Common Product Neurons

The present work develops a comparative performance of artificial neurons obtained in terms of the recently introduced real-valued Jaccard and coincidence indices and respective functionals. The interiority index and classic cross-correlation are also included in our study. After presenting the basic concepts related to multisets and the adopted similarity metrics, including new results about the generalization of the family of real-valued Jaccard and conicidence indices to higher orders, we proceed to studying the response of a single neuron, not taking into account the output non-linearity (e.g.~sigmoid), respectively to the detection of a gaussian stimulus in presence of displacement, magnification, intensity variation, noise and interference from additional patterns. It is shown that the real-valued Jaccard and coincidence approaches are substantially more robust and effective than the interiority index and the classic cross-correlation. The coincidence based neurons are shown to have the best overall performance for the considered type of data and perturbations. The reported concepts, methods, and results, have substantial implications not only for patter recognition and deep learning, but also regarding neurobiology and neuroscience.