AI 中文总结
该研究提出用于保留表示学习的残差代数，通过Fold、FPRC-PQ等实现，在A股数据上显著提升投资收益与夏普比率，增益源于残差所有权与组合的显式建模。
AI 中文摘要
从异构表示中学习通常会简化为特征拼接，这会抹去哪个表示产生了误差。我们转而对残差进行代数化：表示是一种类型化对象，既拥有坐标系，也留有未解决的残差，学习过程是保留或刻意擦除该类型的算子的有序组合。Fold将这些对象实现为10×10秩网格上的时点条件均值场。FPRC-PQ将该代数实现为“松弛-聚合-闭合”：每个场通过适配其自身坐标系中自身残差的校正项进行松弛；校正后的场在固定均值处交汇，该均值是唯一的身份擦除边界；共享学习器仅闭合聚合体的新残差。这种组合精确地 telescoped 为表示、局部残差估计和残差的残差估计。其聚合体是具有总体方差缩减的学习控制变量接口，而沿骨干网络的扰动重新拟合闭合器则产生一阶耦合路径均值正交性。作为分析扩展，反射式沉思算子读取全局重构相对于聚合锚点的位移，对其进行反射，并通过唯一正交投影而非返回调优网格搜索来固定其增益。在2023-2026年367万条中国A股股票日观测值、采用冻结时点协议的条件下，评估的基础代数将成本后净收益从13.52%提升至19.10%，夏普比率从1.42提升至2.09。匹配容量、统一残差、无身份的两阶段及仅成对的对照组均落后于它。因此，该增益无法用更多特征或更多树来解释，而是源于在表示身份仍可用时，明确残差所有权与组合。
英文摘要
Learning from heterogeneous representations is often reduced to feature concatenation, erasing which representation produced each error. We propose residual algebra, in which each representation retains its coordinate system and owns its unresolved residual until an explicit aggregation boundary. Fold instantiates representations as point-in-time conditional-mean fields on 10x10 rank grids, and FPRC-PQ composes them through relax-aggregate-close: each field first fits a correction to its own residual, corrected fields then meet at a fixed mean, and a shared learner closes only the aggregate's fresh residual. We formalize aggregation as a quotient by the zero-sum redistribution kernel, characterizing legal post-aggregation operators as those constant on its cosets. The resulting composition separates representation, local residual estimation, and residual-of-residual estimation, with population variance reduction and first-order coupled-path mean orthogonality. Rumination-B and Rumination-H extend the algebra with quotient-legal finite correction and feedback. On 3.67M Chinese A-share stock-day observations (2023-2026) under a frozen point-in-time protocol, FPRC-PQ raises net-of-cost return from 13.52% to 19.10% and Sharpe from 1.42 to 2.09, outperforming matched-capacity, unified-residual, identity-free two-stage, and pairwise-only controls. The gain is thus attributable to explicit residual ownership and composition rather than additional features or trees.
Comments20 pages, 7 figures
