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Alignment of Vector Fields on Manifolds via Contraction Mappings
Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2018. Vol. 160. No. 2. P. 300–308.
According to the manifold hypothesis, high-dimensional data can be viewed and meaning- fully represented as a lower-dimensional manifold embedded in a higher dimensional feature space. Manifold learning is a part of machine learning where an intrinsic data representation is uncovered based on the manifold hypothesis.
Many manifold learning algorithms were developed. The one called Grassmann&Stiefel eigenmaps (GSE) has been considered in the paper. One of the GSE subproblems is tangent space alignment. The original solution to this problem has been formulated as a generalized eigenvalue problem. In this formulation, it is plagued with numerical instability, resulting in suboptimal solutions to the subproblem and manifold reconstruction problem in general.
We have proposed an iterative algorithm to directly solve the tangent spaces alignment problem. As a result, we have obtained a significant gain in algorithm efficiency and time complexity. We have compared the performance of our method on various model data sets to show that our solution is on par with the approach to vector fields alignment formulated as an optimization on the Stiefel group.
Chikake T. M., Goldengorin B. I., Pardalos P. M., Computer Optics 2025 Vol. 49 No. 6 P. 1191–1201
We present a general-purpose, training-free framework for dimensionality reduction and clustering based on per–sample pseudo–Boolean polynomials (PBP). The method constructs compact, interpreTab. features without model fitting and is evaluated under a standardized protocol that compares PBP to PCA, t-SNE, and UMAP using identical inputs and metrics: clustering alignment (V-measure, Adjusted Rand Index), cluster geometry (Silhouette coefficient, ...
Added: January 2, 2026
Zakharov S., Pisarev V., , in: Supercomputing: 9th Russian Supercomputing Days, RuSCDays 2023, Moscow, Russia, September 25–26, 2023, Revised Selected Papers, Part I.: Springer, 2023. P. 59–73.
We present open-source software for isochoric isothermal phase equilibrium calculation of fluids extensible on custom equations of state. We demonstrate robustness of the solvers by calculation of binodals and equilibrium parameters for a number of mixtures modelled by a cubic and a SAFT-family equations of state. Additionally, we consider multi-threaded computation of a phase diagram. ...
Added: November 11, 2025
Nikita Puchkin, Vladimir Spokoiny, Eugene Stepanov et al., ESAIM - Control, Optimisation and Calculus of Variations 2024 Vol. 30 Article 3
We consider the problem of reconstructing an embedding of a compact connected Riemannian manifold in a Euclidean space up to an almost isometry, given the information on intrinsic distances between points from its “sufficiently large” subset. This is one of the classical manifold learning problems. It happens that the most popular methods to deal with ...
Added: February 2, 2024
Puchkin N., Spokoiny V., Journal of Machine Learning Research 2022 Vol. 23 No. 40 P. 1–62
We consider a problem of manifold estimation from noisy observations. Many manifold learning procedures locally approximate a manifold by a weighted average over a small neighborhood. However, in the presence of large noise, the assigned weights become so corrupted that the averaged estimate shows very poor performance. We suggest a structure-adaptive procedure, which simultaneously reconstructs ...
Added: February 3, 2022
IEEE, 2020.
Dimensionality reduction problem is stated as finding a mapping f:X ∈ R m → Z ∈ R n , where ≪ m while preserving some relevant properties of the data. We formulate topology-preserving dimensionality reduction as finding the optimal orthogonal projection to the lower-dimensional subspace which minimizes discrepancy between persistent diagrams of the original data and the projection. This ...
Added: October 14, 2021
Andreev N. A., Smirnov S. N., Computational Mathematics and Modeling 2021 Vol. 32 P. 22–44
We consider a guaranteed deterministic approach to discrete-time super-replication for guaranteed coverage of contingent claims on options for all possible asset-price scenarios. Price increases during a period are assumed to be contained in a priori specified compacta dependent on price history. A game problem is stated and reduced to the solution of the corresponding Bellman–Isaacs ...
Added: September 30, 2021
Popkov Y., Popkov A., Dubnov Y. A., Информатика и ее применения 2020 Т. 14 № 4 С. 47–54
The work is devoted to development of methods for deterministic and randomized projection aimed at dimensionality reduction problems. In the deterministic case, the authors develop the parallel reduction procedure minimizing Kullback-Leibler cross-entropy target to condition on information capacity based on the gradient projection method. In the randomized case, the authors solve the problem of reduction ...
