Training data reconstruction from KKT conditions has shown striking empirical success, yet it remains unclear when the resulting KKT equations have unique solutions and, even in identifiable regimes, how to reliably recover solutions by optimization. This work hereby focuses on these two complementary questions: identifiability and optimization. On the identifiability side, we discuss the sufficient conditions for KKT system of two-layer networks with polynomial activations to uniquely determine the training data, providing a theoretical explanation of when and why reconstruction is possible. On the optimization side, we introduce sample splitting, a curvature-aware refinement step applicable to general reconstruction objectives (not limited to KKT-based formulations): it creates additional descent directions to escape poor stationary points and refine solutions. Experiments demonstrate that augmenting several existing reconstruction methods with sample splitting consistently improves reconstruction performance.

翻译：基于KKT条件的训练数据重构已展现出显著的实证成功，但由此产生的KKT方程组何时具有唯一解，以及在可识别机制下如何通过优化可靠地获得解，这些问题仍未明晰。本研究聚焦于这两个互补性问题：可识别性与优化。在可识别性方面，我们讨论了具有多项式激活函数的两层网络其KKT系统唯一确定训练数据的充分条件，为重构何时及为何可行提供了理论解释。在优化方面，我们提出样本分割——一种适用于一般重构目标（不限于基于KKT的公式）的曲率感知优化步骤：它通过生成额外的下降方向以逃离不良驻点并优化解。实验表明，在多种现有重构方法中引入样本分割均能持续提升重构性能。