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Sergei Ivanov

Publications and source records attributed to Sergei Ivanov.

At least 37 records · Page 2Linked to original sources
arXiv2022

Fitting a manifold of large reach to noisy data

Let ${\mathcal M}\subset {\mathbb R}^n$ be a $C^2$-smooth compact submanifold of dimension $d$. Assume that the volume of ${\mathcal M}$ is at most $V$ and the reach (i.e. the normal injectivity radius) of ${\mathcal M}$ is greater than $τ$. Moreover, let $μ$ be a probability measure on ${\mathcal M}$ whose density on ${\mathcal M}$ is a strictly positive Lipschitz-smooth function. Let $x_j\in {\mathcal M}$, $j=1,2,\dots,N$ be $N$ independent random samples from distribution $μ$. Also, let $ξ_j$, $j=1,2,\dots, N$ be independent random samples from a Gaussian random variable in ${\mathbb R}^n$ having covariance $σ^2I$, where $σ$ is less than a certain specified function of $d, V$ and $τ$. We assume that we are given the data points $y_j=x_j+ξ_j,$ $j=1,2,\dots,N$, modelling random points of ${\mathcal M}$ with measurement noise. We develop an algorithm which produces from these data, with high probability, a $d$ dimensional submanifold ${\mathcal M}_o\subset {\mathbb R}^n$ whose Hausdorff distance to ${\mathcal M}$ is less than $Cdσ^2/τ$ and whose reach is greater than $cτ/d^6$ with universal constants $C,c > 0$. The number $N$ of random samples required depends almost linearly on $n$, polynomially on $σ^{-1}$ and exponentially on $d$.

math.ST↗
arXiv2022

Approximations of the connection Laplacian spectra

We consider a convolution-type operator on vector bundles over metric-measure spaces. This operator extends the analogous convolution Laplacian on functions in our earlier work to vector bundles, and is a natural extension of the graph connection Laplacian. We prove that for Euclidean or Hermitian connections on closed Riemannian manifolds, the spectrum of this operator and that of the graph connection Laplacian both approximate the spectrum of the connection Laplacian.

math.AP↗
arXiv2021

Key ideas behind perturbing any completely integrable Hamiltonian system obtaining volume entropy non-expansiveness

This paper is an announcement of a result followed with explanations of some ideas behind. The proofs will appear elsewhere. Our goal is to construct a Hamiltonian perturbation of any completely integrable Hamiltonian system with $2n$ degrees of freedom ($n\geq 2$). The perturbation is $C^{\infty}$ small but the resulting flow has positive metric entropy and it satisfies KAM non-degeneracy conditions. The key point is that positive entropy can be generated in an arbitrarily small tubular neighborhood of one trajectory.

math.DS↗
arXiv2021

Terahertz-infrared dielectric properties of lead-aluminum double-cation substituted single-crystalline barium hexaferrite

Hexaferrite materials are highly demanded to develop and manufacture electronic devices operating at radio- and microwave frequencies. In the light of the prospects for their use in the forthcoming terahertz electronics, here, we present our results on the terahertz and infrared dielectric response of a typical representative of hexaferrites family, lead-substituted M-type barium hexaferrite doped with aluminum, Ba0.2Pb0.8AlxFe12-xO19, x(Al)=0.0, 3.0, and 3.3. We studied uniquely large and high-quality single crystals of the. Systematic and detailed investigations of the dependences of terahertz-infrared (frequencies 8 - 8000 cm-1) spectra of complex dielectric permittivity on the temperature, 4 - 300 K, and on the chemical composition, x(Al)=0.0, 1.2, 3.0, 3.3, were performed for polarizations of the electric field E-vector of the probing radiation normal and parallel to the crystallographic c-axis. A number of resonance absorption bands are discovered at infrared-terahertz frequencies and assigned to polar phonons and transitions between energy levels of the fine-structured ground state of Fe2+ (5E) ions. In contrast to undoped BaFe12O19, no softening of the lowest frequency A2u phonon is observed, indicating suppression of a displacive phase transition in substituted compounds. Basing on dielectric data and detailed X-ray experiments, we find that for all concentrations of Al3+ ions, x(Al)=0.0, 1.2, 3.0, and 3.3, they mainly occupy the 2a and 12k octahedral site positions and that the degree of substitution of iron in tetrahedral positions is not substantial. Along with fundamental findings, the obtained data on broad-band dielectric properties of Ba0.2Pb0.8AlxFe12-xO19 crystals provides the information that can be used for development and manufacture of electronic devices with operating frequencies lying in the terahertz spectral band.

