Visualizing Sparse Filtrations

Over the last few years, there have been several approaches to building sparser complexes that still give good approximations to the persistent homology. In this video, we have illustrated a geometric perspective on sparse filtrations that leads to simpler proofs, more general theorems, and a more visual explanation. We hope that as these techniques become easier to understand, they will also become easier to use

A Geometric Perspective on Sparse Filtrations

We present a geometric perspective on sparse filtrations used in topological data analysis. This new perspective leads to much simpler proofs, while also being more general, applying equally to Rips filtrations and Cech filtrations for any convex metric. We also give an algorithm for finding the simplices in such a filtration and prove that the vertex removal can be implemented as a sequence of elementary edge collapses

Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology

Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we propose a novel method using persistent homology to quantify structural changes in time-varying graphs. Specifically, we transform each instance of the time-varying graph into metric spaces, extract topological features using persistent homology, and compare those features over time. We provide a visualization that assists in time-varying graph exploration and helps to identify patterns of behavior within the data. To validate our approach, we conduct several case studies on real world data sets and show how our method can find cyclic patterns, deviations from those patterns, and one-time events in time-varying graphs. We also examine whether persistence-based similarity measure as a graph metric satisfies a set of well-established, desirable properties for graph metrics

Reconstructing forest canopy from the 3D triangulations of airborne laser scanning point data for the visualization and planning of forested landscapes

Key message We present a data-driven technique to visualize forest landscapes and simulate their future development according to alternative management scenarios. Gentle harvesting intensities were preferred for maintaining scenic values in a test of eliciting public's preferences based on the simulated landscapes. Context Visualizations of future forest landscapes according to alternative management scenarios are useful for eliciting stakeholders' preferences on the alternatives. However, conventional computer visualizations require laborious tree-wise measurements or simulators to generate these observations. Aims We describe and evaluate an alternative approach, in which the visualization is based on reconstructing forest canopy from sparse density, leaf-off airborne laser scanning data. Methods Computational geometry was employed to generate filtrations, i.e., ordered sets of simplices belonging to the three-dimensional triangulations of the point data. An appropriate degree of filtering was determined by analyzing the topological persistence of the filtrations. The topology was further utilized to simulate changes to canopy biomass, resembling harvests with varying retention levels. Relative priorities of recreational and scenic values of the harvests were estimated based on pairwise comparisons and analytic hierarchy process (AHP). Results The canopy elements were co-located with the tree stems measured in the field, and the visualizations derived from the entire landscape showed reasonably realistic, despite a low numerical correspondence with plot-level forest attributes. The potential and limitations to improve the proposed parameterization are discussed. Conclusion Although the criteria to evaluate the landscape visualization and simulation models were not conclusive, the results suggest that forest scenes may be feasibly reconstructed based on data already covering broad areas and readily available for practical applications.Peer reviewe

Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph

Data-sensitive metrics adapt distances locally based the density of data points with the goal of aligning distances and some notion of similarity. In this paper, we give the first exact algorithm for computing a data-sensitive metric called the nearest neighbor metric. In fact, we prove the surprising result that a previously published $3$-approximation is an exact algorithm. The nearest neighbor metric can be viewed as a special case of a density-based distance used in machine learning, or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric to compute sparse spanners and persistent homology. We also explore the behavior of the metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest.Comment: 15 page

A Sparse Delaunay Filtration

We show how a filtration of Delaunay complexes can be used to approximate the persistence diagram of the distance to a point set in ?^d. Whereas the full Delaunay complex can be used to compute this persistence diagram exactly, it may have size O(n^?d/2?). In contrast, our construction uses only O(n) simplices. The central idea is to connect Delaunay complexes on progressively denser subsamples by considering the flips in an incremental construction as simplices in d+1 dimensions. This approach leads to a very simple and straightforward proof of correctness in geometric terms, because the final filtration is dual to a (d+1)-dimensional Voronoi construction similar to the standard Delaunay filtration. We also, show how this complex can be efficiently constructed

Visualizing Topological Importance: A Class-Driven Approach

This paper presents the first approach to visualize the importance of topological features that define classes of data. Topological features, with their ability to abstract the fundamental structure of complex data, are an integral component of visualization and analysis pipelines. Although not all topological features present in data are of equal importance. To date, the default definition of feature importance is often assumed and fixed. This work shows how proven explainable deep learning approaches can be adapted for use in topological classification. In doing so, it provides the first technique that illuminates what topological structures are important in each dataset in regards to their class label. In particular, the approach uses a learned metric classifier with a density estimator of the points of a persistence diagram as input. This metric learns how to reweigh this density such that classification accuracy is high. By extracting this weight, an importance field on persistent point density can be created. This provides an intuitive representation of persistence point importance that can be used to drive new visualizations. This work provides two examples: Visualization on each diagram directly and, in the case of sublevel set filtrations on images, directly on the images themselves. This work highlights real-world examples of this approach visualizing the important topological features in graph, 3D shape, and medical image data.Comment: 11 pages, 11 figure

K-Nearest-Neighbors Induced Topological PCA for scRNA Sequence Data Analysis

Single-cell RNA sequencing (scRNA-seq) is widely used to reveal heterogeneity in cells, which has given us insights into cell-cell communication, cell differentiation, and differential gene expression. However, analyzing scRNA-seq data is a challenge due to sparsity and the large number of genes involved. Therefore, dimensionality reduction and feature selection are important for removing spurious signals and enhancing downstream analysis. Traditional PCA, a main workhorse in dimensionality reduction, lacks the ability to capture geometrical structure information embedded in the data, and previous graph Laplacian regularizations are limited by the analysis of only a single scale. We propose a topological Principal Components Analysis (tPCA) method by the combination of persistent Laplacian (PL) technique and L$_{2,1}$ norm regularization to address multiscale and multiclass heterogeneity issues in data. We further introduce a k-Nearest-Neighbor (kNN) persistent Laplacian technique to improve the robustness of our persistent Laplacian method. The proposed kNN-PL is a new algebraic topology technique which addresses the many limitations of the traditional persistent homology. Rather than inducing filtration via the varying of a distance threshold, we introduced kNN-tPCA, where filtrations are achieved by varying the number of neighbors in a kNN network at each step, and find that this framework has significant implications for hyper-parameter tuning. We validate the efficacy of our proposed tPCA and kNN-tPCA methods on 11 diverse benchmark scRNA-seq datasets, and showcase that our methods outperform other unsupervised PCA enhancements from the literature, as well as popular Uniform Manifold Approximation (UMAP), t-Distributed Stochastic Neighbor Embedding (tSNE), and Projection Non-Negative Matrix Factorization (NMF) by significant margins.Comment: 28 pages, 11 figure