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Campus inMoscow

# ­­­Uncertainty quantification in high-dimensional statistics

Priority areas of development: IT and mathematics
2019
The project has been carried out as part of the HSE Program of Fundamental Studies.

#### Goal of research

Development and analysis of new efficient computational statistical algorithms for high- dimensional and challenging statistical problems like Bayesian inference, covariance estimation, community detection, estimation of barycenters.

#### Methodology

Tools from random matrix theory (e.g. Stieltjes transforms methods, special functions approach) and concentration of measure (e.g. Lipschitz concentration and transportation inequalities); methods of high-dimensional approximation based on Stein’s method (and its variations),  Fourier method, theory of Gaussian processes, higher order concentration of measure;  theory of empirical processes and its extension to the case of dependent observations.

#### Empirical base of research

Financial time series (equities, bounds, interest rates, options), model data; applied and theoretical research papers.

#### Results of research

• We proved Gaussian comparison and anti-concentration inequality for norms of non-centred Gaussian elements in separable Hilbert space. Results are published in Bernoulli (Q2 WOS)

• We proved local semicircle law for symmetric random matrices under 4 moment condition (optimal moment conditions). We also provide bounds for rate of localization of eigenvalues, delocalization of eigenvectors and rates of convergence of the empirical spectral distribution to the distribution function of the Wigner semicircle law in the Kolmogorov distance. Results are accepted for publication in Journal of Theoretical Probability (Q3 WOS)

• We proved bootstrap validitiy for procedure to build confidence sets for spectral projectors of sample covariance matrices under condition that observations have normal distribution. Results are published in Probability Thery and Related Fields journal

• We proved central limit theorem for Wasserstein barycenters of Hermitian positively defined matrices and obtained non-asymptotic concentration inequalities in sub-exponential case. Arxiv preprint is prepared.

• We consider a problem of manifold estimation from noisy observations. We suggest a novel adaptive procedure, which simultaneously reconstructs a smooth manifold from the observations and estimates projectors onto the tangent spaces. We also provide a theoretical study of the procedure and prove its optimality deriving both new upper and lower bounds for manifold estimation under the Hausdorff loss. Arxiv preprint is prepared.

• We study the problem of statistical inference for a continuous-time moving average  Le’vy process of the form Z_{t}=\int_{\R}K(t-s) dL_{s}, t \in R,  with a deterministic kernel \K and a  Levy process L. Especially the estimation of  the Levy measure \nu of  L from low-frequency observations of the process Z is considered. We construct a consistent estimator, derive its convergence rates and illustrate its performance by a numerical example.  On the mathematical level, we establish some new results on exponential mixing for continuous-time moving average Le’vy processes. Results are published in Bernoulli (Q2 WOS)

• We study the estimation of the covariance matrix Σ of a p-dimensional normal random vector based on n independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of Σ. We establish an oracle inequality for these estimators under model miss-specification and derive non-asymptotic minimax convergence rates that are shown to be logarithmic in log p/n. We also discuss the estimation of low-rank matrices based on indi- rect observations as well as the generalization to elliptical distributions. The finite sample performance of the threshold estimators is illustrated in a numerical example. Results are published inBernoulli (Q2 WOS)

• Multiple Perron eigenvectors of non-negative matrices occur in applications, where they often become a source of trouble. A usual way to avoid it and to make the Perron eigenvector simple is a regularization of matrix: an initial non-negative matrix A is replaced by 𝐴+𝜀𝑀, where Mis a strictly positive matrix and 𝜀>0is small. However, this operation is numerically unstable and may lead to a significant increase of the Perron eigenvalue, especially in high dimensions. We define a selected Perron eigenvector of A as the limit of normalized Perron eigenvectors of the regularizations 𝐴+𝜀𝑀 as 𝜀→0. It is shown that if the matrix M is rank-one, then the limit eigenvector can be found by an explicit formula and, moreover, is efficiently computed by the power method. The role of the rank-one condition is explained. Results are published in Calcolo (Q1 WOS)

We study sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order d-1 for any d \in N. The bounds are based on dth order derivatives or difference operators. In particular, we consider deviations of functions of independent random variables and differentiable functions over probability measures satisfying a logarithmic Sobolev inequality, and functions on the unit sphere. Applications include concentration inequalities for U-statistics as well as for classes of symmetric functions via polynomial approximations on the sphere (Edgeworth-type expansions). Results are published in Communications in Contemporary Mathematics (Q1 WOS)

Let  F_n  denote the distribution function of the normalized sum of  n  i.i.d. random variables. In this paper, polynomial rates of approximation of  F_n  by the corrected normal laws are considered in the model where the underlying distribution has a convolution structure. As a basic tool, the convergence part of Khinchine’s theorem in metric theory of Diophantine approximations is extended to the class of product characteristic functions. Results are published in Annals of Statistics (Q1 WOS)

