Non-standard Geometries and Data Analysis

Traditional data mining starts with the mapping from entities to points in a Euclidean space. The search for patterns and structure is then framed as a geometric search in this space. Concepts like principal component analysis, regression, clustering, and centrality estimation have natural geometric formulations, and we now understand a great deal about manipulating such (typically high dimensional) spaces. For many domains of interest however, the most natural space to embed data in is not Euclidean. Data might lie on curved manifolds, or even inhabit spaces endowed with different distance structures than l_p spaces. How does one do data analysis in such domains ? In this talk, I'll discuss two specific domains of interest that pose challenges for traditional data mining and geometric methods. One space consists of collections of distributions, and the other is the space of shapes. In both cases, I'll present ongoing work that attempts to interpret and understand clustering in such spaces, driven by different applications.