 Solution

ID: 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/ii/solution

Differentiate the column gradient

2 e^{r^{T} r} r

. The Jacobian matrix product rule gives one term from differentiating the scalar exponential function and one from differentiating

r

H_{f} (x) = e^{∥ r ∥^{2}} (4 r r^{T} + 2 I) .

In particular, for any nonzero vector

v

v^{T} H_{f} (x) v = e^{∥ r ∥^{2}} (4 (r^{T} v)^{2} + 2∥ v ∥^{2}) > 0.

Thus the Hessian matrix is positive definite everywhere. This calculation proves strict convexity and supplies the derivatives needed for sums of these terms.

 New to topics? Read the docs here!