# Capturing Dynamics of Time-Varying Data via Topology

@article{Xian2020CapturingDO, title={Capturing Dynamics of Time-Varying Data via Topology}, author={Lu Xian and Henry Adams and C. Topaz and Lori Ziegelmeier}, journal={ArXiv}, year={2020}, volume={abs/2010.05780} }

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

#### Figures and Tables from this paper

#### 5 Citations

Move Schedules: Fast persistence computations in sparse dynamic settings

- Computer Science, Mathematics
- ArXiv
- 2021

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

- Mathematics
- 2021

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 shape… Expand

Topology Applied to Machine Learning: From Global to Local

- Computer Science, Medicine
- Frontiers in Artificial Intelligence
- 2021

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?

- Computer Science, Mathematics
- ArXiv
- 2021

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

- Computer Science, Mathematics
- ICML
- 2021

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

#### References

SHOWING 1-10 OF 58 REFERENCES

Spatiotemporal Persistent Homology for Dynamic Metric Spaces

- Computer Science, Mathematics
- Discret. Comput. Geom.
- 2021

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

- 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 tool… Expand

Novel type of phase transition in a system of self-driven particles.

- Physics, Medicine
- Physical review letters
- 1995

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

- 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

- Computer Science
- 2009

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

- Computer Science
- 2009

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

- Mathematics, Computer Science
- SCG '09
- 2009

The algorithmic results provide a way to compute zigzag persistence for any sequence of homology groups, but combined with the structural results give a novel algorithm for computing extended persistence that is easily parallelizable and uses (asymptotically) less memory. Expand

Self-propelled particles with soft-core interactions: patterns, stability, and collapse.

- Physics, Medicine
- Physical review letters
- 2006

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

Vines and vineyards by updating persistence in linear time

- Computer Science, Mathematics
- SCG '06
- 2006

The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering and uses the algorithm to compute 1-parameter families of diagrams which are applied to the study of protein folding trajectories. Expand

A Course in Metric Geometry

- 2001

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 graduate… Expand