cond-mat.mtrl-sci↗
arXiv2021

Boost then Convolve: Gradient Boosting Meets Graph Neural Networks

Graph neural networks (GNNs) are powerful models that have been successful in various graph representation learning tasks. Whereas gradient boosted decision trees (GBDT) often outperform other machine learning methods when faced with heterogeneous tabular data. But what approach should be used for graphs with tabular node features? Previous GNN models have mostly focused on networks with homogeneous sparse features and, as we show, are suboptimal in the heterogeneous setting. In this work, we propose a novel architecture that trains GBDT and GNN jointly to get the best of both worlds: the GBDT model deals with heterogeneous features, while GNN accounts for the graph structure. Our model benefits from end-to-end optimization by allowing new trees to fit the gradient updates of GNN. With an extensive experimental comparison to the leading GBDT and GNN models, we demonstrate a significant increase in performance on a variety of graphs with tabular features. The code is available: https://github.com/nd7141/bgnn.

cs.LG↗
arXiv2021

Nearly ideal memristive functionality based on viscous magnetization dynamics

We experimentally demonstrate a proof-of-principle implementation of an almost ideal memristor - a two-terminal circuit element whose resistance is approximately proportional to the integral of the input signal over time. The demonstrated device is based on a thin-film ferromagnet/antiferromagnet bilayer, where magnetic frustration results in viscous magnetization dynamics enabling memristive functionality, while the external magnetic field plays the role of the driving input. The demonstrated memristor concept is amenable to downscaling and can be adapted for electronic driving, making it attractive for applications in neuromorphic circuits.

cond-mat.mtrl-sci↗
arXiv2020

Reinforcement Learning for Combinatorial Optimization: A Survey

Many traditional algorithms for solving combinatorial optimization problems involve using hand-crafted heuristics that sequentially construct a solution. Such heuristics are designed by domain experts and may often be suboptimal due to the hard nature of the problems. Reinforcement learning (RL) proposes a good alternative to automate the search of these heuristics by training an agent in a supervised or self-supervised manner. In this survey, we explore the recent advancements of applying RL frameworks to hard combinatorial problems. Our survey provides the necessary background for operations research and machine learning communities and showcases the works that are moving the field forward. We juxtapose recently proposed RL methods, laying out the timeline of the improvements for each problem, as well as we make a comparison with traditional algorithms, indicating that RL models can become a promising direction for solving combinatorial problems.

cs.LG↗
arXiv2020

Ideal memristor based on viscous magnetization dynamics driven by spin torque

We show that ideal memristors - devices whose resistance is proportional to the charge that flows through them - can be realized using spin torque-driven viscous magnetization dynamics. The latter can be accomplished in the spin liquid state of thin-film heterostructures with frustrated exchange, where memristive response is tunable by proximity to the glass transition, while current-induced Joule heating facilitates nonvolatile operation and second-order memristive functionality beneficial for neuromorphic applications. Ideal memristive behaviors can be achieved in other systems characterized by viscous dynamics of physical, electronic, or magnetic degrees of freedom.

physics.app-ph↗
arXiv2019

Reconstruction and interpolation of manifolds I: The geometric Whitney problem

We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold reconstruction where a smooth $n$-dimensional submanifold $S\subset {\mathbb R}^m$, $m>n$ needs to be constructed to approximate a point cloud in ${\mathbb R}^m$. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov-Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. As an application of the main results we give a new characterisation of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterise the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures.

math.DG↗
arXiv2019

Understanding Isomorphism Bias in Graph Data Sets

In recent years there has been a rapid increase in classification methods on graph structured data. Both in graph kernels and graph neural networks, one of the implicit assumptions of successful state-of-the-art models was that incorporating graph isomorphism features into the architecture leads to better empirical performance. However, as we discover in this work, commonly used data sets for graph classification have repeating instances which cause the problem of isomorphism bias, i.e. artificially increasing the accuracy of the models by memorizing target information from the training set. This prevents fair competition of the algorithms and raises a question of the validity of the obtained results. We analyze 54 data sets, previously extensively used for graph-related tasks, on the existence of isomorphism bias, give a set of recommendations to machine learning practitioners to properly set up their models, and open source new data sets for the future experiments.

