Classification of Manifolds by Single-Layer Neural Networks

The neuronal representation of objects exhibit enormous variability due to changes in the object's physical features such as location, size, orientation, and intensity. How the brain copes with the variability across these manifolds of neuronal states and generates invariant perception of objects remains poorly understood. Here we present a theory of neuronal classification of manifolds, extending Gardner's replica theory of classification of isolated points by a single layer perceptron. We evaluate how the perceptron capacity depends on the dimensionality, size and shape of the classified manifolds.