Recovering Latent Signals from a Mixture of Measurements using a Gaussian Process Prior

TL;DR

This paper introduces GPMM, a Bayesian method using Gaussian processes to recover latent signals from noisy, blurred, and mixed measurements, outperforming standard GP in various signal recovery tasks.

Contribution

The paper proposes GPMM, a novel Bayesian framework that models latent signals as Gaussian processes and handles mixed, noisy measurements for improved signal recovery.

Findings

GPMM outperforms standard GP in estimation accuracy.

GPMM effectively captures uncertainty in signal recovery.

GPMM successfully recovers spectral content of signals.

Abstract

In sensing applications, sensors cannot always measure the latent quantity of interest at the required resolution, sometimes they can only acquire a blurred version of it due the sensor's transfer function. To recover latent signals when only noisy mixed measurements of the signal are available, we propose the Gaussian process mixture of measurements (GPMM), which models the latent signal as a Gaussian process (GP) and allows us to perform Bayesian inference on such signal conditional to a set of noisy mixture of measurements. We describe how to train GPMM, that is, to find the hyperparameters of the GP and the mixing weights, and how to perform inference on the latent signal under GPMM; additionally, we identify the solution to the underdetermined linear system resulting from a sensing application as a particular case of GPMM. The proposed model is validated in the recovery of three signals: a smooth synthetic signal, a real-world heart-rate time series and a step function, where GPMM outperformed the standard GP in terms of estimation error, uncertainty representation and recovery of the spectral content of the latent signal.