

i vxxvxx

Minimizing the error • The total error is

• We already know that the optimal basis is an Eigen vector

• The total error depends on the negative of the corresponding Eigen value

• To minimize error, we must maximize

• i.e. Select the Eigen vector with the largest Eigen value

16 Sep 2014 11-755/18-797 38

i

i

T

iE xx

The typical face

• Compute the correlation matrix for your data – Arrange them in matrix X and compute R = XXT

• Compute the principal Eigen vector of R – The Eigen vector with the largest Eigen value

• This is the typical face

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The typical face

The second typical face

• The first typical face models some of the characteristics of the faces

• Simply by scaling its grey level

• But the approximation has error

• The second typical face must explain some of this error

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?

The second typical face

• Approximation with only the first typical face has error

• The second face must explain this error

• How do we find this this face?

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The first typical face

The second typical face?

Solution: Iterate

• Get the “error” faces by subtracting the first-level approximation from the original image

• Repeat the estimation on the “error” images

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-

-

-

-

=

=

=

=

Solution: Iterate

• Get the “error” faces by subtracting the first-level approximation from the original image

• Repeat the estimation on the “error” images

16 Sep 2014 11-755/18-797 43

-

-

-

-

=

=

=

=

Abstracting the problem: Finding the second typical face

• Each “point” represents an error face in “pixel space”

• Find the vector V2 such that the projection of these error faces on V2 results in the least error

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Pixel 1

Pix

el 2

ERROR

FACES

Minimizing error

• Differentiating w.r.t v and equating to 0

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i

T

ii 022 vvee

1 vvevveeeT

i i

i

TT

ii

T

iE

vvee

i

T

ii

Pixel 1

Pix

el 2

ERROR

FACES

The same math applies

but now to the set

of error data points

Minimizing error

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Pixel 1

Pix

el 2

ERROR

FACES

The same math applies

but now to the set

of error data points

• The minimum-error basis is found by solving

• v2 is an Eigen vector of the correlation matrix Re

corresponding to the largest eigen value of Re

22 vvR e T

e eeR

Which gives us our second typical face

• But approximation with the two faces will still result in error

• So we need more typical faces to explain this error

• We can do this by subtracting the appropriately scaled version of the second “typical” face from the error images and repeating the process

16 Sep 2014 11-755/18-797 47

Solution: Iterate

• Get the second-level “error” faces by subtracting the scaled second typical face from the first-level error

• Repeat the estimation on the second-level “error” images

16 Sep 2014 11-755/18-797 48

-

-

-

-

=

=

=

=

Error face Second-level error

An interesting property

• Each “typical face” will be orthogonal to all other typical faces

– Because each of them is learned to explain what the rest could not

– None of these faces can explain one another!

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To add more faces

• We can continue the process, refining the error each time

– An instance of a procedure is called “Gram-Schmidt” orthogonalization

• OR… we can do it all at once

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With many typical faces

• Approximate every face f as f = wf,1 V1+ wf,2 V2 +... + wf,k Vk

• Here W, V and U are ALL unknown and must be determined – Such that the squared error between U and M is minimum

16 Sep 2014 11-755/18-797 51

M =

U = Approximation

W

V

Typical faces

With multiple bases

• Assumption: all bases v1 v2 v3.. are unit length

• Assumption: all bases are orthogonal to one another: viTvj = 0 if i != j

– We are trying to find the optimal K-dimensional subspace to project the data

– Any set of vectors in this subspace will define the subspace

– Constraining them to be orthogonal does not change this

• I.e. if V = [v1 v2 v3 … ], VTV = I

– Pinv(V) = VT

• Projection matrix for V = VPinv(V) = VVT

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With multiple bases

• Projection for a vector

• Error vector =

• Error length =

16 Sep 2014 11-755/18-797 53

xVVxxxxTTTe )(

xVVxTˆ

xVVxxxT ˆ

x

y

V

x

VVTx

x-VVTx

Represents a

K-dimensional subspace

With multiple bases

• Error for one vector:

• Error for many vectors

• Goal: Estimate V to minimize this error!

16 Sep 2014 11-755/18-797 54

xVVxxxxTTTe )(

x

y

i i

i

TT

ii

T

iE xVVxxx

V

Minimizing error

• With constraint VTV = I, objective to minimize

– Note: now L is a diagonal matrix

– The constraint simply ensures that vTv = 1 for every basis

• Differentiating w.r.t V and equating to 0

16 Sep 2014 11-755/18-797 55

IVVxVVxxx L T

i i

i

TT

ii

T

i traceE

022 L

VVxx

i

T

ii VRV L

i

K

j

ji

T

iE1

xx

Finding the optimal K bases

• Compute the Eigendecompsition of the correlation matrix

• Select K Eigen vectors

• But which K?

