Conventional regularization is designed to control variance, but in small-data regression it can also aggravate underfitting when predictive signal is concentrated in weak directions of a restricted representation. We study a negative-capable ridge family that permits a feasible negative region whenever the estimator remains well posed, and show that negative regularization acts there as controlled anti-shrinkage by increasing effective complexity most strongly along weak eigendirections. Building on this mechanism, we formalize weak-spectrum underfitting, derive a sign-switch result under conservative baseline shrinkage, and study criterion-based automatic selection over the full negative-capable family. Synthetic and semi-synthetic experiments support the theory by verifying feasibility, spectral complexity increase, sign-switch behavior, and effective recovery of negative adjustments in the predicted regimes.

翻译：传统正则化旨在控制方差，但在小样本回归中，当预测信号集中于受限表示的弱方向时，它也可能加剧欠拟合问题。我们研究了一种支持负值的岭家族，只要估计量保持适定性，它便允许进入可行的负区域，并证明此时负向正则化通过沿弱特征方向最强烈地增加有效复杂度，充当了一种受控的反向收缩机制。基于这一机制，我们形式化了弱谱欠拟合现象，在保守基线收缩下推导出符号翻转结果，并研究了在完整负值族基础上的基于准则自动选择方法。合成和半合成实验通过验证可行性、谱复杂度增加、符号翻转行为以及在预测区域内有效恢复负向调整，为理论提供了支持。