Monday, February 5, 2018 - 13:55 , Location: Skiles 005 , Mark A. Davenport , Georgia Institute of Technology , Organizer: Wenjing Liao
The discrete prolate spheroidal sequences (DPSS's) provide an efficient representation for discrete signals that are perfectly timelimited and nearly bandlimited. Due to the high computational complexity of projecting onto the DPSS basis - also known as the Slepian basis - this representation is often overlooked in favor of the fast Fourier transform (FFT). In this talk I will describe novel fast algorithms for computing approximate projections onto the leading Slepian basis elements with a complexity comparable to the FFT. I will also highlight applications of this Fast Slepian Transform in the context of compressive sensing and processing of sampled multiband signals.
Monday, February 5, 2018 - 11:15 , Location: Skiles 005 , Renato Iturriaga , CIMAT , Organizer: Livia Corsi
We present a discrete setting for the viscous Hamilton Jacobi equation, and prove convergence to the continuous case.
Monday, February 5, 2018 - 10:10 , Location: skiles 005 , Riccardo Montalto , University of Zurich , Organizer: Livia Corsi
In this talk I will present some results concerning the existence and the stability of quasi-periodic solutions for quasi-linear and fully nonlinear PDEs. In particular, I will focus on the Water waves equation. The proof is based on a Nash-moser iterative scheme and on the reduction to constant coefficients of the linearized PDE at any approximate solution. Due to the non-local nature of the water waves equation, such a reduction procedure is achieved by using techniques from Harmonic Analysis and microlocal analysis, like Fourier integral operators and Pseudo differential operators.
Series: Research Horizons Seminar
Some basic problems, notions and results of the Ergodic theory will be introduced. Several examples will be discussed. It is also a preparatory talk for the next day colloquium where finite time properties of dynamical and stochastic systems will be discussed rather than traditional questions all dealing with asymptotic in time properties.
Friday, February 2, 2018 - 15:00 , Location: Skiles 271 , Gladston Duarte , University of Barcelona & GT , email@example.com , Organizer: Jiaqi Yang
In a given system of coordinates, the Restricted Three-Body Problem has some interesting dynamical objects, for instance, equilibrium points, periodic orbits, etc. In this work, some connections between the stable and unstable manifolds of periodic orbits of this system are studied. Such connections let one explain the movement of Quasi-Hilda comets, which describe an orbit that sometimes can be approximated by an ellipse of semi-major axis greater than Jupiter's one, sometimes smaller. Using a computer algebra system, one can compute an approximation to those orbits and its manifolds and investigate the above mentioned connections. In addition, the Planar Circular model is used as a base for the fitting of the orbit of comet 39P/Oterma, whose data were collected from the JPL Horizons system. The possibility of using other models is also discussed.
Series: ACO Student Seminar
Physical sensors (thermal, light, motion, etc.) are becoming ubiquitous and offer important benefits to society. However, allowing sensors into our private spaces has resulted in considerable privacy concerns. Differential privacy has been developed to help alleviate these privacy concerns. In this talk, we’ll develop and define a framework for releasing physical data that preserves both utility and provides privacy. Our notion of closeness of physical data will be defined via the Earth Mover Distance and we’ll discuss the implications of this choice. Physical data, such as temperature distributions, are often only accessible to us via a linear transformation of the data. We’ll analyse the implications of our privacy definition for linear inverse problems, focusing on those that are traditionally considered to be "ill-conditioned”. We’ll then instantiate our framework with the heat kernel on graphs and discuss how the privacy parameter relates to the connectivity of the graph. Our work indicates that it is possible to produce locally private sensor measurements that both keep the exact locations of the heat sources private and permit recovery of the ``general geographic vicinity'' of the sources. Joint work with Anna C. Gilbert.
Friday, February 2, 2018 - 10:10 , Location: Skiles 254 , Marc Härkönen , Georgia Tech , firstname.lastname@example.org , Organizer: Kisun Lee
Differential operator rings can be described as polynomial rings over differential operators. We will study two of them: first the relatively simple ring of differential operators R with rational function coefficients, and then the more complicated ring D with polynomial coefficients, or the Weyl algebra. It turns out that these rings are non-commutative because of the way differential operators act on smooth functions. Despite this, with a bit of work we can show properties similar to the regular polynomial rings, such as division, the existence of Gröbner bases, and Macaulay's theorem. As an example application, we will describe the holonomic gradient descent algorithm, and show how it can be used to efficiently solve computationally heavy problems in statistics.
Series: Job Candidate Talk
The mean field variational inference is widely used in statistics and machine learning to approximate posterior distributions. Despite its popularity, there exist remarkably little fundamental theoretical justifications. The success of variational inference mainly lies in its iterative algorithm, which, to the best of our knowledge, has never been investigated for any high-dimensional or complex model. In this talk, we establish computational and statistical guarantees of mean field variational inference. Using community detection problem as a test case, we show that its iterative algorithm has a linear convergence to the optimal statistical accuracy within log n iterations. We are optimistic to go beyond community detection and to understand mean field under a general class of latent variable models. In addition, the technique we develop can be extended to analyzing Expectation-maximization and Gibbs sampler.
Wednesday, January 31, 2018 - 13:55 , Location: Skiles 006 , Sudipta Kolay , GaTech , Organizer: Anubhav Mukherjee
The Jordan curve theorem states that any simple closed curve decomposes the 2-sphere into two connected components and is their common boundary. Schönflies strengthened this result by showing that the closure of either connected component in the 2-sphere is a 2-cell. While the first statement is true in higher dimensions, the latter is not. However under the additional hypothesis of locally flatness, the closure of either connected component is an n-cell. This result is called the Generalized Schönflies theorem, and was proved independently by Morton Brown and Barry Mazur. In this talk, I will describe the proof of due to Morton Brown.
Series: Analysis Seminar
In the recent years, a number of conjectures has appeared, concerning the improvement of the inequalities of Brunn-Minkowski type under the additional assumptions of symmetry; this includes the B-conjecture, the Gardner-Zvavitch conjecture of 2008, the Log-Brunn-Minkowski conjecture of 2012, and some variants. The conjecture of Gardner and Zvavitch, also known as dimensional Brunn-Minkowski conjecture, states that even log-concave measures in R^n are in fact 1/n-concave with respect to the addition of symmetric convex sets. In this talk we shall establish the validity of the Gardner-Zvavitch conjecture asymptotically, and prove that the standard Gaussian measure enjoys 0.3/n concavity with respect to centered convex sets. Some improvements to the case of general log-concave measures shall be discussed as well. This is a joint work with A. Kolesnikov.