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

In this paper, we study the problem of estimating a structured signal $x_0 \in \mathbb{R}^n$ from non-linear Gaussian observations. Supposing that $x_0$ belongs to a certain convex subset $K \subset \mathbb{R}^n$, we prove that an accurate recovery is possible as long as 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. Using the concept of restricted strong convexity, we show that their results for the generalized Lass} can be extended to a fairly large class of convex loss functions. This should be particularly relevant for practical applications, since in many real-world scenarios, adapted loss functions empirically perform better than the classical square loss. To this end, our results provide a unified and general framework for signal reconstruction in high dimensions, covering various challenges from the fields of compressed sensing, signal processing, and statistical learning.