## Data-driven simulation using the nuclear norm heuristic

ELEC seminar room

Speaker: Philippe Dreesen

Abstract: Applications of signal processing and control are classically model-based, involving a two-step procedure for modeling and design: first a model is built from given data, and second, the estimated model is used for filtering, estimation, or control. Both steps typically involve optimization problems, but the combination of both is not necessarily optimal, and the modeling step often ignores the ultimate design objective. Recently, data-driven alternatives [1] are receiving attention, which employ a direct approach combining the modeling and design into a single step. In earlier work, it was shown that data-driven signal processing problems can often be rephrased as missing data completion problems, where the signal of interest is part of an incomplete low-rank mosaic Hankel structured matrix. In this talk, we consider the data-driven simulation problem, in which the output of a linear dynamic system is simulated for a given input, without first building a model of the system. More precisely, we are given an input/output trajectory of the system, and the question is how to generate the output for a given input sequence, such that the input/output data describes a trajectory of the same system. We consider a mosaic Hankel structured matrix that is built from the input/output data. The given input/output dataset defines the left-most part of the Hankel matrix, which is completely known; while the second part of the data contains only the to-be-simulated inputs, and defines the right-most part of the mosaic Hankel matrix, but contains unknown elements for the simulation outputs. The data-driven simulation problem can then be solved by completing the unknown part of the mosaic Hankel matrix such that the matrix has minimal rank, which imposes that both data sets describe trajectories of same system. Minimal rank matrix completion is in general a tough problem, but in the last decade, the use of the nuclear norm heuristic has been proposed as an efficient convex relaxation. Our findings 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 [2]. We will also show how to use the recently proposed tools of Gillard and Usevich [3] to derive bounds on the required scaling factor.

[1] I. Markovsky. A missing data approach to data-driven filtering and control. IEEE Trans. Automat. Contr., 62, pp. 1972-1978, 2017.

[2] P. Dreesen and I. Markovsky. Data-driven simulation using the nuclear norm heuristic. Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2019), Brighton, UK, 2019.

[3] J. Gillard and K. Usevich. Structured low-rank matrix completion for forecasting in time series analysis. International Journal of Forecasting 34 (4), pp. 582-597, 2018.

## A low-rank matrix completion approach to data-driven signal processing

ELEC seminar room

**Speaker:** Ivan Markovsky

**Abstract: **In filtering, control, and other mathematical engineering areas it is common to use a model-based approach, which splits the problem into two steps: 1) model identification and 2) model-based design. Despite its success, the model-based approach has the shortcoming that the design objective is not taken into account at the identification step, i.e., the model is not optimized for its intended use. In this talk, I show a data-driven approach, which combines the identification and the model-based design into one joint problem. The signal of interest is modeled as a missing part of a trajectory of the data generating system. Subsequently, the missing data estimation problem is reformulated as a mosaic-Hankel structured matrix low-rank approximation/completion problem. The missing data estimation approach for data-driven signal processing and a local optimization method for its practical implementation are illustrated on examples of control, state estimation, filtering/smoothing, and prediction. Development of fast algorithms with provable properties in the presence of measurement noise and disturbances is a topic of current research.

## Blind and off-grid acoustic echoes retrieval using multichannel annihilating filters

ELEC seminar room

**Speaker:** Antoine Deleforge, Inria Nancy

**Abstract:** When a sound wave propagates from a point source through a medium and is reflected on surfaces before reaching microphones, the measured signals consist of mixtures of the direct path signal with delayed and attenuated copies of itself. This acoustical phenomenon is referred to as echoes, or reverberation, and is generally considered as a nuisance in audio signal processing. After introducing some basic signal processing and acoustic background, this seminar will present recent works showing how acoustic echoes can be blindly estimated from audio recordings, using a method combining annihilating filter estimation and polynomial rooting. We will then show how the knowledge of such echoes can in fact help some audio signal processing tasks such as beamforming, source separation or sound source localization.