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 (unique) LASSO solution 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 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 LASSO solution.

翻译：我们证明，流行的凸矩阵LASSO问题（核范数惩罚线性最小二乘）的解在经典低秩矩阵感知误差界所需类似条件下具有低秩性。尽管核范数惩罚旨在促进解的低秩性，但迄今为止（据我们所知）仅在极特定情形下得到证明。此外，我们表明该结果对非凸秩约束优化方法具有重要理论意义。具体而言，我们证明：若(a) 真实矩阵具有低秩性，(b) （线性）测量算子满足矩阵受限等距性质(RIP)，且(c) 测量误差相对于核范数惩罚足够小，则（唯一）LASSO解的秩（近似）受真实矩阵秩的约束。由此，我们证明(a) 低秩投影近端梯度下降算法从任意初始化出发均能线性收敛至LASSO解，且(b) 低秩Burer-Monteiro因子化问题形式的非凸景观具有良性特征，即所有二阶临界点均为全局最优解并生成LASSO解。