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Jacobi-Davidson Method on Low-Rank Matrix Manifolds
SIAM Journal of Scientific Computing. 2018. Vol. 40. No. 2. P. A1149–A1170.
Rakhuba M., Oseledets I.
In this work we generalize the Jacobi--Davidson method to the case when the eigenvector can be reshaped into a low-rank matrix. In this setting the proposed method inherits the advantages of the original Jacobi--Davidson method, has lower complexity, and requires less storage. We also introduce a low-rank version of the Rayleigh quotient iteration which naturally arises in the Jacobi--Davidson method.
Piontkovski D., / Series arXiv "math". 2026.
A noncommutative projective variety is defined, following Artin and Zhang, by a graded coherent algebra 𝐴. The category of coherent sheaves is then the quotient qgr(𝐴) of the category of finitely presented graded modules by the subcategory of torsion modules. We consider the categorical and polynomial entropies of the Serre twist, that is, of the ...
Added: June 23, 2026
Piontkovski D., / Series arXiv "math". 2025.
If a symmetric multilinear algebra is weakly nil, then it is Engel. This result may be regarded as an infinite-dimensional analogue of the well-known Jacobian theorem, which states that if a polynomial mapping has a polynomial inverse, then its Jacobian matrix is invertible. This refines a theorem of Gerstenhaber and partially answers a question posed ...
Added: June 23, 2026
Stanislav Morozov, Calcolo 2026 Vol. 63 No. 2 Article 23
The approximation of tensors in a low-para metric format is a crucial component in many mathematical modelling and data analysis tasks. Among the widely used low-parametric representations, the canonical polyadic (CP) decomposition is known to be very efficient. Nowadays, most algorithms for CP approximation aim to construct the approximation in the Frobenius norm; however, some ...
Added: May 22, 2026
Shipilov F., Barnyakov A., Ivanov A. et al., / Series Physics "arxiv.org". 2026.
A fast simulation of the detector response is a vital task in high-energy physics (HEP). Traditional Monte-Carlo methods form the backbone of modern particle physics simulation software but are computationally expensive. We present a machine-learning-based approach to fast simulation of the Focusing Aerogel Ring Imaging Cherenkov (FARICH) detector response. Given a particle track and momentum, ...
Added: May 19, 2026
Vladimir Bogachev, Aletov V., Alexander Molozhavenko et al., , in: The Fourteenth International Conference on Learning Representations (ICLR 2026).: ICLR, 2026. Ch. 20503 P. 1–26.
This work presents a novel, fully Riemannian framework for Low-Rank Adaptation (LoRA) that geometrically treats low-rank adapters by optimizing them directly on the fixed-rank manifold. This formulation eliminates the parametrization ambiguity present in standard Euclidean optimizers. Our framework integrates three key components to achieve this: (1) we derive Riemannion, a new Riemannian optimizer on the fixed-rank ...
Added: April 29, 2026
Derkacheva A., Sakirkina M., Kraev G. et al., /. 2026.
Comprehensive data on natural hazards and their consequences are crucial for effective for risk assessment, adaptation planning, and emergency response. However, many countries face challenges with fragmented, inconsistent, and inaccessible data, particularly regarding local-scale events. To address this data gap in Russia, we developed an end-to-end processing pipeline that scrapes news from various online sources, ...
Added: April 28, 2026
Pilé I., Deng Y., Shchur L., / Series arXiv "math". 2026. No. 2604.10254.
We investigate the spatial overlap of successive spin configurations in Markov chain Monte Carlo simulations using the local Metropolis algorithm and the Svendsen-Wang and Wolff cluster algorithms. We examine the dynamics of these algorithms for two models in different universality classes: the Ising model and the Potts model with three components. The overlap of two ...
Added: April 20, 2026
Gabdullin N., Androsov I., / Series Computer Science "arxiv.org". 2026.
