A characterization of Descartes systems in Haar subspaces

AbstractA characterization of existence of Descartes systems in Haar subspaces is given. Moreover, it is shown that the functions in such systems can be represented as products of piecewise strictly monotone functions

An extremal problem arising in the dynamics of two-phase materials that directly reveals information about the internal geometry

In two phase materials, each phase having a non-local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [-1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [-1,1] and to simplify the analysis Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [-1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov-type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time-dependent boundary potentials are suitably tailored.Comment: 36 pages, 7 figure

Integrable and Chaotic Systems Associated with Fractal Groups

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 80-s of the last century with the purpose to solve some famous problems in mathematics, including the question raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor's question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schr\"odinger operators. One of important developments is the relation of them to the multi-dimensional dynamics, theory of joint spectrum of pencil of operators, and spectral theory of Laplace operator on graphs. The paper gives a quick access to these topics, provide calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains discussion of the dichotomy "integrable-chaotic" in the considered model, and suggests a possible probabilistic approach to the study of discussed problems.Comment: 48 pages, 15 figure

Polynomial Diffusions and Applications in Finance

This paper provides the mathematical foundation for polynomial diffusions. They play an important role in a growing range of applications in finance, including financial market models for interest rates, credit risk, stochastic volatility, commodities and electricity. Uniqueness of polynomial diffusions is established via moment determinacy in combination with pathwise uniqueness. Existence boils down to a stochastic invariance problem that we solve for semialgebraic state spaces. Examples include the unit ball, the product of the unit cube and nonnegative orthant, and the unit simplex.Comment: This article is forthcoming in Finance and Stochastic

Long time behaviour of infinite dimensional stochastic processes

We study two examples of infinite dimensional stochastic processes. Situations and techniques involved are quite varied, however in both cases we achieve a progress in describing their long time behaviour. The first case concerns interacting particle system of diffusions. We construct rigorously the process using finite dimensional approximation and the notion of martingale solution. The existence of invariant measure for the process is proved. The novelty of the results lies in the fact, that our methods enable us to consider such examples, where the generator of the diffusion is subelliptic. The other project is related to stochastic partial differential equations and their stability properties. In particular it is shown that Robbins-Monro procedure can be extended to infinite dimensional setting. Thus we achieve results about pathwise convergence of solution. To be able to define corresponding solution, we rely on so-called variational approach to stochastic partial differential equations as pioneered by E. Pardoux, N. Krylov and B. Rozovskii. Our examples covers situations such as p-Laplace operator or Porous medium operator.Open Acces

Polynomial methods in statistical inference: Theory and practice

Recent advances in genetics, computer vision, and text mining are accompanied by analyzing data coming from a large domain, where the domain size is comparable or larger than the number of samples. In this dissertation, we apply the polynomial methods to several statistical questions with rich history and wide applications. The goal is to understand the fundamental limits of the problems in the large domain regime, and to design sample optimal and time efficient algorithms with provable guarantees. The first part investigates the problem of property estimation. Consider the problem of estimating the Shannon entropy of a distribution over $k$ elements from $n$ independent samples. We obtain the minimax mean-square error within universal multiplicative constant factors if $n$ exceeds a constant factor of $k/\log(k)$; otherwise there exists no consistent estimator. This refines the recent result on the minimal sample size for consistent entropy estimation. The apparatus of best polynomial approximation plays a key role in both the construction of optimal estimators and, via a duality argument, the minimax lower bound. We also consider the problem of estimating the support size of a discrete distribution whose minimum non-zero mass is at least $\frac{1}{k}$. Under the independent sampling model, we show that the sample complexity, i.e., the minimal sample size to achieve an additive error of $\epsilon k$ with probability at least 0.1 is within universal constant factors of $\frac{k}{\log k}\log^2\frac{1}{\epsilon}$, which improves the state-of-the-art result of $\frac{k}{\epsilon^2 \log k}$. Similar characterization of the minimax risk is also obtained. Our procedure is a linear estimator based on the Chebyshev polynomial and its approximation-theoretic properties, which can be evaluated in $O(n+\log^2 k)$ time and attains the sample complexity within constant factors. The superiority of the proposed estimator in terms of accuracy, computational efficiency and scalability is demonstrated in a variety of synthetic and real datasets. When the distribution is supported on a discrete set, estimating the support size is also known as the distinct elements problem, where the goal is to estimate the number of distinct colors in an urn containing $k$ balls based on $n$ samples drawn with replacements. Based on discrete polynomial approximation and interpolation, we propose an estimator with additive error guarantee that achieves the optimal sample complexity within $O(\log\log k)$ factors, and in fact within constant factors for most cases. The estimator can be computed in $O(n)$ time for an accurate estimation. The result also applies to sampling without replacement provided the sample size is a vanishing fraction of the urn size. One of the key auxiliary results is a sharp bound on the minimum singular values of a real rectangular Vandermonde matrix, which might be of independent interest. The second part studies the problem of learning Gaussian mixtures. The method of moments is one of the most widely used methods in statistics for parameter estimation, by means of solving the system of equations that match the population and estimated moments. However, in practice and especially for the important case of mixture models, one frequently needs to contend with the difficulties of non-existence or non-uniqueness of statistically meaningful solutions, as well as the high computational cost of solving large polynomial systems. Moreover, theoretical analysis of the method of moments are mainly confined to asymptotic normality style of results established under strong assumptions. We consider estimating a $k$-component Gaussian location mixture with a common (possibly unknown) variance parameter. To overcome the aforementioned theoretic and algorithmic hurdles, a crucial step is to denoise the moment estimates by projecting to the truncated moment space (via semidefinite programming) before solving the method of moments equations. Not only does this regularization ensures existence and uniqueness of solutions, it also yields fast solvers by means of Gauss quadrature. Furthermore, by proving new moment comparison theorems in the Wasserstein distance via polynomial interpolation and majorization techniques, we establish the statistical guarantees and adaptive optimality of the proposed procedure, as well as oracle inequality in misspecified models. These results can also be viewed as provable algorithms for generalized method of moments which involves non-convex optimization and lacks theoretical guarantees

Relative entropy decay and complete positivity mixing time

We prove that the complete modified logarithmic Sobolev constant of a quantum Markov semigroup is bounded by the inverse of its complete positivity mixing time. For classical Markov semigroups, this implies that every sub-Laplacian given by a H\"ormander system on a compact manifold satisfies a uniform modified log-Sobolev inequality for matrix-valued functions. For quantum Markov semigroups, we obtain that the complete modified logarithmic Sobolev constant is comparable to spectral gap up to a constant as logarithm of dimension constant. This estimate is asymptotically tight for a quantum birth-death process. Our results and the consequence of concentration inequalities apply to GNS-symmetric semigroups on general von Neumann algebras.Comment: 58 pages. Presentation improved. A discussion section added. Comments are very welcome