Data-Driven Structured Noise Filtering via Common Dynamics Estimation

Classical signal from noise separation problems assume that the signal is a trajectory of a low-complexity linear time-invariant system and that the noise is a random process. In this paper, we generalize this classical setup to what we call data-driven structured noise filtering. In the new setup, the noise has two components: structured noise, which is also a trajectory of a low-complexity linear time-invariant system, and unstructured noise, which is a zero-mean white Gaussian process. The key assumption that makes the separation problem in the new setup well posed is that among several experiments, the signal's dynamics remains the same while the structured noise's dynamics varies. The data-driven structured noise filtering problem then becomes a problem of estimation of common linear time-invariant dynamics among several observed signals. We show that this latter problem is a structured low-rank approximation problem with multiple rank constraints and use a subspace identification approach for solving it. The resulting methods allow computationally efficient and numerically robust implementation and have the system theoretic interpretation of finding the intersection of autonomous linear time-invariant behaviors. Statistical analysis providing confidence bounds is a topic for future research.