Publications

Monographs

  • F. Stonyakin, D. Dvinskikh, P. Dvurechensky, A. Kroshnin, O. Kuznetsova, A. Agafonov, A. Gasnikov, A. Tyurin, C. Uribe, D. Pasechnyuk, S. Artamonov, Gradient methods for problems with inexact model of the objective, M. Khachay, Y. Kochetov, P. Pardalos, eds., Mathematical Optimization Theory and Operations Research, Springer International Publishing AG, Cham, Switzerland, 2019, pp. 97--114, (Chapter Published), DOI 10.1007/978-3-030-22629-9_8 .

  • J. Polzehl, K. Tabelow, Magnetic resonance brain imaging: Modeling and data analysis using R, Use R!, Springer Nature, Cham, 2019, 231 pages, (Monograph Published), DOI 10.1007/978-3-030-29184-6 .
    Abstract
    This book discusses the modeling and analysis of magnetic resonance imaging (MRI) data acquired from the human brain. The data processing pipelines described rely on R. The book is intended for readers from two communities: Statisticians who are interested in neuroimaging and looking for an introduction to the acquired data and typical scientific problems in the field; and neuroimaging students wanting to learn about the statistical modeling and analysis of MRI data. Offering a practical introduction to the field, the book focuses on those problems in data analysis for which implementations within R are available. It also includes fully worked examples and as such serves as a tutorial on MRI analysis with R, from which the readers can derive their own data processing scripts. The book starts with a short introduction to MRI and then examines the process of reading and writing common neuroimaging data formats to and from the R session. The main chapters cover three common MR imaging modalities and their data modeling and analysis problems: functional MRI, diffusion MRI, and Multi-Parameter Mapping. The book concludes with extended appendices providing details of the non-parametric statistics used and the resources for R and MRI data.The book also addresses the issues of reproducibility and topics like data organization and description, as well as open data and open science. It relies solely on a dynamic report generation with knitr and uses neuroimaging data publicly available in data repositories. The PDF was created executing the R code in the chunks and then running LaTeX, which means that almost all figures, numbers, and results were generated while producing the PDF from the sources.

  • P. Friz, W. König, Ch. Mukherjee, S. Olla, eds., Probability and Analysis in Interacting Physical Systems. In Honor of S.R.S. Varadhan, Berlin, August, 2016, 283 of Springer Proceedings in Mathematics & Statistics book series, Springer, 2019, pp. 1--294, (Monograph Published), DOI https://doi.org/10.1007/978-3-030-15338-0 .

Articles in Refereed Journals

  • O. Butkovsky, L. Mytnik, Regularization by noise and flows of solutions for a stochastic heat equation, The Annals of Probability, 47 (2019), pp. 165--212.

  • M. Coghi, B. Gess, Stochastic nonlinear Fokker-Planck equations, Nonlinear Analysis. An International Mathematical Journal, 187 (2019), pp. 259--278, DOI 10.1016/j.na.2019.05.003 .
    Abstract
    The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker-Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with common noise. The uniqueness of solutions is obtained without any higher moment assumption on the solution by means of a duality argument to a backward stochastic PDE.

  • P. Pigato, Extreme at-the-money skew in a local volatility model, Finance and Stochastics, 23 (2019), pp. 827--859, DOI 10.1007/s00780-019-00406-2 .

  • D.R. Baimurzina, A. Gasnikov, E.V. Gasnikova, P.E. Dvurechensky, E.I. Ershov, M.B. Kubentaeva, A.A. Lagunovskaya, Universal method of searching for equilibria and stochastic equilibria in transportation networks, Computational Mathematics and Mathematical Physics, 59 (2019), pp. 19--33.

  • H. Bessaih, M. Coghi, F. Flandoli, Mean field limit of interacting filaments for 3D Euler equations, Journal of Statistical Physics, 174 (2019), pp. 562--578, DOI 10.1007/s10955-018-2189-4 .

  • M.F. Callaghan, A. Lutti, J. Ashburner, E. Balteau, N. Corbin, B. Draganski, G. Helms, F. Kherif, T. Leutritz, S. Mohammadi, Ch. Phillips, E. Reimer, L. Ruthotto, M. Seif, K. Tabelow, G. Ziegler, N. Weiskopf, Example Dataset for the hMRI Toolbox, Data in Brief, (2019), published online on 11.06.2019, DOI 10.1016/j.dib.2019.104132 .

  • E.A. Vorontsova, A. Gasnikov, E.A. Gorbunov, P. Dvurechensky, Accelerated gradient-free optimization methods with a non-Euclidean proximal operator, Automation and Remote Control, 80 (2019), pp. 1487--1501.

  • C. Améndola, P. Friz, B. Sturmfels, Varieties of signature tensors, Forum of Mathematics. Sigma, 7 (2019), pp. e10/1--e10/54, DOI 10.1017/fms.2019.3 .
    Abstract
    The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.

  • L. Antoine, P. Pigato, Maximum likelihood drift estimation for a threshold diffusion, , (2019), published online on 23.10.2019, DOI 10.1111/sjos.12417 .
    Abstract
    We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called the drifted Oscillating Brownian motion. The asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results in the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations.

  • D. Belomestny, R. Hildebrand, J.G.M. Schoenmakers, Optimal stopping via pathwise dual empirical maximisation, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 79 (2019), pp. 715--741, DOI 10.1007/s00245-017-9454-9 .
    Abstract
    The optimal stopping problem arising in the pricing of American options can be tackled by the so called dual martingale approach. In this approach, a dual problem is formulated over the space of martingales. A feasible solution of the dual problem yields an upper bound for the solution of the original primal problem. In practice, the optimization is performed over a finite-dimensional subspace of martingales. A sample of paths of the underlying stochastic process is produced by a Monte-Carlo simulation, and the expectation is replaced by the empirical mean. As a rule the resulting optimization problem, which can be written as a linear program, yields a martingale such that the variance of the obtained estimator can be large. In order to decrease this variance, a penalizing term can be added to the objective function of the path-wise optimization problem. In this paper, we provide a rigorous analysis of the optimization problems obtained by adding different penalty functions. In particular, a convergence analysis implies that it is better to minimize the empirical maximum instead of the empirical mean. Numerical simulations confirm the variance reduction effect of the new approach.

