Efficient Box Filtration for Topological Data Analysis

المفاهيم الأساسية

The authors define a new framework called the box filtration that unifies the filtration and mapper approaches from topological data analysis. The box filtration grows hyperrectangles (boxes) non-uniformly and asymmetrically in different dimensions based on the distribution of points in the point cloud data. This approach provides stability guarantees and can produce more accurate topological summaries compared to existing methods like Vietoris-Rips and distance-to-measure filtrations.

الملخص

The authors introduce a new framework called the box filtration that unifies the filtration and mapper approaches from topological data analysis. The key idea is to grow hyperrectangles (boxes) instead of Euclidean balls to capture the topology of a point cloud data (PCD).
The authors present two approaches to handle the boxes: a point cover where each point is assigned its own box at the start, and a pixel cover that works with a pixelization of the PCD space. The boxes are grown non-uniformly and asymmetrically in different dimensions based on the distribution of points, by solving a linear program that balances the cost of expansion and the benefit of including more points.
The authors prove that the box filtrations created using both point and pixel covers satisfy classical stability results based on the Gromov-Hausdorff distance. They also present efficient algorithms for computing the box filtration, with running times of O(m|U(0)| log(mnπ)L(q)) and O(m|U(0)|kL(q)), where m is the number of growth steps, |U(0)| is the initial number of boxes, π is the step length, n is the PCD dimension, q is the number of points times the dimension, and k is the number of steps allowed to find the optimal box.
The authors demonstrate through examples that the box filtration can produce more accurate topological summaries of PCDs compared to Vietoris-Rips and distance-to-measure filtrations, especially in the presence of noise and non-uniform distributions. The box filtration can also function as a mapper framework with stability guarantees.

الإحصائيات

The box filtration algorithm runs in O(m|U(0)| log(mnπ)L(q)) time, where m is number of steps of increments considered for growing the box, |U(0)| is the number of boxes in the initial cover (at most the number of points), π is the step length by which each box dimension is incremented, each linear program is solved in O(L(q)) time, n is the dimension of the PCD, and q = n × |X|.
The authors also present a faster algorithm that runs in O(m|U(0)|kL(q)) where k is the number of steps allowed to find the optimal box.

اقتباسات

"We define a new framework that unifies the filtration and the mapper approaches from topological data analysis, and present efficient algorithms to compute it."
"Using boxes rather than balls as cover elements provides the nice property that all higher order intersections of the boxes are guaranteed as soon as every pair of them intersect, whereas the Vietoris-Rips and Čech filtrations are not the same."
"We demonstrate through multiple examples that the box filtration can produce results that are more resilient to noise and with less symmetry bias than Vietoris-Rips and distance-to-measure filtrations."

الرؤى الأساسية المستخلصة من

Box Filtration

by Enrique Alva... في arxiv.org 04-10-2024

https://arxiv.org/pdf/2404.05859.pdf

استفسارات أعمق

How can the box filtration framework be extended to handle dynamic or streaming point cloud data

To extend the box filtration framework to handle dynamic or streaming point cloud data, we can implement an incremental update mechanism. This mechanism would involve updating the existing box covers as new points are added to the point cloud or as existing points are modified or removed. By dynamically adjusting the boxes based on the incoming data, we can ensure that the box filtration remains accurate and up-to-date. Additionally, we can incorporate algorithms for efficient box resizing and repositioning to adapt to changes in the point cloud in real-time. This way, the box filtration framework can effectively handle the dynamic nature of streaming point cloud data.

What are the potential applications of the box filtration beyond topological data analysis, such as in machine learning or data visualization

The box filtration framework has the potential for various applications beyond topological data analysis. In machine learning, the box filtration can be utilized for feature extraction and dimensionality reduction in high-dimensional datasets. By capturing the topological features of the data using boxes, the framework can provide valuable insights for clustering, classification, and anomaly detection tasks. Moreover, in data visualization, the box filtration can be employed to create interactive and informative visual representations of complex datasets. By mapping the box covers to visual elements, such as hyperrectangles or heatmaps, the framework can help users explore and understand the underlying structures of the data more intuitively.

Can the box filtration be combined with other techniques like kernel density estimation or bifiltrations to further improve its performance on complex or high-dimensional point cloud data

The box filtration can be combined with techniques like kernel density estimation or bifiltrations to enhance its performance on complex or high-dimensional point cloud data. By incorporating kernel density estimation, the framework can capture the density distribution of the points within each box, providing additional information about the local variations in the data. This can lead to more robust topological analysis and improved feature extraction. Additionally, integrating bifiltrations with the box filtration can offer a multi-parameter approach to data analysis, considering both distance and density thresholds simultaneously. This combined approach can provide a more comprehensive understanding of the data's topological properties and enhance the framework's ability to handle intricate structures in the point cloud data.