• Speaker: Arvind Krishna Saibaba, North Carolina State University
• Title: Randomized algorithms for low-rank tensor decompositions
• Date: 10/11/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Many applications in data science and scientific computing require the working with large-scale datasets that are expensive to store and manipulate. These datasets have inherent multidimensional structure that can be exploited in order to efficiently compress and store them in an appropriate tensor format. In recent years, randomized matrix methods have been used to efficiently and accurately compute low-rank matrix decompositions. Motivated by this success, we develop several randomized algorithms for compressing tensor datasets in the Tucker format. We present probabilistic error analysis for our algorithms and numerical results on several datasets: synthetic test tensors, and realistic applications including the compression of facial image samples in the Olivetti database, and word counts in the Enron email dataset. Joint work with Rachel Minster (NC State) and Misha Kilmer (Tufts)

• Speaker: Kritika Singhal, Ohio State University
• Title: Sketching and Clustering Metric Measure Spaces
• Date: 11/08/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Two important optimization problems in the analysis of geometric data sets are clustering and simplification (sketching) of data. Clustering refers to partitioning a dataset, according to some rule, into sets of smaller size with the aim of extracting important information from the data. Sketching, or simplification of data, refers to approximating the input data with another dataset of much smaller size in such a way that properties of the input dataset are retained by the smaller dataset. In this sense, sketching facilitates understanding of the data. There are many clustering methods for metric spaces (mm spaces) already present in literature, such as k-center clustering, k-median clustering, k-means clustering, etc. A natural method for obtaining a k-sketch of a metric space (mm space) is by viewing the space of all metric spaces (mm space) as a metric under Gromov-Hausdorff (Gromov-Wasserstein) distance, and then determining, under this distance, the k point metric space (mm space) closest to the input metric space (mm space). These two problems of sketching and clustering, a priori, look completely unrelated. However, we establish a duality i.e. an equivalence between these notions of sketching and clustering. For metric spaces, we consider the case where the clustering objective is minimizing the maximum cluster diameter. We show that the ratio between the sketching and clustering objectives is constant over compact metric spaces. We extend these results to the setting of metric measure spaces where we prove that the ratio of sketching to clustering objectives is bounded both above and below by some universal constants. In this setting, the clustering objective involves minimizing various notions of the $\ell_p$-diameters of the clusters. We also identify procedures/maps that transform a solution of the sketching problem to a solution of the clustering problem, and vice-versa. These maps give rise to algorithms for performing these transformations and, by virtue of these algorithms, we are able to obtain an approximation to the k-sketch of a metric measure space (metric space) using known approximation algorithms for the respective clustering objectives. This is joint work with Facundo Memoli and Anastasios Sidiropoulos, and is available online at https://arxiv.org/abs/1801.00551.

• Speaker: Yeonhyang Kim, Central Michigan University
• Title: Q-ball imaging using predicted diffusion gradient directions
• Date: 11/22/2019
• Time: 4:10 PM - 5:00 PM
• Place: A203 Wells Hall

We review the theory of q-ball imaging and describe a simple linear matrix formulation for the q-ball reconstruction based on spherical harmonic basis function interpolation. In this talk, we present a novel method to improve the q-ball reconstruction by determining appropriate positions at which a synthesized signal is obtained by combining surrounding signals in given diffusion gradient directions.

• Speaker: Neel Patel, University of Michigan
• Title: TBA
• Date: 12/06/2019
• Time: 4:10 PM - 5:00 PM
• Place: A203 Wells Hall

No abstract available.