Simplicial complex with approximate rotational symmetry: a general class of simplicial complexes

We study the transformation of the vertices of a certain simple simplicial complex in n-dimensional Euclidian space and prove that the resulting set of simplices is a simplicial complex with an approximate rotational symmetry. Such simplicial complexes have applications in computing Lyapunov function for nonlinear dynamical systems using linear optimization and are also of interest for other applications

Persistent Homology of Attractors For Action Recognition

In this paper, we propose a novel framework for dynamical analysis of human actions from 3D motion capture data using topological data analysis. We model human actions using the topological features of the attractor of the dynamical system. We reconstruct the phase-space of time series corresponding to actions using time-delay embedding, and compute the persistent homology of the phase-space reconstruction. In order to better represent the topological properties of the phase-space, we incorporate the temporal adjacency information when computing the homology groups. The persistence of these homology groups encoded using persistence diagrams are used as features for the actions. Our experiments with action recognition using these features demonstrate that the proposed approach outperforms other baseline methods.Comment: 5 pages, Under review in International Conference on Image Processin

A Topology-Based Approach for Nonlinear Time Series with Applications in Computer Performance Analysis

We present a topology-based methodology for the analysis of experimental data generated by a discrete-time, nonlinear dynamical system. This methodology has significant applications in the field of computer performance analysis. Our approach consists of two parts. In the first part, we propose a novel signal separation algorithm that exploits the continuity of the dynamical system being studied. We use established tools from computational topology to test the connectedness of various regions of state space. In particular, a connected region of space that has a disconnected image under the experimental dynamics suggests the presence of multiple signals in the data. Using this as a guideline, we are able to model experimental data as an Iterated Function System (IFS). We demonstrate the success of our algorithm on several synthetic examples--including a Henon-like IFS. Additionally, we successfully model experimental computer performance data as an IFS. In the second part of the analysis, we represent an experimental dynamical system with an algebraic structure that allows for the computation of algebraic topological invariants. Previous work has shown that a cubical grid and the associated cubical complex are effective tools that can be used to identify isolating neighborhoods and compute the corresponding Conley Index--thereby rigorously verifying the existence of periodic orbits and/or chaotic dynamics. Our contribution is to adapt this technique by altering the underlying data structure--improving flexibility and efficiency. We represent the state space of the dynamical system with a simplicial complex and its induced simplicial multivalued map. This contains information about both geometry and dynamics, whereas the cubical complex is restricted by the geometry of the experimental data. This representation has several advantages; most notably, the complexity of the algorithm that generates the associated simplicial multivalued map is linear in the number of data points--as opposed to exponential in dimension for the cubical multivalued map. The synthesis of the two parts of our methodology results in a nonlinear time-series analysis framework that is particularly well suited for computer performance analysis. Complex computer programs naturally switch between `regimes\u27 and are appropriately modeled as IFSs by part one of our program. Part two of our methodology provides the correct tools for analyzing each regime independently

Persistent Homology of Coarse Grained State Space Networks

This work is dedicated to the topological analysis of complex transitional networks for dynamic state detection. Transitional networks are formed from time series data and they leverage graph theory tools to reveal information about the underlying dynamic system. However, traditional tools can fail to summarize the complex topology present in such graphs. In this work, we leverage persistent homology from topological data analysis to study the structure of these networks. We contrast dynamic state detection from time series using CGSSN and TDA to two state of the art approaches: Ordinal Partition Networks (OPNs) combined with TDA, and the standard application of persistent homology to the time-delay embedding of the signal. We show that the CGSSN captures rich information about the dynamic state of the underlying dynamical system as evidenced by a significant improvement in dynamic state detection and noise robustness in comparison to OPNs. We also show that because the computational time of CGSSN is not linearly dependent on the signal's length, it is more computationally efficient than applying TDA to the time-delay embedding of the time series