Research Statement Variational Models and Fast Algorithms for Image Processing and Machine Learning ∗

My research project is to design variational models and fast optimization algorithms to solve efficiently problems arising in image processing, machine learning and other applications such as medical imaging and physics. An important part of my research project is to design convex variational models for basic problems in image processing such as image segmentation and image registration. Indeed, most models published in the last twenty years have used non-convex energy minimization models that capture non-optimal solutions. However, in the last few years, I have developed with my collaborators new convex minimization models along with fast algorithms that, I believe, will provide a new paradigm to solve more effectively basic problems in image processing, computer vision, medical imaging, machine learning and physics. Another part of my research project is to develop a unified framework for image processing. Since several image processing problems such as image denoising, image segmentation and image registration have been defined as variational models, I have developed with my collaborators a new variational model, based on the Polyakov energy, to unify these models. The Polyakov model from the physics of high-energy seems a very promising method to unify image processing models. Unlike standard models, ours can define denoising, segmentation, registration on any smooth and parameterized surface s.a. the sphere. The model is also purely geometric, which means that it does not depend on the choice of the parametrization, which is an undesirable property in several other models. The last part of my research project is to deal with the clustering of a large number of high-dimensional data. This task is called data clustering, and is an important task for machine learning applications such as shape classification, document retrieval, social networks, etc. A standard approach to solve the clustering method is to minimize the balanced ratio cut of a given graph, such as the Cheeger ratio cut. Since minimizing balanced ratio cuts are NP-hard, methods such as spectral clustering have proposed approximate solutions. Recently, I have shown with my collaborators that a continuous relaxation method can provide an optimal solution to the original ratio cut problem. This new technique, which solves parabolic PDEs on graphs, opens the path to several interesting developments in machine learning. To sum up, this research project introduces new ideas that are at the forefront of new emerging models for image processing and that I have been developing through various collaborations at UCLA and EPFL. It is most likely that these new models will have a big impact on the field and will replace standard methods, such as the level set method for image segmentation. These new computational models are mathematically built on inverse problems, continuous relaxation techniques for combinatorial problems, fast optimization techniques for non-differentiable functions, efficient numerical non-linear PDEs schemes, and computational differential geometry. This research project focuses on issues such as convex variational models and fast optimization algorithms, which are keys to further advances. At the end of the report, I also provide the future directions of my research project.