Quiver Laplacians and Feature Selection
Otto Sumray Heather A. Harrington Vidit Nanda
The challenge of selecting the most relevant features of a given dataset
arises ubiquitously in data analysis and dimensionality reduction. However,
features found to be of high importance for the entire dataset may not be
relevant to subsets of interest, and vice versa. Given a feature selector and a
fixed decomposition of the data into subsets, we describe a method for
identifying selected features which are compatible with the decomposition into
subsets. We achieve this by re-framing the problem of finding compatible
features to one of finding sections of a suitable quiver representation. In
order to approximate such sections, we then introduce a Laplacian operator for
quiver representations valued in Hilbert spaces. We provide explicit bounds on
how the spectrum of a quiver Laplacian changes when the representation and the
underlying quiver are modified in certain natural ways. Finally, we apply this
machinery to the study of peak-calling algorithms which measure chromatin
accessibility in single-cell data. We demonstrate that eigenvectors of the
associated quiver Laplacian yield locally and globally compatible features.