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Topological approach to diagonalization algorithms
Sorokin K., Ayzenberg A.
In this paper we prove that there exists an asymptotical diagonalization algorithm for a class of sparse Hermitian (or real symmetric) matrices if and only if the matrices become Hessenberg matrices after some permutation of rows and columns. The proof is based on Morse theory, Roberts’ theorem on indifference graphs, toric topology, and computer-based homological calculations.
Keywords: Topological data analysis
Publication based on the results of:
Alexander Kachura, Vsevolod Chernyshev, Kachan O. et al., Frontiers in Psychiatry 2026 Vol. 16 Article 1677282
Autism spectrum disorder (ASD) is one of the most common neurodevelopmental disorders. Existing studies show that adults with ASD may experience accelerated or altered neurocognitive aging. Consequently, cognitive decline in people with ASD can be delayed if timely measures are taken to treat this disorder. This study focuses on the development of a new algorithm ...
Added: January 21, 2026
Vasilii A. Gromov, Dang Q. N., Asel S. Erbolova, Complexity 2025 Vol. 2025 No. 1 Article 9659172
The present paper employs topological data analysis methods to reveal ‘holes’ (stable persistent homologies) in the semantic spaces of words, bigrams, and trigrams of the English and Russian languages, and to ascertain their boundaries. Furthermore, the paper selects those holes that belong to the large‐scale (coarse‐grained) structure of the language that are not just local ...
Added: November 11, 2025
Абрамов А. С., Chernyshev V. L., Mikhaylets E. et al., / Series Social Science Research Network "Social Science Research Network". 2025.
Computer vision is one of the most relevant modern research areas with broad practical applications. However, traditional solutions based on deep learning have signicant limitations and can be misleading. Topological data analysis, on the other hand, is a modern approach to solving similar problems using mathematically deterministic methods of algebraic topology that reduce the risk ...
Added: September 23, 2025
Kudrjashov S., Karpik O., Klyshinskiy E., , in: Analysis of Images, Social Networks and Texts, 12th International Conference, AIST 2024, Bishkek, Kyrgyzstan, October 17–19, 2024, Revised Selected PapersVol. 15419.: Springer, 2024. P. 120–130.
Added: May 29, 2025
Sergei Kudriashov, Veronika Zykova, Stepanova A. et al., , in: Advances in Neural Computation, Machine Learning, and Cognitive Research VIII, Selected Papers from the XXVI International Conference on Neuroinformatics, October 21-25, 2024, Moscow, RussiaVol. VIII.: Cham: Springer, 2024. P. 13–22.
The interpretation of deep learning models is a rapidly growing field,
with particular interest in language models. There are various approaches to this
task, including training simpler models to replicate neural network predictions and
analyzing the latent space of the model. The latter method allows us to not only
identify patterns in the model’s decision-making process, but also understand ...
Added: October 24, 2024
Vasilii A. Gromov, Nikita S. Borodin, Asel S. Yerbolova, Complexity 2024 Vol. 2024 No. 1 Article 8863360
Te present paper introduces a novel object of study, a language fractal structure; we hypothesize that a set of embeddings of all n-grams of a natural language constitutes a representative sample of this fractal set. (We use the term Hailonakea to refer to the sum total of all language fractal structures, over all n). Te ...
Added: June 29, 2024
Cherniavskii D., Tulchinskii E., Mikhailov V. et al., , in: Findings of the Association for Computational Linguistics: EMNLP 2022.: Association for Computational Linguistics, 2022. Ch. 7 P. 88–107.
Added: February 17, 2023
Sorokin K., Ayzenberg A., Анохин К. В. et al., / Series Computer Science "arxiv.org". 2022.
In present paper we test different approaches to reconstructing the topology of the physical space from neural activity data in A1 fields of mice brains, in particular, having a Cognitome-focused approach in mind. Animals were placed in different new environments and discovered them while their physical and neural activity was recorded. We discuss possible approaches ...
Added: November 17, 2022
Magai German, Ayzenberg A., / Series CS "https://arxiv.org/". 2022.
Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and substantiate the geometric and topological view of the learning process of neural networks. Our attention is focused on ...
Added: November 14, 2022
Society of Photo-Optical Instrumentation Engineers, 2020.
The paper will provide examples of computer vision tasks in which topological data analysis gave new effective solutions. Ideas underlying topological data analysis and its basic methods will be briefly described and illustrated with examples of computer vision problems. No prior knowledge in topological data analysis and computational geometry is assumed, a brief introduction to ...
Added: October 14, 2021
IEEE, 2021.
The increasing use of eye tracking in modern cognitive and clinical psychology, neuroscience, and ophthalmology requires new methods of objective quantitative analysis of complex eye movement data. In the current work, topological data analysis (TDA) is used to extract a new type of features of eye movements to differentiate between two eye movements groups, obtained ...
Added: October 14, 2021