  In this on-going work, I explore certain theoretical and empirical
implications of data transformations under the PCA. In particular, I state and
prove three theorems about PCA, which I paraphrase as follows: 1). PCA without
discarding eigenvector rows is injective, but looses this injectivity when
eigenvector rows are discarded 2). PCA without discarding eigen- vector rows
preserves pair-wise distances, but tends to cause pair-wise distances to shrink
when eigenvector rows are discarded. 3). For any pair of points, the shrinkage
in pair-wise distance is bounded above by an L1 norm reconstruction error
associated with the points. Clearly, 3). suggests that there might exist some
correlation between shrinkages in pair-wise distances and mean square
reconstruction error which is defined as the sum of those eigenvalues
associated with the discarded eigenvectors. I therefore decided to perform
numerical experiments to obtain the corre- lation between the sum of those
eigenvalues and shrinkages in pair-wise distances. In addition, I have also
performed some experiments to check respectively the effect of the sum of those
eigenvalues and the effect of the shrinkages on classification accuracies under
the PCA map. So far, I have obtained the following results on some publicly
available data from the UCI Machine Learning Repository: 1). There seems to be
a strong cor- relation between the sum of those eigenvalues associated with
discarded eigenvectors and shrinkages in pair-wise distances. 2). Neither the
sum of those eigenvalues nor pair-wise distances have any strong correlations
with classification accuracies. 1
