Quiver Laplacians and Feature Selection (2404.06993v1)

Abstract: 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.

• The paper presents a novel framework using quiver representations to identify globally compatible features in high-dimensional, decomposed datasets.
• The paper demonstrates how variations in the Laplacian spectrum effectively correlate with feature selection performance in noisy, overlapping data subsets.
• The paper applies the method to scATAC-seq data, successfully extracting chromatin accessibility peaks across diverse cell types.

Analyzing Feature Selection Algorithms through Quiver Representation and Laplacian Spectra with Applications to scATAC-seq Data

Introduction to Feature Selection in Decomposed Datasets

In the field of high-dimensional data analysis, one often faces the challenge of identifying the most relevant features within a dataset. The field of natural language processing, for instance, has seen the development of techniques for embedding words in a Euclidean space, where coordinates serve as features, to capture semantic relationships. For any given dataset, extracting features that are most informative for the tasks at hand is critical. This process, known as feature selection, relies on various methods to quantify feature relevance. Amidst the diversity of feature selection algorithms, discrepancies often arise when applying these algorithms to subsets of a whole dataset, particularly in scenarios with overlapping subsets, such as biological datasets organized by cell or disease types.

Quiver Representations for Feature Selection across Dataset Decompositions

A novel framework is introduced for the systematic selection of features across decomposed sets of data, accommodating overlaps among subsets. By abstracting feature selectors as deterministic processes and utilizing quiver representations valued in finite-dimensional Hilbert spaces, the method isolates the largest subspace of selected features that remain consistent with respect to the dataset's decomposition into subsets. The construction relies on both local and global forms of feature compatibility, considering restrictions and extensions across subsets. This abstract approach applies generally to any feature selector, urging a compatibility framework that accounts for approximate sections of quiver representations to handle noise and high correlation among features.

The Quiver Laplacian and Approximate Sections

A quiver Laplacian is introduced, serving as a cornerstone of this framework by associating sections of the quiver representation with globally compatible features and defining approximate sections through the eigenspace of the Laplacian. The paper empirically establishes how variations in the spectrum of a quiver Laplacian correspond to changes in feature selection processes, allowing for the efficient identification of relevant features. This methodology bridges the theoretical understanding of feature selection with practical applications, particularly in analyzing single-cell sequencing data to identify relevant genomic features.

Applying the Framework to Single-Cell Chromatin Accessibility Data

The method is applied to single-cell ATAC-seq data for peak calling—a process critical for determining chromatin accessibility across different cell types within a sample. Through the construction of a quiver Laplacian, the paper demonstrates the extraction of locally and globally compatible features (peaks) related to chromatin accessibility. The application efficiently handles the massive dimensionality characteristic of genomic datasets and successfully identifies genomic regions relevant across various cell types, highlighting the versatility of the framework in accommodating the complexity of biological data.

Implications and Future Directions in AI and Data Analysis

The development of a principled framework for feature selection in decomposed datasets paves the way for more accurate and interpretable data analysis across various fields. By providing a robust theoretical foundation and demonstrating applicability to complex biological data, this research opens avenues for future developments in AI and data science. It specifically invites further investigation into the behavior of quiver Laplacians and their applicability in other domains where datasets inherently consist of overlapping subsets. The adaptability of the approach to accommodate noise and feature correlation holds promise for enhancing feature selection methodologies across disciplines.