Pixel Club: Sparse Matrix Models for Finding Good Data Representatives and Constraining the Topology of Networks

Guillermo Sapiro (Duke University)
Tuesday, 25.12.2012, 11:30
EE Meyer Building 1003

We consider the problem of finding a few representatives for a dataset, i.e., a subset of data points that efficiently describes the entire dataset. We assume that each data point can be expressed as a linear combination of the representatives and formulate the problem of finding the representatives as a sparse multiple measurement vector problem. In our formulation, both the dictionary and the measurements are given by the data matrix, and the unknown sparse codes select the representatives via convex optimization. In general, we do not assume that the data are lowrank or distributed around cluster centers. When the data do come from a collection of low-rank models, we show that our method automatically selects a few representatives from each low-rank model. We also analyze the geometry of the representatives and discuss their relationship to the vertices of the convex hull of the data. We show that our framework can be extended to detect and reject outliers in datasets, and to efficiently deal with new observations and large datasets. The proposed framework and theoretical foundations are illustrated with examples in video summarization and image classification using representatives​. Finally, we discuss how to extend this when the data is given as pairwise distances. This is joint work with E. Elhamifar and R. Vidal

Similar mathematical foundations based on sparse modeling lead to the computation of networks and estimation of inverse covariances. I will conclude the talk with some recent results on adding topological constraints to these computations.

This is joint work with M. Fiori and P. Muse.

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