1. A New GPCA Algorithm for Clustering Subspaces by Fitting, Differentiating and Dividing Polynomials
Part I
1 Title
2 Motivation
3 Problem statement
4 Brief review of PCA
5 Extensions: PPA, NLPCA, ICA
6 Problem Statement and Prior Work
7 Our approach
Part II
8 Simple example 1
9 Simple example 2
10 Notation and embedding
11 Solution part1
12 Solution part2
13 Solution part3
14 Algorithm summary
15 Nonlinear case
Part III
16 Simulation results
17 2D Motion Segmentation
18 3D Motion Segmentation
19 Conclusions
René Vidal
Yi Ma
Jacopo Piazzi
Biomedical Engineering Johns Hopkins University
Elect. & Comp. Eng. University of Illinois
Mechanical Engineering Johns Hopkins University

2. Motivation: estimating multiple models
Geometry
Vanishing points
Segmentation
Intensity (black-white)
Texture
Motion (2-D, 3-D)
Scene (host-guest)
Recognition
Faces (Eigenfaces)
Man - Woman
Dynamic Textures
Water-steam
One of the reasons we are interested in GPCA is because there are various problems in computer vision that have to do with the simultaneous estimation of multiple models from visual data.
Consider for example segmenting an image into different regions based on intensity, texture of motion information. Consider also the recognition of various static and dynamic processes such as human faces or human gaits from visual data.
Although in this talk I will only consider the first class of problems, it turns out that, at least from a mathematical perspective, all the above problems can be converted into following generalization of principal component analysis, which we conveniently refer to as GPCA

3. Generalized Principal Component Analysis
Given points on multiple subspaces, identify
The number of subspaces and their dimensions
A basis for each subspace
The segmentation of the data points
“Chicken-and-egg” problem
Given segmentation, estimate subspaces
Given subspaces, segment the data
Prior work
Iterative algorithms: K-subspace (Ho et al. ’03), RANSAC, subspace selection and growing (Leonardis et al. ’02)
Probabilistic approaches learn the parameters of a mixture model using e.g. EM (Tipping-Bishop ‘99):
Initialization?
Geometric approaches: 2 planes in R3 (Shizawa-Maze ’91)
Factorization approaches: independent, orthogonal subspaces of equal dimension (Boult-Brown ‘91, Costeira-Kanade ‘98, Kanatani ’01)
Given a set of data points lying on a collection of linear subspaces, without knowing which point corresponds to which subspace and without even knowing the number of subspaces, estimate the number of subspaces, a basis for each subspace and the segmentation of the data.
As stated, the GPCA problem is one of those challenging CHICKEN-AND-EGG problems in the following sense. If we knew the segmentation of the data points, and hence the number of subspaces, then we could solve the problem by applying standard PCA to each group. On the other hand, if we had a basis for each subspace, we could easily assign each point to each cluster. The main challenge is that we neither know the segmentation of the data points nor a basis for each subspace, and the questions is how to estimate everything from the data only.
Previous geometric approaches to GPCA have been proposed in the context of motion segmentation, and under the assumption that the subspaces do NOT intersect. Such approaches are based on first clustering the data, using for example K-means or spectral clustering techniques, and then applying standard PCA to each group.
Statistical methods, on the other hand, model the data as a mixture model whose parameters are the mixing proportions and the basis for each subspace. The estimation of the mixture is a (usually non-convex) optimization problem that can be solved with various techniques. One of the most popular ones is the expectation maximization algorithm (EM) which alternates between the clustering of the data and the estimation of the subspaces.
Unfortunately, the convergence of such iterative algorithms depends on good initialization. At present, initialization is mostly done either at random, using another iterative algorithm such as K-means, or else in an ad-hoc fashion depending on the particular application.
It is here where our main motivation comes into the picture. We would like to know if we can find a way of initializing iterative algorithms in an algebraic fashion. For example, if you look at the two lines in this picture, there has to be a way of estimating those two lines directly from data in a one shot solution, at least in the ABSENCE of noise.