• Total error =

• Select K eigen vectors corresponding to the K largest Eigen values

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VRV L

Eigen Faces!

• Arrange your input data into a matrix X

• Compute the correlation R = XXT

• Solve the Eigen decomposition: RV = LV

• The Eigen vectors corresponding to the K largest eigen values are our optimal bases

• We will refer to these as eigen faces.

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How many Eigen faces

• How to choose “K” (number of Eigen faces)

• Lay all faces side by side in vector form to form a matrix – In my example: 300 faces. So the matrix is 10000 x 300

• Multiply the matrix by its transpose – The correlation matrix is 10000x10000

16 Sep 2014 11-755/18-797 58

M = Data Matrix

MT =

Tra

nsposed

Data

Matr

ix

Correlation =

10000x300

300x10000

10000x10000

Eigen faces

• Compute the eigen vectors – Only 300 of the 10000 eigen values are non-zero

• Why?

• Retain eigen vectors with high eigen values (>0) – Could use a higher threshold

16 Sep 2014 11-755/18-797 59

[U,S] = eig(correlation)

10000

2

1

.0.0

.....

.....

0.00

0.0.

S

U

eig

enfa

ce1

eig

enfa

ce2

Eigen Faces

• The eigen vector with the highest eigen value is the first typical face

• The vector with the second highest eigen value is the second typical face.

• Etc.

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U

eig

enfa

ce1

eig

enfa

ce2

eigenface1 eigenface2

eigenface3

Representing a face

• The weights with which the eigen faces must be combined to compose the face are used to represent the face!

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= w1 + w2 + w3

Representation = [w1 w2 w3 …. ]T

Energy Compaction Example

• One outcome of the “energy compaction principle”: the approximations are recognizable

• Approximating a face with one basis:

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11vwf



• Approximating a face with one Eigenface:

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11vwf



• Approximating a face with 10 eigenfaces:

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10102211 ... vvv wwwf




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303010102211 ...... vvvv wwwwf




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6060303010102211 ......... vvvvv wwwwwf

How did I do this?

• Hint: only changing weights assigned to Eigen faces..

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Class specificity

• The Eigenimages (bases) are very specific to the class of data they are trained on

– Faces here

• They will not be useful for other classes

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eigenface1 eigenface2

eigenface3

Class specificity

• Eigen bases are class specific

• Composing a fishbowl from Eigenfaces

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Class specificity



• With 1 basis

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11vwf

Class specificity



• With 10 bases

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10102211 ... vvv wwwf

Class specificity



• With 30 bases

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303010102211 ...... vvvv wwwwf

Class specificity



• With 100 bases

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100100303010102211 ......... vvvvv wwwwwf

Universal bases • Universal bases..

• End up looking a lot like discrete cosine transforms!!!! • DCTs are the best “universal” bases

– If you don’t know what your data are, use the DCT

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SVD instead of Eigen

• Do we need to compute a 10000 x 10000 correlation matrix and then perform Eigen analysis? – Will take a very long time on your laptop

• SVD – Only need to perform “Thin” SVD. Very fast

• U = 10000 x 300 – The columns of U are the eigen faces!

– The Us corresponding to the “zero” eigen values are not computed

• S = 300 x 300

• V = 300 x 300

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M = Data Matrix

10000x300

U=10000x300 S=300x300 V=300x300

=

U

eig

enfa

ce1

eig

enfa

ce2

Using SVD to compute Eigenbases

[U, S, V] = SVD(X)

• U will have the Eigenvectors

• Thin SVD for 100 bases:

[U,S,V] = svds(X, 100)

• Much more efficient

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Eigen Decomposition of data

• Nothing magical about faces – can be applied to any data.

– Eigen analysis is one of the key components of data compression and representation

– Represent N-dimensional data by the weights of the K leading Eigen vectors

• Reduces effective dimension of the data from N to K

• But requires knowledge of Eigen vectors

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Eigen decomposition of what?

• Eigen decomposition of the correlation matrix

• Is there an alternate way?

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Linear vs. Affine

• The model we saw

– Approximate every face f as f = wf,1 V1+ wf,2 V2 +... + wf,k Vk

– Linear combination of bases

• If you add a constant

f = wf,1 V1+ wf,2 V2 +... + wf,k Vk + m

– Affine combination of bases

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Estimation with the constant

• Estimate

f = wf,1 V1+ wf,2 V2 +... + wf,k Vk + m

• Lets do this incrementally first:

• f ≈ m

– For every face

– Find m to optimize the approximation

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Estimation with the constant

• Estimate

f ≈ m

– for every f!

