The local convexity of solving systems of quadratic equations

This paper considers the recovery of a rank $r$ positive semidefinite matrix $X X^T\in\mathbb{R}^{n\times n}$ from $m$ scalar measurements of the form $y_i := a_i^T X X^T a_i$ (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, etc. A natural approach to this problem is to minimize the loss function $f(U) = \sum_i (y_i - a_i^TUU^Ta_i)^2$; this is non-convex in the $n\times r$ matrix $U$, but methods like gradient descent are simple and easy to implement (as compared to convex approaches to this problem). In this paper we show that, once we have $m \geq C nr^2 \log^2(nr)$ samples from isotropic gaussian $a_i$, with high probability (a) this function admits a dimension-independent region of local strong convexity, and (b) 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 $X$. We believe that our general technique (local convexity reachable by spectral initialization) should prove applicable to a broader class of nonconvex optimization problems.