An exposition to the finiteness of fibers in matrix completion via Plücker coordinates

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 Cand\`es $\&$ Recht, 2009 and Cand\`es $\&$ Tao, 2010, which asks for what kind of observation patterns of size equal to the dimension of the variety of real $m \times n$ 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 $\operatorname{Gr}(r,m)$ viewed as a projective variety in Pl\"ucker coordinates.