We address some theoretical guarantees for Schatten-$p$ quasi-normminimization ($p \in (0,1]$) in recovering low-rank matrices from compressedlinear measurements. Firstly, using null space properties of the measurementoperator, we provide a sufficient condition for exact recovery of low-rankmatrices. This condition guarantees unique recovery of matrices of ranks equalor larger than what is guaranteed by nuclear norm minimization. Secondly, thissufficient condition leads to a theorem proving that all restricted isometryproperty (RIP) based sufficient conditions for $\ell_p$ quasi-norm minimizationgeneralize to Schatten-$p$ quasi-norm minimization. Based on this theorem, weprovide a few RIP-based recovery conditions.
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机译：在从压缩线性测量中恢复低秩矩阵时，我们为Schatten- $ p $准规范化（$ p \ in（0,1] $）提供了一些理论保证。首先，利用Measurementoperator的零空间属性，我们提供了足够的空间精确恢复低秩矩阵的条件。该条件保证秩的矩阵的唯一恢复等于或大于核规范最小化所保证的矩阵。其次，该充分条件导致一个定理，证明所有基于受限等距性质（RIP）的充分条件对于\ ell_p $准范数最小化广义化为Schatten- $ p $准范数最小化基于此定理，我们提供了一些基于RIP的恢复条件。
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