Label prediction in neural networks (NNs) has O(n) complexity proportional to the number of classes. This holds true for classification using fully connected layers and cosine similarity with some set of class prototypes. In this paper we show that if NN latent space (LS) geometry is known and possesses specific properties, label prediction complexity can ...
Added: April 2, 2026
Sorokin K., Beketov M., Онучин А. et al., / arxiv.org. Серия cs.SI "Social and Information Networks ". 2025.
Community detection in complex networks is a fundamental problem, open to new approaches in various scientific settings. We introduce a novel community detection method, based on Ricci flow on graphs. Our technique iteratively updates edge weights (their metric lengths) according to their (combinatorial) Foster version of Ricci curvature computed from effective resistance distance between the ...
Added: January 15, 2026
Low-rank matrix and tensor approximations for compression of machine-learning interatomic potentials
Vorotnikov I., Romashov F., Rybin N. et al., Journal of Chemical Physics 2025 Vol. 163 No. 24 Article 244112
Machine-learning interatomic potentials (MLIPs) have become a mainstay in computationally guided materials science, surpassing traditional force fields due to their flexible functional form and superior accuracy in reproducing physical properties of materials. This flexibility is achieved through mathematically rigorous basis sets that describe interatomic interactions within a local atomic environment. The number of parameters in ...
Added: January 4, 2026
Petrovanov I., Sergeev A., / Series Computer Science "arxiv.org". 2025. No. 2512.18332.
Transport coding reduces message delay in packet-switched networks by introducing controlled redundancy at the transport layer: original packets are encoded into coded packets, and the message is reconstructed after the first successful deliveries, effectively shifting latency from the maximum packet delay to the -th order statistic. We present a concise, reproducible discrete-event implementation of transport coding in OMNeT++, including ...
Added: December 24, 2025
Alexander Molozhavenko, Rakhuba M., Computational and Applied Mathematics 2026 Vol. 45 No. 6 Article 221
This paper studies tensors that admit decomposition in the Extended Tensor Train (ETT) format, with a key focus on the case where some decomposition factors are constrained to be equal. This factor sharing introduces additional challenges, as it breaks the multilinear structure of the decomposition. Nevertheless, we show that Riemannian optimization methods can naturally handle ...
Added: December 22, 2025
Hessian-based lightweight neural network for brain vessel segmentation on a minimal training dataset
Меньшиков И. А., Бернадотт А. К., Elvimov N. S., / Series arXie "Statistical mechanics". 2025.
Accurate segmentation of blood vessels in brain magnetic resonance angiography (MRA) is essential for successful surgical procedures, such as aneurysm repair or bypass surgery. Currently, annotation is primarily performed through manual segmentation or classical methods, such as the Frangi filter, which often lack sufficient accuracy. Neural networks have emerged as powerful tools for medical image ...
Added: December 1, 2025
Rubchinskiy A., Chubarova D., / Series WP7 "Математические методы анализа решений в экономике, бизнесе и политике". 2025. No. WP7/2025/01.
The article examines one of the most famous examples of socio-economic systems, characterized by significant uncertainty – the S&P-500 stock market, where shares of 500 largest US companies are traded. No assumptions are made about the probabilistic characteristics of the stock market. A flexible algorithm for daily trading has been developed, based on both known fixed data ...
Added: November 9, 2025
Guo Z., Jiang T., Wang G. et al., Journal of Computational and Applied Mathematics 2025 Vol. 463
The introduction of quaternions in the quantum chemistry framework provides new ideas for solving some of the difficulties encountered in traditional approaches and is expected to expand the scope of quantum chemistry research. This paper investigates the eigenvalue problem of a Hermitian quaternion matrix by means of the complex representation of a quaternion matrix. This ...
