Detection of the homotopy type of an object using differential invariants of an approximating map

A method of topological data analysis is proposed that allows one to find out the homotopy type of the object under study. Unlike mature and widely used methods based on persistent homologies, our method is based on computing differential invariants of some map associated with an approximating map. Differential topology tools and the analogy with the main result in Morse theory are used. The approximating map can be constructed in the usual way using a neural network or otherwise. The method allows one to identify the homotopy type of an object in the plane because the number of circles in the homotopy equivalent object representation as a wedge is expressed through the degree of some map associated with the approximating map. The performance of the algorithm is illustrated by examples from the MNIST database and transforms thereof. Generalizations and open questions relating to a higher-dimension case are discussed.

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Optimization of Utility Functions in an Admissible Space of Higher Dimension

Handbook of Research on Military, Aeronautical, and Maritime Logistics and Operations - Advances in Logistics, Operations, and Management Science ◽

10.4018/978-1-4666-9779-9.ch006 ◽

2016 ◽

pp. 102-113

Author(s):

German Almanza ◽

Victor M. Carrillo ◽

Cely C. Ronquillo

Keyword(s):

Graduate Students ◽

Linear Algebra ◽

Algebraic Topology ◽

Morse Theory ◽

Utility Functions ◽

Higher Dimension ◽

Differential Topology ◽

Several Variables

S. Smale published a paper where announce a theorem which optimize a several utility functions at once (cf. Smale, 1975) using Morse Theory, this is a very abstract subject that require high skills in Differential Topology and Algebraic Topology. Our goal in this paper is announce the same theorems in terms of Calculus of Manifolds and Linear Algebra, those subjects are more reachable to engineers and economists whom are concern with maximizing functions in several variables. Moreover, the elements involved in our theorems are accessible to graduate students, also we putting forward the results we consider economically relevant.

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Homotopical decompositions of simplicial and Vietoris Rips complexes

Journal of Applied and Computational Topology ◽

10.1007/s41468-021-00066-2 ◽

2021 ◽

Author(s):

Wojciech Chachólski ◽

Alvin Jin ◽

Martina Scolamiero ◽

Francesca Tombari

Keyword(s):

Data Analysis ◽

Simplicial Complex ◽

Homotopy Type ◽

Topological Data Analysis ◽

The Difference ◽

Topological Data

AbstractMotivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings.

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Classification Of Skin Lesions By Topological Data Analysis Alongside With Neural Network

10.36227/techrxiv.12628088 ◽

2020 ◽

Author(s):

Naiereh Elyasi ◽

mehdi hosseini moghadam

Keyword(s):

Neural Network ◽

Neural Networks ◽

Data Analysis ◽

Skin Diseases ◽

Persistent Homology ◽

Skin Lesions ◽

Topological Data Analysis ◽

Deep Convolutional Neural Networks ◽

Classification Result

In this paper we use TDA mapper alongside with deep convolutional neural networks in the classification of 7 major skin diseases. First we apply kepler mapper with neural network as one of its filter steps to classify the dataset HAM10000. Mapper visualizes the classification result by a simplicial complex, where neural network can not do this alone, but as a filter step neural network helps to classify data better. Furthermore we apply TDA mapper and persistent homology to understand the weights of layers of mobilenet network in different training epochs of HAM10000. Also we use persistent diagrams to visualize the results of analysis of layers of mobilenet network.

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Quantification of the Immune Content in Neuroblastoma: Deep Learning and Topological Data Analysis in Digital Pathology

International Journal of Molecular Sciences ◽

10.3390/ijms22168804 ◽

2021 ◽

Vol 22 (16) ◽

pp. 8804

Author(s):

Nicole Bussola ◽

Bruno Papa ◽

Ombretta Melaiu ◽

Aurora Castellano ◽

Doriana Fruci ◽

...

