Corpus ID: 222290586

Capturing Dynamics of Time-Varying Data via Topology

  title={Capturing Dynamics of Time-Varying Data via Topology},
  author={Lu Xian and Henry Adams and C. Topaz and Lori Ziegelmeier},
One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time. A time-varying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic… Expand
Move Schedules: Fast persistence computations in sparse dynamic settings
This work proposes a coarser strategy for maintaining the decomposition over a discrete 1-parameter family of filtrations, and shows a modification of this technique which maintains only a sublinear number of valid states, as opposed to a quadratic number of states. Expand
Signatures, Lipschitz-free spaces, and paths of persistence diagrams
Paths of persistence diagrams provide a summary of the dynamic topological structure of a oneparameter family of metric spaces. These summaries can be used to study and characterize the dynamic shapeExpand
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The meta-hypothesis is that the short bars are as important as the long bars for many machine learning tasks and work connecting persistent homology to geometric features of spaces, including curvature and fractal dimension is surveyed. Expand
What are higher-order networks?
The goals are to clarify (i) what higher-order networks are, (ii) why these are interesting objects of study, and (iii) how they can be used in applications. Expand
Z-GCNETs: Time Zigzags at Graph Convolutional Networks for Time Series Forecasting
A new topological summary, zigzag persistence image, is developed, which provides a systematic and mathematically rigorous framework to track the most important topological features of the observed data that tend to manifest themselves over time and is validated with state-of-the-art results. Expand


Spatiotemporal Persistent Homology for Dynamic Metric Spaces
This paper extends the Rips filtration stability result for (static) metric spaces to the setting of DMSs and proposes to utilize a certain metric d for comparing these invariants, including the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS. Expand
Principal Component Analysis
  • Heng Tao Shen
  • Computer Science, Mathematics
  • Encyclopedia of Database Systems
  • 2009
The Karhunen-Lo eve basis functions, more frequently referred to as principal components or empirical orthogonal functions (EOFs), of the noise response of the climate system are an important toolExpand
Novel type of phase transition in a system of self-driven particles.
Numerical evidence is presented that this model results in a kinetic phase transition from no transport to finite net transport through spontaneous symmetry breaking of the rotational symmetry. Expand
Statistical topological data analysis using persistence landscapes
  • Peter Bubenik
  • Mathematics, Computer Science
  • J. Mach. Learn. Res.
  • 2015
A new topological summary for data that is easy to combine with tools from statistics and machine learning and obeys a strong law of large numbers and a central limit theorem is defined. Expand
Computational Topology - an Introduction
This book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Expand
Topology and data
This paper will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data, particularly high throughput data from microarray or other sources. Expand
Zigzag persistent homology and real-valued functions
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Self-propelled particles with soft-core interactions: patterns, stability, and collapse.
For the first time, a coherent theory is presented, based on fundamental statistical mechanics, for all possible phases of collective motion of driven particle systems, to predict stability and morphology of organization starting from the shape of the two-body interaction. Expand
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A Course in Metric Geometry
Preface This book is not a research monograph or a reference book (although research interests of the authors influenced it a lot)—this is a textbook. Its structure is similar to that of a graduateExpand