This paper considers the problem of minimizing the sum of a smooth function
and the Schatten-$p$ norm of the matrix. Our contribution involves proposing
accelerated iteratively reweighted nuclear norm methods designed for solving
the nonconvex low-rank minimization problem. Two major novelties characterize
our approach. Firstly, the proposed method possesses a rank identification
property, enabling the provable identification of the "correct" rank of the
stationary point within a finite number of iterations. Secondly, we introduce
an adaptive updating strategy for smoothing parameters. This strategy
automatically fixes parameters associated with zero singular values as
constants upon detecting the "correct" rank while quickly driving the rest
parameters to zero. This adaptive behavior transforms the algorithm into one
that effectively solves smooth problems after a few iterations, setting our
work apart from existing iteratively reweighted methods for low-rank
optimization. We prove the global convergence of the proposed algorithm,
guaranteeing that every limit point of the iterates is a critical point.
Furthermore, a local convergence rate analysis is provided under the
Kurdyka-{\L}ojasiewicz property. We conduct numerical experiments using both
synthetic and real data to showcase our algorithm's efficiency and superiority
over existing methods.

该研究考虑了平滑函数和矩阵的 Schatten-p 范数之和的最小化问题，并提出了用于解决非凸低秩最小化问题的加速迭代重新加权核范数方法，其主要创新包括具有秩识别特性的方法和自适应更新策略，通过快速将参数驱动为零，将算法转化为能有效解决平滑问题的算法。