- The paper introduces a density-sensitive Mapper algorithm that adjusts resolution dynamically based on lens-space density.
- It employs variable-density kernel covers and a scaling factor to capture topological features with minimized parameter tuning.
- Experimental results show that the modified Mapper maintains convergence to the Reeb graph while enhancing robustness.
Improving Mapper's Robustness by Varying Resolution According to Lens-Space Density
The paper presents a novel approach to enhancing the Mapper algorithm's robustness by incorporating lens-space density-dependent resolution. Mapper, a technique for topological data analysis, is traditionally sensitive to its initial parameter settings, particularly the resolution, which determines the detected scale of topological features. This work proposes a method that allows for local variation in resolution based on the density of data in the range of the Morse function used for Mapper, promising more reliable results with less parameter tuning effort.
Methodology
The authors introduce a density-sensitive variant of Mapper by employing variable-density kerneled covers. This approach leverages the density information of the dataset to adjust the resolution dynamically. The method involves using f-kernels that incorporate local density to determine the appropriate width of each cover. Specifically, the width of these kernels varies according to a computed density measure, enabling a more nuanced capture of topological features compared to a uniform resolution setting.
A critical contribution is the integration of a scaling factor that adjusts based on the lens-space density, allowing for a flexible resolution adjustment strategy. The kernel's width is normalized to ensure the statistical properties of Mapper are preserved, particularly its convergence to the Reeb graph.
Theoretical Insights
The paper rigorously proves that the proposed density-based Mapper converges to the Reeb graph similarly to standard Mapper. Using algebraic concepts like persistence diagrams and bottleneck distance, the authors demonstrate that the topological features captured by the Mapper graph remain consistent under their modification. They show that density-based Mapper retains all the topological advantages of regular Mapper while being less sensitive to parameter choices due to its ability to vary resolution locally.
Implementation and Experiments
The paper provides a reference implementation, detailing steps to approximate Morse density and select intervals for open covers. Computational experiments validate the proposed approach, using synthetic data designed to examine the limitations of traditional Mapper. The results indicate that density-based Mapper can effectively identify topological features in datasets with variable density more consistently than the original Mapper algorithm, especially when combined with clustering algorithms like DBSCAN and HDBSCAN.
Implications and Future Work
The introduction of density-sensitive resolution in Mapper holds significant potential for applications in data analysis scenarios characterized by high variability in local data density. This approach could simplify parameter tuning, making topological data analysis more accessible and effective in diverse applied settings, such as computational biology and complex systems modeling.
Future research directions might explore the integration of this method with other Mapper variants or its adaptation to higher-dimensional lens functions. Further computational studies on real-world data could provide additional insights into the practical benefits and limitations of the proposed method.
Overall, this paper contributes a meaningful advancement in topological data analysis, offering a robust and flexible alternative to traditional Mapper approaches.