In this proposal, the investigator and his collaborators address several questions arising from the mathematical analysis of multiscale geometries of sets, and multiscale decomposition of function spaces, that arise from the action of a diffusion semigroup on a manifold, a graph and other rather general metric spaces. While these multiscale geometries are partly implicit (and classical) in differential geometry, in partial differential equations, as well as in many branches in graph theory (with applications to problems in computer science), only recently a very general, yet efficient, coherent and unifying construction has been introduced by the investigator and his collaborators. Multiscale function space decompositions that mirror these multiscale diffusion geometries are also constructed, through the introduction of special wavelet functions. This is a far-reaching, and long sought, generalization of wavelet analysis, both mathematically and computationally. The investigator and his collaborators have shown that algorithms for efficiently computing these multiscale decompositions exist, which generalize the fast wavelet transform and Fast Multipole Methods, yielding fast multiscale algorithms guaranteeing high-precision. The investigator will study the construction of biorthogonal diffusion multiscale decompositions, multiscale function approximation on rough sets, multiscale diffusion analysis of data sets and its relationships with geometric measure theory, multiscale Markov chains, numerical analysis of PDEs, learning theory, hyperspectral imaging and document corpora analysis. The investigator expects this novel multiscale construction to have impact in all these disciplines, in a way similar to the impact wavelet analysis had on low-dimensional signal processing and numerical analysis.
The present proposal stresses the inter-disciplinary nature of several aspects of multiscale analysis, and the vast applicability of the ideas, tools, constructions, to pure and applied mathematics, and to other disciplines such as computer science, physics, engineering, astronomy and statistics, among others. The introduction of these novel multiscale techniques reveals new and interesting multiscale geometric structures of graphs and sets, together with effective computational tools to discover them. The range of applications is very wide, and includes the analysis and organization of large and complex networks (e.g. computer networks, biological regulatory networks etc...), document corpora for information extraction, hyperspectral imagery (for applications to medicine, target recognition etc...), and large datasets in general. It has also applications to the development of new algorithms for learning and artificial intelligence, for the automation of complex tasks. The investigator aims at strenghtening his existing collaborations, and establishing new ones, with other institutions, both in the United States and abroad, across several disciplines, in particular computer science, astronomy, biology, and medicine. He will continue his existing collaborations with companies developing next-generation instrumentation, for applications to hyperspectral imaging. He will continue to actively participate in multi- and inter-disciplinary conferences, workshops and research activities, and effectively communicating and disseminating ideas and techniques to multi-disciplinary audiences, making his work, including papers and computer code for the corresponding algorithms, easily accessible electronically.
