EB-gMCR: Energy-Based Generative Modeling for Signal Unmixing and Multivariate Curve Resolution

A measurement of a mixture is the sum of a few component profiles, each weighted by a concentration, and multivariate curve resolution (MCR) recovers both from many samples. Classical MCR factorizes the data matrix, takes the component count as input, and leaves a rotational ambiguity. This paper instead models the generative process, generative MCR (gMCR): each sample selects a few components from a fixed set and is observed as their linear superposition plus noise. We prove that the data determine which decomposition a solver should reach. Under a spark condition on the component profiles, the decomposition of minimal usage is the true one, up to a relabeling of the components, and under noise the same holds up to a stated conjecture. Components recovered this way decode new samples of the same process. EB-gMCR, an energy-based solver, is one realization of gMCR: a gate selects components per sample and is trained toward minimal usage. With no count given, it recovers the component count to within a few percent on synthetic mixtures of up to 256 components, at a sample-to-component ratio of eight and a signal-to-noise ratio of 30 dB. On two public spectroscopy datasets it recovers the count exactly and reconstructs as well as the best method at that count. The trained model, frozen, decodes mixtures synthesized across the concentration simplex at the noise floor. gMCR applies to any sparse linear mixture of a fixed set of components. The results cover a solver whose candidate pool is at least as large as the true component count.