Nearest Neighbor Representations of Neurons
This work addresses computational complexity in neural modeling, offering potential efficiency gains for representing neurons, though it is incremental in nature.
The paper investigates the trade-off between the number of anchors and resolution in nearest neighbor representations of threshold functions, showing that functions like EQUALITY, COMPARISON, and ODD-MAX-BIT can be represented with polynomially many anchors and O(log n) resolution, improving from O(n log n).
The Nearest Neighbor (NN) Representation is an emerging computational model that is inspired by the brain. We study the complexity of representing a neuron (threshold function) using the NN representations. It is known that two anchors (the points to which NN is computed) are sufficient for a NN representation of a threshold function, however, the resolution (the maximum number of bits required for the entries of an anchor) is $O(n\log{n})$. In this work, the trade-off between the number of anchors and the resolution of a NN representation of threshold functions is investigated. We prove that the well-known threshold functions EQUALITY, COMPARISON, and ODD-MAX-BIT, which require 2 or 3 anchors and resolution of $O(n)$, can be represented by polynomially large number of anchors in $n$ and $O(\log{n})$ resolution. We conjecture that for all threshold functions, there are NN representations with polynomially large size and logarithmic resolution in $n$.