Consistent model selection in the spiked Wigner model via AIC-type criteria
Soumendu Sundar Mukherjee
TL;DR
This work analyzes AIC-type model selection in the spiked Wigner model with GOE noise to consistently estimate the number of spikes. It derives explicit AIC^{(γ)} scores for both known and unknown noise levels and establishes sharp consistency regimes: γ≤2 tends to overfit, γ>2 yields strong consistency when spikes exceed a threshold above BBP, and choosing γ_N=2+δ_N with δ_N→0 and δ_N≫N^{-2/3} achieves weak consistency; a soft-AIC variant attains strong consistency. The authors also extend the framework to stochastic block models, deriving analogous consistency results for estimating the number of communities via an AIC-like criterion applied to a centered, scaled adjacency matrix. Empirical results across GOE, structured noise, and SBM settings, including real networks, corroborate the theoretical findings and demonstrate practical robustness, with data-driven approaches to nuisance parameter estimation ( σ and thresholds ). The results pave the way for flexible, high-dimensional model selection in spectral spiked models and related network problems, with potential extensions to more general noise structures and growing model order.$
Abstract
Consider the spiked Wigner model \[ X = \sum_{i = 1}^k λ_i u_i u_i^\top + σG, \] where $G$ is an $N \times N$ GOE random matrix, and the eigenvalues $λ_i$ are all spiked, i.e. above the Baik-Ben Arous-Péché (BBP) threshold $σ$. We consider AIC-type model selection criteria of the form \[ -2 \, (\text{maximised log-likelihood}) + γ\, (\text{number of parameters}) \] for estimating the number $k$ of spikes. For $γ> 2$, the above criterion is strongly consistent provided $λ_k > λ_γ$, where $λ_γ$ is a threshold strictly above the BBP threshold, whereas for $γ< 2$, it almost surely overestimates $k$. Although AIC (which corresponds to $γ= 2$) is not strongly consistent, we show that taking $γ= 2 + δ_N$, where $δ_N \to 0$ and $δ_N \gg N^{-2/3}$, results in a weakly consistent estimator of $k$. We further show that a soft minimiser of AIC, where one chooses the least complex model whose AIC score is close to the minimum AIC score, is strongly consistent. Based on a spiked (generalised) Wigner representation, we also develop similar model selection criteria for consistently estimating the number of communities in a balanced stochastic block model under some sparsity restrictions.