TALK - Candidate for Probability Search
Area: MCMC, random walks on graphs, randomized algorithms, mixing times for Markov chains... | |
Abstract: Part I: We introduce a probabilistic technique that yields the sharpest bounds obtained on mixing times for Markov chains in terms of isoperimetric properties of the state space. We show that the bounds for mixing time in total variation obtained by Lovasz and Kannan can be refined to apply to the maximum relative deviation $|p^n(x,y)/\pi(y) -1|$ of the distribution at time $n$ from the stationary distribution $\pi$. This is joint work with Yuval Peres.
Part II: We show that the mixing time for random walk on the $n$-dimensional cube truncated by a hyperplane is polynomial in $n$. As a consequence, we obtain a fully-polynomial randomized approximation scheme for counting the feasible solutions of a 0-1 knapsack problem. This is joint work with Alistair Sinclair.
Brown University Center for Statistical Sciences Seminar
Abstract: A class of two-stage group sequential bioequivalence design is developed here using Bayesian decision framework. The resulting stopping rule implements user-specified relative utility of declaring equivalence compared to an unsuccessful continuation to stage-2. All these scenarios are developed while maintaining the nominal error rate protection. Numerical examples are used to illustrate many practical situations where such decision rules are optimal with a negligible impact on overall power.
TALK - Candidate for Probability Search
Area: Machine Learning | |
Abstract: Most of the study of Bayesian prediction procedures is premised on some strong assumptions regarding the prior distribution. In particular, an assumption that always needs to be made is that the prior distribution assigns a non-zero probability to the correct model. In practice, however, we often have to restrict the set of models (in the support of the prior) in order to make it feasable to compute the posterior average. As a result, we often can't assume that the correct model is in our set and the standard Bayesian theory cannot be applied.
In this work we show a classification procedure which uses model averaging and can be interpreted as a Bayesian procedure. We show that this procedure has some desirable properties when it is applied to *any* source of IID examples, regardless of whether or not the source distribution is related to the models in the support of the prior. Our main result is that the predictions made by this procedure are very stable with regard to the choice of the random training set.
This stability property has some far-reaching implications on a variety of issues, including Bias-Variance decomposition of classification error, the "curse of dimensionality" and Bagging.
This is joint work with Yishay Mansour and Rob Schapire
Brown University Joint Materials/Solid Mechanics Seminar Series
Yale University, New Haven, CT 06520, USA | |
Abstract: The ability of the scanning force microscope to study the nanomechanics of materials down to the atomic scale is illustrated with two different examples. First, I will show how data acquired during the sliding of nanometer-sized tip-sample contacts provide new insight into the atomic origins of friction, which lead to a deeper understanding of Amontons? and Coulomb?s phenomenological laws of friction. Interestingly, friction is on the nanometer scale proportional to the contact area, in opposition to the corresponding macroscopic law, whereas the macroscopically observed independence of friction from the sliding velocity remains approximately valid.
In the second part of the talk, I will present an experimental set-up optimized for the operation in the so-called dynamic mode of scanning force microscopy at low temperatures and in ultrahigh vacuum. The instrument achieves a resolution comparable to the one of low-temperature scanning tunneling microscopes, but yields complementary information. Moreover, it provides atomic resolution even on insulating materials, which can otherwise not be investigated. In particular, I will introduce a method that allows the continuous measurement of the tip-sample interaction potential. Comparison of such data with model potentials gives access to a profound analysis of the interactions at and between surfaces, including nano-elastic and nano-contact mechanical effects.
Brown Applied Mathematics Pattern Theory and Vision Seminar
University of Massachusetts at Amherst | |
Abstract: The sequential nature of music makes graphical models well-suited for many musical applications. I will discuss two of these. The first is "rhythmic parsing" in which, given a sequence of note onset times, one infers the associated notated rhythm and the time-varying tempo process. The observed onset times are expressed in terms of two hidden processes: a discrete rhythm process, represented by a Markov chain, and a conditionally Gaussian tempo process.
I will present methodology for computing the globally most likely configuration of tempo and rhythm processes through a dynamic-programming-like technique. The second example is my musical accompaniment system in which a computer plays the role of a musical accompanist. I will give an overview of this work with a number of examples and present a live demonstration.
Special Stochastic Systems Seminar
University of North Carolina at Chapel Hill | |
Abstract: In this work we consider an ergodic control problem for a class of diffusion processes, constrained to take values in a polyhedral cone. The goal is the almost sure minimization of long term cost per unit time. We show that under the assumption of regularity of the Skorohod map and the assumption that the drift vector field takes values in a certain cone of stability, the class of controlled diffusion processes considered have strong, uniform in control, stability properties. Once these properties are available the remaining work lies in identifying weak limits of a certain family of occupation measures. In this regard an extension to the Echeverria-Weiss-Kurtz characterization of invariant measures of Markov processes, to the case of constrained- controlled processes considered in this talk, is proved. Next we characterize the value of the ergodic control problem via a suitable Hamilton-Jacobi-Bellman (HJB) equation. We show that the natural HJB equation for the ergodic control problem admits a unique continuous viscosity solution which enables us to characterize the value function of the control problem. The existence of a solution to this HJB equation is established via the classical vanishing discount argument. The key step is proving the pre-compactness of the family of suitably re-normalized discounted value functions. In this regard we use a recent technique, introduced by Borkar, of using the Athreya-Ney-Nummelin pseudo-atom construction for obtaining a coupling of a pair of embedded, discrete time, controlled Markov chains.
PDE Seminar
Abstract: This is the first part of our Water Wave Bash. It will continue with the Math Colloquium talk at 4:30.
Department of Mathematics Colloquium
Abstract: Consider classical inviscid water waves with non-trivial vorticity, with a free surface under the influence of gravity over a flat bottom. They are modeled by the two-dimensional Euler equations with a nonlinear boundary condition. We prove that there exist global continua of periodic traveling waves. The waves are symmetric with monotone profiles between each crest and trough. The continua extend from flat waves to waves that experience either stagnation (regions of stagnant water) or cavitation (formation of bubbles). Thus there exist many rotational periodic traveling waves of large amplitude. The proof makes use of bifurcation theory, Leray-Schauder degree, and the theory of elliptic PDE.
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