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

Consider the spiked Wigner model [ X = sumi = 1k λi ui ui^top + σ G, ] where G is an N × N GOE random matrix, and the eigenvalues λi are all spiked, i.e. above the Baik-Ben Arous-P'ech'e (BBP) threshold σ. We consider AIC-type model selection criteria of the form [ -2 , (maximised log-likelihood) + γ , (number of parameters) ] for estimating the number k of spikes. For γ > 2, the above criterion is strongly consistent provided λk > λγ, where λγ 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 + δN, where δN → 0 and δN 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.