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- W3018162266 abstract "Low-rank matrix completion is a popular paradigm in machine learning, but little is known about the completion properties of non-random observation patterns. A fundamental open question in this direction is the following: given an observation pattern of a sufficiently generic (e.g. incoherent) $m times n$ real matrix $X$ of rank $r$ with exactly $r(m+n-r)$ entries being observed, this number being the dimension of the space of real rank-$r$ $m times n$ matrices, are there finitely many rank-$r$ completions? This is a challenging problem whose answer is known only for ranks $1$, $2$ and $min{m,n}-1$. In this paper we study this problem by bringing tools from algebraic geometry. In particular, we exploit the fact that both the space of real rank-$r$ $m times n$ matrices as well as the set of $r$-dimensional subspaces of $mathbb{R}^m$, known as the Grassmannian, are algebraic varieties. Our approach is based on a novel formulation of matrix completion in terms of Pl{u}cker coordinates, the latter a traditionally powerful tool in computer vision and graphics and a classical notion in algebraic geometry. This formulation allows us to characterize a large class of minimal (i.e. of size $r(m+n-r)$) observation patterns for which a generic matrix admits finitely many rank-r completions. We conjecture that the converse is also true: any minimal pattern which is generically finitely completable must be of that type. As a consequence, we generalize results that have previously appeared and are being used in the literature, but lack a sufficient theoretical justification. We believe the Pl{u}cker-coordinate based link that we establish between low-rank matrices and the Grassmannian in the context of matrix completion to be of wider significance for matrix and subspace learning problems with incomplete data." @default.
- W3018162266 created "2020-05-01" @default.
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- W3018162266 date "2020-04-26" @default.
- W3018162266 modified "2023-09-24" @default.
- W3018162266 title "An exposition to the finiteness of fibers in matrix completion via Plücker coordinates" @default.
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