Keyword(s):

Neural Network ◽

Data Analysis ◽

Data Augmentation ◽

Digital Pathology ◽

Topological Data Analysis ◽

Main Task ◽

Learning Approaches ◽

Training Process ◽

The Neural Network ◽

Topological Data

We introduce here a novel machine learning (ML) framework to address the issue of the quantitative assessment of the immune content in neuroblastoma (NB) specimens. First, the EUNet, a U-Net with an EfficientNet encoder, is trained to detect lymphocytes on tissue digital slides stained with the CD3 T-cell marker. The training set consists of 3782 images extracted from an original collection of 54 whole slide images (WSIs), manually annotated for a total of 73,751 lymphocytes. Resampling strategies, data augmentation, and transfer learning approaches are adopted to warrant reproducibility and to reduce the risk of overfitting and selection bias. Topological data analysis (TDA) is then used to define activation maps from different layers of the neural network at different stages of the training process, described by persistence diagrams (PD) and Betti curves. TDA is further integrated with the uniform manifold approximation and projection (UMAP) dimensionality reduction and the hierarchical density-based spatial clustering of applications with noise (HDBSCAN) algorithm for clustering, by the deep features, the relevant subgroups and structures, across different levels of the neural network. Finally, the recent TwoNN approach is leveraged to study the variation of the intrinsic dimensionality of the U-Net model. As the main task, the proposed pipeline is employed to evaluate the density of lymphocytes over the whole tissue area of the WSIs. The model achieves good results with mean absolute error 3.1 on test set, showing significant agreement between densities estimated by our EUNet model and by trained pathologists, thus indicating the potentialities of a promising new strategy in the quantification of the immune content in NB specimens. Moreover, the UMAP algorithm unveiled interesting patterns compatible with pathological characteristics, also highlighting novel insights into the dynamics of the intrinsic dataset dimensionality at different stages of the training process. All the experiments were run on the Microsoft Azure cloud platform.

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Model discrepancy of Earth polar motion using topological data analysis and convolutional neural network analysis

International Journal of Modern Physics C ◽

10.1142/s012918312050117x ◽

2020 ◽

Vol 31 (08) ◽

pp. 2050117

Author(s):

Dongjin Lee ◽

Christopher Bresten ◽

Kookhyoun Youm ◽

Ki-Weon Seo ◽

Jae-Hun Jung

Keyword(s):

Neural Network ◽

Data Analysis ◽

Convolutional Neural Network ◽

Polar Motion ◽

Topological Data Analysis ◽

Accurate Analysis ◽

Long Baseline ◽

Motion Signal ◽

Polar Motion Excitation ◽

Topological Data

An accurate analysis of the polar motion variation is essential to understand the global change of the environment and predict useful information about short-term and long-term change in climate. Observation of polar motion excitation using multiple measurements including Very-Long-Baseline-Interferometry (VLBI) provides highly accurate measurement of polar motion variation. The observed polar motion excitation has been modeled with multiple geophysical models, but the discrepancies between observations and models still exist. In this paper, we propose two approaches for detecting the discrepancy of the polar motion excitation: topological data analysis (TDA) and convolutional neural network (CNN) analysis. Our methods clearly show that the observed polar motion has a different topological structure from the model data, and there are time periods that the model fails to represent the polar motion. Numerical results indicate that the proposed methods show promise for applications to polar motion signal analysis.

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The Topology of Neuronal Structures Exposed to Cosmic Radiation

WSEAS TRANSACTIONS ON MATHEMATICS ◽

10.37394/23206.2020.19.26 ◽

2020 ◽

Vol 19 ◽

Keyword(s):

Neural Network ◽

Neural Networks ◽

Data Analysis ◽

Cosmic Radiation ◽

Network Code ◽

Topological Data Analysis ◽

Analysis Approach ◽

The Neural Networks ◽

Topological Data

In this paper, we focus on some leader NASA experiences to explore how cosmic radiation caused significant reductions in dendrite and spine complexity. We adopt a topological data analysis approach and extract more information then the classical methods. Our key idea is to use the NASA images of the neural networks of some mouses that were exposed 12 weeks to cosmic radiation. We associate to this neural network code bares that give us more information, that that given by the original experiences.

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