Improving Mapper's Robustness by Varying Resolution According to Lens-Space Density

TL;DR

This paper addresses Mapper’s sensitivity to variable density in the image of the Morse-type function by introducing density-aware kerneled covers that adapt local resolution to lens-space density. It formalizes kerneled covers, defines density proxies via ρ(x) derived from local k-NN density, and provides practical guidelines for interval selection and parameter tuning. The authors prove convergence guarantees: density-based Mapper preserves convergence to the Reeb graph in bottleneck distance and establish a density-aware zigzag persistence framework to underpin the theory, complemented by a reference implementation and computational experiments. The approach improves robustness to parameter changes and offers a pathway for more reliable topology extraction on heterogeneous data, with potential benefits for applications like temporal topic modelling; code is released for reproducibility.

Abstract

We propose a modification of the Mapper algorithm that removes the assumption of a single resolution scale across semantic space and improves the robustness of the results under change of parameters. Our work is motivated by datasets where the density in the image of the Morse-type function (the lens-space density) varies widely. For such datasets, tuning the resolution parameter of Mapper is difficult because small changes can lead to significant variations in the output. By improving the robustness of the output under these variations, our method makes it easier to tune the resolution for datasets with highly variable lens-space density. This improvement is achieved by generalising the type of permitted cover for Mapper and incorporating the lens-space density into the cover. Furthermore, we prove that for covers satisfying natural assumptions, the graph produced by Mapper still converges in bottleneck distance to the Reeb graph of the Rips complex of the data, while possibly capturing more topological features than a standard Mapper cover. Finally, we discuss implementation details and present the results of computational experiments. We also provide an accompanying reference implementation.

Figures (10)

• Figure 1: Plots of $K(x,t_0,\rho) \propto \exp(-[f(x)-t_0]^2/\rho^2)$, with $t_0=5$ and $\rho$ a constant. On the left, $\rho=1$ is chosen to achieve sufficient width for $(r,\epsilon)=(0.5,0.1)$. On the right, $\rho=1/4$ is chosen to not achieve sufficient width.
• Figure 2: Pullback open set $f^{-1}(U_i)$ of the interval $U_i$, in red.
• Figure 3: Density-sensitive kerneled set corresponding to $U_i$
• Figure 4: Synthetic data for lens-space density computation.
• Figure 5: Left: Colour represents the lens-space density. Right: Points in the kerneled pullback are red.

Theorems & Definitions (39)

• Definition 1: Lens-space, Pullback Cover
• Definition 2: $f$-kernel, sufficient width
• Example 1
• Definition 3
• Proposition 4
• Definition 5
• Proposition 6
• Proposition 7
• Definition 8
• Definition 9