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Series: Stochastics Seminar

Series: Stochastics Seminar

Series: Stochastics Seminar

We study an online algorithm for making a well—equidistributed random set of points in an interval, in the spirit of "power of choice" methods. Suppose finitely many distinct points are placed on an interval in any arbitrary configuration. This configuration of points subdivides the circle into a finite number of intervals. At each time step, two points are sampled uniformly from the interval. Each of these points lands within some pair of intervals formed by the previous configuration. Add the point that falls in the larger interval to the existing configuration of points, discard the other, and then repeat this process. We then study this point configuration in the sense of its largest interval, and discuss other "power of choice" type modifications.
Joint work with Pascal Maillard.

Series: Stochastics Seminar

Series: Stochastics Seminar

The Sherrington-Kirkpatirck (SK) model is
a mean-field spin glass introduced by theoretical physicists in order
to explain the strange behavior of certain alloys, such as CuMn. Despite
of its seemingly simple formulation, it was conjectured to possess a
number of profound properties. This talk will be focused on the energy
landscapes of the SK model and the mixed p-spin model with both Ising
and spherical configuration spaces. We will present Parisi formule for
their maximal energies followed by descriptions of the energy landscapes
near the maximum energy. Based on joint works with A. Auffinger, M. Handschy, G. Lerman, and A. Sen.

Series: Stochastics Seminar

Series: Stochastics Seminar

When considering smooth functionals of dependent data, block bootstrap methods have enjoyed considerable success in theory and application. For nonsmooth functionals of dependent data, such as sample quantiles, the theory is less well-developed. In this talk, I will present a general theory of consistency and optimality, in terms of achieving the fastest convergence rate, for block bootstrap distribution estimation for sample quantiles under mild strong mixing assumptions. The case of density estimation will also be discussed. In contrast to existing results, we study the block bootstrap for varying numbers of blocks. This corresponds to a hybrid between the subsampling bootstrap and the moving block bootstrap (MBB). Examples of `time series’ models illustrate the benefits of optimally choosing the number of blocks. This is joint work with Stephen M.S. Lee (University of Hong Kong) and Alastair Young (Imperial College London).

Series: Stochastics Seminar

Series: Stochastics Seminar

We present a unified framework for sequential low-rank matrix completion and estimation, address the joint goals of uncertainty quantification (UQ) and statistical design. The first goal of UQ aims to provide a measure of uncertainty of estimated entries in the unknown low-rank matrix X, while the second goal of statistical design provides an informed sampling or measurement scheme for observing the entries in X. For UQ, we adopt a Bayesian approach and assume a singular matrix-variate Gaussian prior the low-rank matrix X which enjoys conjugacy. For design, we explore deterministic design from information-theoretic coding theory. The effectiveness of our proposed methodology is then illustrated on applications to collaborative filtering.

Series: Stochastics Seminar