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Agnostic Estimation for Misspecified Phase Retrieval Models
Matey Neykov · Zhaoran Wang · Han Liu

Tue Dec 06 09:00 AM -- 12:30 PM (PST) @ Area 5+6+7+8 #69
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.

Author Information

Matey Neykov (Princeton University)
Zhaoran Wang (Princeton University)
Han Liu (Tencent AI Lab)

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