Kurt Stadlthanner

KPCA denoising and the pre-image problem revisited

Kernel principal component analysis (KPCA) is widely used in classification, feature extraction and denoising applications. In the latter it is unavoidable to deal with the pre-image problem which constitutes the most complex step in the whole processing chain. One of the methods to tackle this problem is an iterative solution based on a fixed-point algorithm. An alternative strategy considers an algebraic approach that relies on the solution of an under-determined system of equations. In this work we present a method that uses this algebraic approach to estimate a good starting point to the fixed-point iteration. We will demonstrate that this hybrid solution for the pre-image shows better performance than the other two methods. Further we extend the applicability of KPCA to one-dimensional signals which occur in many signal processing applications. We show that artefact removal from such data can be treated on the same footing as denoising. We finally apply the algorithm to denoise the famous USPS data set and to extract EOG interferences from single channel EEG recordings.

Hybridizing sparse component analysis with genetic algorithms for microarray analysis

Nonnegative matrix factorization (NMF) has proven to be a useful tool for the analysis of nonnegative multivariate data. However, it is known not to lead to unique results when applied to blind source separation (BSS) problems. In this paper we present an extension of NMF capable of solving the BSS problem when the underlying sources are sufficiently sparse. In contrast to most well-established BSS methods, the devised algorithm is capable of solving the BSS problem in cases where the underlying sources are not independent or uncorrelated. As the proposed fitness function is discontinuous and possesses many local minima, we use a genetic algorithm for its minimization. Finally, we apply the devised algorithm to real world microarray data.

Blind source separation using time-delayed signals

In this work a modified version of AMUSE, called dAMUSE, is proposed. The main modification consists in increasing the dimension of the data vectors by joining delayed versions of the observed mixed signals. With the new data a matrix pencil is computed and its generalized eigendecomposition is performed as in AMUSE. We will show that in this case the output (or independent) signals are filtered versions of the source signals. Some numerical simulations using artificially mixed signals as well as biological data (RR and QT intervals of Electrocardiogram) are presented.

Separation of water artifacts in 2D NOESY protein spectra using congruent matrix pencils

Multidimensional proton nuclear magnetic resonance spectra of biomolecules dissolved in aqueous solutions are usually contaminated by an intense water artifact. We discuss the application of a generalized eigenvalue decomposition (GEVD) method using a matrix pencil to solve the blind source separation (BSS) problem of removing the intense solvent peak and related artifacts. The method explores correlation matrices of the signals and their filtered versions in the frequency domain and implements a two-step algebraic procedure to solve the GEVD. Two-dimensional nuclear Overhauser enhancement spectroscopy (2D NOESY) of dissolved proteins is studied. Results are compared to those obtained with the SOBI [Belouchrani et al., IEEE Trans. Signal Process. 45(2) (1997) 434-444] algorithm which jointly diagonalizes several time-delayed correlation matrices and to those of the fastICA [Hyvarinen and Oja, Neural Comput. 9 (1996) 1483-1492] algorithm which exploits higher order statistical dependencies of random variables.

Extended sparse nonnegative matrix factorization

In sparse nonnegative component analysis (sparse NMF) a given dataset is decomposed into a mixing matrix and a feature data set, which are both nonnegative and fulfill certain sparsity constraints. In this paper, we extend the sparse NMF algorithm to allow for varying sparsity in each feature and discuss the uniqueness of an involved projection step. Furthermore, the eligibility of the extended sparse NMF algorithm for blind source separation is investigated.

Sparse Nonnegative Matrix Factorization with Genetic Algorithms for Microarray Analysis

Nonnegative Matrix Factorization (NMF) has proven to be a useful tool for the analysis of nonnegative multivariate data. Gene expression profiles naturally conform to assumptions about data formats raised by NMF. However, it is known not to lead to unique results concerning the component signals extracted. In this paper we consider an extension of the NMF algorithm which provides unique solutions whenever the underlying component signals are sufficiently sparse. A new sparseness measure is proposed most appropriate to suitably transformed gene expression profiles. The resulting fitness function is discontinuous and exhibits many local minima, hence we use a genetic algorithm for its optimization. The algorithm is applied to toy data to investigate its properties as well as to a microarray data set related to Pseudo-Xanthoma Elasticum (PXE).

Hybridizing sparse component analysis with genetic algorithms for blind source separation

Nonnegative Matrix Factorization (NMF) has proven to be a useful tool for the analysis of nonnegative multivariate data. However, it is known not to lead to unique results when applied to nonnegative Blind Source Separation (BSS) problems. In this paper we present first results of an extension to the NMF algorithm which solves the BSS problem when the underlying sources are sufficiently sparse. As the proposed target function has many local minima, we use a genetic algorithm for its minimization.

dAMUSE---A new tool for denoising and blind source separation

In this work a generalized version of AMUSE, called dAMUSE is proposed. The main modification consists in embedding the observed mixed signals in a high-dimensional feature space of delayed coordinates. With the embedded signals a matrix pencil is formed and its generalized eigendecomposition is computed similar to the algorithm AMUSE. We show that in this case the uncorrelated output signals are filtered versions of the unknown source signals. Further, denoising the data can be achieved conveniently in parallel with the signal separation. Numerical simulations using artificially mixed signals are presented to show the performance of the method. Further results of a heart rate variability (HRV) study are discussed showing that the output signals are related with LF (low frequency) and HF (high frequency) fluctuations. Finally, an application to separate artifacts from 2D NOESY NMR spectra and to denoise the reconstructed artefact-free spectra is presented also.

Denoising Using Local ICA and a Generalized Eigendecomposition with Time-Delayed Signals

We present denoising algorithms based on either local independent component analysis (ICA) and a minimum description length (MDL) estimator or a generalized eigenvalue decomposition (GEVD) using a matrix pencil of time-delayed signals. Both methods are applied to signals embedded in delayed coordinates in a high-dim feature space Ω and denoising is achieved by projecting onto a lower dimensional signal subspace. We discuss the algorithms and provide applications to the analysis of 2D NOESY protein NMR spectra.

Sparse nonnegative matrix factorization applied to microarray data sets

Nonnegative Matrix Factorization (NMF) has proven to be a useful tool for the analysis of nonnegative multivariate data. However, it is known not to lead to unique results when applied to nonnegative Blind Source Separation (BSS) problems. In this paper we present first results of an extension to the NMF algorithm which solves the BSS problem when the underlying sources are sufficiently sparse. As the proposed target function has many local minima, we use a genetic algorithm for its minimization.