Related papers: A persistence landscapes toolbox for topological s…
We define a new topological summary for data that we call the persistence landscape. Since this summary lies in a vector space, it is easy to combine with tools from statistics and machine learning, in contrast to the standard topological…
Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly…
An important problem in the field of Topological Data Analysis is defining topological summaries which can be combined with traditional data analytic tools. In recent work Bubenik introduced the persistence landscape, a stable…
Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using persistence landscapes…
Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects…
Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of…
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…
Phase separation mechanisms can produce a variety of complicated and intricate microstructures, which often can be difficult to characterize in a quantitative way. In recent years, a number of novel topological metrics for microstructures…
Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this…
Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a…
In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on…
Data quality is crucial for the successful training, generalization and performance of machine learning models. We propose to measure the quality of a subset concerning the dataset it represents, using topological data analysis techniques.…
In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals…
Persistent homology probes topological properties from point clouds and functions. By looking at multiple scales simultaneously, one can record the births and deaths of topological features as the scale varies. In this paper we use a…
As the size of data increase, persistence diagrams often exhibit structured asymptotic behavior, converging weakly to a Radon measure. However, conventional vector summaries such as persistence landscapes are not well-behaved in this…
Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules,…
Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and…
Topological data analysis is an approach to study shape of a data set by means of topology. Its main object of study is the persistence diagram, which represents the topological features of the data set at different spatial resolutions.…
We start with a simple introduction to topological data analysis where the most popular tool is called a persistent diagram. Briefly, a persistent diagram is a multiset of points in the plane describing the persistence of topological…
Persistence diagrams have been widely recognized as a compact descriptor for characterizing multiscale topological features in data. When many datasets are available, statistical features embedded in those persistence diagrams can be…