  • Y. Bruned, I. Chevyrev, P. Friz, R. Preiss, A rough path perspective on renormalization, Journal of Functional Analysis, 277 (2019), pp. 108283/1--108283/60, DOI 10.1016/j.jfa.2019.108283 .
    Abstract
    We develop the algebraic theory of rough path translation. Particular attention is given to the case of branched rough paths, whose underlying algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model case of a regularity structure in the sense of Hairer. Pre-Lie structures are seen to play a fundamental rule which allow a direct understanding of the translated (i.e. renormalized) equation under consideration. This construction is also novel with regard to the algebraic renormalization theory for regularity structures due to Bruned-Hairer-Zambotti (2016), the links with which are discussed in detail.

  • I. Chevyrev, P. Friz, Canonical RDEs and general semimartingales as rough paths, The Annals of Probability, 47 (2019), pp. 420--463.

  • K. Efimov, L. Adamyan, V. Spokoiny, Adaptive nonparametric clustering, IEEE Transactions on Information Theory, 65 (2019), pp. 4875--4892, DOI 10.1109/TIT.2019.2903113 .
    Abstract
    This paper presents a new approach to non-parametric cluster analysis called adaptive weights? clustering. The method is fully adaptive and does not require to specify the number of clusters or their structure. The clustering results are not sensitive to noise and outliers, and the procedure is able to recover different clusters with sharp edges or manifold structure. The method is also scalable and computationally feasible. Our intensive numerical study shows a state-of-the-art performance of the method in various artificial examples and applications to text data. The idea of the method is to identify the clustering structure by checking at different points and for different scales on departure from local homogeneity. The proposed procedure describes the clustering structure in terms of weights $w_ij$ , and each of them measures the degree of local inhomogeneity for two neighbor local clusters using statistical tests of ?no gap? between them. The procedure starts from very local scale, and then, the parameter of locality grows by some factor at each step. We also provide a rigorous theoretical study of the procedure and state its optimal sensitivity to deviations from local homogeneity.

  • A. Gasnikov, P. Dvurechensky, F. Stonyakin, A.A. Titov, An adaptive proximal method for variational inequalities, Computational Mathematics and Mathematical Physics, 59 (2019), pp. 836--841.

  • F. Götze, A. Naumov, V. Spokoiny, V. Ulyanov, Large ball probabilities, Gaussian comparison and anti-concentration, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 25 (2019), pp. 2538--2563, DOI 10.3150/18-BEJ1062 .
    Abstract
    We derive tight non-asymptotic bounds for the Kolmogorov distance between the probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimension-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements and on the norm of the mean shift. The obtained bounds significantly improve the bound based on Pinsker?s inequality via the Kullback?Leibler divergence. We also establish an anti-concentration bound for a squared norm of a non-centered Gaussian element in Hilbert space. The paper presents a number of examples motivating our results and applications of the obtained bounds to statistical inference and to high-dimensional CLT.

  • P. Goyal, M. Redmann, Time-limited H2-optimal model order reduction, Applied Mathematics and Computation, 355 (2019), pp. 184--197, DOI 10.1016/j.amc.2019.02.065 .

  • S. Guminov, Y. Nesterov, P. Dvurechensky, A. Gasnikov, Accelerated primal-dual gradient descent with linesearch for convex, nonconvex, and nonsmooth optimization problems, Doklady Mathematics. Maik Nauka/Interperiodica Publishing, Moscow. English. Translation of the Mathematics Section of: Doklady Akademii Nauk. (Formerly: Russian Academy of Sciences. Doklady. Mathematics)., 99 (2019), pp. 125--128.

  • B. Hofmann, S. Kindermann, P. Mathé, Penalty-based smoothness conditions in convex variational regularization, Journal of Inverse and Ill-Posed Problems, 27 (2019), pp. 283--300, DOI 10.1515/jiip-2018-0039 .
    Abstract
    The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.

  • A. Lejay, P. Pigato, A threshold model for local volatility: Evidence of leverage and mean reversion effects on historical data, International Journal of Theoretical and Applied Finance, 22 (2019), published online on 29.05.2019, DOI 10.1142/S0219024919500171 .

  • A. Naumov, V. Spokoiny, V. Ulyanov, Bootstrap confidence sets for spectral projectors of sample covariance, Probability Theory and Related Fields, 174 (2019), pp. 1091--1132, DOI 10.1007/s00440-018-0877-2 .

  • CH. Bayer, J. Häppölä, R. Tempone, Implied stopping rules for American basket options from Markovian projection, Quantitative Finance, 19 (2019), pp. 371--390.
    Abstract
    This work addresses the problem of pricing American basket options in a multivariate setting, which includes among others, the Bachelier and the Black-Scholes models. In high dimensions, nonlinear partial differential equation methods for solving the problem become prohibitively costly due to the curse of dimensionality. Instead, this work proposes to use a stopping rule that depends on the dynamics of a low-dimensional Markovian projection of the given basket of assets. It is shown that the ability to approximate the original value function by a lower-dimensional approximation is a feature of the dynamics of the system and is unaffected by the path-dependent nature of the American basket option. Assuming that we know the density of the forward process and using the Laplace approximation, we first efficiently evaluate the diffusion coefficient corresponding to the low-dimensional Markovian projection of the basket. Then, we approximate the optimal early-exercise boundary of the option by solving a Hamilton-Jacobi-Bellman partial differential equation in the projected, low-dimensional space. The resulting near-optimal early-exercise boundary is used to produce an exercise strategy for the high-dimensional option, thereby providing a lower bound for the price of the American basket option. A corresponding upper bound is also provided. These bounds allow to assess the accuracy of the proposed pricing method. Indeed, our approximate early-exercise strategy provides a straightforward lower bound for the American basket option price. Following a duality argument due to Rogers, we derive a corresponding upper bound solving only the low-dimensional optimal control problem. Numerically, we show the feasibility of the method using baskets with dimensions up to fifty. In these examples, the resulting option price relative errors are only of the order of few percent.

