Instruction-Specific Vector Representations

• Instruction-specific vector representations are defined as modifications in model parameter spaces and hidden states induced by explicit task instructions, enabling targeted control and interpretability.
• They employ techniques such as weight-space arithmetic, activation steering, and contrastive embedding to achieve fine-tuned behavior and enhanced performance.
• Applications include improved model merging in language models, conditioned image retrieval in vision systems, and efficient SIMD optimizations in compiler-level implementations.

Instruction-specific vector representations encode model behaviors that are attributable to the presence or content of instructions in machine learning pipelines. Across modalities and computational systems, these representations allow for targeted control, interpretability, model merging, or fine-grained semantic analysis as a function of explicit task or instruction signals. This article systematically covers the principal definitions, extraction methodologies, applications, and theoretical properties of instruction-specific vectors, with focus on LLMs, vision encoders, and compiler-level vectorization.

1. Definitions and Notational Frameworks

Instruction-specific vector representations arise where model parameters, hidden states, or activations admit structured modifications or characterizations induced by instruction or task prompts.

In LLMs, the canonical “instruction vector” is defined in parameter space: $$\Delta_{\mathrm{inst}} = \Theta_{\mathrm{gi}} - \Theta_{\mathrm{gp}}$$ where $\Theta_{\mathrm{gp}}$ describes base pretrained parameters, and $\Theta_{\mathrm{gi}}$ those after generic instruction tuning (i.e., exposure to instruction-following datasets). $\Delta_{\mathrm{inst}}$ encodes the global weight-space direction responsible for a model’s general instruction-following capability (Hirano et al., 2024).

In the context of hidden activations, “activation steering” methods define a per-layer instruction vector as: $$v_{\mathrm{instr}}^{(\ell)} = a_\ell(x_{\mathrm{with\_instr}}) - a_\ell(x_{\mathrm{no\_instr}})$$ where $a_\ell(\cdot)$ is the residual activation for input $x$ at layer $\ell$ (Stolfo et al., 2024).

For visual encoders, instruction-specific vector representations $f_\theta(I, \tau)$—where $I$ is the image and $\tau$ the instruction—are constructed via joint multimodal encoders that explicitly condition image representations on instructions, supporting zero-shot conditional retrieval and classification (Hsieh et al., 11 Apr 2025).

At the compiler/IR level, “instruction-specific vector representations” can also denote explicit packing of isomorphic scalar instructions from control and dataflow graphs into vector instructions, enabling SIMD/disjoint vectorization (Fang et al., 6 Oct 2025).

2. Extraction and Construction Methodologies

Model Weight-space Task Arithmetic: Extraction of $\Delta_{\mathrm{inst}}$ (LLMs) exploits the difference between instruction-tuned and base model weights. For domain adaptation, a domain vector $$\Delta_{\mathrm{dom}} = \Theta_{\mathrm{dp}} - \Theta_{\mathrm{gp}}$$ is computed analogously, supporting vector arithmetic merging $$\Theta_{\mathrm{gp}} + \Delta_{\mathrm{dom}} + \Delta_{\mathrm{inst}}$$ (Hirano et al., 2024).

Activation Steering Vectors: Layerwise instruction vectors are computed by forward-propagating paired inputs (with and without instruction), averaging the differences across a representative set, and normalizing. Weighting and layer selection are task dependent; steering is enacted by adding $\alpha u_\ell$ to the activation at specified layers (Stolfo et al., 2024).

Head-level IVs and Causal Mediation: The Instruction Vector (IV) framework isolates causal heads in attention blocks, computes average task-conditional activations, and assembles the IV as a sum over these heads. Causal mediation is used to verify functional relevance by intervention on last-token activations (Jiang et al., 2024).

Contrastive Instruction Embedding: Prompt-based instruction embedding (PIE) methods wrap instructions in task-revealing prompts, encode via pretrained LLMs/BERT, and train a supervised contrastive objective to align embeddings of instructions from the same task category while separating negatives. Hard negatives are drawn via verb/noun overlap (Li et al., 2024).

Multimodal Conditional Encoding: Methods such as FocalLens concatenate or fuse instruction and input representations at the embedding or transformer-input level, conditioning the output on both the input (e.g., image patches) and the instruction, and optimize a contrastive loss anchored to the instruction-answer tuple (Hsieh et al., 11 Apr 2025).

Compiler IR Representations: In advanced auto-vectorizing compilers, control/data dependencies are explicated in specialized IRs. Functions are decomposed into layers of Entries, each representing instruction/control-flow triplets, and candidate instruction packs are discovered via isomorphism in opcode/type and control/data independence, guiding vector code generation (Fang et al., 6 Oct 2025).

3. Theoretical Properties and Orthogonality

An essential property for model merging is subspace near-orthogonality between instruction and domain vectors: $$\cos(\Delta_{\mathrm{inst}}, \Delta_{\mathrm{dom}}) \approx 0$$ Measured cosine similarities across transformer layers have empirical mean ≈ 0.002, std ≈ 0.023, confirming high independence (Hirano et al., 2024). This permits linear merging of instruction-following and domain-specificity without destructive interference.

In the hidden-state IV regime, IV-to-IV similarity remains high (around 0.95) even across extensive task-sequential finetuning, while functional ability may be lost due to causal head reallocation, indicating suppression rather than erasure of instruction-specific computation (Jiang et al., 2024).

