破解数据重构的可识别性与优化难题，提升重构精度。
Data Reconstruction: Identifiability and Optimization with Sample Splitting
- 基于KKT条件分析两层网络的可识别性，给出唯一重构的理论条件。
- 引入样本分割法，增强优化过程的下降方向，有效跳出劣解。
- 适用于多种重构方法，实验验证性能稳定提升。
从KKT条件进行训练数据重构已展现出显著的实证效果，但其对应的KKT方程何时具有唯一解仍不明确，即使在可识别情形下，如何通过优化可靠恢复解也尚未解决。本文聚焦这两个互补问题：可识别性与优化。在可识别性方面，研究了带多项式激活函数的两层网络的KKT系统，提出了唯一确定训练数据的充分条件，为重构的可行性提供了理论解释。在优化方面，提出样本分割（sample splitting），一种适用于一般重构目标的曲率感知精炼步骤：通过引入额外下降方向，帮助逃离不良驻点并精化解。实验表明，将该方法应用于多个现有重构算法，均能持续提升重构性能。
原文摘要 · Abstract (English)
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.
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