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

Abstract

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.

Library-assisted nonlinear blind separation and annotation of pure components from a single 1H nuclear magnetic resonance mixture spectra

“…That is referred to in BSS as essential uniqueness [55]. However, even for the linear underdetermined BSS problem hard (sparseness) constraints ought to be imposed on pure components [56,57,58,19] to obtain an essentially unique solution. The quality of separation depends heavily on the degree of sparseness, i.e.…”

“…However, due to the Lorentzian shape of the pure NMR peak, K will be greater than it is the case with mass spectra. Therefore, the role of the EKM-based mapping is even more important than it is for the case with the mass spectra mixtures [19]. In particular, that is the main reason why nonlinear blind separation of pure components from single 1 H NMR mixture spectra has to be performed in multiple RKHSs, as opposed to [19] where only one RKHS was used.…”

“…Algorithms for single-mixture BSS first have to transform single-to pseudo multi-mixture BSS problem [43][44][45][46][47][48][49][50][51][52][53]. For the single-mixture case, EFM (6) reduces to [19]:…”

“…It is, however, hard to know the SNR value in practice. That is why, as opposed to [19], we propose herein mapping of the original 1 H NMR mixture spectra onto multiple RKHS (13). The role of the basis…”

“…Essential differences between the proposed method and single-mixture nonlinear BSS method [19] are: (i) the method [19] maps single mixture spectra onto one RKHS, while the method proposed herein maps single mixture spectra onto multiples RKHSs. As it is shown in experiments conducted on one model 1 H NMR mixture spectra in Section 3.1. as well as two model 1 H NMR mixtures spectra in Supplement Section S3.1, usage of multiple RKHSs enables separation and annotation of more pure components than it is possible from one RKHS only; (ii) after sparseness constrained separation in each RKHS, library of in-house recorded pure components 1 H NMR spectra is used for annotation of separated components.…”