Function Vectors

Function vectors (FVs) have recently emerged as a promising mechanism for steering the behavior of large language models (LLMs) by injecting task-specific latent direction representations derived from in-context demonstrations. While prior studies have shown that FVs can recover task behavior in structured in-context learning settings, their effectiveness on semantically complex tasks and their ability to generalize across languages remain underexplored. We investigate the cross-lingual transferability of FVs using multilingual multi-label emotion recognition as a challenging semantic classification benchmark. Specifically, we examine whether FVs extracted from a source language can steer task behavior in another language under both standard clean and perturbed zero-shot settings without providing demonstrations during inference. Across diverse cross-lingual settings, applying FVs substantially improves performance, suggesting that FVs capture language-agnostic, task-relevant signals rather than purely language-specific lexical patterns, and highlighting their potential as a lightweight and transferable mechanism for multilingual task adaptation. We observe that each LLM exhibits a relatively stable optimal range of attention heads for constructing effective FVs, and the pattern remains consistent across languages. In addition, FVs can partially replicate the task-steering effects of standard few-shot in-context learning while avoiding the computational overhead of processing multiple demonstrations, making them effective for large-scale practical applications. Our code is available at https://github.com/yingjie7/cross_lingual_fvs.

Understanding the meaning of negated sentences remains one of the challenges for language models, even in the era of large language models (LLMs). We analyze systematicity regarding LLM understanding of negation from two perspectives: behavioral systematicity and representational systematicity. For behavioral systematicity, we confirm that through demonstrations and in-context learning, LLMs can recognize negation expressions and scope within sentences to some extent, but they fail to achieve perfect performance. In particular, the difficulty of the negation scope recognition for models varies depending on the output format. For representational systematicity, we analyze the extent to which function vectors can be robustly constructed from in-context examples for tasks that are essential to understanding negation. The experiments suggest that while function vectors can be composed for negation cue extraction tasks, extracting function vectors for recognizing scope is more challenging.

Function vectors (FVs) are task representations elicited during in-context learning that can be used to steer Large Language Models (LLMs). However, design choices in their formulation remain underexplored. In this work, we study the impact of varying FV definitions for instructions along two degrees of freedom: attention head selection and steering. For head selection, using gradient-based attributions with Layer-wise Relevance Propagation (LRP) substantially improves efficiency as well as accuracy. For FV steering, applying it in a distributed manner yields a higher accuracy compared to simple aggregation. Our code is publicly available.

In-context learning lets a language model perform a task specified by examples in its prompt. Function vectors capture task information in a compact activation assembled from attention-head outputs. Across two rule families and three Pythia models, we find two opposed functional populations among candidate function-vector heads. Writers support the rule-correct answer, while cancellers systematically suppress it. These roles predict opposing group effects on held-out prompts, where removing cancellers improves accuracy by 2.4 to 6.8 percentage points in all six settings. The dominant suppressive contributions depend on task content, and the same components can support correct predictions in another task. At the population level, substantial support and suppression can nearly cancel. Separating these roles explains how task-related computation can work against correct predictions and how a small aggregate effect can conceal strong opposing contributions.

In-context learning (ICL) excels at new tasks from minimal examples, yet we still lack a mechanistic explanation of how few-shot prompts shape a model's function vector (FV)--a causal activation direction that drives task behavior on the ICL query. Across tasks and models, an $n$-shot FV is well-approximated by a linear combination of example-level sub-FVs, suggesting additive and composable contributions from individual demonstrations. Beyond additivity, we show that models contextualize individual examples' representations based on prior examples to adaptively reweight which demonstrations dominate the FV: attention shifts toward examples that are more informative and less ambiguous under the context. Finally, a causal decomposition separates Query-Key routing from Value updates, finding that contextualization's most consistent contributions to FV quality arise from Query-Key alignment--particularly in ambiguous settings--while Value-mediated effects are more heterogeneous. Together, these results unify additive superposition with context-dependent attention reweighting into a mechanistic, testable account of how few-shot prompts implement tasks.

Function vectors (FVs) are vector representations of tasks extracted from model activations during in-context learning. While prior work has shown that multilingual model representations can be language-agnostic, it remains unclear whether the same holds for function vectors. We study whether FVs exhibit language-agnosticity, using machine translation as a case study. Across three decoder-only multilingual LLMs, we find that translation FVs extracted from a single English$\to$X direction transfer to other target languages, consistently improving the rank of correct translation tokens across multiple unseen languages. We further find that the highest-gain tokens span multiple languages and that translation FVs across directions share most of their top-ranked heads, indicating that the FV encodes a largely language-agnostic translation signal rather than a language-pair-specific mapping.

Activation steering presupposes that task-relevant behaviors correspond to linear directions in activation space -- directions that should both steer the model and be readable along the unembedding. Function vectors (FVs), extracted as mean differences across ICL demonstrations, are the canonical test case; the prediction: steering and decoding succeed or fail together. Across 12 tasks, 6 models from 3 families, and 4,032 directed cross-template pairs, we find the opposite. FV steering routinely succeeds where the logit lens cannot decode the correct answer at any intermediate layer, while the converse -- decodable without steerable -- is nearly empty (3 of 72). The gap is not representational dialect. A diagonal tuned lens closes 1 of 14 steerable-not-decodable cases; a 2-layer MLP probe with a Hewitt & Liang control closes 5 of 10 via nonlinearly encoded structure but leaves 5 invisible to every decoder tested. Even at $> 0.90$ steering accuracy, projecting the FV through the unembedding yields incoherent token distributions: FVs encode computational instructions, not answer directions. A model-family asymmetry sharpens the picture. Mistral FVs rewrite intermediate representations, while Llama and Gemma FVs steer the final output without leaving a logit-lens-visible trace, corroborated by three signals (post-steering deltas, activation-patching recovery, FV norm-transfer correlations). A previously reported negative cosine-transfer correlation dissolves at scale, adding at most $ΔR^2 = 0.011$ beyond task identity. These results decompose the linear representation hypothesis into linear decodability and linear steerability and show they come apart opposite to intuition, with implications for safety monitoring: vocabulary-projection tools are blind to FV-style interventions on widely deployed model families.

Contemporary studies in mechanistic interpretability have uncovered many puzzling phenomena in the neural information processing of Transformer-based language models, such as induction heads, function vectors, and the Hydra effect. Some of these individual phenomena have been independently tied to different data distributional properties, while some have been loosely associated with model architecture and how Transformers process information. However, a unified understanding of the relationship between data, model architecture, and optimization remains lacking, failing to answer the fundamental question: why do these three phenomena appear universally across different model families and scales, despite their seeming disconnect? In this work, we answer this question by unifying these three phenomena as consequences of hierarchical latent structures in the data generation process, coupled with decorrelated gradients across additive model components and directional concavity in the representation geometry. We validate our theoretical results in a toy model regime and in a large-scale synthetic data regime, comparing them with language models trained on natural language data.