This paper investigates the theoretical recovery capabilities of low-rank approximation models based on the difference of the nuclear norm and the Frobenius norm ( \(L_{*-F}\) ). For the constrained \(L_{*-F}\) minimization model, we establish recovery conditions and quantify the recovery accuracy within the restricted isometry property (RIP) framework. These results improve upon existing RIP-based analyses, and apply to a wide range of underlying matrices, including exactly low-rank, approximately low-rank cases, and scenarios where rank information is unavailable. Notably, we can provide a positive answer to the open problem posed by [T.-H. Ma, Y. Lou, and T.-Z. Huang, SIAM J. Imaging Sci., 10 (2017), pp. 1346–1380], regarding the sufficient conditions that can be sharpened for recovering exactly low-rank matrices through constrained \(L_{*-F}\) minimization. For the unconstrained \(L_{*-F}\) minimization model, commonly employed in numerical algorithms, we provide the first theoretical error bound analysis that does not rely on prior knowledge of the underlying matrix rank. This offers recovery guarantees for unconstrained nonconvex minimization, even when the matrix is only approximately low-rank or its rank is completely unknown.
