Stochastic models seminar 2016-17
Stochastic Models Seminar meets at the department of Mathematics and Statistics at University of Helsinki. The topics touch on various random phenomena and they depend on the interests of the participants. Often the seminar talks work as a 'reading seminar', where we study topics of general interest. A special topic (began fall 2015) is overview of the most most important contributions of Kolmogorov in science - talks on this topic will be given from time to time.
The seminar meets on Thursdays 14-16 in Exactum, room C124. It is run by prof. Kostya Izyurov (firstname.lastname@example.org), prof. Ilkka Norros (email@example.com) and prof.Eero Saksman (firstname.lastname@example.org). Usually the talks are in English.
Thursday 16.3.2017 klo 14-16 C122
Coding of stochastic block models II
Thursday 9.3.2017 klo 14-16 C124 (note the room!)
Coding of stochastic block models
Thursday 23.2.2017 klo 14-16 B321 (note the room!) (note that we have 2 speakers!)
Alain Durmus (Paris Telecom):
"Optimal scaling and convergence of Markov chain"
Monte Carlo methods
Matti Vihola (JY):
"Unbiased estimators and multilevel Monte Carlo"
DURMUS: Sampling over high-dimensional space has become a prerequisite in applications of Bayesian statistics, for example to machine learning problems. The most common methods for dealing with this problem are Markov Chain Monte Carlo methods. In this talk, I will present new insights on the computational complexity of these algorithms. First, I will discuss the optimal scaling problem of the random walk Metropolis algorithm applied to densities which are differentiable in Lp mean but which may be irregular at some points (like Laplace type densities for example) and / or are supported on an interval. The scaling limit is established under assumptions which are much weaker than the one used in the original derivation of (Roberts, Gelman, Gilks, 1997). In the second part of my talk, we will present a method based on the Euler discretization of the Langevin diffusion with either constant or decreasing stepsizes. We will give several new results establishing the convergence to stationarity under different conditions on the log-density. A particular attention of these bounds with respect to the dimension of the state space will be paid.
VIHOLA: Multilevel Monte Carlo (MLMC) is commonly applied, for instance, in numerical approximation of expectations with respect to diffusions. The unbiased estimators recently proposed by McLeish (Monte Carlo Methods Appl., 2011) and Rhee and Glynn (Oper. Res., 2015) are closely related to MLMC. This connection is elaborated by presenting a new general class of unbiased estimators, which admits previous debiasing schemes as special cases. New lower variance estimators are proposed, which are stratified versions of earlier unbiased schemes. Under general conditions, essentially when MLMC admits the canonical square root Monte Carlo error rate, the proposed new schemes are asymptotically as efficient as MLMC, both in terms of variance and cost.
Thursday 12.1.2017 klo 16-18 C124 (joint with the geomeric analysis seminar: note the new time !)
Meng Wu (University of Oulu)
On a conjecture of Furstenberg about intersections of Cantor sets
Abstract: Two compact sets E,F of the real line are said to be strongly transverse if for each u and t, the Hausdorff dimension (dim) of the intersection of E and uF+t is bounded by dim(E)+dim(F)-1 or 0, whichever is larger. In the late 60's, Furstenberg conjectured that two closed sets E,F of [0,1] are strongly transverse if E is invariant under multiplication by 2 (mod 1) and F is invariant under multiplication by 3 (mod 1). In this talk, we will recall some recent progress regarding this conjecture and present a solution.
Thursday 24.11.2016 at 14-16 in D123
Local sets of the Gaussian free field
The Gaussian free field is one the simplest statistical filed theories; it is a Gaussian process that can be viewed as a natural generalization of a Brownian motion to multi-dimensional time. A local set of a GFF is an analog of a stopping time, enabling some sort of a strong Markov property for the field. The theory of local sets is an essential part in the study of relation between GFF and Stochastic Loewner evolution, due to Schramm, Dubédat, Sheffield and Miller. In this talk, I will go through basic facts about the local sets, following lecture notes by Wendelin Werner.
Thursday 17.11.2016 at 14-16 in D123
Statistical models and measurements
Thursday 3.11.2016 at 14-16 in D123
Regular decomposition revisited
Thursday 6.10.2016 at 14-16 in D123
Multiplicative chaos in number theory and random matrix theory
Abstract: I will discuss the statistical behavior of the Riemann zeta function on the critical line. More precisely, I will talk about how in the vicinity of a random point on the critical line, the behavior of the zeta function can be described in terms of a random generalized function known as a Gaussian multiplicative chaos distribution. I will also discuss connections between the Riemann zeta function and random matrix theory in this setting. This is joint work with Eero Saksman.
