When Do Task Vectors Interfere? Mapping the Validity Boundaries of Weight-Space Composition

Task vector composition has emerged as a promising paradigm for editing pre-trained models, enabling model merging through addition and unlearning through subtraction. Fine-tuning in the tangent space of a pre-trained model (linear fine-tuning) has proven effective, as it produces task vectors that are naturally disentangled and resistant to interference. However, linearized models suffer from limited expressivity during training and incur higher computational costs at inference time, which restrict their practical applicability. In this work, we bridge the gap between linear and standard non-linear fine-tuning. We show that linearity with respect to weight perturbations, a property defined in parameter space, can be enforced through constraints in activation space during training. Concretely, we distill hidden representations from a curvature-regularized linearized teacher into a non-linear student trained via conventional fine-tuning. We find that the resulting model inherits key properties of linearized models for task arithmetic, enabling effective composition of task vectors and achieving strong performance across vision and language benchmarks without incurring any inference-time overhead.

Task-arithmetic merging works until it doesn't, and the field diagnoses why by measuring interference inside the merged model. We take the most direct such measure, the exact layerwise activation cross-term of a factorial ledger, establish its causal anatomy, and then ask what it tracks. The anatomy is clean: each block mostly transports and amplifies the cross-term rather than generating it; erased, it is regenerated by the untouched marginal paths to 99% of its norm unless removed late; its output effect varies monotonically with the displacement's angle (orthogonal displacements make interference worse), and a two-assumption model derives the angle law and retro-dicts the dose curve (R^2 >= 0.99). What the measure tracks is not what the field assumes. Behavioural expert-likeness is decoupled from it across four instruments. Its cross-condition behaviour is denominator-dominated: an instruction template pins the main effect to within 1% while the absolute interaction grows 111x from two to six merged tasks, suppressing expressed interference at k=2 and amplifying it at k=6. And where merging actually collapses, the cross-term is a bystander, not the carrier: across two collapse parameterizations at two scales, even erased persistently at every position, removing it entirely repairs none of the collapse. There the output-side ratio carries no method information under a common counterfactual, while two state-space measures the field already uses rank methods correctly at both scales. All 81 predictions were frozen before their data; falsifications are reported as such. Output-side interference measurement reads the gate, the denominator, and the displacement budget, not the interference. What fails a merge is the carrier-bystander split: collapse rides in the marginal displacements while the cross-term merely accompanies it, and only state space sees the carrier.

Full fine-tuning adapts large language models to new tasks but can erode capabilities they already possess. Existing remedies protect through proxies such as parameter distances, importance penalties, output matching, or dominant singular directions of the weights, but none directly asks which activation directions the preserved capability relies on. We argue that a capability is characterized more faithfully by the activation subspace it induces than by the singular geometry of the weight matrix, and develop function-space protection, instantiated as FORA (Function-space Orthogonal Residual Adaptation). From label-free calibration inputs, FORA estimates, per layer, the principal directions $Q$ of the input-activation covariance and forms a right projector $P_Q = I - QQ^T$. Paired with a left projector $P_U$ from the weight SVD, the update is $ΔW = P_U M P_Q + U_2 D_δ V_2^T$: a high-capacity branch structurally barred from reading capability-relevant function directions, plus a narrow spectral channel for controlled plasticity. The construction extends to parameter-efficient adaptation via $M \to (α/r) BA$. Across three settings on Qwen3-1.7B, including COGS and GSM8K learned while preserving translation and translation learned while preserving math, FORA consistently improves preservation over weight-space projection and standard regularization, with only a small new-task trade-off in the math-preservation setting. A controlled ablation isolating the projection source shows that the advantage comes not from projection itself, but from projecting onto capability-derived rather than weight-derived directions. Code is available at https://github.com/zrui239/FORA.