  • CH. Bayer, P. Friz, P. Gassiat, J. Martin, B. Stemper, A regularity structure for rough volatility, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, (2019), online on 19.11.2019, DOI 10.1111/mafi12233 .

  • CH. Bayer, P. Friz, A. Gulisashvili, B. Horvath, B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Quantitative Finance, 19 (2019), pp. 779--798, DOI 10.1080/14697688.2018.1529420 .
    Abstract
    We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter H < ½. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order t1/2 (recent works of Alòs, León & Vives and Fukasawa) to the wider moderate deviations regime.

  • P. Mathé, Bayesian inverse problems with non-commuting operators, Mathematics of Computation, 88 (2019), pp. 2897--2912, DOI 10.1090/mcom/3439 .
    Abstract
    The Bayesian approach to ill-posed operator equations in Hilbert space recently gained attraction. In this context, and when the prior distribution is Gaussian, then two operators play a significant role, the one which governs the operator equation, and the one which describes the prior covariance. Typically it is assumed that these operators commute. Here we extend this analysis to non-commuting operators, replacing the commutativity assumption by a link condition. We discuss its relation to the commuting case, and we indicate that this allows to use interpolation type results to obtain tight bounds for the contraction of the posterior Gaussian distribution towards the data generating element.

  • V. Spokoiny, N. Willrich, Bootstrap tuning in Gaussian ordered model selection, The Annals of Statistics, 47 (2019), pp. 1351--1380, DOI 10.1214/18-AOS1717 .
    Abstract
    In the problem of model selection for a given family of linear estimators, ordered by their variance, we offer a new “smallest accepted” approach motivated by Lepski's device and the multiple testing idea. The procedure selects the smallest model which satisfies the acceptance rule based on comparison with all larger models. The method is completely data-driven and does not use any prior information about the variance structure of the noise: its parameters are adjusted to the underlying possibly heterogeneous noise by the so called “propagation condition” using bootstrap multiplier method. The validity of the bootstrap calibration is proved for finite samples with an explicit error bound. We provide a comprehensive theoretical study of the method and describe in details the set of possible values of the selector ( hatm ). We also establish some precise oracle error bounds for the corresponding estimator ( hattheta = tildetheta_hatm ) which equally applies to estimation of the whole parameter vectors, its subvector or linear mapping, as well as estimation of a linear functional.

  • K. Tabelow, E. Balteau, J. Ashburner, M.F. Callaghan, B. Draganski, G. Helms, F. Kherif, T. Leutritz, A. Lutti, Ch. Phillips, E. Reimer, L. Ruthotto, M. Seif, N. Weiskopf, G. Ziegler, S. Mohammadi, hMRI -- A toolbox for quantitative MRI in neuroscience and clinical research, NeuroImage, 194 (2019), pp. 191--210, DOI 10.1016/j.neuroimage.2019.01.029 .
    Abstract
    Quantitative magnetic resonance imaging (qMRI) finds increasing application in neuroscience and clinical research due to its sensitivity to micro-structural properties of brain tissue, e.g. axon, myelin, iron and water concentration. We introduce the hMRI--toolbox, an easy-to-use open-source tool for handling and processing of qMRI data presented together with an example dataset. This toolbox allows the estimation of high-quality multi-parameter qMRI maps (longitudinal and effective transverse relaxation rates R1 and R2*, proton density PD and magnetisation transfer MT) that can be used for calculation of standard and novel MRI biomarkers of tissue microstructure as well as improved delineation of subcortical brain structures. Embedded in the Statistical Parametric Mapping (SPM) framework, it can be readily combined with existing SPM tools for estimating diffusion MRI parameter maps and benefits from the extensive range of available tools for high-accuracy spatial registration and statistical inference. As such the hMRI--toolbox provides an efficient, robust and simple framework for using qMRI data in neuroscience and clinical research.

Contributions to Collected Editions

  • J. Ebert, V. Spokoiny, A. Suvorikova, Elements of statistical inference in 2-Wasserstein space, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., Springer-CIM Series, Springer Nature Switzerland AG, Cham, 2019, pp. 139--158, DOI 10.1007/978-3-030-33116-0 .

  • TH. Koprucki, A. Maltsi, T. Niermann, T. Streckenbach, K. Tabelow, J. Polzehl, On a database of simulated TEM images for In(Ga)As/GaAs quantum dots with various shapes, in: Proceedings of the 19th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2019), J. Piprek, K. Hinze, eds., IEEE Conference Publications Management Group, Piscataway, NJ, 2019, pp. 13--14 (appeared online 22.08.2019), DOI 10.1109/NUSOD.2019.8807025 .

Preprints, Reports, Technical Reports

  • A. Kroshnin, D. Dvinskikh, P. Dvurechensky, A. Gasnikov, N. Tupitsa, C.A. Uribe, On the complexity of approximating Wasserstein barycenter, Preprint no. 2665, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2665 .
    Abstract, PDF (386 kByte)
    We study the complexity of approximating Wassertein barycenter of discrete measures, or histograms by contrasting two alternative approaches, both using entropic regularization. We provide a novel analysis for our approach based on the Iterative Bregman Projections (IBP) algorithm to approximate the original non-regularized barycenter. We also get the complexity bound for alternative accelerated-gradient-descent-based approach and compare it with the bound obtained for IBP. As a byproduct, we show that the regularization parameter in both approaches has to be proportional to ", which causes instability of both algorithms when the desired accuracy is high. To overcome this issue, we propose a novel proximal-IBP algorithm, which can be seen as a proximal gradient method, which uses IBP on each iteration to make a proximal step. We also consider the question of scalability of these algorithms using approaches from distributed optimization and show that the first algorithm can be implemented in a centralized distributed setting (master/slave), while the second one is amenable to a more general decentralized distributed setting with an arbitrary network topology.

