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First-Order Geometry, Spectral Compression, and Structural Compatibility under Bounded Computation

Changkai Li

TL;DR

This work proposes an operator-theoretic formulation in which computational or feasibility limitations are encoded by self-adjoint operators defining locally reachable subspaces, and demonstrates that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression.

Abstract

Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which computational or feasibility limitations are encoded by self-adjoint operators defining locally reachable subspaces. In this setting, the optimal first-order improvement direction emerges as a pseudoinverse-weighted gradient, revealing how constraints induce a distorted ascent geometry. We further demonstrate that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression, and establish a compatibility principle that characterizes the existence of common admissible directions across multiple objectives. The resulting framework unifies gradient projection, spectral truncation, and multi-objective feasibility within a single geometric structure.

First-Order Geometry, Spectral Compression, and Structural Compatibility under Bounded Computation

TL;DR

This work proposes an operator-theoretic formulation in which computational or feasibility limitations are encoded by self-adjoint operators defining locally reachable subspaces, and demonstrates that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression.

Abstract

Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which computational or feasibility limitations are encoded by self-adjoint operators defining locally reachable subspaces. In this setting, the optimal first-order improvement direction emerges as a pseudoinverse-weighted gradient, revealing how constraints induce a distorted ascent geometry. We further demonstrate that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression, and establish a compatibility principle that characterizes the existence of common admissible directions across multiple objectives. The resulting framework unifies gradient projection, spectral truncation, and multi-objective feasibility within a single geometric structure.
Paper Structure (28 sections, 3 theorems, 42 equations)

This paper contains 28 sections, 3 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Problem Statement and Minimal Setting
  3. Multi-Agent Strategy Structure
  4. Complexity Constraint and Computational Budget
  5. Local Feasibility Structure
  6. Problem Formulation
  7. Geometric Structure of Bounded Computation
  8. Reachable Direction Structure
  9. Computational Geometry Operator
  10. Pseudoinverse Structure
  11. Decomposition of Tangent Space
  12. Theorem 1: First-Order Constrained Optimal Direction
  13. Computational Effort Geometry
  14. First-Order Admissible Strategy Direction Problem
  15. Main Result
  16. ...and 13 more sections

Key Result

Theorem 1

Fix $\theta\in\Theta$ and assume $J$ is differentiable at $\theta$. Let $H_C(\theta)$ satisfy the properties in Section 2 and let $H_C^\dagger(\theta)$ denote its Moore--Penrose pseudoinverse. If $H_C(\theta) g \neq 0$, then the maximizers of eq:varproblem are exactly the rays generated by normalized so that $\mathcal{E}_\theta(\Delta\theta^\star)=1$. If $H_C(\theta) g = 0$, then and no admissib

Theorems & Definitions (7)

  • Theorem 1: First-Order Constrained Optimal Direction
  • proof
  • Theorem 2: Low-Rank Approximation of First-Order Direction
  • proof
  • Proposition 1
  • proof
  • Remark