Robust one-bit compressed sensing under multiplicative noise

Abstract This paper studies one-bit compressed sensing with multiplicative measurement noise $w$. For the pure multiplicative noise model, the zero-threshold sign quantizer reduces the effect of the multiplicative perturbation to its sign. Building on this observation, we characterize the resulting population correlation in terms of the sign-bias coefficient \(\kappa_w:=\mathbb E[\operatorname{sign}(w)]\) and establish nonasymptotic directional recovery guarantees for sparse signals. We further derive minimax sample-complexity lower bounds, demonstrating that the dependence on the sign-bias magnitude is intrinsic up to logarithmic factors. We instantiate this framework for several continuous sign-changing multiplicative noise models, including Gaussian, shifted Laplace, shifted logistic, and shifted Cauchy models, and show how the noise level affects the recovery error through the induced sign bias. For the mixed multiplicative-additive noise model with bounded additive perturbations, the multiplicative perturbation can no longer be reduced to its sign alone. Under Gaussian multiplicative noise, we prove that the effective population correlation is bounded from below by the pure multiplicative correlation minus a perturbation term, yielding a sufficient robustness condition and corresponding recovery guarantees. Numerical experiments for both pure and mixed noise models support the predicted sign-bias dependence and the robustness of the proposed correlation-based method.