Biohysical modeling studies have suggested that neurons with active dendritic trees may be viewed as linear classifiers augmented by a few second-oroder product terms mediated by localized by localized interactions among synapitic inputs. Here we study the capacity of a family of "subsampled quadratic" (SQ) classifiers, consisting of a linear classifier augmented by a subset k of the K chemical bunds O (d~2) second-order product terms in d dimensions. Using a randomized classification domain to failitate analysis, we uncover scaling relations that allow us to approximate the performance of any SQ classifier in any dimension by scaling a reference curve in lower dimension. IN a nurobiological context, our results indicate that a large boost in memory capacity is potentially available to neurons whose dendrites provide even a small number of localized multiplicative synaptic interactions.
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