  Signal extraction out of background noise is a common challenge in high
precision physics experiments, where the measurement output is often a
continuous data stream. To improve the signal to noise ratio of the detection,
witness sensors are often used to independently measure background noises and
subtract them from the main signal. If the noise coupling is linear and
stationary, optimal techniques already exist and are routinely implemented in
many experiments. However, when the noise coupling is non-stationary, linear
techniques often fail or are sub-optimal. Inspired by the properties of the
background noise in gravitational wave detectors, this work develops a novel
algorithm to efficiently characterize and remove non-stationary noise
couplings, provided there exist witnesses of the noise source and of the
modulation. In this work, the algorithm is described in its most general
formulation, and its efficiency is demonstrated with examples from the data of
the Advanced LIGO gravitational wave observatory, where we could obtain an
improvement of the detector gravitational wave reach without introducing any
bias on the source parameter estimation.
