稀疏贝叶斯学习的大系统分析

中文摘要 AI 辅助

稀疏贝叶斯学习广泛用于稀疏线性逆问题，然而由于其所有方差超参数均从同一数据中估计，其大系统平稳行为仍鲜为人知。我们研究经典的稀疏贝叶斯学习，将其表述为证据最大化（第二类最大似然），针对具有独立同分布高斯元素的感知矩阵、高斯测量噪声和未知确定性信号序列的欠定线性模型。超参数重新优化会引入一个不消失的反馈项：一个典型坐标服从重新优化校正的标量高斯定律，其信号系数由归一化自适应响应而非冻结的解析迹控制。单坐标留一法构造给出了精确的条件高斯定律，该定律被转移到一个选定的完整平稳分支，而无需假设约简和完整平稳向量的渐近接近性。将该定律与证据目标的Karush--Kuhn--Tucker条件相结合，产生一个一般集值的标量关系和三个分支式大系统一致性关系。如果模型噪声方差通过证据最大化联合估计，内部联合平稳性产生归一化残差能量与归一化解析迹之间的精确有限维等式。当极限信号定律在零处具有非零质量时，该恒等式进一步产生一个无参数的渐近卡方零分布。在所选标量分支的额外可微性条件下，大系统表征还为后验均值的重建误差提供了封闭关系。该分析基于平稳点，并允许多个平稳分支。

英文摘要

Sparse Bayesian learning is widely used for sparse linear inverse problems, yet its large-system stationary behavior remains poorly understood because all variance hyperparameters are estimated from the same data. We study classical sparse Bayesian learning, formulated as evidence maximization (type-II maximum likelihood), for underdetermined linear models with sensing matrices having independent and identically distributed Gaussian entries, Gaussian measurement noise, and an unknown deterministic signal sequence. Hyperparameter reoptimization induces a nonvanishing feedback term: a typical coordinate obeys a reoptimization-corrected scalar Gaussian law whose signal coefficient is governed by the normalized adaptive response rather than the frozen resolvent trace. A one-coordinate leave-one-out construction gives an exact conditional Gaussian law, which is transferred to a selected full stationary branch without assuming asymptotic closeness of the reduced and full stationary vectors. Combining this law with the Karush--Kuhn--Tucker conditions of the evidence objective yields a generally set-valued scalar relation and three branchwise large-system consistency relations. If the model noise variance is jointly estimated by evidence maximization, interior joint stationarity yields an exact finite-dimensional equality between the normalized residual energy and normalized resolvent trace. When the limiting signal law has nonzero mass at zero, this identity further yields a parameter-free asymptotic chi-square null law. Under an additional differentiability condition on the selected scalar branch, the large-system characterization also gives a closed relation for the reconstruction error of the posterior mean. The analysis is stationary-point based and permits multiple stationary branches.

发表机构

• Yunnan University（云南大学）
• EURECOM（欧洲通信与多媒体研究所）

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