Consistent model selection in the spiked Wigner model via AIC-type criteria
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Consider the spiked Wigner model X = ∑_i = 1^k λ_i u_i u_i^⊤ + σ G, where G is an N × 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 (maximised log-likelihood) + γ (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 → 0 and δ_N ≫ N^-2/3, results in a weakly consistent estimator of k. We also show that a certain soft minimiser of AIC is strongly consistent.
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