Visualizing High-Dimensional Hyperbolic Data

Haoran Guo, Yunhui Guo and Stella Yu

EECS Department
University of California, Berkeley
Technical Report No. UCB/EECS-2022-127
May 14, 2022

http://www2.eecs.berkeley.edu/Pubs/TechRpts/2022/EECS-2022-127.pdf

Hierarchies can often be found in real-world data and semantics. Hyperbolic space can naturally embed hierarchical structures, making it an attractive alternative to traditional Euclidean or spherical space when learning representations. Many popular machine learning methods and architectures have been adapted to embed and use hierarchical data, leading to significant improvements over Euclidean counterparts. Additionally, high-dimensional embeddings can capture more information than low-dimensional ones, which can also lead to performance improvements. While high-dimensional hyperbolic embeddings can lead to better representations, visualizing them in a human understandable way can be challenging. Visualizations of learned embeddings are important for both understanding the representation model and characteristics of the data, so for this and other reasons, many hyperbolic models do not leverage high-dimensional embeddings. To address this problem, we propose CO-SNE which extends the Euclidean space visualization tool, t-SNE to hyperbolic space. CO-SNE is able to deflate high-dimensional embeddings into low-dimensional space without losing their hyperbolic characteristics. We present results that show CO-SNE outperforms previous methods on visualizing high-dimensional hyperbolic data, including real-life biological data and learned hyperbolic embeddings. We also show the the hierarchical nature of image segmentation by visualizing the results of hierarchical semantic segmentation. Overall, CO-SNE is able to successfully visualize high-dimensional hyperbolic data, resulting in visualizations that can present insight on the data and the methods generating the data.

Advisor: Jitendra Malik and Stella Yu


BibTeX citation:

@mastersthesis{Guo:EECS-2022-127,
    Author = {Guo, Haoran and Guo, Yunhui and Yu, Stella},
    Title = {Visualizing High-Dimensional Hyperbolic Data},
    School = {EECS Department, University of California, Berkeley},
    Year = {2022},
    Month = {May},
    URL = {http://www2.eecs.berkeley.edu/Pubs/TechRpts/2022/EECS-2022-127.html},
    Number = {UCB/EECS-2022-127},
    Abstract = {Hierarchies can often be found in real-world data and semantics. Hyperbolic space can
naturally embed hierarchical structures, making it an attractive alternative to traditional
Euclidean or spherical space when learning representations. Many popular machine learning
methods and architectures have been adapted to embed and use hierarchical data, leading
to significant improvements over Euclidean counterparts. Additionally, high-dimensional
embeddings can capture more information than low-dimensional ones, which can also lead
to performance improvements. While high-dimensional hyperbolic embeddings can lead to
better representations, visualizing them in a human understandable way can be challenging.
Visualizations of learned embeddings are important for both understanding the representation model and characteristics of the data, so for this and other reasons, many hyperbolic
models do not leverage high-dimensional embeddings. To address this problem, we propose
CO-SNE which extends the Euclidean space visualization tool, t-SNE to hyperbolic space.
CO-SNE is able to deflate high-dimensional embeddings into low-dimensional space without
losing their hyperbolic characteristics. We present results that show CO-SNE outperforms
previous methods on visualizing high-dimensional hyperbolic data, including real-life biological data and learned hyperbolic embeddings. We also show the the hierarchical nature of
image segmentation by visualizing the results of hierarchical semantic segmentation. Overall, CO-SNE is able to successfully visualize high-dimensional hyperbolic data, resulting in
visualizations that can present insight on the data and the methods generating the data.}
}

EndNote citation:

%0 Thesis
%A Guo, Haoran
%A Guo, Yunhui
%A Yu, Stella
%T Visualizing High-Dimensional Hyperbolic Data
%I EECS Department, University of California, Berkeley
%D 2022
%8 May 14
%@ UCB/EECS-2022-127
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/2022/EECS-2022-127.html
%F Guo:EECS-2022-127