DTM-based Filtrations (1811.04757v3)

Abstract: Despite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. the Cech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from which they are computed. In this paper, we introduce and study a new family of filtrations, the DTM-filtrations, built on top of point clouds in the Euclidean space which are more robust to noise and outliers. The approach adopted in this work relies on the notion of distance-to-measure functions, and extends some previous work on the approximation of such functions.

• The paper introduces DTM-filtrations to extend classical filtrations using distance-to-measure functions, enhancing robustness against noise and outliers.
• It demonstrates that weighted Cech and Rips filtrations remain stable under perturbations measured by both the Hausdorff metric and supremum norm.
• Numerical experiments validate its effectiveness in industrial anomaly detection by improving persistence diagram analysis in topological data.

Insight into DTM-based Filtrations in Topological Data Analysis

The paper "DTM-based Filtrations" by Hirokazu Anai, Frederic Chazal, Marc Glisse, Yuichi Ike, Hiroya Inakoshi, Raphael Tinarrage, and Yuhei Umeda introduces and scrutinizes a new class of filtrations for Topological Data Analysis (TDA), termed DTM-filtrations, predicated on distance-to-measure (DTM) functions. The utility of these filtrations is exemplified in their robustness against noise and outliers, which are significant issues in the analysis of persistent homology of filtrations like Cech or Vietoris-Rips.

The authors define these DTM-filtrations as extensions of classical filtrations on point clouds in Euclidean spaces. Building on past work on distance functions' stability, particularly in the context of geometric inference and kernel density estimates, this research explores new opportunities for noise-resilient topological data analysis.

Key Contributions

1. Weighted Cech and Rips Filtrations:
   • The paper introduces weighted Cech and Rips filtrations using a parameter $p \in [1, \infty)$ and a weight function $f$. They demonstrate the stability of these filtrations concerning the perturbations of the data set $X$ and weight $f$. The stability is analyzed under the Hausdorff metric and the supremum norm.
2. Robustness to Outliers:
   • It is proven that the DTM-filtrations exhibit robust stability when handling noisy datasets containing outliers. This is a marked improvement over previous approaches that were hypersensitive to perturbations.
3. Numerical Results and Applications:
   • The DTM-filtration's applicability is highlighted through industrial applications, such as anomaly detection from sensor data in structural health monitoring. The persistence diagrams derived from these filtrations are shown to better handle noise and highlight topological features more effectively than traditional techniques.

Implications and Theoretical Developments

The introduction of DTM-filtrations expands the toolkit available to those working with TDA, particularly in contexts where traditional methods struggle with data impurities. By leveraging the intuitive concept of DTM functions, this approach aligns better with real-world data analysis scenarios where perfect data cleanliness cannot be assumed. The focus on stability with respect to the Wasserstein metric offers a new dimension of analysis, showing the adaptability of filtrations to approximations of real data.

Future Directions

The theoretical implications suggest exciting avenues for further exploration. The proposed paper paves the way for statistical analyses of persistence diagrams deriving from DTM-filtrations. Additionally, the potential adaptation of the DTM concept using alternative weight functions or in combination with kernel density estimation strategies seems promising. Exploring sparse versions of these filtrations could also enhance computational efficiency while maintaining robustness.

Practical and Computational Considerations

The practical implications focus on the integration of DTM-filtrations in existing computational frameworks, such as those utilizing the GUDHI library. The availability of such open-source tools helps lower the barrier for implementing these techniques in industrial and research applications alike.

In conclusion, this paper enriches the TDA framework by introducing a versatile and stable filtration mechanism. The DTM-based approach holds promise in not only enhancing the theoretical understanding of persistence in data analysis but also in offering practical solutions for real-world data challenges. Researchers and practitioners are likely to find these insights beneficial, especially in applications requiring resilience to noise and outlier presence.