A Mathematical Framework for Feature Selection from Real-World Data with Non-Linear Observations

Abstract: In this paper, we study the challenge of feature selection based on a relatively small collection of sample pairs ${(x_i, y_i)}{1 \leq i \leq m}$. The observations $y_i \in \mathbb{R}$ are thereby supposed to follow a noisy single-index model, depending on a certain set of signal variables. A major difficulty is that these variables usually cannot be observed directly, but rather arise as hidden factors in the actual data vectors $x_i \in \mathbb{R}^d$ (feature variables). We will prove that a successful variable selection is still possible in this setup, even when the applied estimator does not have any knowledge of the underlying model parameters and only takes the 'raw' samples ${(x_i, y_i)}{1 \leq i \leq m}$ as input. The model assumptions of our results will be fairly general, allowing for non-linear observations, arbitrary convex signal structures as well as strictly convex loss functions. This is particularly appealing for practical purposes, since in many applications, already standard methods, e.g., the Lasso or logistic regression, yield surprisingly good outcomes. Apart from a general discussion of the practical scope of our theoretical findings, we will also derive a rigorous guarantee for a specific real-world problem, namely sparse feature extraction from (proteomics-based) mass spectrometry data.