When Dimensionality Reduction Meets Graph (Drawing) Theory: Introducing a Common Framework, Challenges and Opportunities

TL;DR

This work addresses the gap between dimensionality reduction and graph drawing by proposing a formal, four-stage framework that recasts DR as a graph-embedding problem. It formalizes data as 𝒢_D=(𝒱,𝔼,Ω), maps relationships to visual space via GD-inspired mappings, and introduces quality metrics and visualization interactions to validate and explore embeddings. The main contributions include a precise framework for cross-domain integration, discussion of how graph-theoretic techniques (e.g., MST backbones, centrality, force-directed layouts) can augment DR, and experimental illustrations on a digits dataset highlighting the impact of relationship modeling and mapping choices. The framework aims to enable systematic design and validation of DR pipelines using GD concepts, with potential to improve global-local structure preservation and interpretability in visual analytics.

Abstract

In the vast landscape of visualization research, Dimensionality Reduction (DR) and graph analysis are two popular subfields, often essential to most visual data analytics setups. DR aims to create representations to support neighborhood and similarity analysis on complex, large datasets. Graph analysis focuses on identifying the salient topological properties and key actors within networked data, with specialized research on investigating how such features could be presented to the user to ease the comprehension of the underlying structure. Although these two disciplines are typically regarded as disjoint subfields, we argue that both fields share strong similarities and synergies that can potentially benefit both. Therefore, this paper discusses and introduces a unifying framework to help bridge the gap between DR and graph (drawing) theory. Our goal is to use the strongly math-grounded graph theory to improve the overall process of creating DR visual representations. We propose how to break the DR process into well-defined stages, discussing how to match some of the DR state-of-the-art techniques to this framework and presenting ideas on how graph drawing, topology features, and some popular algorithms and strategies used in graph analysis can be employed to improve DR topology extraction, embedding generation, and result validation. We also discuss the challenges and identify opportunities for implementing and using our framework, opening directions for future visualization research.

Figures (4)

• Figure 1: Comparing layouts created using the classical MDS technique and the spring layout of a fully connected distance graph. Both layouts fail to separate the existing classes, indicating modeling pairwise (global) relationships is not indicated if the goal is to see clusters.
• Figure 2: Graphs modeling local relantionships. Compared to the previous example (Figure \ref{['fig:global']}), local relationships are capable of better spliting the groups of data items with same label, suggesting that local reltionships are prefered for group segregation.
• Figure 3: Comparing different t-SNE and UMAP layouts. UMAP and t-SNE modeled relationships are remarkably similar, and when mapped using the same embedding algorithm resulted in very similar outcomes. However, the original t-SNE and UMAP techniques produce very different results, in terms of groups separation and cohesion, indicating that the mapping phase used by both approach have a considerable impact in the produced layouts.
• Figure 4: Nodes' closeness centrality of different t-SNE and UMAP layouts. Groups and sub-groups of nodes presenting low closeness indicates well-defined groups, including the orange sub-group which is well separated from the other oranges nodes considering the modeled relationships.