Factorial coding of natural images: how effective are linear models in removing higher-order dependencies?

The performance of unsupervised learning models for natural images is evaluated quantitatively by means of information theory. We estimate the gain in statistical independence (the multi-information reduction) achieved with independent component analysis (ICA), principal component analysis (PCA), zero-phase whitening, and predictive coding. Predictive coding is translated into the transform coding framework, where it can be characterized by the constraint of a triangular filter matrix. A randomly sampled whitening basis and the Haar wavelet are included in the comparison as well. The comparison of all these methods is carried out for different patch sizes, ranging from 2 X 2 to 16 X 16 pixels. In spite of large differences in the shape of the basis functions, we find only small differences in the multi-information between all decorrelation transforms (5% or less) for all patch sizes. Among the second-order methods, PCA is optimal for small patch sizes and predictive coding performs best for large patch sizes. The extra gain achieved with ICA is always less than 2%. In conclusion. the edge filters found with ICA lead to only a surprisingly small improvement in terms of its actual objective. (C) 2006 Optical Society of America.