Phase transitions for the existence of unregularized M-estimators in single index models
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| Format: | Preprint |
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2025
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| author | Koriyama, Takuya Bellec, Pierre C. |
| author_facet | Koriyama, Takuya Bellec, Pierre C. |
| contents | This paper studies phase transitions for the existence of unregularized M-estimators under proportional asymptotics where the sample size $n$ and feature dimension $p$ grow proportionally with $n/p \to δ\in (1, \infty)$. We study the existence of M-estimators in single-index models where the response $y_i$ depends on covariates $x_i \sim N(0, I_p)$ through an unknown index ${w} \in \mathbb{R}^p$ and an unknown link function. An explicit expression is derived for the critical threshold $δ_\infty$ that determines the phase transition for the existence of the M-estimator, generalizing the results of Candés & Sur (2020) for binary logistic regression to other single-index models.
Furthermore, we investigate the existence of a solution to the nonlinear system of equations governing the asymptotic behavior of the M-estimator when it exists. The existence of solution to this system for $δ> δ_\infty$ remains largely unproven outside the global null in binary logistic regression. We address this gap with a proof that the system admits a solution if and only if $δ> δ_\infty$, providing a comprehensive theoretical foundation for proportional asymptotic results that require as a prerequisite the existence of a solution to the system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_03163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Phase transitions for the existence of unregularized M-estimators in single index models Koriyama, Takuya Bellec, Pierre C. Statistics Theory This paper studies phase transitions for the existence of unregularized M-estimators under proportional asymptotics where the sample size $n$ and feature dimension $p$ grow proportionally with $n/p \to δ\in (1, \infty)$. We study the existence of M-estimators in single-index models where the response $y_i$ depends on covariates $x_i \sim N(0, I_p)$ through an unknown index ${w} \in \mathbb{R}^p$ and an unknown link function. An explicit expression is derived for the critical threshold $δ_\infty$ that determines the phase transition for the existence of the M-estimator, generalizing the results of Candés & Sur (2020) for binary logistic regression to other single-index models. Furthermore, we investigate the existence of a solution to the nonlinear system of equations governing the asymptotic behavior of the M-estimator when it exists. The existence of solution to this system for $δ> δ_\infty$ remains largely unproven outside the global null in binary logistic regression. We address this gap with a proof that the system admits a solution if and only if $δ> δ_\infty$, providing a comprehensive theoretical foundation for proportional asymptotic results that require as a prerequisite the existence of a solution to the system. |
| title | Phase transitions for the existence of unregularized M-estimators in single index models |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2501.03163 |