| Abstract: |
| Low-rank matrix completion is an interdisciplinary problem across statistics, machine learning and signal processing, in which the objective is to complete a partially observed m x n matrix of typically small rank r. A fundamental question is which observation patterns admit finitely many completions of rank r, and among them, which ones admit a unique completion. This can be recast as a problem in algebraic geometry, where one aims to determine whether a system of polynomial equations with coefficients rational functions in the observed entries has finitely many solutions (or even a unique solution). We will report families of finitely and uniquely completable patterns for any matrix size and any rank, which we have obtained by 1) interpreting low-rank matrix completion as a linear section problem on the Grassmannian and 2) using families of local coordinates on the Grassmannian that Sturmfels & Zelevinsky desribed in 1993. |
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