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Hyperedge Representations with Hypergraph Wavelets: Applications to Spatial Transcriptomics

Xingzhi Sun, Charles Xu, João F. Rocha, Chen Liu, Benjamin Hollander-Bodie, Laney Goldman, Marcello DiStasio, Michael Perlmutter, Smita Krishnaswamy

TL;DR

Hypergraph diffusion wavelets are introduced and their utility for biomedical discovery in spatially resolved transcriptomics is demonstrated by applying the method to represent disease-relevant cellular niches for Alzheimer’s disease.

Abstract

In many data-driven applications, higher-order relationships among multiple objects are essential in capturing complex interactions. Hypergraphs, which generalize graphs by allowing edges to connect any number of nodes, provide a flexible and powerful framework for modeling such higher-order relationships. In this work, we introduce hypergraph diffusion wavelets and describe their favorable spectral and spatial properties. We demonstrate their utility for biomedical discovery in spatially resolved transcriptomics by applying the method to represent disease-relevant cellular niches for Alzheimer's disease.

Hyperedge Representations with Hypergraph Wavelets: Applications to Spatial Transcriptomics

TL;DR

Hypergraph diffusion wavelets are introduced and their utility for biomedical discovery in spatially resolved transcriptomics is demonstrated by applying the method to represent disease-relevant cellular niches for Alzheimer’s disease.

Abstract

In many data-driven applications, higher-order relationships among multiple objects are essential in capturing complex interactions. Hypergraphs, which generalize graphs by allowing edges to connect any number of nodes, provide a flexible and powerful framework for modeling such higher-order relationships. In this work, we introduce hypergraph diffusion wavelets and describe their favorable spectral and spatial properties. We demonstrate their utility for biomedical discovery in spatially resolved transcriptomics by applying the method to represent disease-relevant cellular niches for Alzheimer's disease.
Paper Structure (16 sections, 1 theorem, 9 equations, 4 figures, 1 table)

This paper contains 16 sections, 1 theorem, 9 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Hypergraph Preliminaries
  3. Hypergraph representation
  4. Bipartite expansion
  5. Methods
  6. Hypergraph Diffusion
  7. Hypergraph Diffusion Wavelets
  8. Complexity
  9. Modeling Spatial Transcriptomics Data
  10. Hypergraph Modeling of Spatial Transcriptomic
  11. Hyperedge Features
  12. Hyperedge Representations via Hypergraph Wavelets
  13. Results
  14. Representational Diversity and Organization
  15. Visualizing Disease Progression
  16. ...and 1 more sections

Key Result

Proposition 1

The eigenvalues of $\mathbf{P}_H$ are contained in the interval $[0,1]$.

Figures (4)

  • Figure 1: Conceptual illustrations. a. Creation of a hypergraph and the bipartite expansion. b. Diffusion (Lazy Random Walks) on the expanded graph.
  • Figure 2: We represent spatial transcriptomics data as a hypergraph. Physically proximate collections of cells play functional roles, motivating the use of hyperedges to represent cellular niches.
  • Figure 3: A. Hypergraphs clustered over tissue sample. B. PHATE moon2019visualizing space colored by spectral clustering. C. Distribution of cell types on each cluster
  • Figure 4: Visualization of pairs of cellular neighborhood representations derived from hypergraph wavelets at different Braak stages, projected into two dimensions using PHATE moon2019visualizing. Blue points represent cellular neighborhoods at the Braak stage indicated by the column name, while orange points correspond to the stage indicated by the row name.

Theorems & Definitions (2)

  • Proposition 1
  • proof