Speeding up MCR-ALS by selecting the most influential pixels based on Combined Analyte Signal and randomized SVD

Multivariate curve resolution by alternating least squares (MCR-ALS) is arguably one of the most widely used decomposition methods for spectral data. It enables researchers to reveal the contributions of individual chemical compounds and identify these compounds by resolving their spectra. The ALS is an iterative algorithm that loops through instructions (fitting two least squares models and applying a set of constraints) until the convergence criterion is met. This iterative nature makes the algorithm resource-demanding, particularly when non-negative least squares methods are employed for fitting. Consequently, the algorithm can be significantly slow when applied to the resolution of large hyperspectral images with hundreds of thousands of pixels.

This well-known challenge has prompted numerous researchers to address it over the last decade. One recent approach involves employing Singular Value Decomposition (SVD) and constructing a convex hull in a normalized score space (Ghaffari, 2019). The elements of the convex hull—referred to as essential pixels—form a truncated dataset to which MCR-ALS is applied. The number of essential pixels is much smaller than the original number of pixels, resulting in faster computations. While this approach is clear, straightforward, and quite efficient, it has a drawback: the efficiency of the convex hull algorithm diminishes when the number of dimensions exceeds three.

In this work, we propose an alternative solution that also relies on SVD but introduces two principal differences. Firstly, we suggest using randomized SVD decomposition, which accelerates the decomposition by one to two orders of magnitude (Kuckeryavskiy, 2018; Cruz-Tirado, 2022). This solution can be seamlessly integrated with the convex hull-based approach, further improving its speed. Secondly, we propose a novel method for identifying essential pixels—Combined Analyte Signal (CAS), recently introduced for subset selections (Pomerantsev, 2023). CAS is a generalization of the full distance concept from DD-SIMCA, applicable to multiblock datasets. The computation of CAS is rapid and independent of the ALS model's complexity, making it suitable for any number of pure components without sacrificing computational speed.

In this presentation, we will elucidate the theoretical aspects of the proposed approach and present experimental results based on simulated data as well as several real cases.

References
Ghaffari 2019: Ghaffari, M; Omidikia N; Ruckebusch, C. Essential Spectral Pixels for Multivariate Curve Resolution of Chemical Images. Analytical Chemistry 91:17 (2019) 10943-10948. DOI: 10.1021/acs.analchem.9b02890
Kucheryavskiy 2018: Kucheryavskiy S. Blessing of randomness against the curse of dimensionality. Journal of Chemometrics. 2018; 32:e2966. https://doi.org/10.1002/cem.2966
Cruz-Tirado 2022: Cruz-Tirado, J.P.; Amigo, J.M.; Barbin D.F; Kucheryavskiy S. Data reduction by randomization subsampling for the study of large hyperspectral datasets. Analytica Chimica Acta 1209 (2022), 339793. DOI: https://doi.org/10.1016/j.aca.2022.339793.
Pomerantsev 2023: Pomerantsev, A.L.; Rodionova, O.Ye. Subset selection using Combined Analytical Signal, Microchemical Journal. 190 (2023) 108654. DOI: https://doi.org/10.1016/j.microc.2023.108654.