Manifold-tiling Localized Receptive Fields are Optimal in Similarity-preserving Neural Networks

Part of Advances in Neural Information Processing Systems 31 (NeurIPS 2018)

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Anirvan Sengupta, Cengiz Pehlevan, Mariano Tepper, Alexander Genkin, Dmitri Chklovskii


Many neurons in the brain, such as place cells in the rodent hippocampus, have localized receptive fields, i.e., they respond to a small neighborhood of stimulus space. What is the functional significance of such representations and how can they arise? Here, we propose that localized receptive fields emerge in similarity-preserving networks of rectifying neurons that learn low-dimensional manifolds populated by sensory inputs. Numerical simulations of such networks on standard datasets yield manifold-tiling localized receptive fields. More generally, we show analytically that, for data lying on symmetric manifolds, optimal solutions of objectives, from which similarity-preserving networks are derived, have localized receptive fields. Therefore, nonnegative similarity-preserving mapping (NSM) implemented by neural networks can model representations of continuous manifolds in the brain.