低秩矩阵优化的全局最优性
Global Optimality in Low-rank Matrix Optimization
摘要：
本文研究了秩至多为$r$的矩形$n\times m$矩阵集合上的一般目标函数$f(X)$的极小化问题。为了减少计算负担，我们将变量$X$分解为两个较小矩阵的乘积，并在这两个矩阵上进行优化，而不是$X$。尽管结果是非凸的，但最近在矩阵完备化和感知方面的研究表明，因子化问题没有虚假的局部极小值，并遵守所谓的严格鞍性质（函数在所有临界点都有方向负曲率，但局部极小值）。我们分析了一个一般的，但条件良好的目标函数$f(X)$的限制强凸性和限制强光滑常数是可比的全球几何。特别是，我们表明，重新制定的目标函数没有虚假的局部极小值，并遵守严格的鞍性质。这些几何性质意味着一些迭代优化算法（如梯度下降）可以证明解决的因式分解问题的全局收敛性。
英文摘要：
This paper considers the minimization of a general objective function $f(X)$ over the set of rectangular $n\times m$ matrices that have rank at most $r$. To reduce the computational burden, we factorize the variable $X$ into a product of two smaller matrices and optimize over these two matrices instead of $X$. Despite the resulting nonconvexity, recent studies in matrix completion and sensing have shown that the factored problem has no spurious local minima and obeys the so-called strict saddle property (the function has a directional negative curvature at all critical points but local minima). We analyze the global geometry for a general and yet well-conditioned objective function $f(X)$ whose restricted strong convexity and restricted strong smoothness constants are comparable. In particular, we show that the reformulated objective function has no spurious local minima and obeys the strict saddle property. These geometric properties imply that a number of iterative optimization algorithms (such as gradient descent) can provably solve the factored problem with global convergence.
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