We develop a pseudo-likelihood theory for rank one matrix estimation problems in the high dimensional limit. We prove a variational principle for the limiting pseudo-maximum likelihood which also characterizes the performance of the corresponding pseudo-maximum likelihood estimator. We show that this variational principle is universal and depends only on four parameters determined by the corresponding null model. Through this universality, we introduce a notion of equivalence for estimation problems of this type and, in particular, show that a broad class of estimation tasks, including community detection, sparse submatrix detection, and non-linear spiked matrix models, are equivalent to spiked matrix models. As an application, we obtain a complete description of the performance of the least-squares (or ``best rank one'') estimator for any rank one matrix estimation problem.

翻译：我们针对高维极限下的秩一矩阵估计问题发展了一套伪似然理论。我们证明了极限伪极大似然的一个变分原理，该原理同时刻画了相应伪极大似然估计器的性能。我们表明该变分原理具有普适性，仅由对应零模型确定的四个参数决定。通过这种普适性，我们为此类估计问题引入了等价性概念，并特别证明了一大类估计任务——包括社区检测、稀疏子矩阵检测以及非线性尖峰矩阵模型——均等价于尖峰矩阵模型。作为应用，我们获得了最小二乘（或称“最佳秩一”）估计器在任何秩一矩阵估计问题中性能的完整描述。