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

Consider the spiked Wigner model $$X = \sum_{i = 1}^k \lambda_i u_i u_i^\top + \sigma G,$$ 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-P\'ech\'e (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 $\delta_N \gg N^{-2/3}$, 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.