High-Dimensional Estimation of Structured Signals from Non-Linear Observations with General Convex Loss Functions

Understand

In this paper, we study the issue of estimating a structured signal $x_0 \in \mathbb{R}^n$ from non-linear and noisy Gaussian observations.

• Supposing that $x_0$ is contained in a certain convex subset $K \subset \mathbb{R}^n$, we prove that accurate recovery is already feasible if the number of observations exceeds the effective dimension of $K$, which is a common measure for the complexity of signal classes.
• It will turn out that the possibly unknown non-linearity of our model affects the error rate only by a multiplicative constant.
• This achievement is based on recent works by Plan and Vershynin, who have suggested to treat the non-linearity rather as noise which perturbs a linear measurement process.