Related papers: Stabilizing Localization of Representative cycles

We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…

We propose a neural network architecture that can learn discriminative geometric representations of data from persistence diagrams, common descriptors of Topological Data Analysis. The learned representations enjoy Lipschitz stability with…

Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset 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.…

Most algorithms for computing persistent homology do so by tracking cycles that represent homology classes. There are many choices of such cycles, and specific choices have found different uses in applications. Although it is known that…

We apply persistent homology, the dominant tool from the field of topological data analysis, to study electoral redistricting. Our method combines the geographic information from a political districting plan with election data to produce a…

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…

In this article we propose a novel approach for comparing the persistent homology representations of two spaces (filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes,…

We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between…

Persistence diagrams are one of the main tools in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to…

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…

Paths of persistence diagrams provide a summary of the dynamic topological structure of a one-parameter family of metric spaces. These summaries can be used to study and characterize the dynamic shape of data such as swarming behavior in…

The persistence diagram is an increasingly useful tool from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. The most success to date has come from methods that map…

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.…

Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…

Persistent homology studies the birth and death of cycles in a parameterized family of spaces. In this paper, we study the birth and death of cycles in a multifiltration of a chain complex with the goal of producing a persistence diagram…

Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have…

Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their {\em diagram distances}, most of the recent attempts at using persistence diagrams…

Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the (metric) space of…

In this work, we develop a pipeline that associates Persistence Diagrams to digital data via the most appropriate filtration for the type of data considered. Using a grid search approach, this pipeline determines optimal representation…