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Matrix variety: basic geometry
low rank
variety
The set of bounded-rank matrices,
\(\mathbb{R}^{m\times n}_{\leq r}=\{X\in\mathbb{R}^{m\times n}: \mathrm{rank}(X)\leq r\}\)
, is a non-smooth algebraic variety called matrix variety. More specifically, it can be defined by the matrices in which all
\((r+1)\times(r+1)\)
minors are zero, so it is also called a determinantal variety. This post presents the basic geometry of
\(\mathbb{R}^{m\times n}_{\leq r}\)
that is foundational for optimization involving low-rank structures.
Dec 20, 2024
Bin Gao
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