Topology vs. Geometry in Data Analysis/Machine Learning

Topic Information

Dear Colleagues,

Recent years have witnessed a surging interest in the role geometric and topological tools play in machine learning and data science. Indeed topology and geometry have been proven to be natural tools that facilitate a concrete language and offer very useful set tools in solving many longstanding problems in these fields. For instance, geometry has played a key role in (nonlinear) dimension reduction models and geometric deep learning has great potential to revolutionize structure data analysis. Topological data analysis, in particular persistent homology has been overwhelmingly successful at solving a vast array of complex data problems within machine learning and beyond. On the other hand, the role that both geometry and topology play in machine learning is still mostly restricted to techniques that attempt to enhance machine learning models. However, we believe that both geometry and topology can and will play a central role in machine learning and AI in general and an entire set of tools that both geometry and topology can offer are yet to be discovered. Our purpose of this topic call is to invite researchers to contribute to ongoing research interest in a new emerging area that intertwines topological and geometric tools with machine learning. We welcome contributions from theoretical and practical flavors. Specifically, the Topology vs. Geometry in Data Analysis/Machine Learning topic invites papers on theoretical and applied issues including, but not limited to: 

• Persistent Homology
• Generalized Persistence theories
• Applied Graph and Hypergraph Theory
• Dimensional reduction
• Discrete Morse Theory
• Spectral theory (spectral graph/simplicial complex/hypergraph)
• Discrete geometry and discrete exterior calculus
• Reeb graph
• Simplicial complex representations (Dowker complex, Hom complex, Path complex, etc.)
• Cellular Sheaves
• Hyperbolic embedding, Poincaré embedding
• Discrete Ricci curvature
• Hodge theory
• Conformal geometry
• Geometric deep learning
• Geometric signal processing
• Discrete optimal transport

This topic will present the results of research describing recent advances in geometry and topology inspired by or applied to both Data Analysis and Machine Learning.

Prof. Massimo Ferri
Prof. Dr. Kelin Xia
Prof. Dr. Francesco Vaccarino
Prof. Dr. Mustafa Hajij
Prof. Dr. Nežka Mramor Kosta
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Algorithms algorithms	1.8	4.1	2008	15 Days	CHF 1600
Axioms axioms	1.9	-	2012	21 Days	CHF 2400
Machine Learning and Knowledge Extraction make	4.0	6.3	2019	27.1 Days	CHF 1800
Mathematics mathematics	2.3	4.0	2013	17.1 Days	CHF 2600
Symmetry symmetry	2.2	5.4	2009	16.8 Days	CHF 2400

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Published Papers (8 papers)

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All Algorithms Axioms MAKE Mathematics Symmetry

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13 pages, 3297 KiB  

Open AccessArticle

Generalized Persistence for Equivariant Operators in Machine Learning

by Mattia G. Bergomi, Massimo Ferri, Alessandro Mella and Pietro Vertechi

Mach. Learn. Knowl. Extr. 2023, 5(2), 346-358; https://doi.org/10.3390/make5020021 - 24 Mar 2023

Cited by 1 | Viewed by 2396

Abstract

Artificial neural networks can learn complex, salient data features to achieve a given task. On the opposite end of the spectrum, mathematically grounded methods such as topological data analysis allow users to design analysis pipelines fully aware of data constraints and symmetries. We [...] Read more.

Artificial neural networks can learn complex, salient data features to achieve a given task. On the opposite end of the spectrum, mathematically grounded methods such as topological data analysis allow users to design analysis pipelines fully aware of data constraints and symmetries. We introduce an original class of neural network layers based on a generalization of topological persistence. The proposed persistence-based layers allow the users to encode specific data properties (e.g., equivariance) easily. Additionally, these layers can be trained through standard optimization procedures (backpropagation) and composed with classical layers. We test the performance of generalized persistence-based layers as pooling operators in convolutional neural networks for image classification on the MNIST, Fashion-MNIST and CIFAR-10 datasets. Full article

(This article belongs to the Topic Topology vs. Geometry in Data Analysis/Machine Learning)

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38 pages, 13500 KiB  

Open AccessArticle

InvMap and Witness Simplicial Variational Auto-Encoders

by Aniss Aiman Medbouhi, Vladislav Polianskii, Anastasia Varava and Danica Kragic

Mach. Learn. Knowl. Extr. 2023, 5(1), 199-236; https://doi.org/10.3390/make5010014 - 5 Feb 2023

Viewed by 2584

Abstract

Variational auto-encoders (VAEs) are deep generative models used for unsupervised learning, however their standard version is not topology-aware in practice since the data topology may not be taken into consideration. In this paper, we propose two different approaches with the aim to preserve [...] Read more.

