A comparison between three sparse unmixing algorithms using a large library of shortwave infrared mineral spectra

The comparison described in this paper has been motivated by two things: 1) a ''spectral library'' of shortwave infrared reflectance spectra that we have built, consisting of the spectra of 60 nominally pure materials (mostly minerals, but also water, dry vegetation, and several man-made materials) and 2) the needs of users in the mining industry for the use of fast and accurate unmixing software to analyze tens to hundreds of thousands of spectra measured from drill core or chips using HyLogging instruments, and other commercial reflectance spectrometers. Individual samples are typically a mixture of only one, two, three, or occasionally four minerals. Therefore, in order to avoid overfitting, a sparse unmixing algorithm is required. We compare three such algorithms using some real world test data sets: full subset selection (FSS), sparse demixing (SD), and L1 regularization. To aid the comparison, we introduce two novel aspects: 1) the simultaneous fitting of the low frequency background with mineral identification (which provides greater model flexibility) and 2) the combined fitting being carried out using a suitably defined Mahalanobis distance; this has certain optimality properties under an idealized model. Together, these two innovations significantly improve the accuracy of the results. FSS and L1 regularization (suitably optimized) produce similar levels of accuracy, and are superior to SD. Discussion includes possible improvements to the algorithms, and their possible use in other domains.