The aim of matrix recovery is to recover $P\in \mathcal{M}\subset {\mathbb{F}}^{p\times q}$ from ${\mathbf{L}}_{\mathcal{A}}(P)={(\mathrm{Tr}(A_1^{T}P),\mathrm{Tr}(A_2^{T}P),\dots ,\mathrm{Tr}(A_N^{T}P))}^T$ with $A_j\in V_j\subset {\mathbb{F}}^{p\times q}$, which is raised in many areas. In this paper, we build up a framework for almost everywhere matrix recovery which means ${\mathbf{L}}_{\mathcal{A}}$ is almost everywhere injectivity on $\mathcal{M}$. We mainly focus on the following question: how many measurements are needed to recover almost all the matrices in $\mathcal{M}$? For the case where both $\mathcal{M}$ and $V_j$ are algebraic varieties, we use the tools from algebraic geometry to study the question and present some results to address it under many different settings.

中文翻译：

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几乎所有情况下的基质回收问题的注入条件

基质回收的目的是回收 $P\in \mathcal{中号}\subset {\mathbb{F}}^{p\times q}$ 从 ${\mathbf{大号}}_{\mathcal{一种}}（P）={（\mathrm{Tr}（{\mathrm{一种}}_{\mathrm{1个}}^{Ť}P），\mathrm{Tr}（{\mathrm{一种}}_2^{Ť}P），\dots ，\mathrm{Tr}（{\mathrm{一种}}_{ñ}^{Ť}P））}^{Ť}$ 与 ${\mathrm{一种}}_{Ĵ}\in V_{Ĵ}\subset {\mathbb{F}}^{p\times q}$，这在许多领域都有所提出。在本文中，我们建立了一个几乎所有矩阵恢复的框架，这意味着${\mathbf{大号}}_{\mathcal{一种}}$ 几乎无处不在 $\mathcal{中号}$。我们主要关注以下问题：需要多少次测量才能恢复几乎所有矩阵$\mathcal{中号}$？对于两种情况$\mathcal{中号}$ 和 $V_{Ĵ}$ 是代数形式，我们使用代数几何中的工具来研究该问题，并给出一些结果以解决许多不同的情况。