Variational auto-encoders (VAEs) are deep generative models used for unsupervised learning, however their standard version is not topology-aware in practice since the data topology may not be taken into consideration. In this paper, we propose two different approaches with the aim to preserve the topological structure between the input space and the latent representation of a VAE. Firstly, we introduce InvMap-VAE as a way to turn any dimensionality reduction technique, given an embedding it produces, into a generative model within a VAE framework providing an inverse mapping into original space. Secondly, we propose the Witness Simplicial VAE as an extension of the simplicial auto-encoder to the variational setup using a witness complex for computing the simplicial regularization, and we motivate this method theoretically using tools from algebraic topology. The Witness Simplicial VAE is independent of any dimensionality reduction technique and together with its extension, Isolandmarks Witness Simplicial VAE, preserves the persistent Betti numbers of a dataset better than a standard VAE. Full article

(This article belongs to the Topic Topology vs. Geometry in Data Analysis/Machine Learning)

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15 pages, 3276 KiB  

Open AccessArticle

NGD-Transformer: Navigation Geodesic Distance Positional Encoding with Self-Attention Pooling for Graph Transformer on 3D Triangle Mesh

by Jiafu Zhuang, Xiaofeng Liu and Wei Zhuang

Symmetry 2022, 14(10), 2050; https://doi.org/10.3390/sym14102050 - 1 Oct 2022

Cited by 2 | Viewed by 2583

Abstract

Following the significant success of the transformer in NLP and computer vision, this paper attempts to extend it to 3D triangle mesh. The aim is to determine the shape’s global representation using the transformer and capture the inherent manifold information. To this end, [...] Read more.

Following the significant success of the transformer in NLP and computer vision, this paper attempts to extend it to 3D triangle mesh. The aim is to determine the shape’s global representation using the transformer and capture the inherent manifold information. To this end, this paper proposes a novel learning framework named Navigation Geodesic Distance Transformer (NGD-Transformer) for 3D mesh. Specifically, this approach combined farthest point sampling with the Voronoi segmentation algorithm to spawn uniform and non-overlapping manifold patches. However, the vertex number of these patches was inconsistent. Therefore, self-attention graph pooling is employed for sorting the vertices on each patch and screening out the most representative nodes, which were then reorganized according to their scores to generate tokens and their raw feature embeddings. To better exploit the manifold properties of the mesh, this paper further proposed a novel positional encoding called navigation geodesic distance positional encoding (NGD-PE), which encodes the geodesic distance between vertices relatively and spatial symmetrically. Subsequently, the raw feature embeddings and positional encodings were summed as input embeddings fed to the graph transformer encoder to determine the global representation of the shape. Experiments on several datasets were conducted, and the experimental results show the excellent performance of our proposed method. Full article

(This article belongs to the Topic Topology vs. Geometry in Data Analysis/Machine Learning)

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21 pages, 1635 KiB  

Open AccessArticle

Adaptively Promoting Diversity in a Novel Ensemble Method for Imbalanced Credit-Risk Evaluation

by Yitong Guo, Jie Mei, Zhiting Pan, Haonan Liu and Weiwei Li

Mathematics 2022, 10(11), 1790; https://doi.org/10.3390/math10111790 - 24 May 2022

Cited by 3 | Viewed by 1896

Abstract

Ensemble learning techniques are widely applied to classification tasks such as credit-risk evaluation. As for most credit-risk evaluation scenarios in the real world, only imbalanced data are available for model construction, and the performance of ensemble models still needs to be improved. An [...] Read more.

Ensemble learning techniques are widely applied to classification tasks such as credit-risk evaluation. As for most credit-risk evaluation scenarios in the real world, only imbalanced data are available for model construction, and the performance of ensemble models still needs to be improved. An ideal ensemble algorithm is supposed to improve diversity in an effective manner. Therefore, we provide an insight in considering an ensemble diversity-promotion method for imbalanced learning tasks. A novel ensemble structure is proposed, which combines self-adaptive optimization techniques and a diversity-promotion method (SA-DP Forest). Additional artificially constructed samples, generated by a fuzzy sampling method at each iteration, directly create diverse hypotheses and address the imbalanced classification problem while training the proposed model. Meanwhile, the self-adaptive optimization mechanism within the ensemble simultaneously balances the individual accuracy as the diversity increases. The results using the decision tree as a base classifier indicate that SA-DP Forest outperforms the comparative algorithms, as reflected by most evaluation metrics on three credit data sets and seven other imbalanced data sets. Our method is also more suitable for experimental data that are properly constructed with a series of artificial imbalance ratios on the original credit data set. Full article

(This article belongs to the Topic Topology vs. Geometry in Data Analysis/Machine Learning)

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13 pages, 777 KiB  

Open AccessArticle

Generating Soft Topologies via Soft Set Operators

by A. A. Azzam, Zanyar A. Ameen, Tareq M. Al-shami and Mohammed E. El-Shafei

Symmetry 2022, 14(5), 914; https://doi.org/10.3390/sym14050914 - 29 Apr 2022

Cited by 38 | Viewed by 2204

Abstract

As daily problems involve a great deal of data and ambiguity, it has become vital to build new mathematical ways to cope with them, and soft set theory is the greatest tool for doing so. As a result, we study methods of generating [...] Read more.

