Residual Algebra for Representation-Preserving Learning

• URL: http://arxiv.org/abs/2608.07349v1
• Date: Fri, 07 Aug 2026 15:44:55 GMT
• Title: Residual Algebra for Representation-Preserving Learning
• Abstract summary: We algebraize the residual, a typed object that owns both a coordinate system and the residual it leaves unresolved.<n> Fold realizes the objects as point-in-time conditional-mean fields on 10x10 rank grids.<n>On 3.67M Chinese A-share stock-day observations (2023-2026) under a frozen point-in-time protocol, the evaluated base algebra raises net-of-cost return from 13.52% to 19.10% and Sharpe from 1.42 to 2.09.
• Score: 7.900277891102576
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Learning from heterogeneous representations is usually reduced to feature concatenation, which erases which representation produced an error. We instead algebraize the residual: a representation is a typed object that owns both a coordinate system and the residual it leaves unresolved, and learning is an ordered composition of operators that preserve or deliberately erase that type. Fold realizes the objects as point-in-time conditional-mean fields on 10x10 rank grids. FPRC-PQ realizes the algebra as relax-aggregate-close: each field is relaxed by a correction fitted to its own residual in its own coordinates; corrected fields meet at a fixed mean that is the sole identity-erasure boundary; and a shared learner closes only the aggregate's fresh residual. The composition telescopes exactly into representation, local residual estimate, and residual-of-residual estimate. Its aggregate is a learned control-variate interface with population variance reduction, while refitting the closer along perturbations of the backbone yields first-order coupled-path mean orthogonality. As an analytical extension, a reflective rumination operator reads the displacement of a global reconstruction from the aggregate anchor, reflects it, and fixes its gain by a unique orthogonal projection rather than return-tuned grid search. On 3.67M Chinese A-share stock-day observations (2023-2026) under a frozen point-in-time protocol, the evaluated base algebra raises net-of-cost return from 13.52% to 19.10% and Sharpe from 1.42 to 2.09. Matched-capacity, unified-residual, identity-free two-stage, and pairwise-only controls all trail it. The gain is therefore not explained by more features or more trees, but by making residual ownership and composition explicit while representation identity is still available.

