Consistent model selection in the spiked Wigner model via AIC-type criteria

Consider the spiked Wigner model $$X=\sum _{i=1}^k{\lambda }_i{u}_i{u}_i^{\mathrm{\top }}+\sigma G,$$ where $G$ is an $N\times N$ GOE random matrix, and the eigenvalues ${\lambda }_i$ are all spiked, i.e. above the Baik-Ben Arous-Péché (BBP) threshold $\sigma$. We consider AIC-type model selection criteria of the form $$-2(\text{maximised log-likelihood})+\gamma (\text{number of parameters})$$ for estimating the number $k$ of spikes. For $\gamma >2$, the above criterion is strongly consistent provided ${\lambda }_k>{\lambda }_{\gamma }$, where ${\lambda }_{\gamma }$ is a threshold strictly above the BBP threshold, whereas for $\gamma <2$, it almost surely overestimates $k$. Although AIC (which corresponds to $\gamma =2$) is not strongly consistent, we show that taking $\gamma =2+{\delta }_N$, where ${\delta }_N\to 0$ and ${\delta }_N\gg N^{-2\mathrm{/}3}$, 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.