Run Procrustes, Run! On the convergence of accelerated Procrustes Flow
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In this work, we present theoretical results on the convergence of non-convex accelerated gradient descent in matrix factorization models. The technique is applied to matrix sensing problems with squared loss, for the estimation of a rank r optimal solution X^∈R^n × n. We show that the acceleration leads to linear convergence rate, even under non-convex settings where the variable X is represented as U U^ for U ∈R^n × r. Our result has the same dependence on the condition number of the objective --and the optimal solution-- as that of the recent results on non-accelerated algorithms. However, acceleration is observed in practice, both in synthetic examples and in two real applications: neuronal multi-unit activities recovery from single electrode recordings, and quantum state tomography on quantum computing simulators.
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