Low-rank matrix recovery is well-known to exhibit benign nonconvexity under the restricted isometry property (RIP): every second-order critical point is globally optimal, so local methods provably recover the ground truth. Motivated by the strong empirical performance of projected gradient methods for nonnegative low-rank recovery problems, we investigate whether this benign geometry persists when the factor matrices are constrained to be elementwise nonnegative. In the simple setting of a rank-1 nonnegative ground truth, we confirm that benign nonconvexity holds in the fully-observed case with RIP constant $δ=0$. This benign nonconvexity, however, is unstable. It fails to extend to the partially-observed case with any arbitrarily small RIP constant $δ>0$, and to higher-rank ground truths $r^{\star}>1$, regardless of how much the search rank $r\ge r^{\star}$ is overparameterized. Together, these results undermine the standard stability-based explanation for the empirical success of nonconvex methods and suggest that fundamentally different tools are needed to analyze nonnegative low-rank recovery.

翻译：众所周知，在受限等距性质（RIP）条件下，低秩矩阵恢复表现出良性的非凸性：每个二阶临界点都是全局最优的，因此局部方法可证明能恢复真实解。受投影梯度方法在非负低秩恢复问题中优异实证表现的启发，我们研究了当因子矩阵被约束为元素非负时，这种良性几何性质是否仍然存在。在秩为1的非负真实解这一简单设定下，我们证实了在完全观测情况下（RIP常数 $δ=0$）良性非凸性成立。然而，这种良性非凸性是不稳定的。它无法推广到具有任意小RIP常数 $δ>0$ 的部分观测情况，也无法推广到更高秩的真实解 $r^{\star}>1$，无论搜索秩 $r\ge r^{\star}$ 被过参数化到何种程度。这些结果共同削弱了基于稳定性的标准解释对于非凸方法实证成功的说服力，并表明分析非负低秩恢复问题需要 fundamentally different tools。