Robust Uniform Recovery of Structured Signals from Nonlinear Observations

2026-04-21

AI Summary

This work addresses the problem of uniformly robust recovery of structured signals under nonlinear observations—such as those involving noise or discontinuous link functions—by introducing the Restricted Approximate Invertibility Condition (RAIC) and integrating it with projected gradient descent to achieve uniform recovery guarantees for all structured signals. It establishes, for the first time, a unified theoretical framework for uniform recovery that applies broadly to piecewise Lipschitz nonlinearities, extending prior non-uniform results—limited to specific signals—into a general setting with nearly no loss in accuracy in typical sparse regimes. Leveraging covering number and VC-dimension analyses, the approach uniformly handles scenarios including Gaussian single-index models and 1-bit quantization, attaining error rates comparable to existing non-uniform methods (up to logarithmic factors) for both sparse and approximately sparse signals, and achieving log-factor-free uniform recovery in 1-bit quantization via Iterative Hard Thresholding (IHT).

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

While it is well known that the restricted isometry property (RIP) guarantees uniform sparse recovery from noisy linear measurements, uniform recovery of structured signals from nonlinear observations remains much less understood. This paper shows that the restricted approximate invertibility condition (RAIC) provides a unified approach to this end. Particularly, uniform recovery is achieved by projected gradient descent (PGD) with gradients obeying RAIC for all signals. As an application, under a large class of piecewise Lipschitz link functions (possibly discontinuous), we develop a uniform recovery theory for Gaussian single-index model by establishing the uniform RAIC for the gradient of the (scaled) $\ell_2$ loss via a covering argument. The theory generalizes the nonuniform recovery guarantees due to Plan and Vershynin (2016); Oymak and Soltanolkotabi (2017) and exhibits additional error terms that can be interpreted as the cost of uniform recovery. Intriguingly, in the three canonical settings of (a) sparse recovery via PGD with $\ell_0$ projection (i.e., iterative hard thresholding (IHT)), (b) sparse recovery via PGD with $\ell_1$ projection, and (c) recovering approximately sparse signals via PGD with $\ell_1$ projection, the additional error terms are negligible and in turn our uniform recovery error rates are at the same order of existing nonuniform ones, up to log factors. Our results hence improve on Genzel and Stollenwerk (2023). Under the specific nonlinearity of 1-bit quantization, we use a VC dimension argument to show that the uniform recovery error of IHT is at the same order of the nonuniform recovery error, with no loss of log factor. In addition, we show that the robustness of PGD to noise and corruption can be incorporated elegantly by bounding a single additional random process that captures the gradient mismatch.

Problem

• uniform recovery
• structured signals
• nonlinear observations
• restricted approximate invertibility condition
• robust recovery

Innovation

• Restricted Approximate Invertibility Condition
• Uniform Recovery
• Nonlinear Observations
• Projected Gradient Descent
• Single-Index Model