Function Vector

A function vector (FV) is formed by extracting and aggregating the activation deltas produced by a set of in-context examples. It compresses the input-output mapping demonstrated in the prompt into a single, compact representation. This vector captures the abstract function rather than the specific tokens, allowing the model to apply the task to novel inputs.

Function vectors are applied via activation engineering: they are added directly to the residual stream at specific token positions during a forward pass. This is a causal intervention that hijacks the model's computation. Key properties include:

• Summation: The FV is simply added to existing activations.
• Position: Typically applied at the final token position before generation.
• Layer Specificity: Effective FVs are often extracted from and applied to a narrow range of middle layers.

Unlike prompt-based few-shot learning, a function vector is remarkably robust to the specific formatting or lexical choices of the target input. Because it encodes the abstract algorithm (e.g., 'perform antonym generation') rather than surface-level patterns, it generalizes across different input templates, synonyms, and even slight task rephrasings without needing new examples.

A closely related concept is the Natural Task Vector, which is derived not from explicit input-output pairs but from a single, open-ended task definition prompt (e.g., 'Translate to French:'). The NTV is computed as the difference in activations between the task prompt and a neutral baseline. It demonstrates that models internally represent task intent as a directional vector in latent space.

Function vectors differ fundamentally from in-context learning prompts:

• Prompting: Relies on attention heads (specifically induction heads) to dynamically copy and map patterns from the context window at every generation step.
• Function Vectors: Bypass this dynamic attention mechanism by directly setting the model's computational state. The model no longer needs to 'read' the examples; it is already configured to execute the task.

The standard extraction process involves a causal mediation analysis:

1. Run the model on a set of few-shot examples and record the residual stream activations at a target layer.
2. Run the model on a null or baseline prompt.
3. Compute the activation delta (task activation minus baseline activation).
4. Average these deltas across all example tokens to produce the final function vector. This isolates the task-specific signal from the background linguistic processing.

The central information highway in a transformer architecture where each layer reads from and writes its output back to a shared, accumulating state vector. Function vectors are typically injected directly into this stream.

• Structure: A sum of the token embedding, positional encoding, and all previous layer outputs
• Role: Allows layers to communicate by reading from and writing to a common bandwidth
• Connection: Function vectors hijack this pathway by adding a learned direction that biases subsequent computations toward a specific task

The conjecture that high-level concepts are encoded as linear directions in the representation space of a neural network's activation vectors. Function vectors are a direct application of this hypothesis to task-level behaviors.

• Core claim: Features are represented as directions, not isolated neurons
• Evidence: Concepts can be manipulated by adding or subtracting activation vectors
• Implication: A function vector is a linear direction in the residual stream that encodes an entire input-output mapping, supporting the hypothesis that complex behaviors have linear representations