איתי ספרן

We study the approximation of the median of d inputs using ReLU neural networks. We present depth-width tradeoffs under several settings, culminating in a constant-depth, linear-width con¬struction that achieves exponentially small approximation error with respect to the uniform distri¬bution over the unit hypercube. By further establishing a general reduction from the maximum to the median, our results break a barrier suggested by prior work on the maximum function, which indicated that linear width should require depth growing at least as log log d to achieve comparable accuracy. Our construction relies on a multi-stage procedure that iteratively eliminates non-central elements while preserving a candidate set around the median. We overcome obstacles that do not arise for the maximum to yield approximation results that are strictly stronger than those previously known for the maximum itself.