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

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 $x_0 \subset \mathbb{R}^n$, we prove that an accurate recovery is already possible 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. Using the concept of restricted strong convexity, we show that their results for the generalized Lasso can be extended to a fairly large class of convex loss functions. Moreover, we shall allow for the presence of adversarial noise so that even non-random model inaccuracies can be coped with. These generalization should be particularly relevant for real-world applications, since specifically adapted loss functions often perform (empirically) better than the classical square loss. On the other hand, this gives a further evidence of why even standard estimators perform quite well in many practical situations, although they do not rely on any knowledge of the (noisy) output scheme. To this end, our results provide a unified and generic framework for signal reconstruction in high dimensions, covering various challenges from the fields of compressed sensing, signal processing, and statistical learning.