Added: January 26, 2021
Alanov A., Kochurov M., Volkhonskiy D. et al., , in: Proceedings of the 15th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISAPP 2020)Vol. 4.: SciTePress, 2020. P. 214–221.
We propose a novel multi-texture synthesis model based on generative adversarial networks (GANs) with a user-controllable mechanism. The user control ability allows to explicitly specify the texture which should be generated by the model. This property follows from using an encoder part which learns a latent representation for each texture from the dataset. To ensure ...
Added: November 8, 2020
Minabutdinov A., Bouev M., Manaev I., Journal of Computational Finance 2020 Vol. 24 No. 2 P. 103–127
We consider the problem of finding a valid covariance matrix in the foreign
exchange market given an initial non-PSD estimate of such a matrix. The common
no-arbitrage assumption imposes additional linear constraints on such matrices,
whereby inevitably making them singular. As a result, even the most advanced
numerical techniques will predictably balk at a seemingly standard optimization
task. The reason ...
Added: October 31, 2020
Popkov Y., Popkov A., Dubnov Y. A., Математическое моделирование 2020 Т. 32 № 9 С. 35–52
We develop a new method of dimensionality reduction based on direct and inverse projection of data matrix and calculation of projectors minimizing cross-entropy functional. Concept of information capacity of matrix which is used as a restriction in a problem of optimal reduction is introduced. We conduct a comparison of proposed method with known ones based ...
Added: October 31, 2020
Kachan O. N., Yanovich Y., Abramov E., Ученые записки Казанского университета. Серия: Физико-математические науки 2018 Vol. 160 No. 2 P. 300–308
According to the manifold hypothesis, high-dimensional data can be viewed and meaningfully represented as a lower-dimensional manifold embedded in a higher dimensional feature space. Manifold learning is a part of machine learning where an intrinsic data representation is uncovered based on the manifold hypothesis.
Many manifold learning algorithms were developed. The one called Grassmann & Stiefel eigenmaps (GSE) ...
Added: October 29, 2020
Abramov E., Yanovich Y., Ученые записки Казанского университета. Серия: Физико-математические науки 2018 Vol. 160 No. 2 P. 220–228
Real data are usually characterized by high dimensionality. However, real data obtained from real sources, due to the presence of various dependencies between data points and limitations on their possible values, form, as a rule, form a small part of the high-dimensional space of observations. The most common model is based on the hypothesis that ...
Added: October 29, 2020
Kuleshov A. P., Bernstein A. V., Yanovich Y., Ученые записки Казанского университета. Серия: Физико-математические науки 2018 Vol. 160 No. 2 P. 327–338
The problem of unknown high-dimensional density estimation has been considered. It has been suggested that the support of its measure is a low-dimensional data manifold. This problem arises in many data mining tasks. The paper proposes a new geometrically motivated solution to the problem in the framework of manifold learning, including estimation of an unknown ...
Added: October 28, 2020
Kuleshov A. P., Bernstein A. V., Yanovich Y., Ученые записки Казанского университета. Серия: Физико-математические науки 2018 Vol. 160 No. 2 P. 327–338
The problem of unknown high-dimensional density estimation has been considered. It has been suggested that the support of its measure is a low-dimensional data manifold. This problem arises in many data mining tasks. The paper proposes a new geometrically motivated solution to the problem in the framework of manifold learning, including estimation of an unknown ...
Added: October 28, 2020
Denisenko A., Krylov G. O., Biosciences Biotechnology Research Asia 2015 Vol. 12 P. 609–617
We show the way principal components analysis could be used to preprocess the dataset for the purposes of visualization of a particular property of the object. PCA was used to synthesize scores which were visualized in network analysis process. This provides analyst with a holistic picture of the presense of a particular property in the selection of analyzed ...
Added: December 21, 2018
Dubnov Y. A., Информационные технологии и вычислительные системы 2018 № 2 С. 60–69
The paper considers the problem of reducing the dimension of the feature space for describing objects
in data analysis problems using the example of binary classification. The article provides a detailed
overview of existing approaches to solving this problem and proposes several modifications. In which
the dimensionality reduction is considered as the problem of extracting the most relevant ...
Added: July 4, 2018