---
title: "Zero-Complement Extrapolation: Parameter Recovery and Classification in Severely Imbalanced Logistic Regression under Rank-Deficient Regimes"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2610.39194v1"
permalink: "https://feed.astralor.com/items/itm_82ca87a116330ea69c7b42ed"
published: "2026-10-29T02:15:44.000Z"
score: 57
channel: "ai"
tags: ["Zero-Complement Extrapolation","Parameter Recovery","Classification","Severely Imbalanced Logistic","Regression","Rank-Deficient Regimes Parameter","Under","Parameter"]
---

# Zero-Complement Extrapolation: Parameter Recovery and Classification in Severely Imbalanced Logistic Regression under Rank-Deficient Regimes

## 速览

Parameter estimation and classification in binary logistic regression deteriorate severely when the number of minority class observations ($n_1$) is smaller than the feature dimension ($p$). Under such rank-deficient ra…

## 关键点

- Parameter estimation and classification in binary logistic regression deteriorate severely when the number of minority class observations ($n_1$) is smaller than the feature dimension ($p$).
- Under such rank-deficient rarity ($n_1 < p$), maximum likelihood estimators suffer from quasi-complete separation, penalized likelihood methods exhibit severe intercept bias, and convex minority oversampling techniques …
- In this paper, we propose a novel \textit{Self-Calibrated Zero-Complement} estimator that shifts the paradigm from minority interpolation to majority fringe extrapolation.

## 全文

Parameter estimation and classification in binary logistic regression deteriorate severely when the number of minority class observations ($n_1$) is smaller than the feature dimension ($p$). Under such rank-deficient rarity ($n_1 < p$), maximum likelihood estimators suffer from quasi-complete separation, penalized likelihood methods exhibit severe intercept bias, and convex minority oversampling techniques such as SMOTE collapse into an $(n_1 - 1)$-dimensional affine subspace. In this paper, we propose a novel \textit{Self-Calibrated Zero-Complement} estimator that shifts the paradigm from minority interpolation to majority fringe extrapolation. Grounded in Stein's Lemma, we prove that the conditional expectation shift of the abundant majority class ($Y=0$) induced by logistic truncation uniquely identifies the true hyperplane normal $\boldsymbolβ$ after Mahalanobis whitening. We further establish that extrapolating synthetic minority instances outward from the high-quantile density fringe of the majority distribution along the conjugate direction preserves full-rank covariance regularity ($\mathrm{rank} = p$). To accommodate both semi-supervised environments (with an unlabeled background mean) and strictly supervised subsamples, we introduce a hybrid estimator incorporating the intrinsic third-moment skewness of the majority class alongside a location-shift-invariant zero-boundary intercept calibration. Extensive Monte Carlo simulations and real-world benchmark applications (Wisconsin Diagnostic Breast Cancer and Banknote Authentication) under extreme rarity ($n_1 \in \{3, 5\}$) demonstrate that our approach prevents parameter divergence, recovers the oracle hyperplane with high fidelity, and substantially outperforms Firth's regression and SMOTE in F1-score and estimation stability.

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