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Why Inference in Large Models Becomes Decomposable After Training

Jidong Jin

TL;DR

The paper tackles the exponential growth of inference cost in very large models by showing that post-training inference systems naturally decompose into independent sub-operators due to localized gradient updates during training. It introduces a post-training statistical annealing framework that elevates structurally confirmed dependencies to explicit system-level blocks, and a permutation-based restructuring to expose outer block-diagonal and inner Frobenius normal forms. Through edge-wise tests and multi-channel adaptations, the method identifies which connections are meaningfully learned and which remain dormant, enabling modular, parallel inference without changing interfaces. The work further proposes a three-stage deployment framework and a practical algorithm for constructing the permutation that exposes the decomposed structure. Overall, the approach provides a principled, post-training mechanism to reduce inference complexity, improve stability, and guide scalable engineering of large-scale AI systems.

Abstract

Inference in large-scale AI models is typically performed on dense parameter matrices, leading to inference cost and system complexity that scale unsustainably with model size. This limitation does not arise from insufficient model capacity, but from treating post-training inference systems as monolithic operators while ignoring internal structures formed during learning. We show that gradient update events in large models are highly localized and selective, leaving many parameter dependencies statistically indistinguishable from their initialization distribution after training. As a result, post-training inference systems are structurally non-uniform and inherently decomposable. Based on this observation, we introduce a post-training statistical criterion and a structural annealing procedure that removes unsupported dependencies and reveals stable, independent substructures. This work establishes a post-training, model-agnostic structural view of inference systems and enables structured, parallel inference without modifying model functionality or interfaces.

Why Inference in Large Models Becomes Decomposable After Training

TL;DR

The paper tackles the exponential growth of inference cost in very large models by showing that post-training inference systems naturally decompose into independent sub-operators due to localized gradient updates during training. It introduces a post-training statistical annealing framework that elevates structurally confirmed dependencies to explicit system-level blocks, and a permutation-based restructuring to expose outer block-diagonal and inner Frobenius normal forms. Through edge-wise tests and multi-channel adaptations, the method identifies which connections are meaningfully learned and which remain dormant, enabling modular, parallel inference without changing interfaces. The work further proposes a three-stage deployment framework and a practical algorithm for constructing the permutation that exposes the decomposed structure. Overall, the approach provides a principled, post-training mechanism to reduce inference complexity, improve stability, and guide scalable engineering of large-scale AI systems.

Abstract

Inference in large-scale AI models is typically performed on dense parameter matrices, leading to inference cost and system complexity that scale unsustainably with model size. This limitation does not arise from insufficient model capacity, but from treating post-training inference systems as monolithic operators while ignoring internal structures formed during learning. We show that gradient update events in large models are highly localized and selective, leaving many parameter dependencies statistically indistinguishable from their initialization distribution after training. As a result, post-training inference systems are structurally non-uniform and inherently decomposable. Based on this observation, we introduce a post-training statistical criterion and a structural annealing procedure that removes unsupported dependencies and reveals stable, independent substructures. This work establishes a post-training, model-agnostic structural view of inference systems and enables structured, parallel inference without modifying model functionality or interfaces.
Paper Structure (46 sections, 6 theorems, 97 equations, 6 figures, 2 tables)

This paper contains 46 sections, 6 theorems, 97 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Training Dynamics Analysis
  3. Scope and Objects of Study
  4. Local Gradient Updates
  5. Model Decomposability
  6. Implications for the Further Development of Large Models
  7. Inference System Annealing
  8. Motivation and the Choice of Restructuring Units
  9. Statistical Testing of Edge Properties in Simple Neural Networks
  10. Initial Population Test (Neyman Significance Test)
  11. Random-Walk Hypothesis Test (Equiprobability Test)
  12. Statistical Tests for Edge Properties in Multi-Channel Joint-Operator FNNs
  13. Initial Population Test Based on Sequence Norm
  14. Multi-Channel Consistency Test Based on the Random-Walk Hypothesis
  15. Summary of Annealing Rules
  16. ...and 31 more sections

Key Result

Lemma 2.1

Let $w_{ij}$ and $w_{ji}$ be parameters (edges) in the matrix $W$. If gradient updates of the inference system $Y=WX$ follow the local update rule EQ:LOCAL-UPDATE-MATRIX, then a necessary and sufficient condition for $w_{ij}$ or $w_{ji}$ to experience a gradient update event induced by training samp Here $S(t)$ denotes the activated state set during the $t$-th inference, and $x_i(t),x_j(t)$ are th

Figures (6)

  • Figure 4.1: Schematic structure of a large matrix
  • Figure 4.2: Structural principle of inference system restructuring
  • Figure 4.3: Principle of diagonal-block subdivision in inference restructuring
  • Figure A.1: Original graph
  • Figure A.2: Schematic illustration of the permuted structural decomposition
  • ...and 1 more figures

Theorems & Definitions (18)

  • Lemma 2.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.1
  • ...and 8 more