We improve upon the existing literature of denoising techniques studied at the Large Hadron Collider (LHC) for the task of disentangling proton-proton collisions. The primary technique that serves as the foundation for this work is known as Training Optimal Transport using Attention Learning (TOTAL). The TOTAL methodology relies on the use of a transformer architecture using a loss function inspired by optimal transport problems to learn full event descriptions. By comparing matched samples with and without noisy interactions present, the TOTAL network robustly learns an accurate description of noise as a transport function without any need for assumptions of the nature of noise derived from simulations. In this work, we develop an improved version of TOTAL known as Weakly-supervised Optimal Transport Attention-based Noise Mitigation (WOTAN) by reducing the degree of its self-supervision. The reduction of the self-supervision allows us to demonstrate the power of optimal transport-based denoising in being able to use data for particle classification instead of solely simulations. In spite of the reduced supervision, our work still outperforms existing conventional pileup mitigation approaches. Such an extension of the TOTAL methodology allows for more robust denoising, one that would truly be the first fully data-driven machine learning denoising strategy at the LHC.