Convexity Results on Derivative of Negative Fisher Information Along Heat Flow

Recently, it was shown that the Fisher information is log-convex along the heat flow. The main tools involved were the Cauchy-Schwarz inequality and clever observations on the sum of squares. In this work, we reformulate their method as a rank-one constrained semidefinite programming problem. Then we show that the rank-one matrix can be determined through investigating the first diagonal entry. We apply this new approach to recover existing results, as well as to obtain new results on the derivative of the negative Fisher information along the heat flow: the derivative is convex if it is raised to the power of three-eighths, and is log-convex if the input distribution is log-concave.