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Nonparametric Canonical Correlation Analysis

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Presented by: Ev Zisselman

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Motivation

To find correlating variable in multi-view data

      • Johny is happy

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      • Johny is sad

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      • Johny is angry

View 1

View 2

f(X)

g(Y)

L Dim

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Motivation

  • The method proposed extends linear CCA to Non-linear projections.

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  • Previous variants had been proposed, such as
    • KCCA
    • DCCA

Downside: Restrict the projections to a specific family of functions. �

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2-view NCCA

Find f(x) and g(y) such that:

 

 

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2-view NCCA

Consider and as L scalar function:

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We represent:

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2-view NCCA

Express the optimization problem as inner product over one Hilbert space:

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2-view NCCA

Express the optimization problem as inner product over one Hilbert space:

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2-view NCCA

Express the optimization problem as inner product over one Hilbert space:

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2-view NCCA

Define operator by , and the optimization problem can be written as:

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2-view NCCA

Define operator by , and the optimization problem can be written as:

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The optimal projections are the singular function of the SVD of S:

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And the optimal value is .

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2-view NCCA Results

View 1

View 2

View 1

View 2

MNIST

NKCCA

DCCA

PLCCA

NCCA

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M-view NCCA - motivation

View 1

View 2

  • Happy puppies

View 3

L Dim

f(X)

g(Y)

h(z)

Extension of two-view NCCA to three or more views.

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M-view CCA

Consider m sets of variable (m views):

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M-view CCA

Consider m sets of variable (m views):

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First, assume L=1, derive the vector

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M-view CCA

Consider m sets of variable (m views):

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First, assume L=1, derive the vector

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Where is unit variance linear combination of : .

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M-view CCA

  •  

 

 

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M-view CCA

The optimal maximizes in one of the following:

  1. Maximize the sum of elements .
  2. Maximize the largest eigenvalue of .
  3. Maximize the sum of element squares or the sum of the eigenvalue squares.
  4. Minimize the smallest eigenvalue of .
  5. Minimize the product of the eigenvalue .

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M-view CCA

Now for L>1, we concatenate the views:

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M-view CCA

Now for L>1, we concatenate the views:

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Derive the vector of the s-order canonical variants :

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Where is block diagonal matrix:

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M-view CCA

Now for L>1, we concatenate the views:

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Derive the vector of the s-order canonical variants :

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Where is block diagonal matrix:

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And the covariance matrix of

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M-view CCA

Now for L>1, we concatenate the views:

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Derive the vector of the s-order canonical variants :

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Where is block diagonal matrix:

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And the covariance matrix of

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M-view CCA

The non-singular solution satisfies the following restriction for :

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Impose constraint on :

 

 

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Questions?