  • A. Ogaltsov, D. Dvinskikh, P. Dvurechensky, A. Gasnikov, V. Spokoiny, Adaptive gradient descent for convex and non-convex stochastic optimization, Preprint no. 2655, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2655 .
    Abstract, PDF (538 kByte)
    In this paper we propose several adaptive gradient methods for stochastic optimization. Our methods are based on Armijo-type line search and they simultaneously adapt to the unknown Lipschitz constant of the gradient and variance of the stochastic approximation for the gradient. We consider an accelerated gradient descent for convex problems and gradient descent for non-convex problems. In the experiments we demonstrate superiority of our methods to existing adaptive methods, e.g. AdaGrad and Adam.

  • CH. Bayer, Ch.B. Hammouda, R.F. Tempone, Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model, Preprint no. 2652, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2652 .
    Abstract, PDF (646 kByte)
    The rough Bergomi (rBergomi) model, introduced recently in [4], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet exhibits remarkable fit to empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a timeconsuming task. To overcome this issue, we design a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi Monte Carlo (QMC). Both techniques are coupled with Brownian bridge construction and Richardson extrapolation. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.

  • CH. Bayer, R.F. Tempone , S. Wolfers, Pricing American options by exercise rate optimization, Preprint no. 2651, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2651 .
    Abstract, PDF (761 kByte)
    We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model.

  • N. Tapia, L. Zambotti, The geometry of the space of branched rough paths, Preprint no. 2645, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2645 .
    Abstract, PDF (378 kByte)
    We construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between theses two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker-Campbell-Hausdorff formula, on a constructive version of the Lyons-Victoir extension theorem and on the Hairer-Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths.

  • M. Coghi, T. Nilssen, Rough nonlocal diffusions, Preprint no. 2619, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2619 .
    Abstract, PDF (397 kByte)
    We consider a nonlinear Fokker-Planck equation driven by a deterministic rough path which describes the conditional probability of a McKean-Vlasov diffusion with "common" noise. To study the equation we build a self-contained framework of non-linear rough integration theory which we use to study McKean-Vlasov equations perturbed by rough paths. We construct an appropriate notion of solution of the corresponding Fokker-Planck equation and prove well-posedness.

  • M. Coghi, J.-D. Deuschel, P. Friz, M. Maurelli, Pathwise McKean--Vlasov theory with additive noise, Preprint no. 2618, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2618 .
    Abstract, PDF (348 kByte)
    We take a pathwise approach to classical McKean-Vlasov stochastic differential equations with additive noise, as e.g. exposed in Sznitmann [34]. Our study was prompted by some concrete problems in battery modelling [19], and also by recent progress on rough-pathwise McKean-Vlasov theory, notably Cass--Lyons [9], and then Bailleul, Catellier and Delarue [4]. Such a “pathwise McKean-Vlasov theory” can be traced back to Tanaka [36]. This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from [4, 9, 36]. As novel applications we discuss mean field convergence without a priori independence and exchangeability assumption; common noise and reflecting boundaries. Last not least, we generalize Dawson--Gärtner large deviations to a non-Brownian noise setting.

  • D. Belomestny, M. Kaledin, J.G.M. Schoenmakers, Semi-tractability of optimal stopping problems via a weighted stochastic mesh algorithm, Preprint no. 2610, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2610 .
    Abstract, PDF (381 kByte)
    In this article we propose a Weighted Stochastic Mesh (WSM) algorithm for approximating the value of discrete and continuous time optimal stopping problems. It is shown that in the discrete time case the WSM algorithm leads to semi-tractability of the corresponding optimal stopping problem in the sense that its complexity is bounded in order by $varepsilon^-4log^d+2(1/varepsilon)$ with $d$ being the dimension of the underlying Markov chain. Furthermore we study the WSM approach in the context of continuous time optimal stopping problems and derive the corresponding complexity bounds. Although we can not prove semi-tractability in this case, our bounds turn out to be the tightest ones among the complexity bounds known in the literature. We illustrate our theoretical findings by a numerical example.

  • J. Diehl, E.-F. Kurusch, N. Tapia, Time-warping invariants of multidimensional time series, Preprint no. 2603, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2603 .
    Abstract, PDF (325 kByte)
    In data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants.We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties.

Talks, Poster

  • A. Maltsi, Th. Koprucki, T. Streckenbach, K. Tabelow, J. Polzehl, Model-based geometry reconstruction of quantum dots from TEM, ``Microscopy Conference 2019'', Poster session IM 4, Berlin, September 1 - 5, 2019.

  • A. Maltsi, Th. Koprucki, T. Streckenbach, K. Tabelow, J. Polzehl, Model-based geometry reconstruction of quantum dots from TEM, BMS Summer School 2019: Mathematics of Deep Learning, Berlin, August 19 - 30, 2019.

  • V. Avanesov, Nonparametric change point detection in regression, SFB 1294 Spring School 2019, Dierhagen, March 18 - 22, 2019.

  • F. Besold, Adaptive manifold clustering, Rencontres de Statistiques Mathématiques, December 16 - 20, 2019, Centre International de Rencontres Mathématiques (CIRM), Luminy, France, December 19, 2019.

  • F. Besold, Manifold clustering, Pennsylvania State University, Department of Mathematics, University Park, PA, USA, October 28, 2019.

  • F. Besold, Manifold clustering with adaptive weights, Structural Inference in High-Dimensional Models 2, St. Petersburg, Russian Federation, August 26 - 30, 2019.

  • F. Besold, Manifold clustering with adaptive weights, Joint Workshop of BBDC, BZML and RIKEN AIP, September 9 - 10, 2019.

  • F. Besold, Minimax clustering with adaptive weights, New frontiers in high-dimensional probability and statistics 2, February 20 - 23, 2019, Higher School of Economics, Moscow, Russian Federation, February 23, 2019.

  • O. Butkovsky, New coupling techniques for exponential ergodicity of SPDEs in hypoelliptic and effectively elliptic settings, Oberseminar Stochastik, Universität Bonn, Hausdorff Research Center, Institut für Angewandte Mathematik (IAM), November 28, 2019.

