Nonlinear mixture‐wise expansion approach to underdetermined blind separation of nonnegative dependent sources

Abstract

Underdetermined blind separation of nonnegative dependent sources consists in decomposing set of observed mixed signals into greater number of original nonnegative and dependent component (source) signals. That is an important problem for which very few algorithms exist. It is also practically relevant for contemporary metabolic profiling of biological samples, such as biomarker identification studies, where sources (a.k.a. pure components or analytes) are aimed to be extracted from mass spectra of complex multicomponent mixtures. This paper presents method for underdetermined blind separation of nonnegative dependent sources. The method performs nonlinear mixturewise mapping of observed data in high-dimensional reproducible kernel Hilbert space (RKHS) of functions and sparseness constrained nonnegative matrix factorization (NMF) therein. Thus, original problem is converted into new one with increased number of mixtures, increased number of dependent sources and higher-order (error) terms generated by nonlinear mapping. Provided that amplitudes of original components are sparsely distributed, that is the case for mass spectra of analytes, sparseness constrained NMF in RKHS yields, with significant probability, improved accuracy relative to the case when the same NMF algorithm is performed on original problem.The method is exemplified on numerical and experimental examples related respectively to extraction of ten dependent components from five mixtures and to extraction of ten dependent analytes from mass spectra of two to five mixtures. Thereby, analytes mimic complexity of components expected to be found in biological samples.

Empirical kernel map approach to nonlinear underdetermined blind separation of sparse nonnegative dependent sources: pure component extraction from nonlinear mixture mass spectra

“…As emphasized in [4] mass spectrometry is chosen due to its increasing importance in clinical chemistry, safety and quality control as well as biomarker discovery and validation. As in [4,5], we assume that library of reference mass spectra is available to evaluate quality of components extracted by the proposed method.…”

“…Identification of pure components present in the mixture is a traditional problem in spectroscopy (nuclear magnetic resonance-, infrared, Raman) and mass spectrometry [1][2][3][4]. Identification proceeds often by matching separated components spectra with a library of reference compounds [5][6][7], whereas degree of correlation depends on how well pure components are separated from each other.…”

“…Thereby, of interest are blind source separation (BSS) methods that use only the matrix with recorded mixtures spectra as input information [8][9][10][11]. In majority of scenarios, separation of pure components is performed by assuming that mixture spectra are linear combinations of pure components [1][2][3][4]. While linear mixture model is adequate for many scenarios, nonlinear model offers more accurate description of processes and interactions occurring in biological systems.…”

“…To this end, we address underdetermined nonlinear nonnegative BSS (uNNBSS) problem with sparse and dependent sources. As it has been discussed at great length in [4], even linear underdetermined BSS problem comprised of dependent sources is challenging with only few algorithms addressing it. There is basically no method proposed for uNNBSS problem.…”

“…There is basically no method proposed for uNNBSS problem. Herein, we propose method for uNNBSS problem that can be considered as generalization of the method developed in [4] for underdetermined linear nonnegative BSS (uLNBSS) problem comprised of dependent sources. Proposed method constrains sources to be nonnegative and comply with sparse probabilistic model [14,15], that is sources are assumed to be sparse in support and amplitude.…”