# Data-driven signal processing using the nuclear norm heuristic

Applications in signal processing and control theory are typically model-based and proceed in two steps. In the 'modeling' step, a mathematical model is built from the measured noisy data. In the 'design' step, the model is used to solve a specific application problem. In this project, we will explore 'data-driven methods', in which modeling and design are combined into a single task. By combining modelling and design into a single 'direct’ problem, better solutions can be found than by solving the two problems separately.

The tools that we employ are structured low-rank matrix approximation and structured low-rank matrix completion. In the context of linear dynamics, low-rank matrices come with a specific structure that captures time-invariance and linearity, namely the (block) Hankel matrix (having repeated elements along the anti-diagonals). Removing noise from signals is related to finding the nearest low-rank Hankel-structured matrix, viz. the nearest LTI system that explains the observed data. Low-rank matrix completion appears in two ways: Firstly, there can be missing data because of sensor outages or problematic communication channels. Secondly, data completion can be used to find entire unknown signals, which allows us to come up with data-driven variants of classical problems such as system simulation (input u is given, output y is unknown), output tracking control (output y is given, input u is unknown), or for building a Kalman filter which supports missing elements in the data set. We investigate the use of the nuclear norm for the above problems. Our initial findings on the data-driven simulation problem suggest that, when using an adequate rescaling of the given data, the exact data-driven simulation problem can be solved by replacing the original structured low-rank matrix completion problem by a convex optimization problem, using the nuclear norm heuristic. The tools of  allow us to derive a bound on the required scaling factor.

#### Reference

1. P. Dreesen and I. Markovsky. Data-driven simulation using the nuclear norm heuristic, Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2019), pages 8207--8211, Brighton, UK, 2019.
2. J. Gillard and K. Usevich. Structured low-rank matrix completion for forecasting in time series analysis, Int. J. Forecasting, 34(4):582–597, 2018.