Added: December 16, 2024
Peshekhonov I., Aleksey Arzhantsev, Rakhuba M., , in: Proceedings of The 27th International Conference on Artificial Intelligence and Statistics (AISTATS 2024), 2-4 May 2024, Palau de Congressos, Valencia, Spain. PMLR: Volume 238Vol. 238.: Valencia: PMLR, 2024. Ch. 238 P. 3304–3312.
Added: November 29, 2024
Kazeev V., Oseledets I., Maxim V. Rakhuba et al., Multiscale Modeling and Simulation 2022 Vol. 20 No. 3 P. 893–935
Homogenization in terms of multiscale limits transforms a multiscale problem with 𝑛+1n+1 asymptotically separated microscales posed on a physical domain 𝐷⊂ℝ𝑑D⊂Rd into a one-scale problem posed on a product domain of dimension (𝑛+1)𝑑(n+1)d by introducing 𝑛n so-called fast variables. This procedure allows one to convert 𝑛+1n+1 scales in 𝑑d physical dimensions into a single-scale structure in (𝑛+1)𝑑(n+1)ddimensions. We prove here that both the original, physical multiscale problem and ...
Added: October 30, 2022
Marcati C., Rakhuba M., Schwab C., Advances in Computational Mathematics 2022 Vol. 48 No. 3 Article 18
We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in ℝ3. We consider functions in countably normed Sobolev spaces with radial weights and analytic- or Gevrey-type control of weighted semi-norms. Several classes of boundary value and eigenvalue problems from science and engineering are discussed whose solutions belong to the ...
Added: October 30, 2022
Novikov A., Rakhuba M., Oseledets I., SIAM Journal of Scientific Computing 2022 Vol. 44 No. 2 P. A843–A869
In scientific computing and machine learning applications, matrices and more general multidimensional arrays (tensors) can often be approximated with the help of low-rank decompositions. Since matrices and tensors of fixed rank form smooth Riemannian manifolds, one of the popular tools for finding low-rank approximations is to use Riemannian optimization. Nevertheless, efficient implementation of Riemannian gradients ...
Added: October 31, 2021
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
Oseledets I., Rakhuba M., André U., SIAM Journal on Numerical Analysis 2018 Vol. 56 No. 6 P. 3459–3479
In this note we take a new look at the local convergence of alternating optimization methods for low-rank matrices and tensors. Our abstract interpretation as sequential optimization on moving subspaces yields insightful reformulations of some known convergence conditions that focus on the interplay between the contractivity of classical multiplicative Schwarz methods with overlapping subspaces and ...
Added: October 20, 2020
Rakhuba Maxim, Novikov A., Oseledets I., Journal of Computational Physics 2019 Vol. 396 P. 718–737
Such problems as computation of spectra of spin chains and vibrational spectra of molecules can be written as high-dimensional eigenvalue problems, i.e., when the eigenvector can be naturally represented as a multidimensional tensor. Tensor methods have proven to be an efficient tool for the approximation of solutions of high-dimensional eigenvalue problems, however, their performance deteriorates quickly ...
Added: October 19, 2020
Делицын А. Л., Гребеньков Д. С., Quarterly Journal of Mechanics and Applied Mathematics 2018 Vol. 71 No. 4 P. 537–580
We present several applications of mode matching methods in spectral and scattering problems. First, we consider the eigenvalue problem for the Dirichlet Laplacian in a finite cylindrical domain that is split into two subdomains by a ‘perforated’ barrier. Using rather elementary methods, we prove that the first eigenfunction is localized in the larger subdomain, that ...
Added: December 20, 2018
Kurochkin S. V., Computational Mathematics and Mathematical Physics 2018 Vol. 58 No. 12 P. 1937–1947
For the regular Sturm–Liouville boundary value problem with general nonseparated selfadjoint
boundary conditions, conditions for the existence of zero and negative eigenvalues and expressions
for their number are obtained. The conditions are expresses in a closed form, and the coefficient
functions of the original equation appear in these conditions indirectly through a single numerical
characteristic. ...
Added: November 21, 2018