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Data Analysis with Spectral Methods by Joint Approximate Diagonalization and Approximate Commutativity

Speaker: Artiom KovnatskyUniversità della Svizzera italiana

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Abstract:

Analysing data as is often impractical, e.g., it may have inconvenient representation (no vector field structure). Spectral methods are powerful tools for data analysis that relies on analysis of the operator acting on the space of scalar functions on the data rather than data itself. These methods proved to be an important and versatile tool in a wide range of problems in the fields of computer graphics, machine learning, and computer vision. Many approaches rely on a Laplacian operator and its first eigenvectors and eigenvalues (e.g., diffusion distances, dimensionality reduction, and spectral clustering). In this talk, I present the generalisation of spectral methods to a setting with multiple data spaces. Our construction is based on the idea of simultaneous diagonalization of Laplacian matrices. We discuss efficient numerical techniques for finding the first joint bases. I present applications in computer graphics. Additionally, employing the relation between joint diagonalizability and commutativity of matrices, we use image Laplacians commutativity as a criterion of color mapping quality. I show numerous applications of this approach in image processing, including color-to-gray conversion, gamut mapping, multispectra image fusion, and image optimization for color deficient viewers.

The presented work was done with collaboration:

Prof. M. Bronstein, Prof. A. Bronstein, Dr. D. Eynard, Prof. R. Kimmel, Prof. K. Glashoff.

About Artiom Kovnatsky:

Artiom Kovnatsky received the B.Sc. and M.Sc. degrees in Applied Mathematics at the Technion, Israel. He is now towards the end of a Ph.D. degree in Computational Science in the Faculty of Informatics at the University of Lugano (USI), Switzerland under the supervision of Prof. Michael Bronstein. During his Ph.D. research he developed with colleagues mathematical frameworks that were applied to a wide range problems, e.g., multimodal clustering and multimodal data analysis, shape similarity and correspondence, structure preserving color transformations of images, etc. His research interests concern the application of mathematics to problems in the fields of computer vision, computer graphics, data mining, as well as numerical optimisation.

Date: Friday 26th February 2016 - 13h30

Place: Battelle building A - Auditorium ground floor

5 février 2016
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