| Résumé |
This thesis proposes novel computational methods of information geometry with real-time applications in audio signal processing. In this context, we address in parallel the applicative problems of real-time audio segmentation, and of real-time polyphonic music transcription. This is achieved by developing theoretical frame- works respectively for sequential change detection with exponential families, and for non-negative matrix factorization with convex-concave divergences. On the one hand, sequential change detection is studied in the light of the dually flat informa- tion geometry of exponential families. We notably develop a generic and unifying statistical framework relying on multiple hypothesis testing with decision rules based on exact generalized likelihood ratios. This is applied to devise a modular system for real-time audio segmentation with arbitrary types of signals and of homogeneity criteria. The proposed system controls the information rate of the audio stream as it unfolds in time to detect changes. On the other hand, non-negative matrix factorization is investigated by the way of convex-concave divergences on the space of discrete positive measures. In particular, we formulate a generic and unifying optimization framework for non-negative matrix factorization based on variational bounding with auxiliary functions. This is employed to design a real-time system for polyphonic music transcription with an explicit control on the frequency com- promise during the analysis. The developed system decomposes the music signal as it arrives in time onto a dictionary of note spectral templates. These contributions provide interesting insights and directions for future research in the realm of audio signal processing, and more generally of machine learning and signal processing, in the relatively young but nonetheless prolific field of computational information geometry. |
