Canonical Correlation Analysis

1. Canonical Correlation
Analysis: An overview with
application to learning
methods
By David R. Hardoon, Sandor Szedmak, John Shawe-Taylor
School of Electronics and Computer Science, University of
Southampton
Published in Neural Computaion, 2004
Presented by:
Shankar Bhargav

2. Measuring the linear relationship between
two multi dimensional variables
Finding two sets of basis vectors such that
the correlation between the projections of
the variables onto these basis vectors is
maximized
Determine Correlation Coefficients
Canonical Correlation Analysis

3. More than one canonical correlations will
be found each corresponding to a different
set of basis vectors/Canonical variates
Correlations between successively
extracted canonical variates are smaller
and smaller
Correlation coefficients : Proportion of
correlation between the canonical variates
accounted for by the particular variable.
Canonical Correlation Analysis

4. Differences with Correlation
Not dependent on the coordinate system
of variables
Finds direction that yield maximum
correlations

5. Find basis vectors for two sets of variables x, y
such that the correlations between the
projections of the variables onto these basis
vector
Sx = (x.wx) and Sy = (y.wy)
ρ = E[Sx Sy ]
√ E[Sx
2] E[Sy
2]
ρ = E[(xT wx yT wy)]
√E[(xT wx xT wx) ] E[(yT wy yT wy)]

6. ρ = max wx wy E[wx
Tx yT wy]
√E[wx
Tx xT wx ] E[wy
T y yT wy]
ρ = max wx wy wx
TCxy wy
√ wx
TCxxwx wy
TCyy wy
Solving this
with constraint wx
TCxxwx =1
wy
TCyy wy=1

7. Cxx
-1CxyCyy
-1Cyx wx = ρ2 wx
Cyy
-1CyxCxx
-1Cxy wy= ρ2 wy
Cxy wy = ρλx Cxx wx
Cyx wx = ρλy Cyy wy
λx=λy
-1= wy
TCyywy
√ wx
TCxxwx

8. CCA in Matlab
[ A, B, r, U, V ] = canoncorr(x, y)
x, y : set of variables in the form of matrices
 Each row is an observation
 Each column is an attribute/feature
A, B: Matrices containing the correlation coefficient
r : Column matrix containing the canonical
correlations (Successively decreasing)
U, V: Canonical variates/basis vectors for A,B
respectively

9. Interpretation of CCA
Correlation coefficient represents unique
contribution of each variable to relation
Multicollinearity may obscure relationships
Factor Loading : Correlations between the
canonical variates (basis vector) and the
variables in each set
Proportion of variance explained by the
canonical variates can be inferred by
factor loading

10. Redundancy Calculation
Redundancy left =[ ∑ (loadingsleft
2)/p]*Rc
2
Redundancy right =[ ∑ (loadingsright
2)/q]*Rc
2
p – Number of variable in the first (left) set of variables
q – Number of variable in the second (right) set of
variables
Rc2 – Respective squared canonical correlation
Since successively extracted roots are uncorrelated we
can sum the redundancies across all correlations to
get a single index of redundancy.

11. Application
Kernel CCA can be used to find non linear
relationships between multi variates
Two views of the same semantic object to
extract the representation of the semantics
 Speaker Recognition – Audio and Lip
movement
 Image retrieval – Image features (HSV,
Texture) and Associated text

12. Use of KCCA in cross-modal
retrieval
 400 records of JPEG images for each class
with associated text and a total of 3 classes
 Data was split randomly into 2 parts for
training and test
 Features
Image – HSV Color, Gabor texture
Text – Term frequencies
 Results were taken for an average of 10 runs

14. Cross-modal retrieval
Content based retrieval: Retrieve images
in the same class
Tested with 10 and 30 images sets
 where countj
k = 1 if the image k in the set is of
the same label as the text query present in
the set, else countj
k = 0.

15. Comparison of KCCA (with 5 and 30 Eigen
vectors) with GVSM
Content based retrieval

16. `

17. Mate based retrieval
Match the exact image among the
selected retrieved images
Tested with 10 and 30 images sets
 where countj = 1 if the exact matching image
was present in the set else it is 0

19. Comparison of KCCA (with 30 and 150 Eigen
vectors) with GVSM
Mate based retrieval

20. Comments
The good
 Good explanation of CCA and KCCA
 Innovative use of KCCA in image retrieval application
The bad
 The data set and the number of classes used
were small
 The image set size is not taken into account
while calculating accuracy in Mate based
retrieval
 Could have done cross-validation tests

21. Limitations and Assumptions of
CCA
At least 40 to 60 times as many cases as
variables is recommended to get relliable
estimates for two roots– BarciKowski & Stevens(1986)
Outliers can greatly affect the canonical
correlation
Variables in two sets should not be
completely redundant

22. Thank you