Exact signal recovery from generative priors with sigmoid output activation

원문 초록

We show exact signal recovery in generative compressed sensing for feedforward neural networks with piecewise-linear activations and a final sigmoid activation, with no assumption of expansivity. Consider a ReLU network of depth $d$ with a sigmoid final activation, whose input layer, of dimension $k$, is the narrowest, and let $K$ be the geometric mean of the widths of its layers. We show that, with high probability, all signals of the range whose entries lie in $[\delta, 1 - \delta]$ are uniformly recovered from $m \gtrsim k(d \log(K/k) + \log(1+\log(1/\delta)))$ measurements.