AF: Small: Topological Data Analysis for Big and High Dimensional Data

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The computing community is increasingly facing the challenge of processing and interpreting huge amounts of data that is being generated all around us by personal devices, digital sensors, data centers, scientific studies, and social networks. The nature of this data is usually high dimensional, that is, when interpreted as points, they reside in an high dimensional Euclidean space. The curse of `dimensionality' together with `size' puts up a formidable challenge to decipher knowledge from them in a principled way. Among the various approaches proposed to `mine' this data, topological approaches are emerging as robust and global methods that could complement the other approaches, be it statistical or geometrical. How can one deal with the difficulty of `big' data by developing new algorithms in computational topology and geometry is the focus of this proposal. The proposed research aims to investigate three techniques, namely, subsampling, localization, and simplicial collapse to handle the menace of `size' and `dimension'. This requires developing innovative computational tools grounded in algorithmic theory and mathematics from algebraic topology, analysis, and discrete geometry.The proposed topological methods would enhance the understanding of `big' data in general which could originate from images in the medical fields, videos of some events, trends in finance, or connectivities in social networks. The goal is to produce practical algorithms that can be turned into usable software which would be useful for both academia and industry. Other than standard academic forums, there are plans to disseminate the results from this project through course notes, tutorials, and web-pages to reach wider audience. The PI also plans to develop course materials on topological data analysis that will include results obtained in the project. The support from the project will train graduate students. Efforts will be made to recruit students from under-represented groups.