Emergence of Lie Symmetries in Functional Architectures Learned by CNNs

Abstract: Convolutional Neural Networks (CNNs) are a powerful tool providing outstanding performances on image classification tasks, based on an architecture designed in analogy with information processing in biological visual systems. The functional architectures of the early visual pathways have often been described in terms of geometric invariances, and several studies have leveraged this framework to investigate the analogies between CNN models and biological mechanisms. Remarkably, upon learning on natural images, the translation-invariant filters of the first layer of a CNN have been shown to develop as approximate Gabor functions, resembling the orientation-selective receptive profiles found in the primary visual cortex (V1). With a similar approach, we modified a standard CNN architecture to insert computational blocks compatible with specific biological processing stages, and studied the spontaneous development of approximate geometric invariances after training the network on natural images. In particular, inserting a pre-filtering step mimicking the Lateral Geniculate Nucleus (LGN) led to the emergence of a radially symmetric profile well approximated by a Laplacian of Gaussian, which is a well-known model of receptive profiles of LGN cells. Moreover, we introduced a lateral connectivity kernel acting on the feature space of the first network layer. We then studied the learned connectivity as a function of relative tuning of first-layer filters, thus re-mapping it into the roto-translation space. This analysis revealed orientation-specific patterns, which we compared qualitatively and quantitatively with established group-based models of V1 horizontal connectivity.

machine learningMathematics

Audience: researchers in the discipline

Mathematics and Machine Learning

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