• Error over all faces 𝐸 = ||𝑓 − 𝑚||2𝑓

• Minimizing the error with respect to m, we simply get

– 𝑚 = 1

𝑁 𝑓𝑓

• The mean of the data

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Estimation the remaining

• Same procedure as before: – Remaining “typical faces” must model what the constant m

could not

• Subtract the constant from every data point

– 𝑓 = 𝑓 − 𝑚

• Now apply the model: – 𝑓 = wf,1 V1+ wf,2 V2 +... + wf,k Vk

• This is just Eigen analysis of the “mean-normalized” data – Also called the “centered” data

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Estimating the Affine model

f = wf,1 V1+ wf,2 V2 +... + wf,k Vk + m

• First estimate the mean m

𝑚 = 1

𝑁 𝑓

𝑓

• Compute the correlation matrix of the “centered” data 𝑓 = 𝑓 − 𝑚

– 𝑪 = 𝑓 𝑓 𝑇𝑓 = (𝑓 − 𝑚 )(𝑓 − 𝑚 )𝑇

𝑓

– This is the covariance matrix of the set of f

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Estimating the Affine model

f = wf,1 V1+ wf,2 V2 +... + wf,k Vk + m

• First estimate the mean m

𝑚 = 1

𝑁 𝑓

𝑓

• Compute the covariance matrix – 𝑪= (𝑓 − 𝑚 )(𝑓 − 𝑚 )𝑇

𝑓

• Eigen decompose!

• The Eigen vectors corresponding to the top k Eigen values give us the bases Vk

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VCV L

Properties of the affine model

• The bases V1, V2 ,...,Vk are all orthogonal to one another

– Eigen vectors of the symmetric Covariance matrix

• But they are not orthogonal to m

– Because m is an unscaled constant

• We could jointly estimate all V1, V2 ,...,Vk and m by minimizing

||𝑓 − ( 𝑤𝑓,𝑖𝑉𝑖 + 𝑚𝑓 )||2𝑓 +𝑡𝑟𝑎𝑐𝑒(Λ 𝑽𝑇𝑽 − 𝑰 )

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Linear vs. Affine

• The model we saw

– Approximate every face f as f = wf,1 V1+ wf,2 V2 +... + wf,k Vk

– The Karhunen Loeve Transform

– Retains maximum Energy for any order k

• If you add a constant

f = wf,1 V1+ wf,2 V2 +... + wf,k Vk + m

– Principal Component Analysis

– Retains maximum Variance for any order k

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How do they relate

• Relationship between correlation matrix and covariance matrix

R = C + mmT

• Karhunen Loeve bases are Eigen vectors of R

• PCA bases are Eigen vectors of C

• How do they relate

– Not easy to say..

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The Eigen vectors

• The Eigen vectors of C are the major axes of the ellipsoid Cv, where v are the vectors on the unit sphere

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The Eigen vectors

• The Eigen vectors of R are the major axes of the ellipsoid Cv + mmTv

• Note that mmT has rank 1 and mmTv is a line

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mmT

The Eigen vectors

• The principal Eigenvector of R lies between the principal Eigen vector of C and m

• Similarly the principal Eigen value

• Similar logic is not easily extendable to the other Eigenvectors, however

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mmT

||||)1(

m

mee CR

10

2||||)1( m CR

Eigenvectors

• Turns out: Eigenvectors of the correlation matrix represent the major and minor axes of an ellipse centered at the origin which encloses the data most compactly

• The SVD of data matrix X uncovers these vectors

• KLT

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Pixel 1

Pix

el 2

Eigenvectors

• Turns out: Eigenvectors of the covariance represent the major and minor axes of an ellipse centered at the mean which encloses the data most compactly

• PCA uncovers these vectors

• In practice, “Eigen faces” refers to PCA faces, and not KLT faces

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Pixel 1

Pix

el 2

What about sound?

• Finding Eigen bases for speech signals:

• Look like DFT/DCT

• Or wavelets

• DFTs are pretty good most of the time

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Eigen Analysis

• Can often find surprising features in your data

• Trends, relationships, more

• Commonly used in recommender systems

• An interesting example..

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Eigen Analysis

• Cheng Liu’s research on pipes.. • SVD automatically separates useful and uninformative

features

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Eigen Analysis

• But for all of this, we need to “preprocess” data

• Eliminate unnecessary aspects

– E.g. noise, other externally caused variations..

16 Sep 2014 11-755/18-797 96

machine learning for signal...

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