cs.LG↗
arXiv2019

Reconstruction of a Riemannian manifold from noisy intrinsic distances

We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distances) of points in a discrete subset of $M$. In the studied problem the Riemannian manifold $(M,g)$ is considered as an abstract metric space with intrinsic distances, not as an embedded submanifold of an ambient Euclidean space. Let $\{X_1,X_2,\dots,X_N\}$ bea set of $N$ sample points sampled randomly from an unknown Riemannian $M$ manifold. We assume that we are given the numbers $D_{jk}=d_M(X_j,X_k)+η_{jk}$, where $j,k\in \{1,2,\dots,N\}$. Here, $d_M(X_j,X_k)$ are geodesic distances, $η_{jk}$ are independent, identically distributed random variables such that $\mathbb E e^{|η_{jk}|}$ is finite. We show that when $N$ is large enough, it is possible to construct an approximation of the Riemannian manifold $(M,g)$ with a large probability. This problem is a generalization of the geometric Whitney problem with random measurement errors. We consider also the case when the information on noisy distance $D_{jk}$ of points $X_j$ and $X_k$ is missing with some probability. In particular, we consider the case when we have no information on points that are far away.

math.PR↗
arXiv2019

Feeble fish in time-dependent waters and homogenization of the G-equation

We study the following control problem. A fish with bounded aquatic locomotion speed swims in fast waters. Can this fish, under reasonable assumptions, get to a desired destination? It can, even if the flow is time-dependent. Moreover, given a prescribed sufficiently large time $t$, it can be there at exactly the time $t$. The major difference from our previous work is the time-dependence of the flow. We also give an application to homogenization of the G-equation.

math.OC↗
arXiv2019

Distance difference representations of Riemannian manifolds

Let $M$ be a complete Riemannian manifold and $F\subset M$ a set with a nonempty interior. For every $x\in M$, let $D_x$ denote the function on $F\times F$ defined by $D_x(y,z)=d(x,y)-d(x,z)$ where $d$ is the geodesic distance in $M$. The map $x\mapsto D_x$ from $M$ to the space of continuous functions on $F\times F$, denoted by $\mathcal D_F$, is called a distance difference representation of $M$. This representation, introduced recently by M. Lassas and T. Saksala, is motivated by geophysical imaging among other things. We prove that the distance difference representation $\mathcal D_F$ is a locally bi-Lipschitz homeomorphism onto its image $\mathcal D_F(M)$ and that for every open set $U\subset M$ the set $\mathcal D_F(U)$ uniquely determines the Riemannian metric on $U$. Furthermore the determination of $M$ from $\mathcal D_F(M)$ is stable if $M$ has a priori bounds on its diameter, curvature, and injectivity radius. This extends and strengthens earlier results by M. Lassas and T. Saksala.

math.DG↗
arXiv2018

Spectral stability of metric-measure Laplacians

We consider a "convolution mm-Laplacian" operator on metric-measure spaces and study its spectral properties. The definition is based on averaging over small metric balls. For reasonably nice metric-measure spaces we prove stability of convolution Laplacian's spectrum with respect to metric-measure perturbations and obtain Weyl-type estimates on the number of eigenvalues.

math.SP↗
arXiv2018

Rigidity of Busemann convex Finsler metrics

We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane. In the course of the proof we obtain new characterizations of Berwald metrics in terms of the so-called linear parallel transport.

math.DG↗
arXiv2017

Monochromatic Finsler surfaces and a local ellipsoid characterization

We prove the following localized version of a classical ellipsoid characterization: Let $B\subset\mathbb R^3$ be convex body with a smooth strictly convex boundary and 0 in the interior, and suppose that there is an open set of planes through 0 such that all sections of $B$ by these planes are linearly equivalent. Then all these sections are ellipses and the corresponding part of $B$ is a part of an ellipsoid. We apply this to differential geometry of Finsler surfaces in normed spaces and show that in certain cases the intrinsic metric of a surface imposes restrictions on its extrinsic geometry similar to implications of Gauss' Theorema Egregium. As a corollary we construct 2-dimensional Finsler metrics that do not admit local isometric embeddings to dimension~3.

math.DG↗