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Estimation of the signal subspace without estimation of the inverse covariance matrix
Let a high-dimensional random vector $\vX$ be represented as a sum of two components - a signal $\vS$ that belongs to some low-dimensional linear subspace $\S$, and a noise component $\vN$. This paper presents a new approach for estimating the subspace $\S$ based on the ideas of the Non-Gaussian Component Analysis. Our approach avoids the technical difficulties that usually appear in similar methods - it requires neither the estimation of the inverse covariance matrix of $\vX$ nor the estimation of the covariance matrix of $\vN$.
Dorovskiy A., / Series arXiv "math". 2026.
In this paper the structural stability of generic families of vector fields of the PC-HC class on the two-dimensional sphere is proved. A classification of these families up to moderate equivalence in neighborhoods of their large bifurcation supports is presented, based on such invariants as the configuration and the characteristic set. The realization lemma is proved. ...
Added: May 14, 2026
Taletskii D., / Series arXiv "math". 2026.
A vertex subset of a graph is called a \textit{distance-$k$ independent set} if the distance between any two of its distinct vertices is at least $k + 1$. For all $n,k \geq 1$, we determine the minimum possible number of inclusion-wise maximal distance-$k$ independent sets among all $n$-vertex trees. It equals~$n$ if $n \leq k ...
Added: May 1, 2026
Ovcharenko M., / Series arXiv "math". 2026.
We introduce an explicit class of tempered Laurent polynomials in the sense of Villegas and Doran--Kerr in n⩽4 variables including all Landau--Ginzburg models for smooth Fano threefolds with very ample anticanonical class. We check that it contains Landau--Ginzburg models for various Fano fourfolds which are complete intersections in smooth toric varieties and Grassmannians of planes, ...
Added: April 30, 2026
Zlotnik Alexander, / Series arXiv "math". 2026. No. 2602.03481v1.
We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type continuous dependence of the solution $(\eta,u,\theta)$, in a norm slightly stronger than $L^{2,\infty}(Q)\times L^2(Q)\times L^2(Q)$, on the initial data $(\eta^0,u^0,e^0)$ in a ...
Added: April 18, 2026
Medvedev V., / Series arXiv "math". 2026.
We investigate the interplay between the dimension of the space of static potentials and the geometric and topological structure of the underlying static three-manifold. A partial classification of boundaryless static manifolds is obtained in terms of this dimension. We also treat the case of static manifolds with boundary. In particular, we prove that if a ...
Added: April 3, 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
Kolesnikov A., / Series arXiv "math". 2025.
We study Blaschke--Santal{ó}-type inequalities for N>=2 sets (functions) and a special class of cost functions. In particular, we prove new results about reduction of the maximization problem for the Blaschke--Santal{ó}-type functional to homogeneous case (functional inequalities on the sphere) and extend the symmetrization argument to the case of N>2 sets.
We also discuss links to the ...
Added: February 13, 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
Gaianov N., Parusnikova A., / Cornell University. Серия math "arxiv.org". 2025.
An algebraic q-difference equation is considered. A sufficient condition for the existence of a formal power-logarithmic expansion of a solution to such an equation in the neighborhood of zero is proposed. An example of applying this sufficient condition for constructing a formal expansion of a solution to a certain q-difference analogue of the fifth Painlevé equation ...
Added: December 25, 2025
Popov V., / Series arXiv "math". 2025. No. 2502.01539.
We prove that the variety of flexes of algebraic curves
of degree 3 in the projective plane is an ideal theoretic complete
intersection in the product of a two-dimensional and a nine-dimensional projective spaces. ...
Added: December 16, 2025
Gnetov F., Konakov V., / Series arXiv "math". 2025. No. 2512.04667.
We establish a central limit theorem, a local limit theorem, and a law of large numbers for a natural
random walk on a symmetric space M of non-compact type and rank one. This class of spaces, which
includes the complex and quaternionic hyperbolic spaces and the Cayley hyperbolic plane, generalizes
the real hyperbolic space Hn. Our approach introduces ...
Added: December 5, 2025
Kazakov A., Koryakin V., Safonov K. et al., / Series arXiv "math". 2025.
The Lorenz attractor is the first example of a robustly chaotic non-hyperbolic attractor. Each orbit of such an attractor has a positive top Lyapunov exponent, and this property persists under small perturbations despite possible bifurcations of the attractor. In this paper, we study the boundary of the Lorenz attractor existence region in the Shimizu-Morioka model. ...
Added: December 4, 2025
Bitter I., Konakov V., / Cornell University. Серия arXiv "math". 2025. № 2505.24548.
