Olshausen and Field applied the principle of independence maximization by sparse coding to extract features from natural images. This leads to the emergence of oriented linear filters that have simultaneous localization in space and in frequency,thus resembling Gabor functions and simple cell receptive fields. In this paper, we show that the same principle of independence maximization can explain the emergence of phase and shift invariant features, similar to those found in complex cells. Thisnew kind of emergence is obtained by maximizing the independence between norms of projections on linear subspaces (instead of the independence of simple linear filter outputs). The norms of the projections on such "independent feature subspaces" thenindicate the values of invariant features.
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