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P. Comon; *Research Director at CNRS, University of Nice, France*

Saturday, March 19, 13:30 - 16:30, Location: TBA

The problem of identifying linear mixtures of independent random variables only from outputs can be traced back to 1953 with the works of Darmois or Skitovich. They pointed out that when data are non Gaussian, a lot more can be said about the mixture.

In practice, Blind Identification of linear mixtures is useful especially in Factor Analysis, in addition to many other application areas (including signal & image processing, digital communications, biomedical, or complexity theory). Harshman and Carroll provided independently numerical algorithms to decompose a data record stored in a 3-way array into elementary arrays, each representing the contribution of a single underlying factor. The main difference with the well known Principal Component Analysis is that the mixture is not imposed to be a unitary matrix. This is very relevant because the actual mixture often has no reason to have orthogonal columns. The Parafac algorithm, widely used since that time, theoretically does not converge for topological reasons, but yields very usable results after a finite number of iterations under mild conditions.

Independently, the problem of Blind Source Separation (BSS) arose around 1985 and was solved -explicitly or implicitly- with the help of High-Order Statistics (HOS), which are actually tensor objects. It gave rapidly birth to the more general problem of Independent Component Anlalysis (ICA) in 1991. ICA is a tool that can be used to extract factors when the physical diversity does not allow to store efficiently the data in tensor format, in other words when the Parafac algorithm cannot be used.

This tutorial provides a very accessible background on Statistical Independence, High-Order Statistics, and Tensors. Simple examples are given throughout the talk in order to illustrate various concepts and properties. It emphasizes both the usefulness and limitations of Parafac and ICA algorithms. Mathematically advanced topics are only tackled, but striking tensor properties that are not satisfied by matrices are still touched upon. Some reported results show how strange and attractive this research area can be.

**Overview:**
The following topics will be addressed along with demos and numerous simple examples:

- Cumulants and their properties, Circularity, Contrast criteria, Mutual information
- Concept of Independent Component Analysis, possible sets of assumptions to define it
- Various numerical algorithms to compute ICA in dimension 2 and higher, Deflation, Jacobi sweeping
- Tensors as multi-way arrays, definitions, operations, and specific properties. Canonical decomposition

**Target Audience and Prerequisites:**
This tutorial is proposed to researchers, graduate students, or engineers. Participants are expected to be familiar with basic statistics and linear algebra, although all necessary algebraic and statistical tools will be recalled.

**Pierre Comon** received his graduation in 1982, and his Doctorate in 1985,
both from the University of Grenoble, France. He later received the "Habilitation to lead Researches" in 1995, from the University of Nice Sophia-Antipolis UNSA.
His teaching has been mainly in the area of signal processing. His
research interests include High-Order statistics, Tensors, Blind Deconvolution
and Equalization, Digital Communications, Statistical Signal and Array
Processing, and
Numerical analysis. He
has been for nearly 13 years in industry, first with Crouzet/Sextant
between 1982 and 1985, and then with
Thomson
Marconi between 1988 and 1997. In 1987, he spent a post-doc at Stanford University, in the ISL laboratory. He joined Eurecom from March, 1997 to
September, 1998, as a Teaching & Research Associate under contract. He
has also been Research Director Associated with CNRS from January, 1994 to
September, 1998, in the TSASI group of
laboratory I3S. He
holds a position of permanent Research Director at CNRS since
October, 1998, as leader of the ASTRE project.
He has been Associate Editor of the
IEEE Transactions on Signal Processing from September, 1995 to
March, 1998, and a member of the National Committee of scientific
research between September, 1995 and July, 2000. He has been the
coordinator of the 6620 European Basic Research Working Group ATHOS,
from 1992 to 1995. Between Jan 1992 and 1998, he has been a member of
the Technical and Scientific Council of the Thomson Group (multinational Company employing 96,000 persons over 50 countries).
Between July 2001 and January 2004, he acted as launching Associate Editor with the IEEE Transactions on Circuits and
Systems I, in the area of Blind Techniques.
He has about 120 publications to his name, and 5 patents.
He is currently Senior Member of the IEEE and of the SEE.
He has been IEEE distinguished lecturer
over the period 2002-2003. Since 2000,
he is heading the SiCom PhD program.