# Global Optimality in Low-rank Matrix Optimization

## TL;DR

This paper analyzes the geometric landscape of low-rank matrix optimization problems, showing that under certain conditions, the factored formulation has no spurious local minima and satisfies the strict saddle property, enabling global convergence of algorithms.

## Contribution

It provides a geometric analysis of the factored low-rank matrix optimization problem for well-conditioned functions, establishing conditions for no spurious minima and strict saddle property.

## Key findings

- The reformulated problem has no spurious local minima.
- The objective function satisfies the strict saddle property.
- Gradient-based algorithms can provably find global solutions.

## Abstract

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.