NIPS Poster Agnostic Estimation for Misspecified Phase Retrieval Models

Poster

Agnostic Estimation for Misspecified Phase Retrieval Models

Matey Neykov · Zhaoran Wang · Han Liu

Area 5+6+7+8 #69

Keywords: [ Spectral Methods ] [ Information Theory ] [ Learning Theory ] [ Convex Optimization ] [ Sparsity and Feature Selection ]

[ Abstract ]

Abstract: The goal of noisy high-dimensional phase retrieval is to estimate an

s

$s$ -sparse parameter

β^{*} \in R^{d}

$\boldsymbol{\beta}^*\in \mathbb{R}^d$ from

n

$n$ realizations of the model

Y = (X^{⊤} β^{*})^{2} + ε

$Y = (\boldsymbol{X}^{\top} \boldsymbol{\beta}^*)^2 + \varepsilon$ . Based on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which

Y = f (X^{⊤} β^{*}, ε)

$Y = f(\boldsymbol{X}^{\top}\boldsymbol{\beta}^*, \varepsilon)$ with unknown

f

$f$ and

Cov (Y, (X^{⊤} β^{*})^{2}) > 0

$\operatorname{Cov}(Y, (\boldsymbol{X}^{\top}\boldsymbol{\beta}^*)^2) > 0$ . For example, MPR encompasses

Y = h (| X^{⊤} β^{*} |) + ε

$Y = h(|\boldsymbol{X}^{\top} \boldsymbol{\beta}^*|) + \varepsilon$ with increasing

h

$h$ 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

β^{*}

$\boldsymbol{\beta}^*$ . Our theory is backed up by thorough numerical results.

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