Part of Advances in Neural Information Processing Systems 29 (NIPS 2016)
Matey Neykov, Zhaoran Wang, Han Liu
The goal of noisy high-dimensional phase retrieval is to estimate an s-sparse parameter \boldsymbol{\beta}^*\in \mathbb{R}^d from n realizations of the model Y = (\boldsymbol{X}^{\top} \boldsymbol{\beta}^*)^2 + \varepsilon. Based on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which Y = f(\boldsymbol{X}^{\top}\boldsymbol{\beta}^*, \varepsilon) with unknown f and \operatorname{Cov}(Y, (\boldsymbol{X}^{\top}\boldsymbol{\beta}^*)^2) > 0. For example, MPR encompasses Y = h(|\boldsymbol{X}^{\top} \boldsymbol{\beta}^*|) + \varepsilon with increasing h as a special case. Despite the generality of the MPR model, it eludes the reach of most existing semi-parametric estimators. In this paper, we propose an estimation procedure, which consists of solving a cascade of two convex programs and provably recovers the direction of \boldsymbol{\beta}^*. Our theory is backed up by thorough numerical results.