arXiv subjects
Publications and source records attributed to Sergey Bobkov.
Energy bounds for Kantorovich transport distances are developed for convex cost functions. The main results extend estimates due to M. Ledoux for the Kantorovich distances $W_p$.
The paper is devoted to the investigation of Esscher's transform on high dimensional Euclidean spaces in the light of its application to the central limit theorem. With this tool, we explore necessary and sufficient conditions of normal approximation for normalized sums of i.i.d. random vectors in terms of the R\'enyi divergence of infinite order, extending recent one dimensional results.
Berry-Esseen-type bounds are developed in the multidimensional local limit theorem in terms of the Lyapunov coefficients and maxima of involved densities.
We give a detailed exposition of the proof of Richter's local limit theorem in a refined form, and establish the stability of the remainder term in this theorem under small perturbations of the underlying distribution (including smoothing). We also discuss related quantitative bounds for characteristic functions and Laplace transforms.
Single particle imaging (SPI) at X-ray free electron lasers (XFELs) is particularly well suited to determine the 3D structure of particles in their native environment. For a successful reconstruction, diffraction patterns originating from a single hit must be isolated from a large number of acquired patterns. We propose to formulate this task as an image classification problem and solve it using convolutional neural network (CNN) architectures. Two CNN configurations are developed: one that maximises the F1-score and one that emphasises high recall. We also combine the CNNs with expectation maximization (EM) selection as well as size filtering. We observed that our CNN selections have lower contrast in power spectral density functions relative to the EM selection, used in our previous work. However, the reconstruction of our CNN-based selections gives similar results. Introducing CNNs into SPI experiments allows streamlining the reconstruction pipeline, enables researchers to classify patterns on the fly, and, as a consequence, enables them to tightly control the duration of their experiments. We think that bringing non-standard artificial intelligence (AI) based solutions in a well-described SPI analysis workflow may be beneficial for the future development of the SPI experiments.
An improved analysis for single particle imaging (SPI) experiments, using the limited data, is presented here. Results are based on a study of bacteriophage PR772 performed at the AMO instrument at the Linac Coherent Light Source (LCLS) as part of the SPI initiative. Existing methods were modified to cope with the shortcomings of the experimental data: inaccessibility of information from the half of the detector and small fraction of single hits. General SPI analysis workflow was upgraded with the expectation-maximization based classification of diffraction patterns and mode decomposition on the final virus structure determination step. The presented processing pipeline allowed us to determine the three-dimensional structure of the bacteriophage PR772 without symmetry constraints with a spatial resolution of 6.9 nm. The obtained resolution was limited by the scattering intensity during the experiment and the relatively small number of single hits.
The emergence of high repetition-rate X-ray free-electron lasers (XFELs) powered by superconducting accelerator technology enables the measurement of significantly more experimental data per day than was previously possible. The European XFEL will soon provide 27,000 pulses per second, more than two orders of magnitude more than any other XFEL. The increased pulse rate is a key enabling factor for single-particle X-ray diffractive imaging, which relies on averaging the weak diffraction signal from single biological particles. Taking full advantage of this new capability requires that all experimental steps, from sample preparation and delivery to the acquisition of diffraction patterns, are compatible with the increased pulse repetition rate. Here, we show that single-particle imaging can be performed using X-ray pulses at megahertz repetition rates. The obtained results pave the way towards exploiting high repetition-rate X-ray free-electron lasers for single-particle imaging at their full repetition rate.
We present a short and elementary proof of the Ajtai-Komlós-Tusnády (AKT) optimal matching theorem in dimension 2 via Fourier analysis and a smoothing argument. The upper bound applies to more general families of samples, including dependent variables, of interest in the study of rates of convergence for empirical measures. Following the recent pde approach by L. Ambrosio, F. Stra and D. Trevisan, we also adapt a simple proof of the lower bound.
We prove several estimates for the moments of arbitrary measures on convex bodies. We apply these estimates to show a new slicing inequality for measures on convex bodies. We also deduce estimates for the outer volume ratio distance from an arbitrary centrally-symmetric convex body in R^n to the class of unit balls of n-dimensional subspaces of L_p-spaces. Finally, we prove a result of the Busemann-Petty type for these moments.
An extension of the entropy power inequality to the form $N_r^α(X+Y) \geq N_r^α(X) + N_r^α(Y)$ with arbitrary independent summands $X$ and $Y$ in $\mathbb{R}^n$ is obtained for the Rényi entropy and powers $α\geq (r+1)/2$.
Under correlation-type conditions, we derive upper bounds of order $\frac{1}{\sqrt{n}}$ for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law.
We show that for any metric probability space $(M,d,μ)$ with a subgaussian constant $σ^2(μ)$ and any set $A \subset M$ we have $σ^2(μ_A) \leq c \log\left(e/μ(A)\right)\,σ^2(μ)$, where $μ_A$ is a restriction of $μ$ to the set $A$ and $c$ is a universal constant. As a consequence we deduce concentration inequalities for non-Lipschitz functions.
The de cit in the logarithmic Sobolev inequality for the Gaussian measure is considered and estimated by means of transport and information-theoretic distances.
A concentration property of the functional ${-}\log f(X)$ is demonstrated, when a random vector X has a log-concave density f on $\mathbb{R}^n$. This concentration property implies in particular an extension of the Shannon-McMillan-Breiman strong ergodic theorem to the class of discrete-time stochastic processes with log-concave marginals.
We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by $α$-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then, we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prékopa-Leindler and Brascamp-Lieb inequalities. Further issues that we transpose to this functional setting are: integral-geometric formulae of Cauchy-Kubota type, valuation property and isoperimetric/Uryshon-like inequalities.
We develop a reverse entropy power inequality for convex measures, which may be seen as an affine-geometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this inequality to log-concave measures may be seen as a version of Milman's reverse Brunn-Minkowski inequality. The proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies, and on a study of effective supports of convex measures, both of which are of independent interest, as well as on Milman's deep technology of $M$-ellipsoids and on certain information-theoretic inequalities. As a by-product, we also give a continuous analogue of some Plünnecke-Ruzsa inequalities from additive combinatorics.
We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian comparison results for log-concave measures, an entropic formulation of the hyperplane conjecture, and a new reverse entropy power inequality for log-concave measures analogous to V. Milman's reverse Brunn-Minkowski inequality.
A generalization of Young's inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified proof of recent entropy power inequalities of Barron and Madiman, as well as of a (conjectured) generalization of the Brunn-Minkowski inequality. It is shown that the generalized Brunn-Minkowski conjecture is true for convex sets; an application of this to the law of large numbers for random sets is described.