# Residual Algebra for Representation-Preserving Learning

arXiv: https://arxiv.org/abs/2608.07349 · HTML: https://mlsanity.com/paper/2608.07349/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 34 references that are in the map.
- First posted on arXiv 7 Aug 2026.

## Builds on (most cited first, 10 of 10)

- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Revisiting Deep Learning Models for Tabular Data](https://mlsanity.com/paper/2106.11959/) (2021)
- [TabNet: Attentive Interpretable Tabular Learning](https://mlsanity.com/paper/1908.07442/) (2019)
- [Tabular Data: Deep Learning is Not All You Need](https://mlsanity.com/paper/2106.03253/) (2021)
- [Deep Neural Networks and Tabular Data: A Survey](https://mlsanity.com/paper/2110.01889/) (2021)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [When Do Neural Nets Outperform Boosted Trees on Tabular Data?](https://mlsanity.com/paper/2305.02997/) (2023)
- [On Embeddings for Numerical Features in Tabular Deep Learning](https://mlsanity.com/paper/2203.05556/) (2022)
- [Factor Fields: A Unified Framework for Neural Fields and Beyond](https://mlsanity.com/paper/2302.01226/) (2023)

## Abstract (arXiv)

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.