Learning Visual Spatial Pooling by Strong PCA Dimension Reduction

In visual modeling, invariance properties of visual cells are often explained by a pooling mechanism, in which outputs of neurons with similar selectivities to some stimulus parameters are integrated so as to gain some extent of invariance to other parameters. For example, the classical energy model of phase-invariant V1 complex cells pools model simple cells preferring similar orientation but different phases. Prior studies, such as independent subspace analysis, have shown that phase-invariance properties of V1 complex cells can be learned from spatial statistics of natural inputs. However, those previous approaches assumed a squaring nonlinearity on the neural outputs to capture energy correlation; such nonlinearity is arguably unnatural from a neurobiological viewpoint but hard to change due to its tight integration into their formalisms. Moreover, they used somewhat complicated objective functions requiring expensive computations for optimization. In this study, we show that visual spatial pooling can be learned in a much simpler way using strong dimension reduction based on principal component analysis. This approach learns to ignore a large part of detailed spatial structure of the input and thereby estimates a linear pooling matrix. Using this framework, we demonstrate that pooling of model V1 simple cells learned in this way, even with nonlinearities other than squaring, can reproduce standard tuning properties of V1 complex cells. For further understanding, we analyze several variants of the pooling model and argue that a reasonable pooling can generally be obtained from any kind of linear transformation that retains several of the first principal components and suppresses the remaining ones. In particular, we show how the classic Wiener filtering theory leads to one such variant.