Thursday 22. 9. 2016 at 16-18 in C124
On non-Gaussian multiplicative chaos
Talks in the past (2015-16)
Torstai/Thursday, 28.4. 2016 klo 16-18 C124
A category of Markov processes
Torstai/Thursday, 7.4. 2016 klo 16-18 C124
Kolmogorov and Turbulence (KOLMOGOROV-sarjaa)
Torstai/Thursday, 17.3. 2016 klo 16-18 C124
Stability and instability of file-sharing systems
Torstai/Thursday, 25.2. 2016 klo 16-18 C124
Projections of Mandelbrot measures and phase transitions
Torstai/Thursday, 11.2. 2016 klo 16-18 C124
Torstai/Thursday, 4.2. 2016 klo 16-18 C124
Torstai/Thursday, 28.1. 2016 klo 16-18 C124
Abstract: Data analysis has been a driving force behind recent advances in randomized linear algebra. The methods for computing partial matrix decompositions efficiently use random sampling to identify a subspace that captures most of the action of a matrix. In fact these methods rely on the concentration of measure phenomenon: the random input leads to very small variance on the quantities of interest.In computational mechanics the interest lies in the smallest eigenvalues of generalized eigenvalue problems which is exactly opposite to data analysis where the focus is on largest singular values. Remarkably, for realistic classes of symmetric and positive definite systems the randomized algorithms perform better than the state-of-the-art general purpose eigensolvers. This result appears to be new.
Torstai/Thursday, 14.1. 2016 klo 16-18 C124
Torstai/Thursday, 3.12. 2015 klo 16-18 C124
Torstai/Thursday, 19.11. 2015 klo 16-18 C124
Torstai/Thursday, 12.11. 2015 klo 16-18 C124
Torstai/Thursday, 05.11. 2015 klo 16-18 C124
On the uniqueness of the multiplicative Gaussian chaos
Torstai/Thursday, 29.10. 2015 klo 16-18 C124
Torstai/Thursday, 22.10. 2015 klo 16-18 C124
Optimal Berry-Esseen bounds on the Poisson space
Torstai/Thursday, 8.10. 2015 klo 16-18 C124
Divergent Fourier series [ ]
Torstai/Thursday, 1.10. 2015 klo 16-18 C124
Aiempia esitelmiä kaudella 2014-15
Torstai/Thursday, 28.5.2015 klo 15-17 C129
Torstai/Thursday, 30.4.2015 klo 14-16 C129 (HUOM aika!)
Torstai/Thursday, 9.4.2015 klo 15-17 C129
The Candès-Tao Theory of Compressed Sensing: Essentials, Strategies, Systems, ... , and Proofs
The last decade has seen the growth of a new theory called "compressed sensing". Due to this theory it is now well-known that one can reconstruct sparse or compressible signals accurately from a very limited number of measurements, possibly contaminated with noise. This lecture follows the progress of Candès and Tao to state the essentials of compressed sensing. Recent efforts from industry to implement this theory to massive data acquisition systems will be also covered. In the last part, the lecture reformulates the "surprising" classical recovery strategy of compressed sensing in terms of sparsity and entropy.
Torstai/Thursday, 19.3.2015 klo 15-17 C129
Torstai/Thursday, 5.3.2015 klo 15-17 C129
ABSTRACT: The class of planar quadrangulations is an important subclass of the class of planar maps. In addition to their nice graph theoretical properties they also can be seen as discretizations of the two dimensional sphere, for instance. The latter property is a key observation in the theory of 2d quantum gravity, first conjectured by physicists and lately made rigorous by mathematicians.In this presentation I will show the construction of a bijection between the set of labelled plane trees with n vertices and an additional coding parameter and the set of rooted and pointed planar quadrangulations with n faces. The construction was first done by R. Cori and B. Vauquelin in 1981 and later simplified by G. Schaeffer. Thus the bijection is called the CVS bijection.
The CVS bijection provides a nice way to compute the number of rooted quadrangulations with n faces. This is a result of W.T. Tutte from 1963 obtained by a different method. Second, the bijection allows us to deal with the metric properties of a quadrangulation by the means of the labelling of the corresponding tree. This is needed in analysing the scaling limit of uniformly distributed random quadrangulations. Time permitting, I will briefly introduce these properties.
Torstai/Thursday, 26.2.2015 . klo 15-17 C129
Eric Moulines and Alain Durmus (Paris Telecom)
Moulines: "Subgeometric rates of convergence in Wasserstein distance for Markov chains"
Durmus: "New Langevin-based Metropolis algorithm "
ABSTRACTS: (Moulines) We provide sufficient conditions for the existence of the invariant distribution and for subgeometric rates of convergence in Wasserstein distance for general state-space Markov chains which are (possibly) not irreducible. Compared to (Butkovsky, 2013, AoAP) our approach is based on a purely probabilistic coupling construction which allows to retrieve rates of convergence matching those previously reported for convergence in total variation in (Douc, M., Soulier, 2007). Our results are applied to establish the subgeometric ergodicity in Wasserstein distance of non-linear autoregressive models and of the pre-conditioned Crank-Nicolson Markov chain Monte Carlo algorithm in Hilbert space.
