Distortion-based achievability conditions for joint estimation of sparse signals and measurement parameters from undersampled acquisitions

In this paper, we consider an undersampling system model of the form y = A(T(x, θ)) + n, where x is a k-sparse signal, T(·, · is a (possibly non-linear) function specified by a parameter vector θ and acting on x, A is a sensing matrix, and n is additive measurement noise. We consider an information theoretic decoder that aims to recover the sparse signal and the transformation parameter vector jointly, and study the achievability conditions for estimating the underlying signal within a specified ℓ₂ distortion for Gaussian sensing matrices. We compare the achievable distortion of the joint estimation process to that of the standard noisy compressed sensing model, where the sparse signal is directly measured with a sensing matrix with the same number of measurements. We also provide a numerical example to illustrate potential applications.