  • O. Butkovsky, Numerical methods for SDEs: A stochastic sewing approach, 12th Oxford-Berlin Young Researchers Meeting onApplied Stochastic Analysis, December 5 - 6, 2019, University of Oxford, Mathematical Institute, UK, December 6, 2019.

  • O. Butkovsky, Regularization by noise for SDEs and SPDEs with applications to numerical methods, Seminar Wahrscheinlichkeitstheorie, Universität Mannheim, Probability & Statistics Group, October 16, 2019.

  • O. Butkovsky, Regularization by noise for SDEs and related systems: A tale of two approaches, Hausdorff Junior Trimester, Universität Bonn, Hausdorff Research Institute for Mathematics (HIM), November 26, 2019.

  • M. Coghi, Mean field limit of interacting filaments for 3D Euler equations, Second Italian Meeting on Probability and Mathematical Statistics, June 17 - 20, 2019, Università degli Studi di Salerno, Dipartimento di Matematica, Vietri sul Mare, Italy, June 20, 2019.

  • M. Coghi, Pathwise McKean-Vlasov theory, Oberseminar Partielle Differentialgleichungen, Universität Konstanz, Fachbereich Mathmatik und Statistik, February 6, 2019.

  • M. Coghi, Rough nonlocal diffusions, Recent Trends in Stochastic Analysis and SPDEs, July 17 - 20, 2019, University of Pisa, Department of Mathematics, Italy, July 18, 2019.

  • M. Coghi, Stochastic nonlinear Fokker-Planck equations, 11th Annual ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, May 23 - 25, 2019, WIAS Berlin, May 23, 2019.

  • P. Pigato, Applications of stochastic analysis to volatility modelling, Università degli Studi di Roma Tor Vergata, Italy, September 27, 2019.

  • P. Pigato, Density and tube estimates for diffusion processes under Hormander-type conditions, Statistik Seminar, University of Bologna, Italy, February 28, 2019.

  • P. Pigato, Parameters estimation in a threshold diffusion, 62nd ISI World Statistics Congress 2019, August 18 - 23, 2019, Kuala Lumpur, Malaysia, August 21, 2019.

  • P. Pigato, Precise asymptotics of rough stochastic volatility models, 11th Annual ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, May 23 - 25, 2019, WIAS Berlin, May 23, 2019.

  • P. Pigato, Precise asymptotics: Robust stochastic volatility models, Probability and Statistic Seminar, University of Potsdam, July 1, 2019.

  • P. Pigato, Rough stochastic volatility models, University of Rome Tor Vergata, Department of Economics and Finance, Italy, June 26, 2019.

  • M. Redmann, Energy estimates and model order reduction for stochastic bilinear systems, 12th International Workshop on Stochastic Models and Control, March 19 - 22, 2019, Cottbus, March 21, 2019.

  • M. Redmann, Model reduction for stochastic bilinear systems, International Congress on Industrial and Applied Mathematics, July 15 - 19, 2019, Valencia, Spain, July 17, 2019.

  • M. Redmann, Numerical approximations for rough and stochastic differential equations, Technische Universität Bergakademie Freiberg, April 1, 2019.

  • M. Redmann, Numerical approximations for rough and stochastic differential equations, Technische Universität Dresden, April 12, 2019.

  • N. Tapia, Non-commutative Wick polynomials, Rencontre GDR Renormalisation, September 30 - October 4, 2019, L'Université du Littoral Côte d'Opale, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, Calais, France, October 3, 2019.

  • N. Tapia, Algebraic aspects of signatures, SciCADE 2019, International Conference on Scientific Computationand Differential Equations, July 22 - 26, 2019, Innsbruck, Austria, July 24, 2019.

  • N. Tapia, Iterated-sums signature, quasi-symmetric functions and time series analysis, 12th Oxford-Berlin Young Researchers Meeting onApplied Stochastic Analysis, December 4 - 6, 2019, University of Oxford, Mathematical Institute, UK, December 5, 2019.

  • N. Tapia, Signatures in shape analysis, 4th International Conference, GSI 2019, August 27 - 29, 2019, École nationale de l'aviation civile, Toulouse, France, August 27, 2019.

  • A. Gasnikov, P. Dvurechensky, E. Gorbunov, E. Vorontsova, D. Selikhanovych, C.A. Uribe, Near-optimal method for highly smooth convex optimization, Conference on Learning Theory, COLT 2019, Phoenix, Arizona, USA, June 24 - 28, 2019.

  • A. Kroshnin, N. Tupitsa, D. Dvinskikh, P. Dvurechensky, A. Gasnikov, C.A. Uribe , On the complexity of approximating Wasserstein barycenters, Thirty-sixth International Conference on Machine Learning, ICML 2019, Long Beach, CA, USA, June 9 - 15, 2019.

  • M. Opper, S. Reich, V. Spokoiny, V. Avanesov, D. Maoutsa , P. Rozdeba, Approximative Bayesian inference and model selection for stochastic differential equations, SFB 1294 Annual Meeting 2019, September 23, 2019.

  • D. Dvinskikh, Complexity bounds for optimal distributed primal and dual methods for finite sum minimization problems, New frontiers in high-dimensional probability and statistics 2, February 22 - 23, 2019, Higher School of Economics, Moskau, Russian Federation, February 23, 2019.

  • D. Dvinskikh, Complexity rates for accelerated primal-dual gradient method for stochastic optimisation problem, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Primal-Dual Methods for Structured Optimization'', August 5 - 8, 2019, Berlin, August 7, 2019.

  • D. Dvinskikh, Decentralized and parallelized primal and dual accelerated methods, Structural Inference in High-Dimensional Models 2, St. Petersburg, Russian Federation, August 26 - 30, 2019.

  • D. Dvinskikh, Distributed decentralized (stochastic) optimization for dual friendly functions, Optimization and Statistical Learning, Les Houches, France, March 24 - 29, 2019.

  • D. Dvinskikh, Introduction to decentralized optimization, Summer School, July 15 - 18, 2019, Sirius Educational Centre, Sochi, Russian Federation, July 16, 2019.

