In ICASSP09, pages: 3285-3288, IEEE Service Center, Piscataway, NJ, USA, 34th International Conference on Acoustics, Speech, and Signal Processing, April 2009 (inproceedings)
In this paper, we study the problem of non-orthogonal joint diagonalisation of a set of real symmetric matrices via simultaneous conjugation. A family of block Jacobi-type methods are proposed to optimise two popular cost functions for the non-orthogonal joint diagonalisation, namely, the off-norm function and the log-likelihood function. By exploiting the appropriate underlying manifold, namely the so-called oblique manifold, rigorous analysis shows that, under the exact non-orthogonal joint diagonalisation setting, the proposed methods converge locally quadratically fast to a joint diagonaliser. Finally, performance of our methods is investigated by numerical experiments for both exact and approximate non-orthogonal joint diagonalisation.
IEEE Transactions on Signal Processing, 57(9):3498-3511, September 2009 (article)
Recent approaches to independent component analysis (ICA) have used kernel independence measures to obtain highly accurate solutions, particularly where classical methods experience difficulty (for instance, sources with near-zero kurtosis). FastKICA (fast HSIC-based kernel ICA) is a new optimization method for one such kernel independence measure, the Hilbert-Schmidt Independence Criterion (HSIC). The high computational efficiency of this approach is achieved by combining geometric optimization techniques, specifically an approximate Newton-like method on the orthogonal group, with accurate estimates of the gradient and Hessian based on an incomplete Cholesky decomposition. In contrast to other efficient kernel-based ICA algorithms, FastKICA is applicable to any twice differentiable kernel function. Experimental results for problems with large numbers of sources and observations indicate that FastKICA provides more accurate solutions at a given cost than gradient descent on HSIC. Comparing with other recently published ICA methods, FastKICA is competitive in terms of accuracy, relatively insensitive to local minima when initialized far from independence, and more robust towards outliers. An analysis of the local convergence properties of FastKICA is provided.
In JMLR Workshop and Conference Proceedings Volume 2: AISTATS 2007, pages: 476-483, (Editors: Meila, M. , X. Shen), MIT Press, Cambridge, MA, USA, 11th International Conference on Artificial Intelligence and Statistics, March 2007 (inproceedings)
Recent approaches to independent component analysis (ICA) have used
kernel independence measures to obtain very good performance,
particularly where classical methods experience difficulty (for
instance, sources with near-zero kurtosis). We present Fast Kernel ICA
(FastKICA), a novel optimisation technique for one such kernel
independence measure, the Hilbert-Schmidt independence criterion
(HSIC). Our search procedure uses an approximate Newton method on the
special orthogonal group, where we estimate the Hessian locally about
independence. We employ incomplete Cholesky decomposition to
efficiently compute the gradient and approximate Hessian. FastKICA results in more accurate solutions at a given cost
compared with gradient descent, and is relatively insensitive to local minima
when initialised far from independence. These properties allow kernel approaches to be
extended to problems with larger numbers of sources and observations.
Our method is competitive with other modern and classical ICA
approaches in both speed and accuracy.
Our goal is to understand the principles of Perception, Action and Learning in autonomous systems that successfully interact with complex environments and to use this understanding to design future systems