# The emptiness inside: Finding gaps, valleys, and lacunae with geometric data analysis

**Authors:** Gabriella Contardo, David W. Hogg, Jason A. S. Hunt, Joshua E. G. Peek, Yen-Chi Chen  
**arXiv:** [2201.10674](https://arxiv.org/abs/2201.10674)  
**Date:** 2022-01-25  
**Summary page:** https://www.summarizepaper.com/en/paper/2201.10674v1/

## Key points

- Gaps in data are crucial in astrophysics research
- Gaps can take various forms, such as kinematic gaps or rare exoplanets with specific radii
- Gaps represent anomalous regions compared to their surroundings
- Identifying and characterizing gaps can be challenging, especially when they have non-trivial shapes
- The authors present a methodological approach to identify critical points and select the most relevant ones to trace the "valleys" in the density field
- They propose a novel gappiness criterion based on observed properties of critical points that can be computed at any point in the data space
- This allows for identifying a broader variety of gaps by highlighting regions of the data-space that are "gappy"
- The authors explore methodological ways to make detected gaps robust to changes in density estimation and noise in the data
- The methodology is applied to velocity distribution of nearby stars in Milky Way disk plane, which exhibits gaps that could originate from different processes
- Identifying and characterizing these gaps could help determine their origins
- Geometric data analysis can be used to find and characterize gaps, valleys, and lacunae within complex datasets
- Proposed methodology provides a powerful tool for identifying anomalous regions within large datasets with broad applications beyond astrophysics research.

## Summary

The discovery of gaps in data has been a crucial aspect of astrophysics research. These gaps can take various forms, such as kinematic gaps opened by resonances in dynamical systems or exoplanets with a specific radius that are empirically rare. Unlike traditional anomalies, which are single outlier data points far from other data points, gaps represent regions or sets of points that are anomalous compared to their surroundings. Identifying and characterizing these gaps can be challenging, particularly when they have non-trivial shapes. To address this problem, the authors present a methodological approach to identify critical points and select the most relevant ones to trace the "valleys" in the density field. They then propose a novel gappiness criterion based on the observed properties of critical points that can be computed at any point in the data space. This allows for identifying a broader variety of gaps by highlighting regions of the data-space that are "gappy" or selecting data points that lie in local under densities. The authors also explore methodological ways to make the detected gaps robust to changes in density estimation and noise in the data. To illustrate their methods, they apply them to the velocity distribution of nearby stars in the Milky Way disk plane, which exhibits gaps that could originate from different processes. Identifying and characterizing these gaps could help determine their origins. Overall, this study demonstrates how geometric data analysis can be used to find and characterize gaps, valleys, and lacunae within complex datasets. The proposed methodology provides a powerful tool for identifying anomalous regions within large datasets and has broad applications beyond astrophysics research.