The generalized Lasso with non-linear observations

TL;DR

This paper develops a theoretical framework for signal estimation from non-linear observations using the generalized Lasso, accommodating complex non-linearities and unknown measurement covariances, with applications to 1-bit compressed sensing.

Contribution

It extends the generalized Lasso approach to handle non-linear, discontinuous, and unknown measurement models, providing near-optimal guarantees and new insights for 1-bit compressed sensing.

Findings

The theory applies to non-linear, discontinuous observations.

Provides near-optimal guarantees for noisy linear models.

First accuracy guarantee for 1-bit compressed sensing with unknown covariance.

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.