在正交不变广义线性模型中的最优估计：谱初始化与近似消息传递
Optimal Estimation in Orthogonally Invariant Generalized Linear Models: Spectral Initialization and Approximate Message Passing
AI总结：
本文提出基于谱初始化和近似消息传递算法的最优估计方法，用于正交不变广义线性模型中的参数估计，理论和实验均验证了其有效性。
摘要：
我们考虑从具有随机设计矩阵的广义线性模型中进行参数估计的问题，该设计矩阵在分布上正交不变。此类模型允许设计具有任意的奇异值分布，并且仅假设其奇异向量是通用的。这是理论文献中通常考虑的独立同分布高斯设计的广泛扩展，因为现实数据往往具有复杂的相关结构，因此依赖于i.i.d假设的方法可能非常次优。基于谱初始化迭代优化的范式，本文提出了最优谱估计器，并将其与近似消息传递（AMP）算法结合，建立了这两个算法步骤的严格性能保证。谱初始化和随后的AMP都满足现有关于估计根本限制的猜想——前者关于高效弱恢复的最优样本复杂度，后者关于最优误差。数值实验表明我们的方法在超越正交不变数据时的有效性和理论准确性。
英文摘要：
We consider the problem of parameter estimation from a generalized linear model with a random design matrix that is orthogonally invariant in law. Such a model allows the design have an arbitrary distribution of singular values and only assumes that its singular vectors are generic. It is a vast generalization of the i.i.d. Gaussian design typically considered in the theoretical literature, and is motivated by the fact that real data often have a complex correlation structure so that methods relying on i.i.d. assumptions can be highly suboptimal. Building on the paradigm of spectrally-initialized iterative optimization, this paper proposes optimal spectral estimators and combines them with an approximate message passing (AMP) algorithm, establishing rigorous performance guarantees for these two algorithmic steps. Both the spectral initialization and the subsequent AMP meet existing conjectures on the fundamental limits to estimation -- the former on the optimal sample complexity for efficient weak recovery, and the latter on the optimal errors. Numerical experiments suggest the effectiveness of our methods and accuracy of our theory beyond orthogonally invariant data.
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