Low solution rank of the matrix LASSO under RIP with consequences for rank-constrained algorithms
证明了凸矩阵LASSO问题的解在类似低秩矩阵感知误差界的条件下具有低秩，并由此推导出非凸秩约束优化算法（如低秩投影近端梯度下降和Burer-Monteiro分解）的收敛性与良性景观。
Abstract We show that solutions to the popular convex matrix LASSO problem (nuclear-norm–penalized linear least-squares) have low rank under similar assumptions as required by classical low-rank matrix sensing error bounds. Although the purpose of the nuclear norm penalty is to promote low solution rank, a proof has not yet (to our knowledge) been provided outside very specific circumstances. Furthermore, we show that this result has significant theoretical consequences for nonconvex rank-constrained optimization approaches. Specifically, we show that if (a) the ground truth matrix has low rank, (b) the (linear) measurement operator has the matrix restricted isometry property (RIP), and (c) the measurement error is small enough relative to the nuclear norm penalty, then the LASSO solution is unique and has rank (approximately) bounded by that of the ground truth. From this, we show (a) that a low-rank–projected proximal gradient descent algorithm will converge linearly to the unique LASSO solution from any initialization, and (b) that the nonconvex landscape of the low-rank Burer-Monteiro–factored problem formulation is benign in the sense that all second-order critical points are globally optimal and yield the unique LASSO solution.