Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Xiao, Fangqing, Slock, Dirk T. M.
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212008566
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