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• W55130788 abstract "This thesis presents a novel algebraic geometric framework for simultaneous data segmentation and model estimation, with the hope of providing a theoretical footing for the problem as well as an algorithm for initializing iterative techniques. The algebraic geometric approach presented in this thesis is based on eliminating the data segmentation part algebraically and then solving the model estimation part directly using all the data and without having to iterate between data segmentation and model estimation. The algebraic elimination of the data segmentation part is achieved by finding algebraic equations that are segmentation independent, that is equations that are satisfied by all the data regardless of the group or model associated with each point. For the classes of problems considered in this thesis, such segmentation independent constraints are polynomials of a certain degree in several variables. The degree of the polynomials corresponds to the number of groups and the factors of the polynomials encode the model parameters associated with each group. The problem is then reduced to (1) Computing the number of groups from data: this question is answered by looking for polynomials with the smallest possible degree that fit all the data points. This leads to simple rank constraints on the data from which one can estimate the number of groups after embedding the data into a higher-dimensional linear space. (2) Estimating the polynomials representing all the groups from data: this question is trivially answered by showing that the coefficients of the polynomials representing the data lie in the null space of the embedded data matrix. (3) Factoring such polynomials to obtain the model for each group: this question is answered with a novel polynomial factorization technique based on computing roots of univariate polynomials, plus a combination of linear algebra with multivariate polynomial differentiation and division. The solution can be obtained in closed form if and only if the number of groups is less than or equal to four. (Abstract shortened by UMI.)" @default.
• W55130788 created "2016-06-24" @default.
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• W55130788 date "2003-01-01" @default.
• W55130788 modified "2023-10-09" @default.
• W55130788 title "Generalized principal component analysis (gpca): an algebraic geometric approach to subspace clustering and motion segmentation" @default.
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