The discrete empirical interpolation method in class identification and data summarization

The discrete empirical interpolation method (DEIM) is well-established as a means of performing model order reduction in approximating solutions to differential equations, but it has also more recently demonstrated potential in performing data class detection through subset selection. Leveraging the singular value decomposition for dimension reduction, DEIM uses interpolatory projection to identify the representative rows and/or columns of a data matrix. This approach has been adapted to develop additional algorithms, including a CUR matrix factorization for performing dimension reduction while preserving the interpretability of the data. DEIM-oversampling techniques have also been developed expressly for the purpose of index selection in identifying more DEIM representatives than would typically be allowed by the matrix rank. Even with these developments, there is still a relatively large gap in the literature regarding the use of DEIM in performing unsupervised learning tasks to analyze large data sets. Known examples of DEIM's demonstrated applicability include contexts such as physics-based modeling/monitoring, electrocardiogram data summarization and classification, and document term subset selection. This overview presents a description of DEIM and some DEIM-related algorithms, discusses existing results from the literature with an emphasis on more statistical-learning-based tasks, and identifies areas for further exploration moving forward.