Principal component analysis

Questions/things to explain

• Analogously to the covariance matrix one can define a correlation matrix. What happens if you run SVD on the correlation matrix?
• multiple ways to look at PCA:
  • SVD on the covariance matrix (this is probably the same as the maximum variance interpretation, or rather a sub-interpretation of that; if you view the covariance matrix as a transformation that takes white noise to your data set, then the principal components = axes of the ellipsoid = the views that maximize variance)
  • maximum variance (see Bishop). This one uses the Lagrange multiplier and derivative of a quadratic form.
  • minimum-error (see Bishop)
  • the best linear compression-recovery of data to a lower dimension (see Shalev-Shwartz and Ben-David). Is this the same as minimum-error interpretation?
  • maximum-variance and minimum-error are related by the Pythagorean theorem, see page 16 of these slides. There's a similar picture in this post.
  • once you've done PCA, how do you calculate the percentage of variance captured by a principal component? what is the relationship between the percentage variance and the size of the eigenvalue (the larger the eigenvalue, the larger the variance, but what is the specific relationship)?