Toward a Flexible Framework for Linear Representation Hypothesis Using Maximum Likelihood Estimation

TL;DR

This paper introduces SAND, a new method using maximum likelihood estimation and activation differences to derive concept directions in LLMs, overcoming previous limitations and improving flexibility and performance.

Contribution

We propose a novel MLE-based approach, SAND, that models activation differences as vMF distributions to compute concept directions without relying on unembedding or single-token pairs.

Findings

SAND outperforms previous methods in activation engineering tasks.

The approach is more flexible and applicable to complex, context-dependent concepts.

Experiments show improved monitoring and manipulation of LLM representations.

Abstract

Linear representation hypothesis posits that high-level concepts are encoded as linear directions in the representation spaces of LLMs. Park et al. (2024) formalize this notion by unifying multiple interpretations of linear representation, such as 1-dimensional subspace representation and interventions, using a causal inner product. However, their framework relies on single-token counterfactual pairs and cannot handle ambiguous contrasting pairs, limiting its applicability to complex or context-dependent concepts. We introduce a new notion of binary concepts as unit vectors in a canonical representation space, and utilize LLMs' (neural) activation differences along with maximum likelihood estimation (MLE) to compute concept directions (i.e., steering vectors). Our method, Sum of Activation-base Normalized Difference (SAND), formalizes the use of activation differences modeled as samples from a von Mises-Fisher (vMF) distribution, providing a principled approach to derive concept directions. We extend the applicability of Park et al. (2024) by eliminating the dependency on unembedding representations and single-token pairs. Through experiments with LLaMA models across diverse concepts and benchmarks, we demonstrate that our lightweight approach offers greater flexibility, superior performance in activation engineering tasks like monitoring and manipulation.