**Organizers**

Rene Vidal (Johns Hopkins University, USA)

Yi Ma (University of Illinois, Urbana-Champaign, USA)

**Additional Participants**

Allen Yang (University of California, Berkeley, USA)

**Description**

Over the past two decades, we have seen tremendous advances on the simultaneous segmentation and estimation of a collection of models from sample data points, without knowing which points correspond to which model. These advances have been motivated and constantly driven by numerous potential applications in hybrid system identification, computer vision, image processing, systems theory, robotics, and more recently, also in biological systems.

Most existing hybrid model identification methods treat the data segmentation problem as "chicken-and-egg problem". This is because in order to estimate a mixture of models one needs to first segment the data. Conversely, 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 tutorial will show that for a wide variety of hybrid model identification problems (e.g. mixtures of subspaces, mixtures of rigid-body motions, mixtures of linear dynamical models), the "chicken-and-egg" dilemma can be tackled using an algebraic geometric technique called Generalized Principal Component Analysis (GPCA). The main idea behind GPCA is to eliminate the data segmentation step algebraically and then use all the data to recover all the models without previously segmenting the data as follows:

1. Fit a set of polynomials to all data points, without clustering the data.

2. Obtain the model parameters for each group from the derivatives of these polynomials.

The workshop will include several applications of GPCA to hybrid system identification and computer vision problems such as image/video segmentation, 3-D motion segmentation, and dynamic texture segmentation.

**List of topics**

I Introduction to Generalized Principal Component Analysis, 8.30-8.45am
(slides)

II Basic GPCA Theory and Algorithms, 8.45-9.45am
(slides)

- Review of Principal Component Analysis (PCA)
- Introductory Cases: Line, Plane and Hyperplane Segmentation
- Segmentation with Known Number of Subspaces
- Segmentation with Unknown Number of Subspaces

COFFEE BREAK, 10.00-10.30

III Advanced Statistical and Algebraic Methods for GPCA, 10.30-11.15am
(slides)

- Model Selection for Subspace Arrangements
- Robust Sampling Techniques for Subspace Segmentation
- Voting Techniques for Subspace Segmentation

IV Applications to Hybrid System Identification, 11.15-12.00 noon
(slides)

- Batch Identification of Switched ARX Models in Input-Output Form
- Recursive Identification of Switched ARX Models in Input-Output Form

LUNCH BREAK, 12.00 noon - 1.30pm

V Applications to Motion and Video Segmentation, 1.30-2.15pm
(slides)

- 2-D and 3-D Motion Segmentation
Representation
- Temporal Video Segmentation
- Segmentation of Dynamic Textures

VI Applications to Image Representation and Segmentation, 2.15-3.00pm
(slides)

- Multi-Scale Hybrid Linear Models for Sparse Image Representation
- Multi-Scale Hybrid Linear Models in Wavelet Domain
- Hybrid Linear Models for Image Segmentation