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Title: Symmetric Component Analysis and Functional Independent Component Analysis

Abstract: With the increase in measurement precision, functional data is becoming common practice. Relatively few techniques for analysing such data have been developed, however, and a first step often consists in reducing the dimension via Functional PCA, which amounts to diagonalising the covariance operator. Joint diagonalisation of a pair of scatter functionals has proved useful in many different setups, such has Independent Component Analysis (ICA), Invariant Coordinate Selection (ICS), etc.

After an introduction to classical ICA techniques (and a look into several extensions of ICS), the main part of this talk will consist in extending the Fourth Order Blind Identification procedure to the case of data on a separable Hilbert space (with classical FDA setting being the go-to example). In the finite-dimensional setup, this procedure provides a matrix Γ such that ΓX has independent components, if one assumes that the random vector X satisfies X = ΨZ, where Z has independent marginals and Ψ is an invertible mixing matrix. When dealing with distributions on Hilbert spaces, two major problems arise: (i) the notion of marginals is not naturally defined and (ii) the covariance operator is, in general, non invertible. These limitations are tackled by reformulating the problem in a coordinate-free manner and by imposing natural restrictions on the mixing model.

The proposed procedure is shown to be Fisher consistent and affine invariant. A sample estimator is provided and its convergence rates are derived. The procedure is amply illustrated on simulated and real datasets.

References

[1] J.-F. Cardoso, Source Separation Using Higher Moments Proceedings of IEEE international conference on acoustics, speech and signal processing 2109-2112.

[2] D. Tyler, F. Critchley, L. Dumbgen and H. Oja, Invariant Co-ordinate Selection J. R. Statist. Soc. B., 2009, 71, 549–592.

[3] J. Ramsay and B.W. Silverman Functional Data Analysis 2nd edn. Springer, New York, 2006.