NCP: Neighborhood-Preserving Non-Uniform Circle Packing for Visualization

TL;DR

NCP addresses the challenge of visualizing high-dimensional data by encoding a quantitative attribute in non-uniform circle radii while preserving neighborhood relationships. It recasts circle packing as a maximal planar graph embedding and solves a multi-objective optimization via a continuation method, using a three-stage pipeline: neighborhood-preserving planar graph initialization (t-SNE-based), power-diagram-based refinement to boost compactness, and force-directed refinement to improve convexity. The approach yields superior neighborhood preservation and convexity compared with baselines, with comparable compactness, and scales efficiently to around 1,000 items thanks to parallelized computations. Demonstrated through Clothing and Boston Housing use cases and a user study, NCP offers practical utility for data analysis and lays groundwork for interactive and integrated visualization workflows, with open-source code available for reuse.

Abstract

Circle packing is widely used in visualization due to its aesthetic appeal and simplicity, particularly in tasks where the spatial arrangement and relationships between data are of interest, such as understanding proximity relationships (e.g., images with categories) or analyzing quantitative data (e.g., housing prices). Many applications require preserving neighborhood relationships while encoding a quantitative attribute using radii for data analysis. To meet these two requirements simultaneously, we present a neighborhood-preserving non-uniform circle packing method, NCP. This method preserves neighborhood relationships between the data represented by non-uniform circles to comprehensively analyze similar data and an attribute of interest. We formulate neighborhood-preserving non-uniform circle packing as a planar graph embedding problem based on the circle packing theorem. This formulation leads to a non-convex optimization problem, which can be solved by the continuation method. We conduct a quantitative evaluation and present two use cases to demonstrate that our NCP method can effectively generate non-uniform circle packing results.

Figures (23)

• Figure 1: Using neighborhood-preserving non-uniform circle packing to identify noisy labels in the sampled Clothing dataset: (a) analyzing images with random noise; (b) analyzing images with content-ambiguity-related noise.
• Figure 2: Optimization objectives for neighborhood-preserving non-uniform circle packing.
• Figure 3: Correspondence between the circle packing result and the planar graph: (a) example circle packing result, (b) corresponding planar graph. Red circle: referenced circle, green circles: its $1$-hop neighbors, blue circles: its $2$-hop neighbors.
• Figure 4: The continuation method's basic idea is to transform the original non-convex problem into a sequence of smoother ones and progressively solves them. This strategy guides optimization toward better regions in the solution space and leads to a better solution.
• Figure 5: Pipeline of the $\emph{NCP}$ method: (a) neighborhood-preserving planar graph initialization generates an initial planar graph that connects similar data items, (b) power-diagram-based planar graph layout refines the planar graph and produces an intermediate result preserving both compactness and neighborhood, (c) force-directed refinement compacts the circles and improves convexity.