Spectral Methods for Multimodal Data Analysis
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Spectral methods have proven themselves as an important and versatile tool in a wide range of problems in the fields of computer graphics, machine learning, pattern recognition, and computer vision, where many important problems boil down to constructing a Laplacian operator and finding a few of its eigenvalues and eigenfunctions. Classical examples include the computation of diffusion distances on manifolds in computer graphics, Laplacian eigenmaps, and spectral clustering in machine learning. In many cases, one has to deal with multiple data spaces simultaneously. For example, clustering multimedia data in machine learning applications involves various modalities or “views” (e.g., text and images), and finding correspondence between shapes in computer graphics problems is an operation performed between two or more modalities. In this thesis, we develop a generalization of spectral methods to deal with multiple data spaces and apply them to problems from the domains of computer graphics, machine learning, and image processing. Our main construction is based on simultaneous diagonalization of Laplacian operators. We present an efficient numerical technique for computing joint approximate eigenvectors of two or more Laplacians in challenging noisy scenarios, which also appears to be the first general non-smooth manifold optimization method. Finally, we use the relation between joint approximate diagonalizability and approximate commutativity of operators to define a structural similarity measure for images. We use this measure to perform structure-preserving color manipulations of a given image. To the best of our knowledge, the original contributions of this work are the following: 1 Introduction of joint diagonalization methods to the fields of machine learning, computer vision, pattern recognition, image processing, and graphics; 2 Formulation of the coupled approximate diagonalization problem that extends the joint diagonalization to cases with no bijective correspondence between the domains, and its application in a wide range of problems in the above fields; 3 Introduction of a new structural similarity measure of images based on the approximate commutativity of their respective Laplacians, and its application in image processing problems such as color-to-gray conversion, colors adaptation for color-blind viewers, gamut mapping, and multispectral image fusion; 4 Development of Manifold Alternating Direction Method of Multipliers (MADMM), the first general method for non-smooth optimization with manifold constraints, and its applications to several problems.