Quiver Laplacians and Feature Selection
This addresses the challenge of identifying relevant features that are consistent across subsets in data analysis, particularly for single-cell genomics, but is incremental as it builds on existing quiver theory.
The paper tackles the problem of feature selection across data subsets by reframing it as finding sections of a quiver representation and introducing a quiver Laplacian operator, demonstrating its application to chromatin accessibility data to yield compatible features.
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.