Optimal transport meets speech: a tutorial review

Optimal Transport (OT) provides a principled framework for comparing and transforming probability distributions while preserving geometric structure. Recently, OT has gained significant attention in machine learning due to its ability to measure discrepancies between distributions, even when their supports do not overlap, making it effective for tasks such as generative modeling, domain adaptation, and transfer learning. Despite its success in fields such as computer vision and natural language processing, OT remains relatively underexplored in speech research. Speech signals present unique challenges, including temporal dynamics, speaker variability, noise, reverberation, and heterogeneous multimodal representations involving audio, text, and visual information. These factors often lead to distribution mismatches, where OT offers a natural framework for alignment and interpretation. This work aims to promote broader adoption of OT in speech processing by: (1) reviewing OT foundations through intuitive physical interpretations and highlighting connections to modern generative models; (2) presenting computational algorithms suitable for deep learning frameworks; and (3) demonstrating OT applications in cross-domain and cross-modal speech tasks, including speech enhancement, automatic speech recognition, language and speaker recognition, and audio spoof detection. We highlight OT's strong potential for addressing distributional variations in real-world speech applications.