Topological data analysis and machine learning

Leykam, Daniel; Angelakis, Dimitris G.

doi:10.1080/23746149.2023.2202331

Cited by 7 publications

(4 citation statements)

References 109 publications

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“…Using TDA, we were able to overcome the limitation of the parameter S d to characterize generated lattices. One can build several metrics to describe persistence diagrams, which can be used as simpler descriptors of the topology of data sets or as inputs of more refined machine learning based models. , In this work, we use two statistical descriptors based on lattices’ persistence diagrams in order to describe both the typical distance between each point of the lattices and their positional disorder. The first numerical descriptor is normalized structural heterogeneity of degree 0 (n SH 0 ) and is the sum of the lifetime of the topological features of homology 0, the connected components, divided by the number of points of the lattice, N : normaln S H 0 = 1 N ∑ ( b , d ) ∈ H 0 ⊂ scriptD d − b with b and d the birth and death of each topological feature of degree 0, H 0 , of the persistence diagram scriptD .…”

Section: Resultsmentioning

confidence: 99%

Topological Learning for the Classification of Disorder: An Application to the Design of Metasurfaces

Madeleine,

Podoliak,

Buchnev

et al. 2023

ACS Nano

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Structural disorder can improve the optical properties of metasurfaces, whether it is emerging from some large-scale fabrication methods or explicitly designed and built lithographically. For example, correlated disorder, induced by a minimum inter-nanostructure distance or by hyperuniformity properties, is particularly beneficial for light extraction. Inspired by topology, we introduce numerical descriptors to provide quantitative measures of disorder with universal properties, suitable to treat both uncorrelated and correlated disorder at all length scales. The accuracy of these topological descriptors is illustrated both theoretically and experimentally by using them to design plasmonic metasurfaces with controlled disorder that we then correlate to the strength of their surface lattice resonances. These descriptors are an example of topological tools that can be used for the fast and accurate design of disordered structures or as aid in improving their fabrication methods.

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Section: Resultsmentioning

confidence: 99%

Topological Learning for the Classification of Disorder: An Application to the Design of Metasurfaces

Madeleine,

Podoliak,

Buchnev

et al. 2023

ACS Nano

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“…As a result, datasets that might be initially hard to describe can have their descriptions condensed into several key features of the latent manifold which allow for easier visualization and effective learning. Within the context of many-body physics, a few recent works have applied dimensionality reduction techniques from machine learning for facilitating ground-state searches and identifying distinct phases of spin models [17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Unsupervised learning of quantum many-body scars using intrinsic dimension

Cao,

Angelakis,

Leykam

2024

Mach. Learn.: Sci. Technol.

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Quantum many-body scarred systems contain both thermal and non-thermal scar eigenstates in their spectra. When these systems are quenched from special initial states which share high overlap with scar eigenstates, the system undergoes dynamics with atypically slow relaxation and periodic revival. This scarring phenomenon poses a potential avenue for circumventing decoherence in various quantum engineering applications. Given access to an unknown scar system, current approaches for identification of special states leading to non-thermal dynamics rely on costly measures such as entanglement entropy. In this work, we show how two dimensionality reduction techniques, multidimensional scaling and intrinsic dimension estimation, can be used to learn structural properties of dynamics in the PXP model and distinguish between thermal and scar initial states. The latter method is shown to be robust against limited sample sizes and experimental measurement errors.

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“…Topological data analysis (TDA) has witnessed many important advances over the last twenty years that aim to unravel and provide insight to the "shape" of the data (Edelsbrunner et al, 2002;Edelsbrunner and Harer, 2008;Wasserman, 2018;Chazal and Michel, 2021). The development of TDA tools such as barcodes and persistence diagrams (Ghrist, 2008;Bubenik, 2015;Adams et al, 2017) have opened many new perspectives for analyzing various types of data (Umeda, 2017;Gholizadeh and Zadrozny, 2018;Motta, 2018;Xu et al, 2021;Leykam and Angelakis, 2023). These tools enable practitioners to grasp the topological characteristics inherent in high-dimensional data, which often remain beyond the reach of classical data analysis methods.…”

Section: Introductionmentioning

confidence: 99%

Statistical inference for dependence networks in topological data analysis

El-Yaagoubi,

Chung,

Ombao

2023

Front. Artif. Intell.

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Topological data analysis (TDA) provide tools that are becoming increasingly popular for analyzing multivariate time series data. One key aspect in analyzing multivariate time series is dependence between components. One application is on brain signal analysis. In particular, various dependence patterns in brain networks may be linked to specific tasks and cognitive processes. These dependence patterns may be altered by various neurological and cognitive impairments such as Alzheimer's and Parkinson's diseases, as well as attention deficit hyperactivity disorder (ADHD). Because there is no ground-truth with known dependence patterns in real brain signals, testing new TDA methods on multivariate time series is still a challenge. Our goal here is to develop novel statistical inference procedures via simulations. Simulations are useful for generating some null distributions of a test statistic (for hypothesis testing), forming confidence regions, and for evaluating the performance of proposed TDA methods. To the best of our knowledge, there are no methods that simulate multivariate time series data with potentially complex user-specified connectivity patterns. In this paper we present a novel approach to simulate multivariate time series with specific number of cycles/holes in its dependence network. Furthermore, we also provide a procedure for generating higher dimensional topological features.

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Topological data analysis and machine learning

Cited by 7 publications

References 109 publications

Topological Learning for the Classification of Disorder: An Application to the Design of Metasurfaces

Topological Learning for the Classification of Disorder: An Application to the Design of Metasurfaces

Unsupervised learning of quantum many-body scars using intrinsic dimension

Statistical inference for dependence networks in topological data analysis

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