  • CH. Bayer, A regularity structure for rough volatility, Vienna Seminar in Mathematical Finance and Probability, TU Wien, Research Unit of Financial and Actuarial Mathematics, Austria, January 10, 2019.

  • CH. Bayer, Calibration of rough volatility models by deep learning, Rough Workshop 2019, September 4 - 6, 2019, TU Wien, Financial and Actuarial Mathematics, Austria.

  • CH. Bayer, Deep calibration of rough volatility models, New Directions in Stochastic Analysis: Rough Paths, SPDEs and Related Topics, WIAS und TU Berlin, March 18, 2019.

  • CH. Bayer, Deep calibration of rough volatility models, SIAM Conference on Financial Mathematics & Engineering, June 4 - 7, 2019, Society for Industrial and Applied Mathematics, Toronto, Ontario, Canada, June 7, 2019.

  • CH. Bayer, Learning rough volatility, Algebraic and Analytic Perspectives in the theory of Rough Paths and Signatures, November 14 - 15, 2019, University of Oslo, Department of Mathematics, Oslo, Norway, November 14, 2019.

  • CH. Bayer, Numerics for rough volatility, Stochastic Processes and related topics, February 21 - 22, 2019, Kansai University, Senriyama Campus, Osaka, Japan, February 22, 2019.

  • CH. Bayer, Pricing American options by exercise rate optimization, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Stochastic Optimization and Its Applications (Part III)'', August 5 - 8, 2019, Berlin, August 7, 2019.

  • CH. Bayer, Pricing american options by exercise rate optimization, Workshop on Financial Risks and Their Management, February 19 - 20, 2019, Ryukoku University, Wagenkan, Kyoto, Japan, February 19, 2019.

  • P. Dvurechensky, A unifying framework for accelerated randomized optimization methods, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Large-Scale Stochastic First-Order Optimization (Part I)'', August 5 - 8, 2019, Berlin, August 6, 2019.

  • P. Dvurechensky, Distributed calculation of Wasserstein barycenters, Huawei, Shanghai, China, June 6, 2019.

  • P. Dvurechensky, Near-optimal method for highly smooth convex optimization, Conference on Learning Theory, COLT 2019, June 24 - 28, 2019, Phoenix, Arizona, USA, June 27, 2019.

  • P. Dvurechensky, On the complexity of approximating Wasserstein barycenters, Thirty-sixth International Conference on Machine Learning, ICML 2019, June 9 - 15, 2019, Long Beach, CA, USA, June 12, 2019.

  • P. Dvurechensky, On the complexity of optimal transport problems, Computational and Mathematical Methods in Data Science, Berlin, October 24 - 25, 2019.

  • P. Dvurechensky, On the complexity of optimal transport problems, Optimal transportation meeting, September 23 - 27, 2019, Higher School of Economics, Moskau, Russian Federation, September 26, 2019.

  • P. Dvurechensky, Optimization and Statistical Learning, Les Houches, France, March 24 - 29, 2019.

  • P. Friz, Multiscale systems, homogenization and rough paths, CRC 1114 Colloquium & Lecture, Freie Universität Berlin, June 13, 2019.

  • P. Friz, On differential equations with singular forcing, Berliner Oberseminar Nichtlineare partielle Differentialgleichungen (Langenbach-Seminar), WIAS Berlin, January 9, 2019.

  • P. Friz, Rough paths, rough volatility and regularity structures, Mini-course consisting of two sessions, Mathematics and CS Seminar, July 4 - 5, 2019, Institute of Science and Technology Austria, Klosterneuburg, Austria.

  • P. Friz, Rough paths, rough volatility, regularity structures, Rough Workshop 2019, September 4 - 6, 2019, TU Wien, Financial and Actuarial Mathematics, Austria.

  • P. Friz, Rough semimartingales, Paths between Probability, PDEs, and Physics: Conference 2019, July 1 - 5, 2019, Imperial College London, July 2, 2019.

  • P. Mathé, Relating direct and inverse Bayesian problems via the modulus of continue, Stochastic Computation and Complexity, April 15 - 16, 2019, Institut Henri Poincaré, Paris, France, April 16, 2019.

  • P. Mathé, Relating direct and inverse problems via the modulus of continuity, The Chemnitz Symposium on Inverse Problems 2019, September 30 - October 2, 2019, TU Chemnitz, Fakultät für Mathematik, Frankfurt a. M., October 1, 2019.

  • P. Mathé, The role of the modulus of continuity in inverse problems, Forschungsseminar Inverse Probleme, Technische Universität Chemnitz, Fachbereich Mathematik, August 13, 2019.

  • J. Polzehl, K. Tabelow, Analyzing neuroimaging experiments within R, 2019 OHBM Annual Meeting, Rom, Italy, June 9 - 13, 2019.

  • J. Polzehl, R Introduction, visualization and package management / Exploring functional data, Leibniz MMS Summer School 2019, October 28 - November 1, 2019, Mathematisches Forschungsinstitut Oberwolfach.

  • J.G.M. Schoenmakers, Tractability of continuous time optimal stopping problems, DynStoch 2019, June 12 - 15, 2019, Delft University of Technology, Institute of Applied Mathematics, Netherlands, June 14, 2019.

  • J.G.M. Schoenmakers, Tractability of continuous time optimal stopping problems, Weekly Workshop on Mathematical Finance and Numerical Probabilty, Université Paris Diderot, École Doctorale de Sciences Mathématiques de Paris Centre, France, June 27, 2019.

  • V. Spokoiny, Advanced statistical methods, April 9 - 11, 2019, Higher School of Economics (HSE), Moskau, Russian Federation.

  • V. Spokoiny, Bayesian inference for nonlinear inverse problems, Rencontres de Statistiques Mathématiques, December 16 - 20, 2019, Centre International de Rencontres Mathématiques (CIRM), Luminy, France, December 19, 2019.

  • V. Spokoiny, Inference for spectral projectors, RTG Kolloquium, Universität Heidelberg, Institut ür angewandte Mathematik, January 10, 2019.

