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Presented by: Shankar Bhargav

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.

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Presented by: Shankar Bhargav

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  1. Canonical Correlation Analysis: An overview with application to learning methodsBy David R. Hardoon, Sandor Szedmak, John Shawe-TaylorSchool of Electronics and Computer Science, University of SouthamptonPublished in Neural Computaion, 2004 Presented by: Shankar Bhargav

  2. Canonical Correlation Analysis • 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

  3. Canonical Correlation Analysis • 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.

  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[Sx2] E[Sy2] ρ = E[(xT wx yT wy)] √E[(xT wx xT wx) ] E[(yT wy yT wy)]

  6. ρ = max wx wy E[wxTxyT wy] √E[wxTx xT wx ] E[wyT yyT wy] ρ = max wx wy wxTCxywy √ wxTCxxwx wyTCyywy Solving this with constraint wxTCxxwx=1 wyTCyywy=1

  7. Cxx-1CxyCyy-1Cyx wx = ρ2 wx Cyy-1CyxCxx-1Cxy wy= ρ2 wy Cxy wy = ρλxCxx wx Cyx wx = ρλy Cyywy λx=λy-1= wyTCyywy √wxTCxxwx

  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 =[ ∑ (loadingsleft2)/p]*Rc2 • Redundancy right =[ ∑ (loadingsright2)/q]*Rc2 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

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

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

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  16. 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

  17. Comparison of KCCA (with 30 and 150 Eigen vectors) with GVSMMate based retrieval

  18. 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

  19. 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

  20. Thank you

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