Seminars and Colloquia by Series

Distributionally robust demand forecasting and inventory control with martingale uncertainty sets

Series
Stochastics Seminar
Time
Thursday, January 19, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dave Goldberg – ISyE, GaTech
Demand forecasting plays an important role in many inventory control problems. To mitigate the potential harms of model misspecification, various forms of distributionally robust optimization have been applied. Although many of these methodologies suffer from the problem of time-inconsistency, the work of Klabjan et al. established a general time-consistent framework for such problems by connecting to the literature on robust Markov decision processes. Motivated by the fact that many forecasting models exhibit very special structure, as well as a desire to understand the impact of positing different dependency structures, in this talk we formulate and solve a time-consistent distributionally robust multi-stage newsvendor model which naturally unifies and robustifies several inventory models with demand forecasting. In particular, many simple models of demand forecasting have the feature that demand evolves as a martingale (i.e. expected demand tomorrow equals realized demand today). We consider a robust variant of such models, in which the sequence of future demands may be any martingale with given mean and support. Under such a model, past realizations of demand are naturally incorporated into the structure of the uncertainty set going forwards. We explicitly compute the minimax optimal policy (and worst-case distribution) in closed form, by combining ideas from convex analysis, probability, and dynamic programming. We prove that at optimality the worst-case demand distribution corresponds to the setting in which inventory may become obsolete at a random time, a scenario of practical interest. To gain further insight, we prove weak convergence (as the time horizon grows large) to a simple and intuitive process. We also compare to the analogous setting in which demand is independent across periods (analyzed previously by Shapiro), and identify interesting differences between these models, in the spirit of the price of correlations studied by Agrawal et al. This is joint work with Linwei Xin, and the paper is available on arxiv at https://arxiv.org/abs/1511.09437v1

Multiscale adaptive approximations to data and functions near low-dimensional sets

Series
Job Candidate Talk
Time
Thursday, January 19, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wenjing Liao – Johns Hopkins University –
High-dimensional data arise in many fields of contemporary science and introduce new challenges in statistical learning due to the well-known curse of dimensionality. Many data sets in image analysis and signal processing are in a high-dimensional space but exhibit a low-dimensional structure. We are interested in building efficient representations of these data for the purpose of compression and inference, and giving performance guarantees that are only cursed by the intrinsic dimension of data. Specifically, in the setting where a data set in $R^D$ consists of samples from a probability measure concentrated on or near an unknown $d$-dimensional manifold with $d$ much smaller than $D$, we consider two sets of problems: low-dimensional geometric approximation to the manifold and regression of a function on the manifold. In the first case we construct multiscale low-dimensional empirical approximations to the manifold and give finite-sample performance guarantees. In the second case we exploit these empirical geometric approximations of the manifold to construct multiscale approximations to the function. We prove finite-sample guarantees showing that we attain the same learning rates as if the function was defined on a Euclidean domain of dimension $d$. In both cases our approximations can adapt to the regularity of the manifold or the function even when this varies at different scales or locations. All algorithms have complexity $C n\log (n)$ where $n$ is the number of samples, and the constant $C$ is linear in $D$ and exponential in $d$.

Manifolds on the verge of breakdown

Series
School of Mathematics Colloquium
Time
Thursday, January 19, 2017 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex Haro – Univ. of Barcelona –
The long term behavior of dynamical systems can be understood by studying invariant manifolds that act as landmarks that anchor the orbits. It is important to understand which invariant manifolds persist under modifications of the system. A deep mathematical theory, developed in the 70's shows that invariant manifolds which persist under changes are those that have sharp expansion (in the future or in the past) in the the normal directions. A deep question is what happens in the boundary of these theorems of persistence. This question requires to understand the interplay between the geometric properties and the functional analysis of the functional equations involved.In this talk we present several mechanisms in which properties of normal hyperbolicity degenerate, so leading to the breakdown of the invariant manifold. Numerical studies lead to surprising conjectures relating the breakdown to phenomena in phase transitions. The results have been obtained combining numerical exploration and rigorous reasoning.

