---
title: "Linear Representation Hypothesis — Why LLM Concepts Are Directions"
description: "High-level concepts like refusal, sentiment, and honesty are encoded as linear directions in activation space. This hypothesis underpins abliteration, steering, and probing."
keywords: "linear representation hypothesis, linear probing, concept directions, LLM interpretability, activation space geometry"
---

<!-- SPDX-FileCopyrightText: 2026 Teilo Millet -->
<!-- SPDX-License-Identifier: Apache-2.0 -->

# Linear Representation Hypothesis

The theoretical foundation for why activation-space interventions work.

## The hypothesis

High-level concepts in neural networks — refusal, sentiment, truthfulness, style, toxicity — are represented as **linear directions** in activation space. A concept is not smeared across the network in an inscrutable way; it corresponds to a vector, and the strength of that concept in a given activation is the scalar projection onto that vector.

Formally, for a concept $c$ with associated direction $\hat{d}_c \in \mathbb{R}^{d_{\text{model}}}$ and an activation $a$:

$$\text{strength}(c, a) = \langle a, \hat{d}_c \rangle$$

This is a strong claim. It does not say that *everything* is linear — only that many behaviorally important properties are. Refusal turns out to be one of the cleanest examples, nearly perfectly rank-1.

> **What does "linear" mean here?** — Think of a compass needle. No matter how complex the terrain, the compass gives you a single reading: how much you are aligned with north. A linear direction works the same way — it reduces a high-dimensional activation to a single number (the projection) that tells you "how much of concept X is present." The concept itself may emerge from complex interactions across layers, but its *readout* is a dot product. A non-linear concept would be more like a winding mountain path — you cannot summarize your position with one number. Refusal is a compass; some behaviors are paths.

## Evidence

The evidence for linear representation comes from multiple independent lines of work:

**Probing classifiers.** Train a linear classifier on activations to predict a concept label (e.g., "is this prompt harmful?"). If a linear probe achieves high accuracy, the concept is linearly decodable. This has been shown for sentiment, factuality, part-of-speech, entity type, and many other properties.

> **Linear probe** — a simple classifier (essentially a weighted sum) trained on top of a model's internal activations. If a linear probe can accurately predict a property (like "is this harmful?"), that property must be encoded in the activations in a straightforward, extractable way — it is not buried in complex nonlinear patterns.

**Abliteration.** Arditi et al. (2024) showed that removing a single direction from weight matrices eliminates refusal behavior. If refusal were non-linear, a rank-1 projection would not suffice — yet it does, cleanly and reliably.

**Linearly decodable refused knowledge.** Shrivastava & Holtzman (2025) demonstrated that even when a model refuses to answer, the correct answer is linearly decodable from its hidden states. Simple linear probes extract the refused content. This confirms the [refusal direction](refusal-direction.md) is a gate, not an eraser.

**Steering vectors.** Adding or subtracting a direction from activations reliably shifts model behavior in the expected direction. If the representation were non-linear, linear perturbations would produce unpredictable results.

## Three operations

If a concept is linear, three operations become possible:

### Read: projection

Compute $\langle a, \hat{d}_c \rangle$ to measure the concept's strength in an activation. This is what Vauban's `probe` mode does — it runs a forward pass and reports per-layer projections onto a direction.

### Remove: orthogonal projection

Project the direction out of weight matrices:

$$W' = W - \alpha \cdot \hat{d} \cdot \hat{d}^\top W$$

The modified weight matrix $W'$ cannot write the removed direction into the [residual stream](activation-geometry.md). This is [abliteration](abliteration.md).

### Add: vector addition

During generation, add a scaled direction to activations:

$$a' = a + \beta \cdot \hat{d}$$

Positive $\beta$ amplifies the concept; negative $\beta$ suppresses it. This is [steering](steering.md).

All three operations are linear algebra on vectors in $\mathbb{R}^{d_{\text{model}}}$. No gradient computation, no fine-tuning, no retraining.

## Limitations

Not everything is linear. The hypothesis has known boundaries:

**Subspace, not direction.** Some behaviors are better described by a $k$-dimensional subspace than a single direction. Vauban's subspace [measurement](measurement.md) mode extracts the top-$k$ singular vectors to capture richer structure. If a concept requires $k > 1$ dimensions, a single direction misses the off-axis components.

> **SVD (Singular Value Decomposition)** — a technique that breaks a matrix into its most important "axes of variation," ranked by importance. Think of it like finding the main ingredients in a recipe: the first singular vector captures the biggest pattern, the second captures the next biggest, and so on. In Vauban, SVD finds the top directions along which harmful and harmless activations differ most.

**Detection vs. execution.** DBDI shows that "refusal" is really two things: harm *detection* (HDD) and refusal *execution* (RED). These correspond to different directions at different token positions. Treating refusal as a single direction conflates them. Separating them allows finer-grained intervention — cut RED while preserving HDD.

**Distributed effects across layers.** A direction extracted at one layer may not capture contributions from other layers. Weight-diff measurement (LoX, Lermen et al.) operates on weight matrices directly, capturing distributed safety effects that activation-based measurement may miss.

**Non-linear interactions.** Compositional behaviors — "be helpful *unless* the request is harmful" — involve conditional logic that a single linear direction cannot represent. The linear representation hypothesis describes the components; the interactions between components may be non-linear.

!!! tip "Practical implication"
    When a rank-1 direction does not fully capture a behavior (e.g., cut reduces refusal but also damages quality), try subspace or DBDI measurement. If the behavior is not well-approximated by *any* low-rank subspace, activation-space intervention may not be the right tool.
