Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 6 / a

Write $r=x-a$. The squared Euclidean norm is $r^{T}r$, whose gradient is $2r$. The chain rule for the exponential function therefore gives the column gradientThe row derivative, if that convention is used for $df/dx$, is its transpose.

Differentiate the column gradient $2e^{r^{T}r}r$. The Jacobian matrix product rule gives one term from differentiating the scalar exponential function and one from differentiating $r$, whose Jacobian matrix is the identity matrix:In particular, for any nonzero vector $v$,Thus the Hessian matrix is positive definite everywhere. This calculation proves strict convexity and supplies the derivatives needed for sums of these terms.