The HRT Conjecture

Series
Analysis Seminar
Time
Wednesday, January 18, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chris Heil – Georgia Tech
The Linear Independence of Time-Frequency Translates Conjecture, also known as the HRT conjecture, states that any finite set of time-frequency translates of a given $L^2$ function must be linearly independent. This conjecture, which was first stated in print in 1996, remains open today. We will discuss this conjecture, its relation to the Zero Divisor Conjecture in abstract algebra, and the (frustratingly few) partial results that are currently available.

Critical Percolation

Series
Research Horizons Seminar
Time
Wednesday, January 18, 2017 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael Damron – Georgia Institute of Technology
On the two-dimensional square grid, remove each nearest-neighbor edge independently with probability 1/2 and consider the graph induced by the remaining edges. What is the structure of its connected components? It is a famous theorem of Kesten that 1/2 is the ``critical value.'' In other words, if we remove edges with probability p \in [0,1], then for p < 1/2, there is an infinite component remaining, and for p > 1/2, there is no infinite component remaining. We will describe some of the differences in these phases in terms of crossings of large boxes: for p < 1/2, there are relatively straight crossings of large boxes, for p = 1/2, there are crossings, but they are very circuitous, and for p > 1/2, there are no crossings.

Variational inequalities and mean-field games

Series
PDE Seminar
Time
Tuesday, January 17, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Diogo Gomes – KAUST –
We consider stationary monotone mean-field games (MFGs) and study the existence of weak solutions. We introduce a regularized problem that preserves the monotonicity and prove the existence of solutions to the regularized problem. Next, using Minty's method, we establish the existence of solutions for the original MFGs. Finally, we examine the properties of these weak solutions in several examples.

Spectral Submanifolds and Exact Model Reduction for Nonlinear Beam Dynamics

Series
CDSNS Colloquium
Time
Friday, January 13, 2017 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Florian Kogelbauer – ETH (Zurich)
We use invariant manifold results on Banach spaces to conclude the existence of spectral submanifolds (SSMs) in a class of nonlinear, externally forced beam oscillations . Reduction of the governing PDE to the SSM provides an exact low-dimensional model which we compute explicitly. This model captures the correct asymptotics of the full, infinite-dimensional dynamics. Our approach is general enough to admit extensions to other types of continuum vibrations. The model-reduction procedure we employ also gives guidelines for a mathematically self-consistent modeling of damping in PDEs describing structural vibrations.

Computational Concerns in Statistical Inference and Learning for Network Data Analysis

Series
Job Candidate Talk
Time
Thursday, January 12, 2017 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tengyuan Liang – University of Pennsylvania –
Network data analysis has wide applications in computational social science, computational biology, online social media, and data visualization. For many of these network inference questions, the brute-force (yet statistically optimal) methods involve combinatorial optimization, which is computationally prohibitive when faced with large scale networks. Therefore, it is important to understand the effect on statistical inference when focusing on computationally tractable methods. In this talk, we will discuss three closely related statistical models for different network inference problems. These models answer inference questions on cliques, communities, and ties, respectively. For each particular model, we will describe the statistical model, propose new computationally efficient algorithms, and study the theoretical properties and numerical performance of the algorithms. Further, we will quantify the computational optimality through describing the intrinsic barrier for certain efficient algorithm classes, and investigate the computational-to-statistical gap theoretically. A key feature shared by our studies is that, as the parameters of the model changes, the problems exhibit different phases of computational difficulty.

Alpert multiwavelets and Legendre-Angelesco multiple orthogonal polynomials

Series
Analysis Seminar
Time
Wednesday, January 11, 2017 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Walter Van Assche – Katholieke University Lueven
We show that the multiwavelets, introduced by Alpert in 1993, are related to type I Legendre-Angelesco multiple orthogonal polynomials. We give explicit formulas for these Legendre-Angelesco polynomials and for the Alpert multiwavelets. The multiresolution analysis can be done entirely using Legendre polynomials, and we give an algorithm, using Cholesky factorization, to compute the multiwavelets and a method, using the Jacobi matrix for Legendre polynomials, to compute the matrices in the scaling relation for any size of the multiplicity of the multiwavelets.Based on joint work with J.S. Geronimo and P. Iliev

Pages