#### Publications:

Naumov A., Moulines E., Каледин М. Л., Tadic V., Wai H. Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise // Working papers by Cornell University. Series math "arxiv.org". 2020. P. 1-61.
Belomestny D., GUGUSHVILI S., SCHAUER M., SPREIJ P. NONPARAMETRIC BAYESIAN INFERENCE FOR GAMMA-TYPE LEVY SUBORDINATORS // Communications in Mathematical Sciences. 2019. Vol. 17. No. 3. P. 781-816. doi
Vorontsova E., Gasnikov A., Dvurechensky P., Gorbunov E. Accelerated Gradient-Free Optimization Methods with a Non-Euclidean Proximal Operator / Пер. с рус. // Automation and Remote Control. 2019. Vol. 80. No. 8. P. 1487-1501. doi
Gasnikov A., Tyurin A. Fast Gradient Descent for Convex Minimization Problems with an Oracle Producing a (δ, L)-Model of Function at the Requested Point / Пер. с рус. // Computational Mathematics and Mathematical Physics. 2019. Vol. 59. No. 7. P. 1085-1097. doi
Беломестный Д. В., Иосипой Л. Об оценке плотности распределения с помощью ряда Фурье // Управление большими системами: сборник трудов. 2019. № 82. С. 28-43. doi
Гасников А. В., Тюрин А. И. Быстрый градиентный спуск для задач выпуклой минимизации с оракулом, выдающим (δ, L)-модель функции в запрошенной точке // Журнал вычислительной математики и математической физики. 2019. Т. 59. № 7. С. 1137-1150.
Гасников А. В., Двуреченский П. Е., Воронцова Е., Горбунов Э. Ускоренные безградиентные методы оптимизации с неевклидовым проксимальным оператором // Автоматика и телемеханика. 2019. Т. 80. № 8. С. 1487-1501. doi
Kolesnikov A., Gladkov N., Zimin A. On multistochastic Monge–Kantorovich problem, bitwise operations, and fractals // Calculus of Variations and Partial Differential Equations. 2019. Vol. 58. No. 173. P. 1-33. doi
Protasov V. Y., Shirokov M. On Mutually Inverse Transforms of Functions on a Half-Line / Пер. с рус. // Doklady Mathematics. 2019. Vol. 100. No. 3. P. 1-4.
Moulines E., Robin G., Klopp O., Josse J., Tibshirani R. Main Effects and Interactions in Mixed and Incomplete Data Frames // Journal of the American Statistical Association. 2019. P. 1-12. doi
Goetze F., Naumov A., Spokoiny V., Ulyanov V. V. Large ball probability, Gaussian comparison and anti-concentration // Bernoulli: a journal of mathematical statistics and probability. 2019. Vol. 25. No. 4(A). P. 2538-2563. doi
Naumov A., Spokoiny V., Ulyanov V. V. Bootstrap confidence sets for spectral projectors of sample covariance // Probability Theory and Related Fields. 2019. Vol. 174. No. 3-4. P. 1091-1132. doi
Bobkov S., Goetze F., Sambale H. Higher order concentration of measure // Communications in Contemporary Mathematics. 2019. Vol. 21. No. 3. P. 1-31. doi
Belomestny D., Panov V., Woerner J. Low-frequency estimation of continuous-time moving average Levy processes // Bernoulli: a journal of mathematical statistics and probability. 2019. Vol. 25. No. 2. P. 902-931. doi
Gasnikov A. Optimal Tensor Methods in Smooth Convex and Uniformly Convex Optimization, in: Proceedings of Machine Learning Research Vol. 99: Conference on Learning Theory, 25-28 June 2019, Phoenix, AZ, USA. PMLR, 2019.. , 2019.
Гасников А. В. АДАПТИВНЫЙ ПРОКСИМАЛЬНЫЙ МЕТОД ДЛЯ ВАРИАЦИОННЫХ НЕРАВЕНСТВ // Журнал вычислительной математики и математической физики. 2019. Т. 59. № 5. С. 889-894. doi
Belomestny D., Trabs M., Tsybakov A. Sparse covariance matrix estimation in high-dimensional deconvolution // Bernoulli: a journal of mathematical statistics and probability. 2019. Vol. 25. No. 3. P. 1901-1938. doi
Belomestny D., Kraetschmer V., Hübner T., Nolte S. Minimax theorems for American options without time-consistency // Finance and Stochastics. 2019. Vol. 23. P. 209-238. doi
Protasov V. Y. How to make the Perron eigenvector simple // Calcolo. 2019. Vol. 56. No. 2. P. 1-11. doi
Belomestny D., Comte F., Genon-Catalot V. Sobolev-Hermite versus Sobolev nonparametric density estimation on R // Annals of the Institute of Statistical Mathematics. 2019. Vol. 71. No. 1. P. 29-62. doi
Bobkov S. KHINCHINE’S THEOREM AND EDGEWORTH APPROXIMATIONS FOR WEIGHTED SUMS // Annals of Statistics. 2019. Vol. 47. No. 3. P. 1616-1633. doi
Puchkin N., Spokoiny V. Structure-adaptive manifold estimation / Cornell University. Series arxive "math". 2019.