Related papers: A Fast and Robust Method for Global Topological Fu…
Across many scientific domains, practitioners rely on coarse, discretized summaries to track the evolving structure of complex systems under noise, measurement error, and changing system size. Understanding when such summaries are reliable…
We introduce diffusion geometry as a new framework for geometric and topological data analysis. Diffusion geometry uses the Bakry-Emery $\Gamma$-calculus of Markov diffusion operators to define objects from Riemannian geometry on a wide…
Network topology inference is a cornerstone problem in statistical analyses of complex systems. In this context, the fresh look advocated here permeates benefits from convex optimization and graph signal processing, to identify the…
As the field of Topological Data Analysis continues to show success in theory and in applications, there has been increasing interest in using tools from this field with methods for machine learning. Using persistent homology, specifically…
This paper presents a practical approach for the optimization of topological simplification, a central pre-processing step for the analysis and visualization of scalar data. Given an input scalar field f and a set of "signal" persistence…
Topological loss based on persistent homology has shown promise in various applications. A topological loss enforces the model to achieve certain desired topological property. Despite its empirical success, less is known about the…
Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics of data sets which reflect global geometric properties. In order to be useful in practice, for example for feature generation in…
Persistence diagrams are important tools in the field of topological data analysis that describe the presence and magnitude of features in a filtered topological space. However, current approaches for comparing a persistence diagram to a…
This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the…
In this work we consider algorithms for reconstructing time-varying data into a finite sum of discrete trajectories, alternatively, an off-the-grid sparse-spikes decomposition which is continuous in time. Recent work showed that this…
Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including…
How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? The answer is with a coordinate projection. Neural networks based on topological data analysis (TDA) use…
We introduce a framework for analyzing topological tipping in time-evolutionary point clouds by extending the recently proposed Topological Optimal Transport (TpOT) distance. While TpOT unifies geometric, homological, and higher-order…
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…
The topological (or graph) structures of real-world networks are known to be predictive of multiple dynamic properties of the networks. Conventionally, a graph structure is represented using an adjacency matrix or a set of hand-crafted…
Topological Data Analysis (TDA) is a novel, and relatively new approach to analysing high-dimensional data sets. It does this by focussing on global properties like the shape and connectivity of the data giving it a significant advantage…
Recurrence plot based methods are highly efficient and widely accepted tools for the investigation of time series or one-dimensional data. We present an extension of the recurrence plots and their quantifications in order to study recurrent…
Inferring network topology from dynamical observations is a fundamental problem pervading research on complex systems. Here, we present a simple, direct method to infer the structural connection topology of a network, given an observation…
Networks are important representations in computer science to communicate structural aspects of a given system of interacting components. The evolution of a network has several topological properties that can provide us information on the…
0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the…