Computational Methods of Information Geometry

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Table of contents

Abstract
Preface
Introduction
Context of the thesis
From information geometry theory
To computational information geometry
Outcomes of the thesis
Outline and main contributions
Related publications and communications
I. Computational Methods of Information Geometry
1. Preliminaries on Information Geometry
1.1. Exponential families of probability distributions
1.1.1. Basic notions
1.1.2. First properties
1.1.3. Convex duality
1.1.4. Maximum likelihood
1.1.5. Dually flat geometry
1.2. Separable divergences on the space of discrete positive measures
1.2.1. Basic notions
1.2.2. Csiszár divergences
1.2.3. Skew Jeffreys-Bregman divergences
1.2.4. Skew Jensen-Bregman divergences
1.2.5. Skew (, , )-divergences
2. Sequential Change Detection with Exponential Families
2.1. Context
2.1.1. Background
2.1.2. Motivations
2.1.3. Contributions
2.2. Statistical framework
2.2.1. Multiple hypothesis problem
2.2.2. Test statistics and decision rules
2.3. Methods for exponential families
2.3.1. Generic scheme
2.3.2. Case of a known parameter before change
2.3.3. Case of unknown parameters before and after change
2.3.4. Generic scheme revisited through convex duality
2.3.5. Case of unknown parameters and maximum likelihood
2.4. Discussion
3. Non-Negative Matrix Factorization with Convex-Concave Divergences
3.1. Context
3.1.1. Background
3.1.2. Motivations
3.1.3. Contributions
3.2. Optimization framework
3.2.1. Cost function minimization problem
3.2.2. Variational bounding and auxiliary functions
3.3. Methods for convex-concave divergences
3.3.1. Generic updates
3.3.2. Case of Csiszár divergences
3.3.3. Case of skew Jeffreys-Bregman divergences
3.3.4. Case of skew Jensen-Bregman divergences
3.3.5. Case of skew (, , )-divergences
3.4. Discussion
II. Real-Time Applications in Audio Signal Processing
4. Real-Time Audio Segmentation
4.1. Context
4.1.1. Background
4.1.2. Motivations
4.1.3. Contributions
4.2. Proposed approach
4.2.1. System architecture
4.2.2. Change detection
4.3. Experimental results
4.3.1. Segmentation into silence and activity
4.3.2. Segmentation into music and speech
4.3.3. Segmentation into different speakers
4.3.4. Segmentation into polyphonic note slices
4.3.5. Evaluation on musical onset detection
4.4. Discussion
5. Real-Time Polyphonic Music Transcription
5.1. Context
5.1.1. Background
5.1.2. Motivations
5.1.3. Contributions
5.2. Proposed approach
5.2.1. System architecture
5.2.2. Non-negative decomposition
5.3. Experimental results
5.3.1. Sample example of piano music
5.3.2. Evaluation on multiple fundamental frequency estimation
5.3.3. Evaluation on multiple fundamental frequency tracking
5.4. Discussion
Conclusion
Summary of the present work
From sequential change detection to audio segmentation
From non-negative decomposition to polyphonic music transcription
Perspectives for future work
Change detection and audio segmentation
Non-negative matrix factorization and music transcription
General directions of investigation
Bibliography