An exposition to the finiteness of fibers in matrix completion via Plücker coordinates
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Matrix completion is a popular paradigm in machine learning and data science, but little is known about the geometric properties of non-random observation patterns. In this paper we study a fundamental geometric analogue of the seminal work of Candès & Recht, 2009 and Candès & Tao, 2010, which asks for what kind of observation patterns of size equal to the dimension of the variety of real m × n rank-r matrices there are finitely many rank-r completions. Our main device is to formulate matrix completion as a hyperplane sections problem on the Grassmannian Gr(r,m) viewed as a projective variety in Plücker coordinates.
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