Agnostic matrix phase retrieval (AMPR) is a general low-rank matrix recovery problem given a set of noisy high-dimensional data samples. To be specific, AMPR is targeting at recovering an $r$ -rank matrix $\mathbf {M}^*\in \mathbb {R}^{d_1\times d_2}$ as the parametric component from $n$ instantiations/samples of a semi-parametric model $y=f(\langle \mathbf {M}^*, \mathbf {X}\rangle, \epsilon)$ , where the predictor matrix is denoted as $\mathbf {X}\in \mathbb {R}^{d_1\times d_2}$ , link function $f(\cdot, \epsilon)$ is agnostic under some mild distribution assumptions on $\mathbf {X}$ , and $\epsilon$ represents the noise. In this paper, we formulate AMPR as a rank-restricted largest eigenvalue problem by applying the second-order Stein's identity and propose a new spectrum truncation power iteration (STPower) method to obtain the desired matrix efficiently. Also, we show a favorable rank recovery result by adopting the STPower method, i.e. , a near-optimal statistical convergence rate under some relatively general model assumption from a wide range of applications. Extensive simulations verify our theoretical analysis and showcase the strength of STPower compared with the other existing counterparts.

中文翻译：

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不可知矩阵相位检索的频谱截断幂迭代

不可知矩阵相位检索 (AMPR) 是给定一组嘈杂的高维数据样本的一般低秩矩阵恢复问题。具体来说，AMPR 的目标是恢复$ r $ -秩矩阵 $ \ mathbf {M} ^ * \ in \ mathbb {R} ^ {d_1 \ times d_2} $ 作为参数组件从 $ n $ 半参数模型的实例/样本 $y=f(\langle\mathbf{M}^*,\mathbf{X}\rangle,\epsilon)$ ，其中预测矩阵表示为 $ \ mathbf {X} \ in \ mathbb {R} ^ {d_1 \ times d_2} $ , 链接功能 $ f(\cdot,\epsilon) $ 在一些温和的分布假设下是不可知的 $ \ mathbf {X} $ ， 和 $\epsilon $代表噪音。在本文中，我们通过应用二阶 Stein 恒等式将 AMPR 公式化为一个秩受限的最大特征值问题，并提出了一种新的频谱截断功率迭代 (STPower) 方法来有效地获得所需矩阵。此外，我们通过采用 STPower 方法显示出良好的秩恢复结果，即，在来自广泛应用的一些相对一般的模型假设下的接近最优的统计收敛率。广泛的模拟验证了我们的理论分析，并展示了 STPower 与其他现有同行相比的实力。