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 :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 O( k log(Ln) m ) as long as m ≥ O(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.

Theoretical Perspectives on Deep Learning Methods in Inverse Problems

“…Non-linear measurement models. While Theorem 1 concerns linear observation models, analogous guarantees have been provided for a variety of non-linear measurement models, including 1-bit observations [85], [105], [69], spiked matrix models [7], [28], phase retrieval [84], [53], principal component analysis [87], and general single-index models [88], [83], [86]. While these each come with their own challenges, the intuition behind their associated results is often similar to that discussed above for the linear model, with the m = O k log Lr δ scaling typically remaining.…”

Theoretical Perspectives on Deep Learning Methods in Inverse Problems

“…Non-linear measurement models. While Theorem 1 concerns linear observation models, analogous guarantees have been provided for a variety of non-linear measurement models, including 1-bit observations [78], [98], [65], spiked matrix models [6], [25], phase retrieval [77], [49], principal component analysis [80], and general single-index models [81], [76], [79]. While these each come with their own challenges, the intuition behind their associated results is often similar to that discussed above for the linear model, with the m = O k log Lr δ scaling typically remaining.…”