Convergence of Alternating Gradient Descent for Matrix Factorization

We consider alternating gradient descent (AGD) with fixed step size applied to the asymmetric matrix factorization objective. We show that, for a rank-$r$ matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$, $T = C (\frac{\sigma_1(\mathbf{A})}{\sigma_r(\mathbf{A})})^2 \log(1/\epsilon)$ iterations of alternating gradient descent suffice to reach an $\epsilon$-optimal factorization $\| \mathbf{A} - \mathbf{X} \mathbf{Y}^{T} \|^2 \leq \epsilon \| \mathbf{A}\|^2$ with high probability starting from an atypical random initialization. The factors have rank $d \geq r$ so that $\mathbf{X}_{T}\in\mathbb{R}^{m \times d}$ and $\mathbf{Y}_{T} \in\mathbb{R}^{n \times …

convergence cs.lg factorization gradient math.oc matrix show stat.ml