Approximating Probability Distributions by ReLU Networks
This work offers incremental theoretical insights into neural network approximation for probability distributions, relevant to researchers in machine learning theory.
The paper addresses how many neurons are needed in ReLU networks to approximate histogram distributions with uniform input, achieving a new upper bound that improves on previous results and providing a lower bound.
How many neurons are needed to approximate a target probability distribution using a neural network with a given input distribution and approximation error? This paper examines this question for the case when the input distribution is uniform, and the target distribution belongs to the class of histogram distributions. We obtain a new upper bound on the number of required neurons, which is strictly better than previously existing upper bounds. The key ingredient in this improvement is an efficient construction of the neural nets representing piecewise linear functions. We also obtain a lower bound on the minimum number of neurons needed to approximate the histogram distributions.