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In this on-going work, I explore certain theoretical and empiricalimplications of data transformations under the PCA. In particular, I state andprove three theorems about PCA, which I paraphrase as follows: 1). PCA withoutdiscarding eigenvector rows is injective, but looses this injectivity wheneigenvector rows are discarded 2). PCA without discarding eigen- vector rowspreserves pair-wise distances, but tends to cause pair-wise distances to shrinkwhen eigenvector rows are discarded. 3). For any pair of points, the shrinkagein pair-wise distance is bounded above by an L1 norm reconstruction errorassociated with the points. Clearly, 3). suggests that there might exist somecorrelation between shrinkages in pair-wise distances and mean squarereconstruction error which is defined as the sum of those eigenvaluesassociated with the discarded eigenvectors. I therefore decided to performnumerical experiments to obtain the corre- lation between the sum of thoseeigenvalues and shrinkages in pair-wise distances. In addition, I have alsoperformed some experiments to check respectively the effect of the sum of thoseeigenvalues and the effect of the shrinkages on classification accuracies underthe PCA map. So far, I have obtained the following results on some publiclyavailable data from the UCI Machine Learning Repository: 1). There seems to bea strong cor- relation between the sum of those eigenvalues associated withdiscarded eigenvectors and shrinkages in pair-wise distances. 2). Neither thesum of those eigenvalues nor pair-wise distances have any strong correlationswith classification accuracies. 1

Correlation of Data Reconstruction Error and Shrinkages in Pair-wise Distances under Principal Component Analysis (PCA)