Just Least Squares: Binary Compressive Sampling with Low Generative Intrinsic Dimension

Abstract

In this paper, we consider recovering $$n-$$ dimensional signals from m binary measurements corrupted by noises and sign flips under the assumption that the target signals have low generative intrinsic dimension, i.e., the target signals can be approximately generated via an L-Lipschitz generator $$G: \mathbb {R}^k\rightarrow \mathbb {R}^{n}, k\ll n$$ . Although the binary measurements model is highly nonlinear, we propose a least square decoder and prove that, up to a constant c, with high probability, the least square decoder achieves a sharp estimation error $$C\sqrt{\frac{k\log (Ln)}{m}}$$ as long as $$m\ge C( k\log (Ln))$$ . Extensive numerical simulations and comparisons with state-of-the-art methods demonstrated the least square decoder is robust to noise and sign flips, as indicated by our theory. By constructing a ReLU network with properly chosen depth and width, we verify the (approximately) deep generative prior, which is of independent interest.