Georgia Institute of Technology College of Sciences

One of the most fundamental steps in data analysis and dimensionality reduction consists of approximating a given dataset by a single low-dimensional subspace, which is classically achieved via Principal Component Analysis (PCA). However, in many applications, the data often lie near a union of low-dimensional subspaces, reflecting the multiple categories or classes a set of observations may belong to. In this talk we discuss the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes. Simply stated the task is to assign each data point to a cluster so as to recover all the hidden subspaces. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information about their dimensions. We present a novel geometric analysis of an algorithm named sparse subspace clustering (SSC), which significantly broadens the range of problems where it is provably effective. For instance, we show that SSC can recover multiple subspaces, each of dimension comparable to the ambient dimension. We also show that SSC can correctly cluster data points even when the subspaces of interest intersect. Further, we develop an extension of SSC that succeeds when the data set is corrupted with possibly overwhelmingly many outliers. Underlying our analysis are clear geometric insights, which may bear on other sparse recovery problems. We will also demonstrate the effectiveness of these methods by various numerical studies.

This work is concerned with the Almost Axisymmetric Flows with Forcing Terms which are derived from the inviscid Boussinesq equations. It is our hope that these flows will be useful in Meteorology to describe tropical cyclones. We show that these flows give rise to a collection of Monge-Ampere equations for which we prove an existence and uniqueness result. What makes these equations unusual is the boundary conditions they are expected to satisfy and the fact that the boundary is part of the unknown. Our study allows us to make inferences in a toy model of the Almost Axisymmetric Flows with Forcing Terms.

Slow Feature Analysis (SFA) is a method for extracting slowly varying features from input signals. In this talk, we generalize SFA to vector-valued functions of multivariables and apply it to the problem of blind source separation, in particular image separation. When the sources are correlated, we apply the following technique called decorrelation filtering: use a linear filter to decorrelate the sources and their derivatives, then apply the separating matrix obtained on the filtered sources to the original sources. We show that if the filtered sources are perfectly separated by this matrix, then so are the original sources.We show how to numerically obtain such a decorrelation filter by solving a nonlinear optimization problem. This technique can also be applied to other linear separation methods, whose output signals are uncorrelated, such as ICA.This is joint work with Laurenz Wiskott (Proceedings of the 13th IEEE International Conference in Computer Vision, ICCV 2011, Barcelona, Spain).