## Multi-kernel unmixing and super-resolution using the Modified Matrix Pencil method

ELEC seminar room (Building K, 7th floor)

Speaker: Stephane Chretien (National Physical Laboratory, London)

Abstract: In this talk, we consider a generalization of the usual super-resolution problem that we call the multi-kernel unmixing super-resolution problem. Assuming access to Fourier samples, we derive an algorithm for this problem which is able to estimate the source parameters of each group, along with precise non-asymptotic guarantees.

Our approach involves estimating the group parameters sequentially in the order of increasing scale parameters. Each step involves Moitra's modified matrix pencil method, and a fine study of perturbation bounds for generalised eigenvectors. Applications to various engineering problems will also be presented.

Bio: Stephane Chretien studied at Ecole Normale Superieure (Cachan) and obtained his PhD in Electrical Engineering from the Universite Paris Sud Orsay in 1996, where he worked on successive projection methods for nonconvex set theoretic feasibility problems in signal processing. He then joined the University of Michigan (Ann Arbor) as a post-doctorate where he developed a Kullback-Proximal framework for the analysis of estimation algorithms in statistics and machine learning with application to Positron Emission Tomography. He then went back to France as a researcher in the NUMOPT team at INRIA where he studied numerical methods for nonsmooth optimization and EM-types algorithms for clustering. In 1999, he joined the Mathematics for Decision team in Brussels, where he studied network flow problems and convex relaxations for urban traffic modelling and control. In 2000, he was appointed Assistant Professor in the Mathematics Laboratory (Probability and Statistics team) at the Universite de Franche Comte, Besancon where he studied efficient algorithms for Compressed Sensing, time series analysis and clustering, and contributed theoretical results on sparse recovery and finite random matrices. He joined the National Physical Laboratory (Mathematics and Modelling) in September 2015. He is now with the newly created Data Science Division, working on medical and industrial applications. His research focuses on time series analysis, machine learning, clustering, image segmentation, genetics, scheduling, combinatorial optimization, etc.

## Identification of autonomous Wiener systems

ELEC seminar room (Building K, 7th floor)

Speaker: Ivan Markovsky (ELEC)

Abstract: We show that the behavior of an autonomous Wiener system, consisting of an order-n linear time-invariant (LTI) subsystem and a degree-d polynomial nonlinearity, is included in the behavior of an LTI system of order (n + d)-choose-d --- the number of combinations with repetitions of d elements out of n elements. The eigenvalues of the embedding system are products of up to d eigenvalues of the linear subsystem. Based on this result, we propose the following three-step identification procedure:

1) identify the embedding LTI system from the given output data,

2) compute the linear subsystem, and

3) compute the nonlinear subsystem.

Step 1 is a classical LTI identification problem, however, there are two practical challenges. First, the order of the system is high even when n and d are small. Second, due to a wide range of dampings and frequencies, some eigenvalues of the system can not be computed numerically in a finite precision arithmetic, even when the data is exact (noise free).

Step 2 is a rank-1 factorization of a symmetric d-way tensor constructed from the eigenvalues of the identified LTI system. Although this problem can be solved by d singular value decompositions, there is a practical challenge of dealing with missing values due to the numerically unidentifiable eigenvalues.

Step 3 of the identification procedure is a structured data fusion problem, for which there are existing theoretical results and computational methods.