Generalized principal component analysis (GPCA)

We propose an algebraic geometric approach to the problem of estimating a mixture of linear subspaces from sample data points, the so-called generalized principal component analysis (GPCA) problem. In the absence of noise, we show that GPCA is equivalent to factoring a homogeneous polynomial whose degree is the number of subspaces and whose factors (roots) represent normal vectors to each subspace. We derive a formula for the number of subspaces n and provide an analytic solution to the factorization problem using linear algebraic techniques. The solution is closed form if and only if n ≤ 4. In the presence of noise, we cast GPCA as a constrained nonlinear least squares problem and derive an optimal function from which the subspaces can be directly recovered using standard nonlinear optimization techniques. We apply GPCA to the motion segmentation problem in computer vision, i.e. the problem of estimating a mixture of motion models from 2D imagery.