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Factorization-based gradient descent is a scalable and efficient algorithmfor solving low-rank matrix completion. Recent progress in structurednon-convex optimization has offered global convergence guarantees for gradientdescent under certain statistical assumptions on the low-rank matrix and thesampling set. However, while the theory suggests gradient descent enjoys fastlinear convergence to a global solution of the problem, the universal nature ofthe bounding technique prevents it from obtaining an accurate estimate of therate of convergence. In this paper, we perform a local analysis of the exactlinear convergence rate of gradient descent for factorization-based matrixcompletion for symmetric matrices. Without any additional assumptions on theunderlying model, we identify the deterministic condition for local convergenceof gradient descent, which only depends on the solution matrix and the samplingset. More crucially, our analysis provides a closed-form expression of theasymptotic rate of convergence that matches exactly with the linear convergenceobserved in practice. To the best of our knowledge, our result is the first onethat offers the exact rate of convergence of gradient descent for matrixfactorization in Euclidean space for matrix completion.

Exact Linear Convergence Rate Analysis for Low-Rank Symmetric Matrix Completion via Gradient Descent