As daily problems involve a great deal of data and ambiguity, it has become vital to build new mathematical ways to cope with them, and soft set theory is the greatest tool for doing so. As a result, we study methods of generating soft topologies through several soft set operators. A soft topology is known to be determined by the system of special soft sets, which are called soft open (dually soft closed) sets. The relationship between specific types of soft topologies and their classical topologies (known as parametric topologies) is linked to the idea of symmetry. Under this symmetry, we can study the behaviors and properties of classical topological concepts via soft settings and vice versa. In this paper, we show that soft topological spaces can be characterized by soft closure, soft interior, soft boundary, soft exterior, soft derived set, or co-derived set operators. All of the soft topologies that result from such operators are equivalent, as well as being identical to their classical counterparts under enriched (extended) conditions. Moreover, some of the soft topologies are the systems of all fixed points of specific soft operators. Multiple examples are presented to show the implementation of these operators. Some of the examples show that, by removing any axiom, we will miss the uniqueness of the resulting soft topology. Full article

(This article belongs to the Topic Topology vs. Geometry in Data Analysis/Machine Learning)

21 pages, 3259 KiB  

Open AccessArticle

LogLS: Research on System Log Anomaly Detection Method Based on Dual LSTM

by Yiyong Chen, Nurbol Luktarhan and Dan Lv

Symmetry 2022, 14(3), 454; https://doi.org/10.3390/sym14030454 - 24 Feb 2022

Cited by 16 | Viewed by 4702

Abstract

System logs record the status and important events of the system at different time periods. They are important resources for administrators to understand and manage the system. Detecting anomalies in logs is critical to identifying system faults in time. However, with the increasing [...] Read more.

System logs record the status and important events of the system at different time periods. They are important resources for administrators to understand and manage the system. Detecting anomalies in logs is critical to identifying system faults in time. However, with the increasing size and complexity of today’s software systems, the number of logs has exploded. In many cases, the traditional manual log-checking method becomes impractical and time-consuming. On the other hand, existing automatic log anomaly detection methods are error-prone and often use indices or log templates. In this work, we propose LogLS, a system log anomaly detection method based on dual long short-term memory (LSTM) with symmetric structure, which regarded the system log as a natural-language sequence and modeled the log according to the preorder relationship and postorder relationship. LogLS is optimized based on the DeepLog method to solve the problem of poor prediction performance of LSTM on long sequences. By providing a feedback mechanism, it implements the prediction of logs that do not appear. To evaluate LogLS, we conducted experiments on two real datasets, and the experimental results demonstrate the effectiveness of our proposed method in log anomaly detection. Full article

(This article belongs to the Topic Topology vs. Geometry in Data Analysis/Machine Learning)

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26 pages, 860 KiB  

Open AccessArticle

A Novel Twin Support Vector Machine with Generalized Pinball Loss Function for Pattern Classification

by Wanida Panup, Wachirapong Ratipapongton and Rabian Wangkeeree

Symmetry 2022, 14(2), 289; https://doi.org/10.3390/sym14020289 - 31 Jan 2022

Cited by 11 | Viewed by 3077

Abstract

We introduce a novel twin support vector machine with the generalized pinball loss function (GPin-TSVM) for solving data classification problems that are less sensitive to noise and preserve the sparsity of the solution. In addition, we use a symmetric kernel trick to enlarge [...] Read more.

We introduce a novel twin support vector machine with the generalized pinball loss function (GPin-TSVM) for solving data classification problems that are less sensitive to noise and preserve the sparsity of the solution. In addition, we use a symmetric kernel trick to enlarge GPin-TSVM to nonlinear classification problems. The developed approach is tested on numerous UCI benchmark datasets, as well as synthetic datasets in the experiments. The comparisons demonstrate that our proposed algorithm outperforms existing classifiers in terms of accuracy. Furthermore, this employed approach in handwritten digit recognition applications is examined, and the automatic feature extractor employs a convolution neural network. Full article

(This article belongs to the Topic Topology vs. Geometry in Data Analysis/Machine Learning)

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24 pages, 645 KiB  

Open AccessArticle

Some Topological Approaches for Generalized Rough Sets and Their Decision-Making Applications

by Radwan Abu-Gdairi, Mostafa A. El-Gayar, Tareq M. Al-shami, Ashraf S. Nawar and Mostafa K. El-Bably

Symmetry 2022, 14(1), 95; https://doi.org/10.3390/sym14010095 - 7 Jan 2022

Cited by 30 | Viewed by 2619

Abstract

The rough set principle was proposed as a methodology to cope with vagueness or uncertainty of data in the information systems. Day by day, this theory has proven its efficiency in handling and modeling many real-life problems. To contribute to this area, we [...] Read more.

The rough set principle was proposed as a methodology to cope with vagueness or uncertainty of data in the information systems. Day by day, this theory has proven its efficiency in handling and modeling many real-life problems. To contribute to this area, we present new topological approaches as a generalization of Pawlak’s theory by using j-adhesion neighborhoods and elucidate the relationship between them and some other types of approximations with the aid of examples. Topologically, we give another generalized rough approximation using near open sets. Also, we generate generalized approximations created from the topological models of j-adhesion approximations. Eventually, we compare the approaches given herein with previous ones to obtain a more affirmative solution for decision-making problems. Full article

(This article belongs to the Topic Topology vs. Geometry in Data Analysis/Machine Learning)

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