Adapting large language models to multiple tasks can cause cross-skill interference, where improvements for one skill degrade another. While methods such as LoRA impose orthogonality constraints at the weight level, they do not fully address interference in hidden-state representations. We propose Compositional Subspace Representation Fine-tuning (CS-ReFT), a novel representation-based approach that learns multiple orthonormal subspace transformations, each specializing in a distinct skill, and composes them via a lightweight router. By isolating these subspace edits in the hidden state, rather than weight matrices, CS-ReFT prevents cross-task conflicts more effectively. On the AlpacaEval benchmark, applying CS-ReFT to Llama-2-7B achieves a 93.94% win rate, surpassing GPT-3.5 Turbo (86.30%) while requiring only 0.0098% of model parameters. These findings show that specialized representation edits, composed via a simple router, significantly enhance multi-task instruction following with minimal overhead.

A multilinear subspace regression model based on so called latent variable decomposition is introduced. Unlike standard regression methods which typically employ matrix (2D) data representations followed by vector subspace transformations, the proposed approach uses tensor subspace transformations to model common latent variables across both the independent and dependent data. The proposed approach aims to maximize the correlation between the so derived latent variables and is shown to be suitable for the prediction of multidimensional dependent data from multidimensional independent data, where for the estimation of the latent variables we introduce an algorithm based on Multilinear Singular Value Decomposition (MSVD) on a specially defined cross-covariance tensor. It is next shown that in this way we are also able to unify the existing Partial Least Squares (PLS) and N-way PLS regression algorithms within the same framework. Simulations on benchmark synthetic data confirm the advantages of the proposed approach, in terms of its predictive ability and robustness, especially for small sample sizes.