  • V. Spokoiny, Optimal stopping and control via reinforced regression, Optimization and Statistical Learning, March 25 - 28, 2019, Les Houches School of Physics, France, March 26, 2019.

  • V. Spokoiny, Optimal stopping via reinforced regression, HUB-NUS FinTech Workshop, March 18 - 21, 2019, National University of Singapore, Institute for Mathematical Science, Singapore, March 21, 2019.

  • V. Spokoiny, Statistical inference for barycenters, Optimal transportation meeting, September 23 - 27, 2019, Higher School of Economics, Moskau, Russian Federation, September 26, 2019.

  • K. Tabelow, Adaptive smoothing data from multi-parameter mapping, 7th Nordic-Baltic Biometric Conference, June 3 - 5, 2019, Vilnius University, Faculty of Medicine, Lithuania, June 5, 2019.

  • K. Tabelow, Model-based imaging for quantitative MRI, KoMSO Challenge-Workshop Mathematical Modeling of Biomedical Problems, December 12 - 13, 2019, Friedrich-Alexander-University Erlangen-Nuremberg (FAU), December 12, 2019.

  • K. Tabelow, Neuroimaging workshop, Advanced Statistics, February 13 - 14, 2019, University of Zurich, Center for Reproducible Science, Switzerland.

  • K. Tabelow, Quantitative MRI for in-vivo histology, Neuroimmunological Colloquium, Charité-Universitätsmedizin Berlin, November 11, 2019.

  • K. Tabelow, Quantitative MRI for in-vivo histology, Doktorandenseminar, Berlin School of Mind and Brain, April 1, 2019.

  • K. Tabelow, Version control using git / Dynamic documents in R, Leibniz MMS Summer School 2019, October 28 - November 1, 2019, Mathematisches Forschungsinstitut Oberwolfach.

External Preprints

  • V. Avanesov, How to gamble with non-stationary x-armed bandits and have no regrets, Preprint no. arXiv:1908.07636, Cornell University Library, arXiv.org, 2019.
    Abstract
    In X-armed bandit problem an agent sequentially interacts with environment which yields a reward based on the vector input the agent provides. The agent's goal is to maximise the sum of these rewards across some number of time steps. The problem and its variations have been a subject of numerous studies, suggesting sub-linear and some times optimal strategies. The given paper introduces a novel variation of the problem. We consider an environment, which can abruptly change its behaviour an unknown number of times. To that end we propose a novel strategy and prove it attains sub-linear cumulative regret. Moreover, in case of highly smooth relation between an action and the corresponding reward, the method is nearly optimal. The theoretical result are supported by experimental study.

  • V. Avanesov, Nonparametric change point detection in regression, Preprint no. arXiv:1903.02603, Cornell University Library, arXiv.org, 2019.
    Abstract
    This paper considers the prominent problem of change-point detection in regression. The study suggests a novel testing procedure featuring a fully data-driven calibration scheme. The method is essentially a black box, requiring no tuning from the practitioner. The approach is investigated from both theoretical and practical points of view. The theoretical study demonstrates proper control of first-type error rate under H0 and power approaching 1 under H1. The experiments conducted on synthetic data fully support the theoretical claims. In conclusion, the method is applied to financial data, where it detects sensible change-points. Techniques for change-point localization are also suggested and investigated.

  • F. Besold, V. Spokoiny, Adaptive manifold clustering, Preprint no. arXiv:1912.04869, Cornell University Library, arXiv.org, 2019.

  • O. Butkovsky, K. Dareiotis, M. Gerencsér, Approximation of SDEs -- a stochastic sewing approach, Preprint no. arXiv:1909.07961, Cornell University Library, arXiv.org, 2019.

  • O. Butkovsky, A. Kulik, M. Scheutzow, Generalized couplings and ergodic rates for SPDEs and other Markov models, Preprint no. arXiv:1806.00395, Cornell University Library, arXiv.org, 2019.
    Abstract
    We establish verifiable general sufficient conditions for exponential or subexponential ergodicity of Markov processes that may lack the strong Feller property. We apply the obtained results to show exponential ergodicity of a variety of nonlinear stochastic partial differential equations with additive forcing, including 2D stochastic Navier-Stokes equations. Our main tool is a new version of the generalized coupling method.

  • O. Butkovsky, M. Scheutzow, Couplings via comparison principle and exponential ergodicity of SPDEs in the hypoelliptic setting, Preprint no. arXiv:1907.03725, Cornell University Library, arXiv.org, 2019.

  • O. Butkovsky, F. Wunderlich, Asymptotic strong Feller property and local weak irreducibility via generalized couplings, Preprint no. arXiv:1912.06121, Cornell University Library, arXiv.org, 2019.
    Abstract
    In this short note we show how the asymptotic strong Feller property (ASF) and local weak irreducibility can be established via generalized couplings. We also prove that a stronger form of ASF together with local weak irreducibility implies uniqueness of an invariant measure. The latter result is optimal in a certain sense and complements some of the corresponding results of Hairer, Mattingly (2008).

  • M. Redmann, An $L^2_T$-error bound for time-limited balanced truncation, Preprint no. arXiv:1907.05478, Cornell University Library, arXiv.org, 2019.

  • Y.-W. Sun, K. Papagiannouli, V. Spokoiny, Online graph-based change-point detection for high dimensional data, Preprint no. arXiv:1906.03001, Cornell University Library, arXiv.org, 2019.
    Abstract
    Online change-point detection (OCPD) is important for application in various areas such as finance, biology, and the Internet of Things (IoT). However, OCPD faces major challenges due to high-dimensionality, and it is still rarely studied in literature. In this paper, we propose a novel, online, graph-based, change-point detection algorithm to detect change of distribution in low- to high-dimensional data. We introduce a similarity measure, which is derived from the graph-spanning ratio, to test statistically if a change occurs. Through numerical study using artificial online datasets, our data-driven approach demonstrates high detection power for high-dimensional data, while the false alarm rate (type I error) is controlled at a nominal significant level. In particular, our graph-spanning approach has desirable power with small and multiple scanning window, which allows timely detection of change-point in the online setting.

