  Given a length $n$ sample from $\mathbb{R}^d$ and a neural network with a
fixed architecture with $W$ weights, $k$ neurons, linear threshold activation
functions, and binary outputs on each neuron, we study the problem of uniformly
sampling from all possible labelings on the sample corresponding to different
choices of weights. We provide an algorithm that runs in time polynomial both
in $n$ and $W$ such that any labeling appears with probability at least
$\left(\frac{W}{2ekn}\right)^W$ for $W<n$. For a single neuron, we also provide
a random walk based algorithm that samples exactly uniformly.
