Rigorous cubical approximation and persistent homology of continuous functions

International audienceThe interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions defined on finite-dimensional Euclidean spaces in such a way that the discretization error is bounded by a pre-specified small constant. While the approximation scheme has a number of potential applications, we consider its usefulness in the context of computational homology. More precisely, we demonstrate that our approximation procedure can be used to rigorously compute the persistent homology of the original continuous function on a compact domain, up to small explicitly known and verified errors. In contrast to other work in this area, our approach requires minimal smoothness assumptions on the underlying function

Topological data analysis and geometry in quantum field dynamics

Many non-perturbative phenomena in quantum field theories are driven or accompanied by non-local excitations, whose dynamical effects can be intricate but difficult to study. Amongst others, this includes diverse phases of matter, anomalous chiral behavior, and non-equilibrium phenomena such as non-thermal fixed points and thermalization. Topological data analysis can provide non-local order parameters sensitive to numerous such collective effects, giving access to the topology of a hierarchy of complexes constructed from given data. This dissertation contributes to the study of topological data analysis and geometry in quantum field dynamics. A first part is devoted to far-from-equilibrium time evolutions and the thermalization of quantum many-body systems. We discuss the observation of dynamical condensation and thermalization of an easy-plane ferromagnet in a spinor Bose gas, which goes along with the build-up of long-range order and superfluidity. In real-time simulations of an over-occupied gluonic plasma we show that observables based on persistent homology provide versatile probes for universal dynamics off equilibrium. Related mathematical effects such as a packing relation between the occurring persistent homology scaling exponents are proven in a probabilistic setting. In a second part, non-Abelian features of gauge theories are studied via topological data analysis and geometry. The structure of confining and deconfining phases in non-Abelian lattice gauge theory is investigated using persistent homology, which allows for a comprehensive picture of confinement. More fundamentally, four-dimensional space-time geometries are considered within real projective geometry, to which canonical quantum field theory constructions can be extended. This leads to a derivation of much of the particle content of the Standard Model. The works discussed in this dissertation provide a step towards a geometric understanding of non-perturbative phenomena in quantum field theories, and showcase the promising versatility of topological data analysis for statistical and quantum physics studies

Persistence Bag-of-Words for Topological Data Analysis

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex structure and are difficult to integrate in today's machine learning workflows. This paper introduces persistence bag-of-words: a novel and stable vectorized representation of PDs that enables the seamless integration with machine learning. Comprehensive experiments show that the new representation achieves state-of-the-art performance and beyond in much less time than alternative approaches.Comment: Accepted for the Twenty-Eight International Joint Conference on Artificial Intelligence (IJCAI-19). arXiv admin note: substantial text overlap with arXiv:1802.0485

Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories

This thesis motivates and examines the use of methods from topological data analysis in detecting and analysing topological features relevant to models from sta-tistical physics and particle physics.In statistical physics, we use persistent homology as an observable of three dif-ferent variants of the two-dimensional XY model in order to identify relevant topo-logical features and study their relation to the phase transitions undergone by each model. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and demonstrate its use in detecting topological defects called vortices. By considering the fluctuations in the output of logistic regression and k-nearest neighbours mod-els trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.In particle physics, we investigate the use of persistent homology as a means to detect and quantitatively describe center vortices in SU(2) lattice gauge theory in a gauge-invariant manner. The sensitivity of our method to vortices in the deconfined phase is confirmed by using twisted boundary conditions which inspires the definition of a new phase indicator for the deconfinement phase transition. We also construct a phase indicator without reference to twisted boundary conditions using a k-nearest neighbours classifier. Finite-size scaling analyses of both persistence-based indicators yield accurate estimates of the critical β and critical exponent of correlation length for the deconfinement phase transition. We also use persistent homology to study the stability of vortices under gradient flow and the classification of different vortex surface geometries

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