RepICL: Reusable In-Context Prediction Across Heterogeneous Representation Spaces

In-context learning (ICL) can be amortized into latent objects (task vectors, function vectors, context vectors) that recover few-shot behavior at zero-shot inference cost, but recent theory shows a static vector acts as a single synthetic demonstration and must fail on high-rank mappings such as word-level bijections. We ask a linguistic version of this question: which linguistic operations can be amortized out of the prompt? We train a 2.6M-parameter network that reads the geometry of a few-shot support set (centroid, principal subspace, spectrum, computed once and cached) and produces an input-conditioned additive update to the query's residual stream at a mid-depth layer of a frozen GPT-2-large/XL. Across eight inflectional directions and one lexical relation, under a canonical split that bars inverted-pair leakage between directions, three regimes emerge. On forward inflection, where 10-shot ICL is strong (0.67-0.89) and extracted task vectors collapse (<=0.06), the transform matches ICL at strictly zero-shot per-query cost. On lemmatization directions, which frozen GPT-2 can execute but 10 demonstrations systematically fail to convey (ICL 0.13-0.48 at 1.5B), the transform is not capped by ICL at all: it reaches 0.78-0.92, up to +72 points over ICL (past to present: 0.85 vs. 0.13). On arbitrary pairings (antonymy) every amortizer plateaus near half of ICL at every scale, capacity, and seed tested. Controls show the support manifold acts as a causally necessary task fingerprint: wrong-task manifolds collapse accuracy to <=0.06, query-only variants cannot disambiguate tasks sharing an input space, and leave-one-task-out transfer is zero. Productive rules amortize into latent task representations, sometimes better than prompting can convey them; memorized pairings do not.

Transformers achieve remarkable performance by jointly learning broad families of tasks during pretraining and adapting to unseen tasks from only a short prompt. Yet a rigorous mathematical and statistical understanding of this phenomenon remains limited. This paper aims to study how Transformers exploit shared cross-task structure and how this structure affects the sample complexity of in-context learning (ICL). Specifically, we characterize task-space complexity through covering numbers under a prescribed metric, thereby quantifying the low-dimensional cross-task structure without requiring an explicit parametric representation. The resulting cover provides a set of anchor functions, which we use to introduce a task-identification-and-evaluation procedure: context observations localize an unseen task among the anchor functions, and the response at a query is predicted by aggregating the corresponding anchor function query evaluations. For approximation, we explicitly construct a Transformer with Softmax attention to approximate this procedure. For generalization, we derive an error bound that separates the effects of the number of pretraining tasks and the prompt length. The scaling with respect to the number of pretraining tasks is governed by the intrinsic dimensions of the task space and input domain; once sufficiently many tasks are available, the dependence on the prompt context length becomes dimension-free. To the best of our knowledge, this is the first work to quantify cross-task complexity for general nonlinear task families and explicitly construct a Transformer that exploits their low-dimensional structure to perform ICL. Our theory provides a quantitative explanation of how joint pretraining across related tasks improves in-context generalization.

Few-shot recognition with frozen visual features is especially fragile under domain shift and one-shot supervision, where a single labelled image is an unreliable estimate of its class. We ask how far this fragility can be reduced purely at test time, without retraining the encoder or augmenting the source domain. We present Spectral Transductive Refinement (STR), a training-free transductive inference rule that exploits the geometry of the complete support-query episode. Given frozen embeddings, STR builds a joint k-nearest-neighbour graph, maps the episode into a normalized-Laplacian spectral coordinate system, initializes class representatives from the labelled support, and iteratively refines them using pseudo-labelled queries. We evaluate STR under two protocols. A controlled component study with frozen ResNet-18 features shows that spectral refinement consistently improves over single-prototype spectral initialization across five shifted domains, with the largest gains in the one-shot regime where support estimates are weakest. We then benchmark STR against recent Cross-Domain Few-Shot Learning (CD-FSL) methods using the standard miniImageNet-pretrained ResNet-10 backbone over eight established target domains. Operating entirely at inference time, STR attains the highest 1-shot average among compared methods and remains competitive at 5-shot, rivalling approaches relying on heavy source-domain meta-training augmentations. Because STR is transductive, we report its setting explicitly. Diagnostics attribute its gains to iterative refinement in spectral coordinates rather than added prototype capacity, which remains inactive in our configuration.