Combinatorics Seminar
Apoorva Khare
Stanford University
The critical exponent of a graph
Abstract: We classify the powers that preserve positive semidefiniteness, when applied entrywise to matrices with rank and sparsity constraints. This is part of
a broad program to study entrywise functions preserving positivity on distinguished submanifolds of the cone. In our first main result, we completely
classify the powers preserving Loewner properties on positive semidefinite matrices with fixed dimension and rank. This includes the case where the
matrices have negative entries.
Our second main result characterizes powers preserving positivity on matrices with zeros according to a chordal graph. We show how preserving positivity
relates to the geometry of the graph, thus providing interesting connections between combinatorics and analysis. The work has applications in
regularizing covariance/correlation matrices, where entrywise powers are used to separate signal from noise, while minimally modifying the entries of
the original matrix. (Based on joint work with D. Guillot and B. Rajaratnam.)
Monday September 28, 2015 at 3:00 PM in SEO 427