Deep data analysis via physically constrained linear unmixing: universal framework, domain examples, and a community-wide platform

Abstract

Many spectral responses in materials science, physics, and chemistry experiments can be characterized as resulting from the superposition of a number of more basic individual spectra. In this context, unmixing is defined as the problem of determining the individual spectra, given measurements of multiple spectra that are spatially resolved across samples, as well as the determination of the corresponding abundance maps indicating the local weighting of each individual spectrum. Matrix factorization is a popular linear unmixing technique that considers that the mixture model between the individual spectra and the spatial maps is linear. Here, we present a tutorial paper targeted at domain scientists to introduce linear unmixing techniques, to facilitate greater understanding of spectroscopic imaging data. We detail a matrix factorization framework that can incorporate different domain information through various parameters of the matrix factorization method. We demonstrate many domain-specific examples to explain the expressivity of the matrix factorization framework and show how the appropriate use of domain-specific constraints such as non-negativity and sum-to-one abundance result in physically meaningful spectral decompositions that are more readily interpretable. Our aim is not only to explain the off-the-shelf available tools, but to add additional constraints when ready-made algorithms are unavailable for the task. All examples use the scalable open source implementation from https://github.com/ramkikannan/nmflibrary that can run from small laptops to supercomputers, creating a user-wide platform for rapid dissemination and adoption across scientific disciplines.

A machine perspective of atomic defects in scanning transmission electron microscopy

“…Therefore, the issue of structure representation is conceptually equivalent to the dimension reduction problem . Dimension reduction can be achieved via different techniques including matrix factorization, descriptor‐based methods, manifold learning, and so on. To facilitate the subsequent structure identification process, the following criteria are summarized to evaluate the effectiveness and usefulness of the dimension reduction scheme: (a) Interpretability: the reduced feature vectors can convey specific structure information in a physics‐based sense; (b) Separability: the reduced feature vectors can be easily separated into different groups in the low dimensional space; (c) Flexibility: the reduced feature vectors can be combined to represent high‐level structure information such as rotation‐invariant features, and symmetry measurement, and so on.…”

“…H, Three components obtained by reshaping row vectors of matrix V after NMF. F‐H, Reproduced with permission from Reference Copyright 2018, Springer Nature…”

“…Therefore, the issue of structure representation is conceptually equivalent to the dimension reduction problem. 90 Dimension reduction can be achieved via different techniques including matrix factorization, 50,91 descriptor-based methods, manifold learning, and so on. To facilitate the subsequent structure identification process, the following criteria are summarized to evaluate the effectiveness and usefulness of the dimension reduction scheme: (a) Interpretability: the reduced feature vectors can convey specific structure information in a physics-based sense;…”

Machine Learning for Challenging EELS and EDS Spectral Decomposition

“…One way to combat this effect is to impose additional constraints, or costs, on the minimization algorithm to reduce the likelihood of settling at a local minimum. Another approach is to use an improved initial guess, which increases the likelihood of the algorithm reaching the global minimum [1].Here, we combine these techniques to improve the quality of results by using the non-linear "perfect pixel" algorithm, ATGP [2], to generate initial guesses for the joint-non-negative matrix factorization [3] algorithm that augments the cost function of NMF and encourages sparsity in the pixels and smooth transitions. Figures 1 and 2 show that Joint-NMF identified the palladium nanocube in the second component and differentiated two carbon components.…”

“…Here, we combine these techniques to improve the quality of results by using the non-linear "perfect pixel" algorithm, ATGP [2], to generate initial guesses for the joint-non-negative matrix factorization [3] algorithm that augments the cost function of NMF and encourages sparsity in the pixels and smooth transitions. Figures 1 and 2 show that Joint-NMF identified the palladium nanocube in the second component and differentiated two carbon components.…”