In-Context Learning of Anonymous Matrix Operators

We study how to infer a shared distribution transformation from unpaired input and output samples and apply it to a new input distribution. The samples form matrices with unordered rows and unlabeled but aligned columns. The new output distribution can be identifiable even when the operator itself is not. For shared Markov kernels, we characterize input backgrounds that uniformly identify the query and derive a prediction lower bound when contexts are indistinguishable. For affine-Gaussian responses, we establish constructive recovery and stability conditions. We also derive the Bayes response law under squared MMD and analyze conditional flow matching with finite pretraining. ACEMO-ICL implements this approach with a conditional generator that respects the matrix symmetries. Small-budget experiments show that it uses demonstrations and benefits from informative backgrounds.