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

Understand

We consider a rank-one symmetric matrix corrupted by additive noise.

• The rank-one matrix is formed by an $n$-component unknown vector on the sphere of radius $\sqrt{n}$, and we consider the problem of estimating this vector from the corrupted matrix in the high dimensional limit of $n$ large, by gradient descent for a quadratic cost function on the sphere.
• Explicit formulas for the whole 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 recover the well known spectral phase transition, as a function of the signal-to-noise ratio.