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

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 mixture-wise 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

Explicit–implicit mapping approach to nonlinear blind separation of sparse nonnegative dependent sources from a single mixture: pure component extraction from nonlinear mixture mass spectra

The nonlinear, nonnegative single-mixture blind source separation (BSS) problem consists of decomposing observed nonlinearly mixed multicomponent signal into nonnegative dependent component (source) signals. The problem is difficult and is a special case of the underdetermined BSS problem. However, it is practically relevant for the contemporary metabolic profiling of biological samples when only one sample is available for acquiring mass spectra ; afterwards, the pure components are extracted. Herein, we present a method for the blind separation of nonnegative dependent sources from a single, nonlinear mixture. First, an explicit feature map is used to map a single mixture into a pseudo multi-mixture. Second, an empirical kernel map is used for implicit mapping of a pseudo multi-mixture into a high-dimensional reproducible kernel Hilbert space (RKHS). Under sparse probabilistic conditions that were previously imposed on sources, the single-mixture nonlinear problem is converted into an equivalent linear, multiple-mixture problem that consists of the original sources and their higher order monomials. These monomials are suppressed by robust principal component analysis, hard-, soft- and trimmed thresholding. Sparseness constrained nonnegative matrix factorizations in RKHS yield sets of separated components. Afterwards, separated components are annotated with the pure components from the library using the maximal correlation criterion. The proposed method is depicted with a numerical example that is related to the extraction of 8 dependent components from 1 nonlinear mixture. The method is further demonstrated on 3 nonlinear chemical reactions of peptide synthesis in which 25, 19 and 28 dependent analytes are extracted from 1 nonlinear mixture mass spectra. The goal application of the proposed method is, in combination with other separation techniques, mass spectrometry-based non-targeted metabolic profiling, such as biomarker identification studies