(Durmus) The Metropolis Hastings algorithm provides a generic and efficient way to sample from a given distribution. Since the pioneering result of (Roberts, Gelman and Gilks 97'), there is an extensive work on optimal scaling of various Metropolis Hastings algorithms in different contexts. The interest of such results is to obtain an asymptotic number of steps needed by the algorithm to explore the state space in function of the dimension. The fewer this number is the better the algorithm is.
Maybe the most well known results are concerning the optimal scaling of the Random Walk Metropolis algorithm and the Metropolis Adjusted Langevin algorithm. Whereas the first needs O(d) steps to explore the space, where d is the dimension, the second on only needs O(d^(1/3)).We will present in this talk a new class of proposals for the Metropolis Hastings algorithm which leads to an exploration in O(d^(1/5)) number of steps. Also, in a second part, we will expose positive and negative ergodicity results for these new algorithms. Finally if time permits, we will discuss about practical solutions to reduce the computational cost of these algorithms and their link with the Hamiltoninan Monte Carlo algorithm. It is a joint work with Gareth O. Roberts, Gilles Vilmart, and Konstantinos Zygalakis.
Torstai/Thursday, 5.2.2015 ja 19.2. klo 15-17 C129
Aldous' continuum random tree, part I and II
Torstai/Thursday, 29.1.2015 klo 15-17 C129
A Multivariate Hill Estimator
ABSTRACT: We propose an estimator of the tail index of regularly varying elliptical random vector. The estimator is based on the distance between a tail probability contour and the observations outside of this contour. The proposed estimator is a multivariate extension of the well-known univariate Hill estimator. We provide the multivariate estimator, we consider its asymptotical properties, and we present large simulation study. We illustrate the practical use of the new multivariate estimator by a real data example.
Torstai/Thursday, 22.1.2015 klo 15-17 C129
Lorentz Gas Problem: Faint Echos of a Proof
Torstai/Thursday, 11.12.2014 klo 15-17 C129
Algorithm Based Fault Tolerant Computation
Torstai/Thursday, 27.11.2014 klo 17.00-18.00 C129 (Huom poikk. aika!)
Ehsan Azmoodeh (Luxenburg)
The fourth moment theorem: a revisit and generalization
Torstai/Thursday, 20.11.2014 klo 15-17 C129
Introduction to Malliavin Calculus in the Poisson space (part II)
Torstai/Thursday, 13.11.2014 klo 15-17 C129
Introduction to Malliavin Calculus in the Poisson space (part I)
Torstai/Thursday, 6.11.2014 klo 15-17 C129
Improving extreme value analysis
Torstai/Thursday, 23.10.2014 klo 15-17 C129
On the correctness of the Regular decomposition algorithm of graph compression
Torstai/Thursday, 9.10.2014 klo 15-17 C129
Random variables on finitely additive measures
Torstai/Thursday, 2.10.2014 klo 15-17 C129
On the absolute continuity of finitely additive measures
Torstai/Thursday, 25.9.2014 klo 15-17 C129
Proving ergodicity with help of zombies
Torstai/Thursday, 18.9.2014 klo 15-17 C129
The renewal theorem revisited: an analytical approach
Esitelmiä kaudella 2013-14
Torstai/Thursday, 8.5.2014 klo 15-17 C129
Introduction to Kahane's multiplicative chaos
Torstai/Thursday, 3.4.2014 klo 15-17 C129
Simo Särkkä (Aalto)
Theory and Practice of Particle Filtering for State Space Models Bayesian networks
Torstai/Thursday, 27.3.2014 klo 15-17 C129
Learning Bayesian networks by Minimum Description Length
Torstai/Thursday, 20.3.2014 klo 15-17 C129
The Gaussian Correlation Conjecture Proof
Torstai/Thursday, 27.2.2014 klo 15-17 C129
Large graph or matrix compression method inspired by Szemerédi's Regularity Lemma and Rissanen's Minimum Description Length Principle
Torstai/Thursday, 13.2.2014 klo 15-17 C129
A local limit theorem for random walks in balanced environments
Torstai/Thursday, 6.2.2014 klo 15-17 C129
Tensor trains and high-dimensional integration
Torstai/Thursday, 30.1.2014 klo 15-17 C129
Stationary Apollonian packings
Comparability, monotonicity and asymptotic properties of typical distances in connected random geometric graphs
Torstai/Thursday, 23.1.2014 klo 15-17 C129
On stochastic Moduli of Quadrilaterals
Torstai/Thursday, 5.12.2013 klo 15-17 CK111
Data Stream Model
Torstai/Thursday, 28.11.2013 klo 15-17 CK111
Torstai/Thursday, 31.10., 7.11., 14.11. klo 15-17 CK111
Combinatorics in free probability I-III
Torstai/Thursday, 17.10.2013 klo 15-17 CK111
Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms
Torstai/Thursday, 3.10.2013 klo 15-17 CK111
Torstai/Thursday, 26.9.2013 klo 15-17 CK111
Torstai/Thursday, 19.9.2013 klo 15-17 CK111
Torstai/Thursday, 29.8.2013 klo 15-17 C129