Instruction embeddings trained via PIE/contrastive objectives yield high clustering purity and ARI (up to 0.91), robustly reflecting task-level semantics even as instruction surface forms vary greatly (Li et al., 2024).

Multimodal instruction-conditional representations generalize zero-shot to previously unseen tasks by aligning representation space across instruction and modality (Hsieh et al., 11 Apr 2025).

4. Applications and Empirical Outcomes

Domain	Vector Formulation	Empirical Benefit
LLM Weight-Space	$\Delta_{\mathrm{inst}}$, $\Delta_{\mathrm{dom}}$	Model merging: domain+instruction LLMs w/o new data (Hirano et al., 2024)
LLM Hidden Activations	$v_{\mathrm{instr}}^{(\ell)}$	Inference-time format/length/word steering (accuracy +20–60 points) (Stolfo et al., 2024)
Causal IV (Attention)	$\theta_c$ constructed over heads	IV-guided training mitigates catastrophic forgetting (IP boosts +10–20 points) (Jiang et al., 2024)
Instruction Embedding	PIE, contrastive task vectors	Shrink-bench selection, demo retrieval, data selection (Li et al., 2024)
Vision	$f_\theta(I, \tau)$	Conditioned image retrieval/classif. (avg +5–10 pts) (Hsieh et al., 11 Apr 2025)
Compiler Vector Packs	VIR-layer isomorphic instruction	>50% speedup in custom SIMD pipelines (Fang et al., 6 Oct 2025)

In LLMs, merged models produced by vector arithmetic on pretrained and instruction-tuned weights outperform singly-tuned variants on all but translation tasks in the tested financial benchmark suite (e.g., PFMT-Bench-Fin-JA overall: 2.58 vs 1.11–0.19) (Hirano et al., 2024).

Activation steering enhances compliance with output constraints, including format, length, and content, modulating model behaviors continuously at inference, and supporting compositional instruction combination (Stolfo et al., 2024).

The IV-guided training regime, which explicitly preserves the computation graph associated with instruction vector subspaces, recovers or surpasses original "held-out" performance in continual learning, undoing catastrophic forgetting not by regularization alone, but also by maintaining head-level access to instruction subspaces (Jiang et al., 2024).

In instruction-embedding approaches, PIE-trained embeddings yield ARI $>$0.9 and outperform generic text embeddings on data selection, demo retrieval, and dataset correlation tasks; for example, error rates in tiny testbench approximations drop from 18.4% to 6.9% (Li et al., 2024).

FocalLens achieves +8–12 mAP points on conditional visual retrieval—e.g., SugarCrepe and MMVP-VLM average accuracies +4.7 and +9.7 points, respectively, over standard CLIP (Hsieh et al., 11 Apr 2025).

5. Practical Implementation Considerations

LLM vector merging is a single matrix operation given aligned architectures, requiring no new training data or hyperparameters; continual pretraining employs standard optimizers, learning rate schedules, and gradient checkpointing (Hirano et al., 2024).

Activation steering in LLMs entails 2×$N$ forward passes per instruction for vector extraction, but incurs minimal inference overhead—a single vector addition per relevant layer; primary sensitivity is to layer/weight selection (Stolfo et al., 2024).

IV extraction necessitates per-head activation logging and causal effect scoring, followed by progressive intervention and regularization in training, but demonstrates high utility across diverse benchmarks and baselines (Jiang et al., 2024).

Instruction embedding via PIE employs prompt-wrapping and contrastive loss. Architecture-specific readout (BERT: [CLS]; Llama2: avg last-2 layer final token) is critical; prompt engineering (task-centric vs semantic) substantially modulates embedding efficacy (Li et al., 2024).

Vision encoders with instruction conditioning require additional fusion or concatenation modules and must be trained on multimodal triplets (image, instruction, answer), usually via a contrastive objective projecting into a CLIP-aligned latent space. Zero-shot conditional generalization is supported by the nature of the instruction-answer alignment (Hsieh et al., 11 Apr 2025).

Compiler frameworks implementing instruction-specific vector packing need formal IR structures (VIR/SIR), explicit control over code-lists and dependence graphs, and suitable cost models to identify beneficial vectorization opportunities (Fang et al., 6 Oct 2025).

6. Limitations and Open Challenges

Instruction-specific vectors in LLMs require model architectural alignment for arithmetic-based merging; over-steering in activation or causal IV-based methods can lead to unnatural outputs. Modular interpretability is sensitive to the definition of instruction (surface vs. latent distinction) and the presence of hard negatives in embedding learning (Li et al., 2024, Stolfo et al., 2024).

Limited pretraining in conditional vision encoders constrains the attainable conceptual scope compared to web-scale methods. Extending from global to local/spatialized conditional features and broadening instruction diversity (multi-hop, temporal, compositional queries) remain open (Hsieh et al., 11 Apr 2025).

In compiler-level vectorization, correct identification and isomorphism of instruction packs depend on the fidelity of control/data dependencies and abstraction of loops/branches, limiting applicability amid complex or indirect memory access patterns (Fang et al., 6 Oct 2025).

A plausible implication is that as instruction-specific vector methods mature, joint study of vector orthogonality, compositionality, and transferability will increasingly determine scalable, domain-adaptable architecture and pipeline design in both model-centric and system-level applications.