论文标题
旋转不变设计的随机线性估计：高温下渐近性
Random linear estimation with rotationally-invariant designs: Asymptotics at high temperature
论文作者
论文摘要
我们在线性模型中研究估计$ y =aβ^\ star+ε$，在贝叶斯环境中，$β^\ star $具有入口处i.i.d.先验和设计$ a $在法律上是旋转不变的。在大型系统限制中，随着尺寸和样本量的比例增加，已经假定一组相关的猜想是针对渐近互信息，贝叶斯最佳的平均平方误差和点击平均值的平均场方程，这些方程是$β^\ star $的贝叶斯后均值。在这项工作中，我们证明了这些猜想的一般信号先验和任意旋转不变的设计$ a $，在“高温”条件下限制了$ a^\ top a $的特征值范围。我们的证明使用有条件的第二矩方法参数，在其中我们根据迭代的迭代条件来求解TAP均值场方程。
We study estimation in the linear model $y=Aβ^\star+ε$, in a Bayesian setting where $β^\star$ has an entrywise i.i.d. prior and the design $A$ is rotationally-invariant in law. In the large system limit as dimension and sample size increase proportionally, a set of related conjectures have been postulated for the asymptotic mutual information, Bayes-optimal mean squared error, and TAP mean-field equations that characterize the Bayes posterior mean of $β^\star$. In this work, we prove these conjectures for a general class of signal priors and for arbitrary rotationally-invariant designs $A$, under a "high-temperature" condition that restricts the range of eigenvalues of $A^\top A$. Our proof uses a conditional second-moment method argument, where we condition on the iterates of a version of the Vector AMP algorithm for solving the TAP mean-field equations.