4. Our approach to segmentation (GPCA)
We consider the most general case
Subspaces of unknown and possibly different dimensions
Subspaces may intersect arbitrarily (not only at the origin)
Subspaces do not need to be orthogonal
Multiple subspaces can be represented with multiple homogeneous polynomials
Number of subspaces = degree of a polynomial
Subspace basis = derivatives of a polynomial
Subspace clustering is algebraically equivalent to
Polynomial fitting
Polynomial differentiation
Polynomial division
We are therefore interested in finding an analytic solution to the above segmentation problem. In particular we would like to know if we can estimate all the subspaces simultaneously from all the data without having to iterate between data clustering and subspace estimation. Furthermore we would like to know if we can do so in an ANALYTIC fashion, and even more, we would like to understand under what conditions we can do it in closed form and whether there is a formula for the number of subspaces.
It turns out that all these questions can be answered using tools from algebraic geometry. Since at first data segmentation and algebraic geometry may seem to be totally disconnected, let me first tell you what the connection is. We will model the number of subspaces as the degree of a polynomial and the subspace parameters (basis) as the roots of a polynomial, or more precisely as the factors of the polynomial.
With this algebraic geometric algebraic geometric interpretation in mind, we will be able to prove the following the GPCA problem.
First, there are some geometric constraints that do not depend on the segmentation of the data. Therefore, the chicken-and-egg dilemma can be resolved since the segmentation of the dada can be algebraically eliminated.
Second, one can prove that the problem has a unique solution, which can be obtained in closed form if and only if the number of subspaces is less than or equal to four
And third, the solution can be computed using linear algebraic techniques.
In the presence of noise, one can use the algebraic solution to GPCA to initialize any iterative algorithm, for example EM. In the case of zero-mean Gaussian noise, we will show that the E-step can be algebraically eliminated, since we will derive an objective function that depends on the motion parameters only.

5. Representing one subspace
One plane
One line
One subspace can be represented with
Set of linear equations
Set of polynomials of degree 1

6. Representing n subspaces
Two planes
One plane and one line
Plane:
Line:
A union of n subspaces can be represented with a set of homogeneous polynomials of degree n
De Morgan’s rule

7. Polynomial fiting Veronese Map
Null space of Ln contains information about all the polynomials.

8. Polynomial differentiation
The information of the mixture of subspaces can be obtained via polynomial differentiation.

9. Hybrid decoupling polynomial

10. Fitting polynomials to data points
Polynomials can be written linearly in terms of the vector of coefficients by using polynomial embedding
Coefficients of the polynomials can be computed from nullspace of embedded data
Solve using least squares
N = #data points
Veronese map

11. Dealing with high-dimensional data
Minimum number of points
K = dimension of ambient space
n = number of subspaces
In practice the dimension of each subspace ki is much smaller than K
Number and dimension of the subspaces is preserved by a linear projection onto a subspace of dimension
Can remove outliers by robustly fitting the subspace
Open problem: how to choose projection?
PCA?
Subspace 1
Subspace 2

12. Finding a basis for each subspace
Case of hyperplanes:
There is only one polynomial
Basis are normal vectors
Problems
Computing roots may be sensitive to noise
The estimated polynomial may not perfectly factor with noisy
Cannot be applied to subspaces of different dimensions
Polynomials are estimated up to change of basis, hence they may not factor, even with perfect data
Polynomial Factorization Alg. (PFA) [CVPR 2003]
Find roots of polynomial of degree in one variable
Solve linear systems in variables
Solution obtained in closed form for

13. Finding a basis for each subspace
To learn a mixture of subspaces we just need one positive example per class
Theorem: GPCA by differentiation

14. Choosing points by polynomial division
With noise and outliers
Polynomials may not be a perfect union of subspaces
Normals can estimated correctly by choosing points optimally
Distance to closest subspace without knowing segmentation?

