Generalized Principal Component Analysis
Data segmentation is usually though of as a "chicken-and-egg" problem. In order to estimate a mixture of models one needs to first segment the data and in order to segment the data one needs to know the model parameters. Therefore, data segmentation is usually solved in two stages (1) data clustering and (2) model fitting, or else iteratively using, e.g. the Expectation Maximization (EM) algorithm.
This talk will show that for a wide class of segmentation problems (eigenvector segmentation, mixtures of subspaces, mixtures of fundamental matrices/trifocal tensors, mixtures of linear dynamical models), the "chicken-and-egg" dilemma can be tackled using algebraic geometric techniques. In fact, it is possible to eliminate the data segmentation step algebraically and then use all the data to recover all the models without previously segmenting the data. The solution can be obtained in closed form using linear algebraic techniques, and is a natural extension of classical PCA from one to multiple subspaces.
The talk will also include several applications of GPCA to computer vision problems such as image/video segmentation, 3-D motion segmentation, and dynamic texture segmentation.