The local convexity of solving systems of quadratic equations

This paper considers the recovery of a rank r positive semidefinite matrix X XT∈ℝn× n from m scalar measurements of the form yi := aiT X XT ai (i.e., quadratic measurements of X). Such problems arise in a variety of applications, including covariance sketching of high-dimensional data streams, quadratic regression, quantum state tomography, among others. A natural approach to this problem is to minimize the loss function f(U) = sumi (yi - aiTUUTai)2 which has an entire manifold of solutions given by XOO∈Or where Or is the orthogonal group of r× r orthogonal matrices; this is it non-convex in the n× r matrix U, but methods like gradient descent are simple and easy to implement (as compared to semidefinite relaxation approaches). In this paper we show that once we have m ≥ C nr log2(n) samples from isotropic gaussian ai, with high probability em (a) this function admits a dimension-independent region of em local strong convexity on lines perpendicular to the solution manifold, and em (b) with an additional polynomial factor of r samples, a simple spectral initialization will land within the region of convexity with high probability. Together, this implies that gradient descent with initialization (but no re-sampling) will converge linearly to the correct X, up to an orthogonal transformation. We believe that this general technique (local convexity reachable by spectral initialization) should prove applicable to a broader class of nonconvex optimization problems.