Graduate Student Seminar

Speaker 1: Teresa Lebair

Title

The L-infinity-norm of the Spline Projector is Bounded Independently of the Knot Sequence

Abstract

In 1972, Carl de Boor conjectured that for a fixed spline order, the orthogonal projection onto the space of splines was bounded in the L-infinity norm, independent of the B-spline knot sequence.  Twenty-seven years later, Alexi Yu Shardin gave a proof of de Boor's conjecture.  In this presentation, we will explore the history of this famous question in approximation theory, and consider some applications and ramifications of Shardin's result.  Most of the presentation will be devoted to important background information on B-splines and the projection onto the space of splines.  A very brief sketch of Shardin's proof will also be discussed.

Speaker 2: Maria Barouti

Title

Adaptive Clustering for Monitoring Distributed Data Streams

Abstract

Clustering is a task of exploratory data mining that groups a set of objects. There are two types of clustering algorithms, partitional and hierarchical. Some desirable properties of clustering algorithms are scalability as well as the ability to deal with different data types.  Assuming that we have a set of vectors changing with time and we want to monitor their mean without computing it. Our goal is to reduce communication load by clustering. Therefore, clustering can be formulated as an optimization problem by partitioning this set into clusters, so that the norm of the cluster’s mean is minimized. Now the question is which vectors should be clustered together, so that the mean of the cluster is bounded. A correct choice of clusters yields a reduction in communication load. Unlikely many clustering algorithms that attempt to collect together similar data items, monitoring requires clusters with dissimilar vectors canceling each other as much as possible. Finally, even if clustering is no universal remedy, there are cases where clustering dissimilar vectors helps.