Related papers

• The concentration game: Bayesian updating, regret, and information [0.42671348654734564]
  We give a two-player zero-sum repeated game between a learner and nature.<n>The terminal payoff is the most a comparator can gain at fixed relative entropy from the prior.<n>With the learner's move otherwise unrestricted, Gibbs/Bayes weights emerge as its unique Bellman equalizer.
  arXiv  Detail & Related papers  (2026-08-18T17:52:26Z)
• Feature Priming in Online Linear Regression: Sparse-Regret Lower Bounds and a Tight Univariate Rate [47.36630149538994]
  In high-dimensional online prediction, the best predictor may depend on only a few features, so regret should scale with sparsity rather than the ambient dimension.<n>Warmuth and Amid asked at COLT 2023 whether any of three such rules admits a competitive online regret guarantee.<n>Our analysis identifies a common obstruction: cheap nuisance causes the refit to underweight the truly predictive coordinate.
  arXiv  Detail & Related papers  (2026-08-18T09:32:03Z)
• A Deep Learning Model for Spatially Clustered Data via Differentiable Cluster Assignment [2.5782420501870296]
  A nonparametric estimator is proposed to learn the partition and the cluster-specific regression functions jointly.<n>A real data analysis is provided to demonstrate the validity and effectiveness of the proposed method.
  arXiv  Detail & Related papers  (2026-08-15T01:39:56Z)
• Residual Dominance as a Structural Account of Last-Item Reliance in Causal Self-Attention Recommenders [47.8060666024071]
  We show that SASRec-style models exhibit highly localized last-item reliance.<n>We then find that residual addition sharply shifts the full-block representation toward same-position contributions.<n>Changing the residual strength induces a monotonic trade-off between structural mixing and last-item reliance.
  arXiv  Detail & Related papers  (2026-08-14T07:09:58Z)
• Simplex Relaxation for Discrete Diffusion [58.33696160279345]
  We introduce Simplax, an exact Dirichlet--categorical augmentation that couples each corrupted categorical state with an auxiliary simplex-valued variable.<n>Simplax improves the generative perplexity--entropy tradeoff on unconditional OpenWebText generation.
  arXiv  Detail & Related papers  (2026-08-11T07:58:17Z)
• CoCurve: Cross-Module Co-Pruning Curvature for Training-Free Structured LLM Pruning [47.406985340436925]
  CoCurve is a calibration-only, fine-tuning-free method that prunes attention and FFN units jointly.<n>It reduces to one budgeted quadratic program, solved in a single shot under a shared attention--FFN budget, with no labels, fine-tuning, or recovery.
  arXiv  Detail & Related papers  (2026-07-20T05:27:57Z)
• The Resolution of Causal Heterogeneity [5.222979968110479]
  Causal subgroup analyses often report a small number of groups summarizing treatment effect heterogeneity, as if that number were a well-defined estimand.<n>We replace it with a new population estimand, the resolution profile, a functional of the causal feature law.<n>Inference is organized around one cross-fitted Bayesian-bootstrap posterior for a single structured moment process.
  arXiv  Detail & Related papers  (2026-07-19T14:56:06Z)
• Right Makes Might: Aligning Verified Hidden States Empowers RL Reasoning [55.264863369127774]
  Current methods reduce each correct rollout to a single reward bit, ignoring the geometric structure shared among their hidden states.<n>We propose Hidden-Align, an auxiliary loss function that aligns the last-layer hidden states of correct rollouts at the anchor token during RL training, with zero overhead in both training and inference.<n>On eight mathematical reasoning benchmarks, Hidden-Align improves average pass@1 over the DAPO baseline by 3.8, 6.2, and 5.4 percentage points on Qwen3-1.7B, 4B, and 14B respectively, with consistent pass@k gains across all three scales.
  arXiv  Detail & Related papers  (2026-06-02T06:51:15Z)
• A Geometric View of SRC: Learning Representations for Stable Residual Inference [0.0]
  Reconstruction-based inference assigns a class by comparing class-wise reconstruction residuals.<n>Sparse Representation Classification (SRC) is a canonical instance whose reliability depends on the geometry of the learned representation.<n>We formalize residual-ordering stability through a residual margin and characterize geometric obstructions.
  arXiv  Detail & Related papers  (2026-05-28T09:33:18Z)
• Quotient-Categorical Representations for Bellman-Compatible Average-Reward Distributional Reinforcement Learning [8.988097534171995]
  We introduce a quotient-space formulation in which state-indexed bias laws are identified up to a common translation.<n>We show that synchronous exact updates are gain-independent at the quotient-law level.
  arXiv  Detail & Related papers  (2026-05-11T22:17:09Z)
• In-Context Symbolic Regression for Robustness-Improved Kolmogorov-Arnold Networks [3.7163623763519733]
  Symbolic regression aims to replace blackbox predictors with analytical expressions that can be inspected and validated in machine learning.<n> Kolmogorov-Arnold Pursuit Networks (KANs) are well suited to this goal because each connection between adjacent units (an "edge") ismetd by a univariate function.<n>In practice, however, symbolic extraction is a bottleneck: the standard KAN-to-symbol approach fits operators to each learned edge function in fine isolation.
  arXiv  Detail & Related papers  (2026-03-16T13:21:26Z)
• Deterministic Zeroth-Order Mirror Descent via Vector Fields with A Posteriori Certification [45.85698554568285]
  We develop a deterministic zeroth-order mirror descent framework by replacing gradients with a general vector field.<n>Our analysis provides an evaluation template for last-rate inequality evaluations.<n>Together, these results expose a hidden geometric linking Bregman identities, deterministic certification, and robust conic geometry in zeroth-order mirror descent.
  arXiv  Detail & Related papers  (2026-01-31T10:05:05Z)
• Improve Representation for Imbalanced Regression through Geometric Constraints [8.903197320328164]
  We focus on ensuring uniformity in the latent space for imbalanced regression through two key losses.<n>The enveloping loss encourages the induced trace to uniformly occupy the surface of a hypersphere, while the homogeneity loss ensures smoothness.<n>Our method integrates these geometric principles into the data representations via a Surrogate-driven Representation Learning framework.
  arXiv  Detail & Related papers  (2025-03-02T12:31:34Z)
• PseudoNeg-MAE: Self-Supervised Point Cloud Learning using Conditional Pseudo-Negative Embeddings [55.55445978692678]
  PseudoNeg-MAE enhances global feature representation of point cloud masked autoencoders by making them both discriminative and sensitive to transformations.<n>We propose a novel loss that explicitly penalizes invariant collapse, enabling the network to capture richer transformation cues while preserving discriminative representations.
  arXiv  Detail & Related papers  (2024-09-24T07:57:21Z)
• Self-Supervised Learning Disentangled Group Representation as Feature [82.07737719232972]
  We show that existing Self-Supervised Learning (SSL) only disentangles simple augmentation features such as rotation and colorization. We propose an iterative SSL algorithm: Iterative Partition-based Invariant Risk Minimization (IP-IRM) We prove that IP-IRM converges to a fully disentangled representation and show its effectiveness on various benchmarks.
  arXiv  Detail & Related papers  (2021-10-28T16:12:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.