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A long standing question of biological vision research is to identify the computational goal underlying the response properties of sensory neurons in the early visual system. Some response properties of visual neurons such as bandpass filtering and contrast gain control have been shown to exhibit a clear advantage in terms of redundancy reduction. The situation is less clear in the case of complex cells whose defining property is that of phase invariance. While it has been shown that complex cells can be learned based on the redundancy reduction principle by means of subspace ICA [Hyvärinen Hoyer 2000], the resulting gain in redundancy reduction is very small [Sinz, Simoncelli, Bethge 2010]. Slow feature analysis (SFA, [WiskottSejnowski 2002]) advocates an alternative objective function which does not seek to fit a density model but constitutes a special case of oriented PCA by maximizing the signal to noise ratio when treating temporal changes as noise.Here we set out to evaluate SFA with respect to two important empirical properties of complex cells RFs: 1) locality (i.e. finite, non-zero RF bandwidth) and 2) the relationship between RF bandwidth and RF spatial frequency (wavelet scaling). To this end we use an approach similar to that employed by [Field 1987] for sparse coding. Instead of single Gabor functions, however, we use the energy model of complex cells which is built with a (quadrature) pair of even and odd symmetric Gabor filters. We evaluate the objective function of SFA on the energy model responses to motion sequences of natural images for different spatial frequencies and envelope sizeswith patch sizes ranging from 16x16 to 512x512.We find that the objective function of SFA grows without bound for increasing envelope size and is only limited by a finite patch size (see Figure, solid line). Consequently, SFA by itself cannot explain spatially localized RFs but would need to evoke other mechanisms such as anatomical wiring constraints to limit the RF bandwidth. It is unlikely, however, that such anatomical constraints are able to reproduce the relationship between bandwidth and spatial frequency.In contrast to SFA, the objective function of subspace ICA yields a clear optimum for finite, non-zerobandwidth, regardless of assumed patch size (see Figure, dashed line). In particular, the optimum bandwidth is proportional to spatial frequency - just as observed for physiologically measured RFs in primary visual cortex of cat [Field Tolhust 1986] and monkey ([Ringach 2002], histogram see Figure).We conclude that SFA fails to reproduce important features of complex cells. In contrast, the RF bandwidth predicted by subspace ICA lies well within the range of physiologically measured receptive field bandwidths. As a consequence, if we interpret complex cell coding as a step towards building an invariant representation, the underlying algorithm is more likely to resemble a sparse coding strategy as employed by subspace ICA than the covariance based learning rule employed by SFA.
