Robust Topological Inference: Distance To a Measure and Kernel Distance (1412.7197v1)

Abstract: Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topological features of the sublevel sets of the distance function (the distance of any point x to S). Given a sample from P we can infer the persistent homology using an empirical version of the distance function. However, the empirical distance function is highly non-robust to noise and outliers. Even one outlier is deadly. The distance-to-a-measure (DTM), introduced by Chazal et al. (2011), and the kernel distance, introduced by Phillips et al. (2014), are smooth functions that provide useful topological information but are robust to noise and outliers. Chazal et al. (2014) derived concentration bounds for DTM. Building on these results, we derive limiting distributions and confidence sets, and we propose a method for choosing tuning parameters.

• The paper introduces and analyzes the Distance-to-a-measure (DTM) and Kernel Distance as robust statistical methods for topological inference using persistent homology.
• These methods offer enhanced robustness against noise and outliers compared to traditional empirical distance functions, providing theoretical guarantees and connections to machine learning.
• Practical evaluations through simulations demonstrate that DTM and Kernel Distance effectively capture relevant topological features in noisy environments, crucial for applying TDA to real-world data.

Robust Topological Inference: Distance To a Measure and Kernel Distance

The paper "Robust Topological Inference: Distance To a Measure and Kernel Distance" provides an in-depth examination of robust statistical methods for topological data analysis (TDA), focusing on the extraction of significant topological features from data embedded potentially in high-dimensional stochastic environments. The authors discuss two robust methods for topological inference, namely the Distance-to-a-measure (DTM) and the Kernel Distance, and their statistical properties concerning persistent homology. Persistent homology is utilized to summarize multiscale topological features of a data set, such as connected components and loops, and is a central tool in TDA. This paper outlines how DTM and Kernel Distance offer robustness against noise and outliers, which are critical factors that affect the empirical distance functions traditionally used in TDA.

The paper begins by asserting the significance of persistent homology in quantifying the salient features of data sampled from a distribution with support $S$. Persistent homology tracks topological features across various scales, allowing for the inference of the evolution of these features. This is particularly advantageous for complex point cloud data, such as the astronomical data provided as an example, where visual similarities might overshadow underlying structural differences.

The authors introduce the concept of DTM, which is constructed to mitigate the limitations of the empirical distance function's sensitivity to noise and outliers. Unlike traditional measures, DTM is less affected by boundary noise due to its statistical robustness, delivered through a smoothing parameter, $m$. For the theoretical framework, mass concentration bounds for DTM had been initiated by prior research, which this paper builds upon, establishing limiting distributions and confidence sets. It also proposes a method for optimally selecting the smoothing parameter $m$. The derivation shows that as $m \rightarrow 0$, DTM consistently estimates the distance function of $S$, even in the presence of noise, by leveraging the Wasserstein distance properties.

The Kernel Distance is analyzed in parallel, particularly under conditions typical of density estimation using Gaussian kernels, offering a viable alternative to DTM. The Kernel Distance also facilitates robust topological inference by utilizing properties from reproducing kernel Hilbert spaces (RKHS). This connection to machine learning positions the kernel-based method as a significant counterpart in contexts where integrating TDA into machine learning systems is essential.

The theoretical depth is complemented by practical evaluations illustrated through simulations, such as with Voronoi models. Each model is tested against both environmental perturbations and to determine feature significance using the methods described. By adapting bootstrap methodologies, the authors establish statistical significance for homological features derived from these robust measures. Their experimental evidence suggests that DTM, in particular, effectively resolves the inherent trade-off between sensitivity to noise and the capacity to capture relevant topological features across different data sets.

In a broader sense, the paper encourages the thoughtful selection of parameters for robust TDA, which remains an open problem but hints at a balance between maximizing topological feature information and mitigating noise contribution. Expanding on current statistical guarantees, the authors introduce adaptations that improve model robustness to boundary bias and encourage further exploration into noise-resistant data sharpening techniques.

This paper is crucial for researchers aiming to deploy TDA in noisy environments or in large-scale data contexts, where understanding the true topological structure encoded in the data is paramount. In future work, this approach of inferring persistent homology could extend beyond the confines of compact supports to scenarios such as manifold learning, contributing insight into how high-dimensional spaces can be effectively characterized using robust topological summaries.