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SubXPCA versus PCA: A Theoretical Investigation

Principal Component Analysis (PCA) is a widely accepted dimensionality reduction technique that is optimal in a MSE sense. PCA extracts `global' variations and is insensitive to `local' variations in sub patterns. Recently, we have proposed a novel approach, SubXPCA, which was more effective computationally than PCA and also effective in computing principal components with both global and local information across sub patterns. In this paper, we show the near-optimality of SubXPCA (in terms of summarization of variance) by proving analytically that `SubXPCA approaches PCA with increase in number of local principal components of sub patterns.' This is demonstrated empirically upon CMU Face Data.