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Shorting Dynamics and Structured Kernel Regularization

James Tian

TL;DR

The paper introduces a nonlinear operator shorting dynamic that progressively suppresses a prescribed nuisance subspace while preserving structure elsewhere, extending classical shorted operators to a dynamical setting and to reproducing kernel Hilbert spaces. It establishes monotone, convergent flows for operators and kernels, with a residual additive decomposition that clarifies what is removed at each step. In finite samples, the dynamics induce a structured path for Gram operators and kernel ridge predictors, enabling invariance to nuisance factors and a maximal invariant kernel under dominance constraints. The framework supports task-driven design (greedy, covariance-based, or nuisance-based shorting) and yields interpretable decompositions and flexible regularization. Overall, it provides a unified operator-analytic approach to invariant kernel construction and structured regularization in data analysis, with a clear geometric and algorithmic interpretation of each step.

Abstract

This paper develops a nonlinear operator dynamic that progressively removes the influence of a prescribed feature subspace while retaining maximal structure elsewhere. The induced sequence of positive operators is monotone, admits an exact residual decomposition, and converges to the classical shorted operator. Transporting this dynamic to reproducing kernel Hilbert spaces yields a corresponding family of kernels that converges to the largest kernel dominated by the original one and annihilating the given subspace. In the finite-sample setting, the associated Gram operators inherit a structured residual decomposition that leads to a canonical form of kernel ridge regression and a principled way to enforce nuisance invariance. This gives a unified operator-analytic approach to invariant kernel construction and structured regularization in data analysis.

Shorting Dynamics and Structured Kernel Regularization

TL;DR

The paper introduces a nonlinear operator shorting dynamic that progressively suppresses a prescribed nuisance subspace while preserving structure elsewhere, extending classical shorted operators to a dynamical setting and to reproducing kernel Hilbert spaces. It establishes monotone, convergent flows for operators and kernels, with a residual additive decomposition that clarifies what is removed at each step. In finite samples, the dynamics induce a structured path for Gram operators and kernel ridge predictors, enabling invariance to nuisance factors and a maximal invariant kernel under dominance constraints. The framework supports task-driven design (greedy, covariance-based, or nuisance-based shorting) and yields interpretable decompositions and flexible regularization. Overall, it provides a unified operator-analytic approach to invariant kernel construction and structured regularization in data analysis, with a clear geometric and algorithmic interpretation of each step.

Abstract

This paper develops a nonlinear operator dynamic that progressively removes the influence of a prescribed feature subspace while retaining maximal structure elsewhere. The induced sequence of positive operators is monotone, admits an exact residual decomposition, and converges to the classical shorted operator. Transporting this dynamic to reproducing kernel Hilbert spaces yields a corresponding family of kernels that converges to the largest kernel dominated by the original one and annihilating the given subspace. In the finite-sample setting, the associated Gram operators inherit a structured residual decomposition that leads to a canonical form of kernel ridge regression and a principled way to enforce nuisance invariance. This gives a unified operator-analytic approach to invariant kernel construction and structured regularization in data analysis.

Paper Structure

This paper contains 17 sections, 14 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Kernel Shorting Dynamics
  3. Invariance and Maximality of the Limit Kernel
  4. Dynamic Kernel Regularization on Finite Samples
  5. Gram Operators and Their Dynamic Evolution
  6. RKHS Norms and Dynamic Geometry
  7. Dynamic Kernel Ridge Regression on Finite Samples
  8. Interpretation
  9. Task-Driven Shorting and Learning Dynamics
  10. Task-Driven Design of the Effects
  11. Shorting-Based Energy Decomposition
  12. Greedy Rank-One Shorting
  13. Specialization to Kernel Ridge Regression
  14. Nuisance-Invariant Limits
  15. Summary
  16. ...and 2 more sections

Key Result

Lemma 2.1

For each $n\ge0$, $R_{n}$ is a positive operator on $\mathcal{H}$, and

Figures (1)

  • Figure 6.1: Evolution of the decision boundary under shorting dynamics. As $n$ increases, the kernel $K_{n}$ increasingly suppresses the nuisance coordinate $u$, causing the decision boundary (solid lines) to rotate from the tilted data-generating direction (at $n=0$) to the vertical invariant limit (at $n=\infty$).

Theorems & Definitions (29)

  • Lemma 2.1: Positivity and monotonicity
  • proof
  • Lemma 2.2: Residual identity
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 19 more