Compressed Sensing with Nonlinear Observations and Related Nonlinear Optimisation Problems (1205.1650v1)

Abstract: Non-convex constraints have recently proven a valuable tool in many optimisation problems. In particular sparsity constraints have had a significant impact on sampling theory, where they are used in Compressed Sensing and allow structured signals to be sampled far below the rate traditionally prescribed. Nearly all of the theory developed for Compressed Sensing signal recovery assumes that samples are taken using linear measurements. In this paper we instead address the Compressed Sensing recovery problem in a setting where the observations are non-linear. We show that, under conditions similar to those required in the linear setting, the Iterative Hard Thresholding algorithm can be used to accurately recover sparse or structured signals from few non-linear observations. Similar ideas can also be developed in a more general non-linear optimisation framework. In the second part of this paper we therefore present related result that show how this can be done under sparsity and union of subspaces constraints, whenever a generalisation of the Restricted Isometry Property traditionally imposed on the Compressed Sensing system holds.

• The paper extends Compressed Sensing (CS) theory to scenarios involving nonlinear observations and non-convex optimization problems.
• It demonstrates that sparse signal recovery is possible with nonlinear measurements by applying concepts like the Restricted Isometry Property (RIP) to the system's Jacobian.
• The work has practical implications for real-world applications where linear measurements are challenging, enabling wider use of CS in fields like medical imaging and sensor networks.

Compressed Sensing with Nonlinear Observations and Related Nonlinear Optimisation Problems

This paper by Thomas Blumensath extends the traditional theory of Compressed Sensing (CS) into the field of nonlinear observations and optimization under non-convex constraints. Traditional CS methods assume linear measurements, where sparsity constraints are used to reconstruct signals sampled below the Nyquist rate. This paper challenges the linearity assumption by exploring scenarios where samples are acquired through nonlinear measurements. The research focuses on establishing that recovery of sparse or structured signals remains feasible under certain conditions when the Iterative Hard Thresholding (IHT) algorithm is applied.

Key contributions of the paper lie in generalizing CS by developing a framework for nonlinear observations, thereby broadening the applicability of compressed sensing techniques. The paper demonstrates that by utilizing concepts from the linear CS theory, notably the Restricted Isometry Property (RIP), accurate signal recovery is achievable even with nonlinear systems, provided these systems adhere to similar constraints as their linear counterparts. Specifically, the author explores conditions under which a nonlinear measurement system's Jacobian satisfies the RIP, allowing the IHT algorithm to function effectively.

The paper further explores a generalized nonlinear optimization context, introducing the notion of a Union of Subspaces and extending the applicability of IHT in these environments. By examining general non-convex constraints, the author provides theoretical guarantees for the recovery of signals, which may not be strictly sparse but lie in specified subspaces. The interplay between nonlinearity and non-convex constraints is treated rigorously, with the work showing that results akin to those in linear settings can be obtained if the nonlinear function adheres to a generalized RIP.

Critical results assert that if a nonlinear function $f(x) = ||y - \Phi(x)||$, where $\Phi$ is a nonlinear mapping, satisfies a form of the RIP, then signal recovery via IHT can be as robust as traditional linear scenarios. Importantly, the author provides a detailed analysis of error bounds, conditions for convergence, and the computational implications of extending CS into the nonlinear domain. The paper showcases numerical stability and convergence behavior of the IHT algorithm under these broadened conditions.

Furthermore, the paper investigates the Restricted Strong Convexity Property for general nonlinear optimization under non-convex constraints. The author demonstrates that with this property, the iterative approach can still approximate the optimal solution with provable bounds, thus opening new avenues for solving inverse problems and optimization tasks beyond the classical CS framework.

Blumensath's work has practical implications for scenarios where precise linear measurements are challenging or infeasible, such as in real-world applications involving sensor non-linearities and hardware limitations. This extension of CS promises wider applicability in fields such as medical imaging, telecommunication, and sensor networks where measurements cannot always be ensured to align linearly.

Looking forward, these findings invite further exploration into the robustness and adaptability of CS techniques across diverse and complex signal acquisition contexts. Moreover, the methodological integration of nonlinear optimization with non-convex constraints enriches this field's theoretical landscape, potentially inspiring novel algorithmic developments tailored to emerging technological demands. As applied technology and computational methodologies advance, the fusion of nonlinear theory with CS principles stands to significantly enhance the fidelity and efficiency of signal processing systems across a spectrum of disciplines.