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Quiver Laplacians and Feature Selection

Otto Sumray, Heather A. Harrington, Vidit Nanda

TL;DR

The paper introduces quiver Laplacians to identify features that are simultaneously compatible with a prescribed decomposition of data into overlapping subsets. It recasts feature selection as finding (approximately) globally or locally compatible sections of a quiver representation, with the quiver Laplacian $L$ yielding low-energy eigenvectors that approximate these sections. It establishes spectral stability and eigenvalue-interlacing results under quiver and representation modifications, and provides a reduction framework to make computation tractable on large-scale data. The approach is demonstrated on single-cell chromatin accessibility (ATAC-seq) data, showing that eigenvectors of the quiver Laplacian reveal both globally shared and subset-specific peaks, enabling robust, scale-aware feature discovery for complex biological datasets.

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.

Quiver Laplacians and Feature Selection

TL;DR

The paper introduces quiver Laplacians to identify features that are simultaneously compatible with a prescribed decomposition of data into overlapping subsets. It recasts feature selection as finding (approximately) globally or locally compatible sections of a quiver representation, with the quiver Laplacian yielding low-energy eigenvectors that approximate these sections. It establishes spectral stability and eigenvalue-interlacing results under quiver and representation modifications, and provides a reduction framework to make computation tractable on large-scale data. The approach is demonstrated on single-cell chromatin accessibility (ATAC-seq) data, showing that eigenvectors of the quiver Laplacian reveal both globally shared and subset-specific peaks, enabling robust, scale-aware feature discovery for complex biological datasets.

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.
Paper Structure (23 sections, 22 theorems, 104 equations, 7 figures)

This paper contains 23 sections, 22 theorems, 104 equations, 7 figures.

Table of Contents

  1. Introduction
  2. From Feature Selection to Quiver Representations
  3. Compatibility and Sections
  4. Global Compatibility and Sections
  5. Local Compatibility
  6. Bicompatible features
  7. Dual Representations
  8. The Quiver Laplacian and Approximate Sections
  9. The Quiver Laplacian
  10. Approximate Sections
  11. Spectral Stability of Feature Selectors
  12. Eigenvalue Inequalities for Quiver Operations
  13. Sheaves over Quivers
  14. Operations on the Quiver
  15. Reduction of the Feature Selector Problem
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\mathbf{A}_\bullet$ and $\mathbf{A}'_\bullet$ be two representations of $Q$, and consider any vertex-indexed collection $\tau = \left\{{\tau_v:\mathbf{A}_v \to \mathbf{A}'_v \mid v \in Q_0}\right\}$ of linear maps. Write $q$ for the dimension of $\ker \tau$ viewed as a (block diagonal) map $\ma holds for every $k$ in $\left\{{1,2,\ldots,n-q}\right\}$.

Figures (7)

  • Figure 1: Two features defined on a dataset equipped with a decomposition into two overlapping subsets.
  • Figure 2: Example of different compatibility conditions.
  • Figure 3: An illustration of the floret based at $u$ and its representation.
  • Figure 4: Illustration of the merged quiver
  • Figure 5: An example of the reduction of a quiver $Q$ with $V = \left\{{v}\right\}$.
  • ...and 2 more figures

Theorems & Definitions (55)

  • Theorem : I
  • Theorem : II
  • Theorem : III
  • Theorem : IV
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Definition 3.2
  • ...and 45 more