Necessary and Sufficient Conditions for Autoencoder-based Style Transfer

While style transfer and controllable generation have achieved strong empirical performance across audio, vision, and multimodal learning, most existing methods remain largely heuristic and offer limited theoretical guarantees. This paper studies autoencoder-based attribute transformation from a theoretical perspective and characterizes when exact conversion is achievable. We introduce a formal framework that isolates a task-relevant factor from a controllable attribute and ask when a learned encoder–decoder pair can implement the desired transformation while preserving all non-target information. Our main contributions are theoretical: under minimal and interpretable assumptions on the data-generating process, we establish both sufficient and necessary conditions for exact attribute conversion. We further move beyond idealized settings by deriving explicit error bounds that quantify how imperfect reconstruction and residual dependence in the learned representation translate into transfer degradation. Overall, our analysis shifts the focus from latent recovery to synthesis-oriented guarantees, providing a principled foundation for understanding when reliable task-driven style transfer is possible.