Seminars and Colloquia by Series

Non-Asymptotic Bounds for Prediction Problems and Density Estimation

Series
Dissertation Defense
Time
Tuesday, May 1, 2012 - 15:00 for 2 hours
Location
Skiles 005
Speaker
Stanislav Minsker – School of Mathematics, Georgia Tech
This dissertation investigates the statistical learning scenarios where a high-dimensional parameter has to be estimated from a given sample of fixed size, often smaller than the dimension of the problem. The first part answers some open questions for the binary classification problem in the framework of active learning. Given a random couple (X,Y)\in R^d\times {\pm 1} with unknown distribution P, the goal of binary classification is to predict a label Y based on the observation X. The prediction rule is constructed based on the observations (X_i,Y_i)_{i=1}^n sampled from P. The concept of active learning can be informally characterized as follows: on every iteration, the algorithm is allowed to request a label Y for any instance X which it considers to be the most informative. The contribution of this work consists of two parts: first, we provide the minimax lower bounds for performance of the active learning methods under certain assumptions. Second, we propose an active learning algorithm which attains nearly optimal rates over a broad class of underlying distributions and is adaptive with respect to the unknown parameters of the problem. The second part of this work is related to sparse recovery in the framework of dictionary learning. Let (X,Y) be a random couple with unknown distribution P, with X taking its values in some metric space S and Y - in a bounded subset of R. Given a collection of functions H={h_t}_{t\in \mb T} mapping S to R, the goal of dictionary learning is to construct a prediction rule for Y given by a linear (or convex) combination of the elements of H. The problem is sparse if there exists a good prediction rule that depends on a small number of functions from H. We propose an estimator of the unknown optimal prediction rule based on penalized empirical risk minimization algorithm. We show that proposed estimator is able to take advantage of the possible sparse structure of the problem by providing probabilistic bounds for its performance. Finally, we provide similar bounds in the density estimation framework.

Kac's program in Kinetic Theory

Series
PDE Seminar
Time
Tuesday, May 1, 2012 - 10:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Clement Mouhot – University of Cambridge
Mark Kac proposed in 1956 a program for deriving the spatially homogeneous Boltzmann equation from a many-particle jump collision process. The goal was to justify in this context the molecular chaos, as well as the H-theorem on the relaxation to equilibrium. We give answers to several questions of Kac concerning the connexion between dissipativity of the many-particle process and the limit equation; we prove relaxation rates independent of the number of particles as well as the propagation of entropic chaos. This crucially relies on a new method for obtaining quantitative uniform in time estimates of propagation of chaos. This is a joint work with S. Mischler.

Stability for the relative isoperimetric inequality inside an open, convex cone

Series
Math Physics Seminar
Time
Monday, April 30, 2012 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Emanuel Indrei – University of Texas
The relative isoperimetric inequality inside an open, convex cone C states that under a volume constraint, the ball intersected the cone minimizes the perimeter inside C. In this talk, we will show how one can use optimal transport theory to obtain this inequality, and we will prove a corresponding sharp stability result. This is joint work with Alessio Figalli.

Discrete Mathematical Biology Working Seminar

Series
Other Talks
Time
Monday, April 30, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Martin Copenhaver – Georgia Tech
A discussion of the paper "Modeling and automation of sequencing-based characterization of RNA structure" by Aviran et al (PNAS, 2011).

Southeast Geometry Seminar

Series
Other Talks
Time
Sunday, April 29, 2012 - 08:30 for 8 hours (full day)
Location
Skiles 005
Speaker
Southeast Geometry Seminar – School of Mathematics, Georgia Tech

Please Note: The general public lecture will be presented by Jason Cantarella (University of Georgia) entitled The Square Peg Theorems or What does it mean to solve simultaneous equations? to take place in Klaus 1116 at 5:00PM

The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions: The University of Alabama at Birmingham;  The Georgia Institute of Technology;  Emory University;  The University of Tennessee Knoxville.  The following five speakers will give presentations on topics that include geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology. Jason Cantarella (University of Georgia);   Meredith Casey (The Georga Institute of Technology);  Kirk Lancaster (Wichita State University); Junfang Li ( University of Alabama at Birmingham)  Jason Parsley (Wake Forest University);

Towards Sarkozy's Problem

Series
Combinatorics Seminar
Time
Friday, April 27, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ernie Croot – School of Math, Ga Tech –
Sarkozy's problem is a classical problem in additive number theory, which asks for the size of the largest subset A of {1,2,...,n} such that the difference set A-A does not contain a (non-zero) square. I will discuss the history of this problem, some recent progress that I and several collaborators have made on it, and our future research plans.

5-List-Coloring Graphs on Surfaces

Series
ACO Student Seminar
Time
Friday, April 27, 2012 - 13:00 for 1 hour (actually 50 minutes)
Location
Executive classroom, ISyE Main Building
Speaker
Luke Postle – School of Math., Georgia Tech
Thomassen proved that there are only finitely many 6-critical graphs embeddable on a fixed surface. He also showed that planar graphs are 5-list-colorable. We develop new techniques to prove a general theorem for 5-list-coloring graphs embedded on a fixed surface. Namely, for every surface S and every integer C > 0, there exists D such that the following holds: Let G be a graph embedded in a surface S with edge-width at least D and a list assignment L such that, for every vertex v in G, L(v) has size at least five. If F is a collection of any number of facial cycles of length at most C such that every two cycles in F are distance at least D apart and every cycle in F has a locally planar neighborhood up to distance D/2, then any proper L-coloring of F extends to an L-coloring of G. This theorem implies the following results. In what follows, let S be a fixed surface, G be a graph embedded in S (except in 4, where G is drawn in S) and L a list assignment such that, for every vertex v of G, L(v) has size at least five. 1. If G has large edge-width, then G is 5-list-colorable. (Devos, Kawarabayashi and Mohar) 2. There exists only finitely many 6-list-critical graphs embeddable in S. (Conjectured by Thomassen, Proof announced by Kawarabayashi and Mohar) As a corollary, there exists a linear-time algorithm for deciding 5-list-colorability of graphs embeddable on S. Furthermore, we exhibit an explicit bound on the size of such graphs. 3. There exists D(S) such that the following holds: If X is a subset of the vertices of G that are pairwise distance at least D(S) apart, then any L-coloring of X extends to an L-coloring G. For planar graphs, this was conjectured by Albertson and recently proved by Dvorak, Lidicky, Mohar, and Postle. 4. There exists D(S) such that the following holds: If G is a graph drawn in S with face-width at least D(S) such that any pair of crossings is distance at least D apart, then G is L-colorable. For planar graphs, this was recently proved by Dvorak, Lidicky and Mohar. Joint work with Robin Thomas.

Pages