Hyperbolic Diffusion Distance

Prerequisites

• Numpy
• SciPy
• NetworkX
• scikit-learn
• Optional repository for distance evaluation: MAP & Distortion [1]
• Optional packages for evaluating downstream classification: Nearest centroid classifier
  • random seed: 1234

Usage Example

Graph

from hyperbolic_diffusion_distance import *
from diffusion_operator_util import *

# graph_data is edge connectivity
ev, left_evv, right_evv = diffusion_operator_graph(graph_data, if_full_spec = True)
hde, hdd = hyperbolic_diffusion(ev, left_evv, right_evv, K)

Data Graph

from hyperbolic_diffusion_distance import *
from diffusion_operator_util import *

# data is the high-dimensional observation
# cosine distance used here, other distance can be used to explore in diffusion geometry 
dis_mat = pairwise_distances(data, metric = 'cosine') 
ev, left_evv, right_evv = diffusion_operator_normalized_data(data, dis_mat, if_full_spec = True)
hde, hdd = hyperbolic_diffusion(ev, left_evv, right_evv, K)

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Experiments

Download the data below and run hyperbolic diffusion distance experiments using the exp_main.py script

Hierarchical Graph Embedding Learning

• Data: the small balanced tree, the phylogenetic tree, the disease, the CS-PHD, and the Gr-Qc graphs
• K: balanced tree (K=3), phylogenetic tree (K=3), the disease (K=3), the CS-PHD (K=4), and the Gr-Qc graphs (K=10)

Single-Cell Gene Expression Data

• Data [2]: Zeisel [3] and CBMC [4]
• Hidden hierarchical structure [2]
• K: Zeisel (K=9), CBMC (K=13)

Unsupervised Hierarchical Metric Learning

• Data [5]: the Zoo, the Iris, the Glass, and the Image Segmentation datasets
• K: Zoo (K=4), Iris(K=6), Glass(K=5), and Image Segmentation (K=8)

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Reference

[1] Sala, F., De Sa, C., Gu, A., and Re, C. Representation tradeoffs for hyperbolic embeddings. In International conference on machine learning, pp. 4460–4469. PMLR, 2018.

[2] Dumitrascu, B., Villar, S., Mixon, D. G., and Engelhardt, B. E. Optimal marker gene selection for cell type discrim- ination in single cell analyses. Nature communications, 12(1):1–8, 2021.

[3] Zeisel, A., Mun ̃oz-Manchado, A. B., Codeluppi, S., Lo ̈nnerberg, P., La Manno, G., Jure ́us, A., Marques, S., Munguba, H., He, L., Betsholtz, C., et al. Cell types in the mouse cortex and hippocampus revealed by single-cell rna-seq. Science, 347(6226):1138–1142, 2015.

[4] Stoeckius, M., Hafemeister, C., Stephenson, W., Houck- Loomis, B., Chattopadhyay, P. K., Swerdlow, H., Satija, R., and Smibert, P. Simultaneous epitope and transcrip- tome measurement in single cells. Nature methods, 14 (9):865–868, 2017.

[5] Dua, D. and Graff, C. UCI machine learning repository, 2017. URL http://archive.ics.uci.edu/ml.