  • M. Alkousa, D. Dvinskikh, F. Stonyakin, A. Gasnikov, Accelerated methods for composite non-bilinear saddle point problem, Preprint no. arXiv:1906.03620, Cornell University Library, arXiv.org, 2019.

  • S. Athreya, O. Butkovsky, L. Mytnik, Strong existence and uniqueness for stable stochastic differential equations with distributional drift, Preprint no. arXiv:1801.03473, Cornell University Library, arXiv.org, 2019.

  • J. Diehl, K. Ebrahimi-Fard, N. Tapia, Time warping invariants of multidimensional time series, Preprint no. arXiv:1906.05823, Cornell University Library, arXiv.org, 2019.
    Abstract
    In data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants. We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties.

  • E. Gorbunov, D. Dvinskikh, A. Gasnikov, Optimal decentralized distributed algorithms for stochastic convex optimization, Preprint no. arXiv:1911.07363, Cornell University Library, arXiv.org, 2019.

  • A. Kroshnin, D. Dvinskikh, P. Dvurechensky, N. Tupitsa, C. Uribe, On the complexity of approximating Wasserstein barycenter, Preprint no. arXiv:1901.08686, Cornell University Library, arXiv.org, 2019.

  • A. Kroshnin, V. Spokoiny, A. Suvorikova, Statistical inference for Bures-Wasserstein barycenters, Preprint no. arXiv:1901.00226, Cornell University Library, arXiv.org, 2019.

  • A. Olgatsov, D. Dvinskikh, P. Dvurechensky, A. Gasnikov, V. Spokoiny, Adaptive gradient descent for convex and non-convex stochastic optimization, Preprint no. arXiv:1911.08380, Cornell University Library, arXiv.org, 2019.

  • N. Puchkin, V. Spokoiny, Structure-adaptive manifold estimation, Preprint no. arXiv:1906.05014, Cornell University Library, arXiv.org, 2019.
    Abstract
    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 novel computationally efficient structure-adaptive procedure, which simultaneously reconstructs a smooth manifold and estimates projections of the point cloud onto this manifold. The proposed approach iteratively refines the weights on each step, using the structural information obtained at previous steps. After several iterations, we obtain nearly öracle" weights, so that the final estimates are nearly efficient even in the presence of relatively large noise. In our theoretical study we establish tight lower and upper bounds proving asymptotic optimality of the method for manifold estimation under the Hausdorff loss. Our finite sample study confirms a very reasonable performance of the procedure in comparison with the other methods of manifold estimation.

  • A. Rastogi, G. Blanchard, P. Mathé, Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems, Preprint no. arXiv:1902.05404, Cornell University Library, arXiv.org, 2019.
    Abstract
    We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization, MOR) approach to reconstruct the estimator of the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the ansatz of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.

  • F. Stonyakin, A. Gasnikov, A. Tyurin, D. Pasechnyuk, A. Agafonov, P. Dvurechensky, D. Dvinskikh, A. Kroshnin, V. Piskunova, Inexact Model: A framework for optimization and variational inequalities, Preprint no. arXiv:1902.00990, Cornell University Library, arXiv.org, 2019.

  • F. Stonyakin, D. Dvinskikh, P. Dvurechensky, A. Kroshnin, O. Kuznetsova, A. Agafonov, A. Gasnikov, A. Tyurin, C.A. Uribe, D. Pasechnyuk, S. Artamonov, Gradient methods for problems with inexact model of the objective, Preprint no. arXiv:1902.09001, Cornell University Library, arXiv.org, 2019.

  • N. Tupitsa, P. Dvurechensky, A. Gasnikov, S. Guminov, Alternating minimization methods for strongly convex optimization, Preprint no. arXiv:1911.08987, Cornell University Library, arXiv.org, 2019.
    Abstract
    We consider alternating minimization procedures for convex optimization problems with variable divided in many block, each block being amenable for minimization with respect to its variable with freezed other variables blocks. In the case of two blocks, we prove a linear convergence rate for alternating minimization procedure under Polyak-Łojasiewicz condition, which can be seen as a relaxation of the strong convexity assumption. Under strong convexity assumption in many-blocks setting we provide an accelerated alternating minimization procedure with linear rate depending on the square root of the condition number as opposed to condition number for the non-accelerated method.

  • D. Dvinskikh, A. Gasnikov, Decentralized and parallelized primal and dual accelerated methods for stochastic convex programming problems, Preprint no. arXiv:1904.09015, Cornell University Library, arXiv.org, 2019.
    Abstract
    We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. The proposed methods are optimal in terms of communication steps for primal and dual oracles. However, optimality in terms of oracle calls per node takes place in all the cases up to a logarithmic factor and the notion of smoothness (the worst case vs the average one). All the methods for stochastic oracle can be additionally parallelized on each node due to the batching technique.

  • D. Dvinskikh, E. Gorbunov, A. Gasnikov, P. Dvurechensky, C.A. Uribe, On dual approach for distributed stochastic convex optimization over networks, Preprint no. arXiv:1903.09844, Cornell University Library, arXiv.org, 2019.
    Abstract
    We introduce dual stochastic gradient oracle methods for distributed stochastic convex optimization problems over networks. We estimate the complexity of the proposed method in terms of probability of large deviations. This analysis is based on a new technique that allows to bound the distance between the iteration sequence and the solution point. By the proper choice of batch size, we can guarantee that this distance equals (up to a constant) to the distance between the starting point and the solution.

  • P. Dvurechensky, A. Gasnikov, P. Ostroukhov, C.A. Uribe, A. Ivanova, Near-optimal tensor methods for minimizing the gradient norm of convex function, Preprint no. arXiv:1912.03381, Cornell University Library, arXiv.org, 2019.

  • V. Spokoiny, M. Panov, Accuracy of Gaussian approximation in nonparametric Bernstein -- von Mises theorem, Preprint no. arXiv:1910.06028, Cornell University Library, arXiv.org, 2019.