Consistent model selection in the spiked Wigner model via AIC-type criteria
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917915563917312 |
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| author | Mukherjee, Soumendu Sundar |
| author_facet | Mukherjee, Soumendu Sundar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12982 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Consistent model selection in the spiked Wigner model via AIC-type criteria Mukherjee, Soumendu Sundar Statistics Theory Information Theory Methodology Machine Learning 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. |
| title | Consistent model selection in the spiked Wigner model via AIC-type criteria |
| topic | Statistics Theory Information Theory Methodology Machine Learning |
| url | https://arxiv.org/abs/2307.12982 |