A Low-Rank Subspace Analysis of LLM Interventions

2026-06-12

AI Summary

This work addresses the challenge of precisely controlling specific behaviors—such as refusal or sycophancy—in large language models, where targeted interventions often produce unintended side effects. The authors propose a low-rank subspace diagnostic framework that reveals, for the first time, that distinct behaviors share internal representations in activation space. Through geometric analysis of decision subspaces and the mean squared cosine of principal angles, they demonstrate that intervention effects propagate asymmetrically, depending on the degree of subspace overlap and the angular proximity to the decision subspace. Experiments across multiple instruction-tuned models (7B–70B) show that behaviors exhibiting high representational overlap and closer alignment with the decision subspace are more susceptible to intervention, thereby explaining the fundamental difficulty in achieving independent behavioral control.

Abstract

Interventions designed to modify a particular behavior in LLMs, such as refusal or sycophancy, often produce unintended changes in other behaviors. This lack of targeted control makes it difficult to design and implement reliable safety controls. To understand these side-effects, we introduce a diagnostic framework for analyzing interacting behaviors in LLMs. We model behaviors as low-rank subspaces in activation space, and study how interventions influence across behaviors. Across multiple instruction-tuned models (7B-70B) and across refusal, jailbreak, and sycophancy settings, we find that different behaviors share internal representations, and intervening on one behavior alters others in asymmetric ways. Some behaviors act as upstream control points whose interventions propagate broadly across other behaviors, while others remain more isolated. We relate these effects to two geometric quantities: (i) the overlap between behavior subspaces, measured as the average squared cosine of principal angles, and (ii) the angle between each behavior subspace and the decision subspace (capturing the model's final decision e.g., refuse vs. comply). Empirically, intervention effects on other behaviors tend to be larger for behavior pairs with higher subspace overlap, and for source behaviors whose subspaces lie closer (smaller angle) to the decision subspace. These findings highlight a challenge for targeted behavior control: behaviors are difficult to modify independently, as interventions can propagate through shared representations and asymmetric interactions.

Problem

• behavioral interference
• large language models
• targeted control
• unintended side effects
• safety interventions

Innovation

• low-rank subspace
• behavior intervention
• activation space
• subspace overlap
• decision subspace