The generalized Lasso with non-linear observations (1502.04071v2)

Abstract: We study the problem of signal estimation from non-linear observations when the signal belongs to a low-dimensional set buried in a high-dimensional space. A rough heuristic often used in practice postulates that non-linear observations may be treated as noisy linear observations, and thus the signal may be estimated using the generalized Lasso. This is appealing because of the abundance of efficient, specialized solvers for this program. Just as noise may be diminished by projecting onto the lower dimensional space, the error from modeling non-linear observations with linear observations will be greatly reduced when using the signal structure in the reconstruction. We allow general signal structure, only assuming that the signal belongs to some set K in R^n. We consider the single-index model of non-linearity. Our theory allows the non-linearity to be discontinuous, not one-to-one and even unknown. We assume a random Gaussian model for the measurement matrix, but allow the rows to have an unknown covariance matrix. As special cases of our results, we recover near-optimal theory for noisy linear observations, and also give the first theoretical accuracy guarantee for 1-bit compressed sensing with unknown covariance matrix of the measurement vectors.

• The paper analyzes signal estimation via generalized Lasso using non-linear observations by approximating them as noisy linear measurements.
• Theoretical results show recovery error bounds depend on the signal's structural complexity and the number of observations through Gaussian mean width.
• The work extends compressed sensing theory to handle various non-linearities and provides theoretical backing for applied signal processing methods.

An Analysis of The Generalized Lasso Under Non-Linear Observations

The paper by Yaniv Plan and Roman Vershynin investigates the problem of signal estimation through the generalized Lasso in scenarios where observations are non-linear, yet the underlying signal structure is embedded within a low-dimensional subspace of a high-dimensional space. This work addresses a significant challenge in signal processing and statistical estimation, particularly when the measurement processes introduce non-linearities that could hinder classical linear recovery methods.

At the core of the paper is the heuristic whereby non-linear observations are approximated as noisy linear observations. This approximation is practically appealing as it leverages the extensive toolkit available for solving linear inverse problems, particularly through the generalized Lasso framework. The authors focus on how the nature of non-linear measurements translates into a scaled and perturbed linear observation model, subsequently analyzed using robust statistical and geometric methods.

Theoretical results are established under the assumption that the measurement matrix follows a Gaussian distribution with potential unknown covariance. This includes models where data is obtained through structured linear models, a premise notable for its applicability across numerous domains. Within this context, the estimation of the signal is facilitated via minimization of a modified $\ell_2$ loss, constrained within a defined set $K$, reflecting the signal's structural assumptions such as sparsity or low-rank properties.

The paper's key insight lies in its ability to characterize the signal recovery process through the lens of Gaussian mean width, a concept deeply rooted in convex geometry and statistical dimension theory. The results elucidate how the error in signal recovery through the generalized Lasso is dictated by the complexity of the set $K$, as captured by the tangent cone's mean width and how this links to the number of observations. Specifically, the analysis demonstrates that the generalized Lasso can effectively diminish errors introduced by treating non-linear observations as linear when combined with an appropriate structuring of the signal.

A particular strength of the paper is its rigorous extension of current linear compressed sensing frameworks to encompass discontinuous, non-one-to-one, and even unknown non-linearities using a minimal set of assumptions. This adaptability significantly broadens the applicability of compressed sensing techniques to more general settings, including those in 1-bit compressed sensing, often constrained by hardware limitations requiring binary data collection.

Moreover, the authors do not only restate known theory under linear conditions as a special case but also provide novel insights into scenarios involving non-linear measurements. This extension represents a crucial step in strengthening the theoretical foundations of signal processing techniques in such adversarial setups and paves the way for analyzing problems with inherent non-linearities more comprehensively.

The implications of these findings are broad, providing a theoretical backing for practitioners who have been implicitly using linear techniques for inherently non-linear datasets. It additional holds promise for practical applications in sectors where data is naturally or otherwise converted into non-linear forms, such as communications engineering and medical imaging.

This paper establishes a pathway for future works in areas like robust signal estimation in unknown possibly adversarial environments and the development of strategies for optimizing quantization schemes using theoretical performance bounds derived from these results. Future developments could further explore generalizations to other stochastic process-driven measurement matrices and assessment of these results' empirical universality across varying non-linear model classes.