Bengisu Ozbay’s PhD Proposal Review

“Fast Identification via Subspace Clustering and Applications to Dynamic and Geometric Scene Understanding”

Abstract:

More and more data is needed in order to build new machine learning and computer vision techniques. Using human operators to identify these vast datasets would be too expensive, hence the use of unsupervised learning has grown more common. Piecewise linear or affine models can be used in a broad range of applications connected to system identification and computer vision.
In this proposal, we suggest an efficient method that only requires singular value decomposition of matrices whose size is unaffected by the total number of points. This method only has to be performed (number of clusters) times. We discovered that it is feasible to find the polynomials that represent the hyperplanes by doing a singular value decomposition (SVD) on the empirical moments matrix containing the data. In this approach, the notion of using polynomials and Christoffel functions to conduct SVDs in order to partition data into sets, each of which originates from a different cluster, is central. Data may be segmented and then the parameters of each group can be extracted using application-specific techniques. In particular, the problems that are taken into consideration in this proposal include identification of Auto-regressive with Extra Input (SARX) models, affine linear subspace clustering, two-view motion segmentation, and identification of a group of nonlinear systems known as Wiener systems.
This proposal is structured as follows: to begin with, we offer a semi-algebraic clustering framework for locating reliable subsets from the data, which belongs in a union of varieties and segments the data sequentially using Christoffel polynomials. We employ this strategy for switched system identification and affine subspace clustering challenges. In both instances, the data resides in linear affine varieties. To expand the given approach beyond linear affine arrangements, we reformulate it for quadratic surfaces and further apply it to the two-view motion segmentation task. Finally, using this suggested semi-algebraic formulation, we are able to detect a class of nonlinearities, namely Wiener systems with an even nonlinearity, which is indeed an NP-hard issue.