15. GPCA: subspaces of different dimensions
There are multiple polynomials fitting the data
The derivative of each polynomial gives a different normal vector
Can obtain a basis for the subspace by applying PCA to normal vectors

16. GPCA: Algorithm summary
Apply polynomial embedding to projected data
Obtain multiple subspace model by polynomial fitting
Solve to obtain
Need to know number of subspaces
Obtain bases & dimensions by polynomial differentiation
Optimally choose one point per subspace using distance

17. Combining GPCA with spectral clustering
Build a similarity matrix between pairs of points
Use eigenvectors to cluster data
How to define a similarity for subspaces?
Want points in the same subspace to be close
Want points in different subspaces to be far
Use GPCA to get basis
Distance: subspace angles

18. News video segmentation
Segmenting N=30 frames of a sequence containing n=3 scenes
Host
Guest
Both
Apply PCA to obtain 2 principal components
Apply GPCA to fit n=3 subspaces

19. News video segmentation
Segmenting N=60 frames of a sequence containing n=3 scenes
Burning wheel
Burnt car with people
Burning car
Apply PCA to obtain 2 principal components
Apply GPCA to fit n=3 subspaces

20. Face clustering under varying illumination
Yale Face Database B
n = 3 faces
N = 64 views per face
K = 30x40 pixels per face
Apply PCA to obtain 3 principal components
Apply GPCA in homoge-neous coordinates ( )
n=3 subspaces
dimensions 3, 2 and 2

21. 2-D and 3-D motion segmentation
Given point correspondences in multiple views, determine
Motion model: (homography, fundamental matrix, affine)
Data segmentation: model associated with each pixel
Problem depends on
Number of frames (2, 3, …)
Projection (affine, persp.)
Motion model
3-D structure (planar or not)
Multiframe optical flow segmentation
All cases can be solved using complex GPCA (ECCV’04)

22. Conclusions New GPCA algorithm for clustering subspaces
Deals with unknown and possibly different dimensions
Deals with arbitrary intersections among the subspaces
Our approach is based on
Projecting data onto a low-dimensional subspace
Fitting polynomials to projected subspaces
Differentiating polynomials to obtain a basis
Open issues and ongoing work
Choosing a projection
Dealing with outliers: projection & polynomial fitting
Robust estimation of number and dimension of subspaces
Combining GPCA with model selection (Huang et al. CVPR’04)
Dealing with mixtures of nonlinear spaces (bilinear, trilinear)
In this paper, we have presented a new way of looking at segmentation problems that is based on algebraic geometric techniques. In essence we showed that segmentation is equivalent to polynomial factorization and presented an algebraic solution for the case of mixtures of subspaces, which is a natural generalization of the well known Principal Component Analysis (PCA).
Of course, as this theory is fairly new, there are still a lot of open problems. For example, how to do polynomial factorization in the presence of noise, or how to deal with outliers. We are currently working on such problems and expect to report our results in the near future. Also, we are currently working on generalizing the ideas in this paper to the cases of subspaces of different dimensions and to other types of data.
In fact, I cannot resist doing some advertising for my talk tomorrow, where I will talk about the segmentation of multiple rigid motions, which corresponds to the case of bilinear data.
Finally, before taking your questions, I would like to say that I think it is time that we start thinking about putting segmentation problems in a theoretical footing. I think this can be done by building a mathematical theory of segmentation that not only incorporates well known statistical modeling techniques, but also the geometric constraints inherent in various applications in computer vision. We hope that by combining the algebraic geometric approach to segmentation presented in this paper with the more standard statistical approaches, we should be able to build a new mathematical theory of segmentation in the followi should be able to build such a theory, which a nice mathematical theory of segmentation. Such a theory should find applications in various disciplines, including of course lots of segmentation and recognition problems in computer vision.

23. Segmentation of dynamic textures
Dynamic textures: output of a linear dynamical model (Soatto et al., 2001)
Segmentation of dynamic textures using GPCA
A dynamic textures lives in the observability subspace
Multiple textures live in multiple subspaces
Segmentation of dynamic textures using level-sets (Doretto et al. 2003)
PCA+GPCA segmentation
Apply PCA to intensities
Apply GPCA to projected data
dynamics x t + 1 = A v y C w
images
appearance

24. Texture-based image segmentation
GPCA
Human

25. Texture-based image segmentation

26. Segmentation of 3-D translational motions

27. Comparison of PFA, PDA, K-sub, EM

28. Computation time GPCA, KMEANS and EM