Reconstruction of Signals Drawn from a Gaussian Mixture from Noisy Compressive Measurements

TL;DR

This paper establishes the minimum number of measurements needed to reconstruct signals from a Gaussian mixture model in noisy compressive sensing, providing bounds that depend on the geometric properties of the mixture components.

Contribution

It introduces a new framework for determining measurement bounds based on the geometry of Gaussian mixture components and extends analysis to optimal measurement kernels.

Findings

Bounds are tighter than standard sparse recovery bounds.

Optimal kernels do not reduce the number of measurements for phase transition.

The method reveals the MMSE phase transition and decay characteristics.

Abstract

This paper determines to within a single measurement the minimum number of measurements required to successfully reconstruct a signal drawn from a Gaussian mixture model in the low-noise regime. The method is to develop upper and lower bounds that are a function of the maximum dimension of the linear subspaces spanned by the Gaussian mixture components. The method not only reveals the existence or absence of a minimum mean-squared error (MMSE) error floor (phase transition) but also provides insight into the MMSE decay via multivariate generalizations of the MMSE dimension and the MMSE power offset, which are a function of the interaction between the geometrical properties of the kernel and the Gaussian mixture. These results apply not only to standard linear random Gaussian measurements but also to linear kernels that minimize the MMSE. It is shown that optimal kernels do not change the number of measurements associated with the MMSE phase transition, rather they affect the sensed power required to achieve a target MMSE in the low-noise regime. Overall, our bounds are tighter and sharper than standard bounds on the minimum number of measurements needed to recover sparse signals associated with a union of subspaces model, as they are not asymptotic in the signal dimension or signal sparsity.