NonnegativeRank · Geometry of Nonnegative Rank

Geometry of Nonnegative Rank

This project is centered on nonnegative rank - a notion that modifies the definition of matrix rank. Our goal is to do basic research aimed at better understanding the geometry of matrices and tensors of nonnegative rank at most r. Nonnegative rank appears in various applications such as statistics, machine learning, audio processing, image compression and document analysis. Understanding the geometry of nonnegative rank is fundamental in its own right and important for the theory and algorithms behind the applications. The overall objectives of this project are investigation of zero patterns in nonnegative matrix factorizations, understanding the boundaries and semialgebraic descriptions of the set of matrices and tensors of nonnegative rank r for small r, and implications of this theory to applications in statistics and other fields. We concluded that tools from (algebraic) geometry, rigidity theory and optimization are essential for studying nonnegative rank and related notions, and that especially the study of uniqueness and identifiability aspects benefits enormously from geometric tools.

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