Slowness and Sparseness Have Diverging Effects on Complex Cell Learning

Abstract

Following earlier studies which showed that a sparse coding principle may explain the receptive field properties of complex cells in primary visual cortex, it has been concluded that the same properties may be equally derived from a slowness principle. In contrast to this claim, we here show that slowness and sparsity drive the representations towards substantially different receptive field properties. To do so, we present complete sets of basis functions learned with slow subspace analysis (SSA) in case of natural movies as well as translations, rotations, and scalings of natural images. SSA directly parallels independent subspace analysis (ISA) with the only difference that SSA maximizes slowness instead of sparsity. We find a large discrepancy between the filter shapes learned with SSA and ISA. We argue that SSA can be understood as a generalization of the Fourier transform where the power spectrum corresponds to the maximally slow subspace energies in SSA. Finally, we investigate the trade-off between slowness and sparseness when combined in one objective function. A key question in visual neuroscience is how neural representations achieve invariance against appearance changes of objects. In particular, the invariance of complex cell responses in primary visual cortex against small translations is commonly interpreted as a signature of an invariant coding strategy possibly originating from an unsupervised learning principle. Various models have been proposed to explain the response properties of complex cells using a sparsity or a slowness criterion and it has been concluded that physiologically plausible receptive field properties can be derived from either criterion. Here, we show that the effect of the two objectives on the resulting receptive field properties is in fact very different. We conclude that slowness alone cannot explain the filter shapes of complex cells and discuss what kind of experimental measurements could help us to better asses the role of slowness and sparsity for complex cell representations.