What functions preserve positive semidefiniteness (psd) when applied entrywise to psd matrices? This question has a long history beginning with Schur, Schoenberg, and Rudin, and has also recently received renewed attention due to its applications in high-dimensional statistics. However, effective characterizations of entrywise functions preserving positivity in a fixed dimension remain elusive to date.

I will present recent progress on this question, obtained by: (a) imposing rank and sparsity constraints, (b) restricting to structured matrices, and (c) restricting the class of functions to special families such as polynomials or power functions. These constraints arise in theory as well as applications, and provide natural ways to relax the elusive original problem. Moreover, novel connections to symmetric function theory, matrix analysis, and combinatorics emerge out of these refinements.

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Last updated: 17 May 2024