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We consider a rank-one symmetric matrix corrupted by additive noise. Therank-one matrix is formed by an $n$-component unknown vector on the sphere ofradius $\sqrt{n}$, and we consider the problem of estimating this vector fromthe corrupted matrix in the high dimensional limit of $n$ large, by gradientdescent for a quadratic cost function on the sphere. Explicit formulas for thewhole time evolution of the overlap between the estimator and unknown vector,as well as the cost, are rigorously derived. In the long time limit we recoverthe well known spectral phase transition, as a function of the signal-to-noiseratio. The explicit formulas also allow to point out interesting transientfeatures of the time evolution. Our analysis technique is based on recentprogress in random matrix theory and uses local versions of the semi-circlelaw.

Rank-one matrix estimation: analytic time evolution of gradient descent dynamics