В работе приводится обобщение локальной предельной теоремы о сходимости неоднородных цепей Маркова к диффузионному пределу на случай, когда соответ- ствующие коэффициенты процессов удовлетворяют слабым условиям регулярности и совпадают лишь асимптотически. В частности, рассматриваемые нами коэффици- енты сноса могут быть неограниченными с не более чем линейным ростом, а оценки отражают перенос терминального состояния неограниченным трендом через ...
Added: December 3, 2025
Bogomolov F. A., Schrandt S., / Series arXiv "math". 2025.
We discuss phenomena of stabilization for direct images of line bundles over projective curves mapping onto the projective line, for maps of sufficiently big degree. ...
Added: December 1, 2025
Deviatov R., Baek S., / Series arXiv "math". 2025.
The torsion index of split simple groups has been extensively studied, notably by Totaro, who calculated the torsion indexes of the spin groups and $E_{8}$ in [5] and [6], respectively. The aim of this paper is to provide upper bounds for the torsion index of half-spin groups, the only remaining case in the calculation of ...
Added: December 1, 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
Prokhorov Y., / Series arXiv "math". 2025.
A $\mathbf{Q}$-conic bundle is a contraction $f: X\to Z$ of a three-dimensional algebraic variety $X$ to a surface~$Z$ such that the variety~$X$ has only terminal $\mathbf{Q}$-factorial singularities, the anticanonical divisor $-K_X$ is~$f$-ample, and $\uprho(X/Z)=1$. We provide an algorithm to transform a $\mathbf{Q}$-conic bundle to its standard form. ...
Added: December 1, 2025
Amerik E., Verbitsky M., Soldatenkov A., / Series arXiv "math". 2025.
Wierzba and Wisniewski proved that in dimension 4, every bimeromorphic map of hyperkahler manifolds is represented as a composition of Mukai flops. Hu and Yau conjectured that this result can be generalized to arbitrary dimension. They defined ``Mukai's elementary transformation'' as the blow-up of a subvariety ruled by complex projective spaces, composed with the contraction ...
Added: December 1, 2025
Popkov Y., Popkov A. Y., Dubnov Y. A., Mathematical Models and Computer Simulations 2021 Vol. 13 No. 3 P. 382–394
© 2021, Pleiades Publishing, Ltd.Abstract: We develop/propose the method reducing the dimension of a data matrix, based on its direct and inverse projection, and the calculation of projectors that minimize the cross-entropy functional, remove. We introduce the concept of information capacity of a matrix, which is used as a constraint in the optimal reduction problem, ...
Added: October 28, 2022
Вельдяйкин Николай, Yanovich Y., В кн.: Сборник статей конференции «Информационные технологии и системы» (ИТиС'17).: ИППИ РАН, 2017. С. 74–80.
Методы снижения размерности позволяют заменить многомерные описания данных на их низкоразмерные аналоги почти без потери информации, что способно упростить построение моделей по ним в рамках машинного обучения. Как правило, программные реализации алгоритмов снижения размерности строят лишь низкоразмерные описания для точек из обучающих выборок. Однако для последующего решения задач классификации и регрессии важно уметь строить вложение ...
Added: July 10, 2021
Denisenko A., Крылов Г. О., Корнев И. А., Известия Самарского научного центра Российской академии наук 2015 Т. 17 № 2 С. 1131–1140
В работе представлены результаты анализа методом главных компонент данных по финансовым потокам компаний и аффилированных с ними лиц в отрасли российской экономики, синтезированы рейтинговые оценки активности компаний в этой отрасли экономики в разрезе регионов. ...
Added: December 21, 2018
Panov V., / Series Discussion paper SFB 649 "Economic risk". 2010. No. 2010-026.
In this article, we present new ideas concerning Non-Gaussian Component Analysis (NGCA). We use the structural assumption that a high-dimensional random vector $\vX$ can be represented as a sum of two components - a low-dimensional signal $\vS$ and a noise component $\vN$. We show that this assumption enables us for a special representation for the ...
Added: September 3, 2015
Karasev M., Novikova E., Mathematical notes 2014 Vol. 96 No. 6 P. 965–970
We discuss the general opportunity to create (asymptotically) a comletely integrable system from the original perturbed system by inserting additional perturbing terms. After such an artificial insertion, there appears an opportunity to make the secondary averaging and secondary reduction of the original system. Thus, by this way, the $3D$-system becomes $1$-dimensional. We demonstrate this approach ...
Added: November 25, 2014
Belomestny D., Panov V., Spokoiny V., Journal of machine learning and data analysis 2012 Vol. 1 No. 3 P. 140–147
Let a high-dimensional random vector $\vX$ be represented as a sum of two components - a signal $\vS$ that belongs to some low-dimensional linear subspace $\S$, and a noise component $\vN$. This paper presents a new approach for estimating the subspace $\S$ based on the ideas of the Non-Gaussian Component Analysis. Our approach avoids the ...
Added: September 23, 2013