Algorithm 1 Gradient descent on the manifold of positive definite matrices.

Input:$f\left(X\right)$    initialize  X    while no convergence do       $\eta :$ step size       $\mathrm{grad}\leftarrow X\left(\frac{\partial f\left(X\right)}{\partial X}\right)X$       $X\leftarrow {exp}_X\left(\eta \mathrm{grad}\right)=X\mathrm{Exp}\left(\eta X^{-1}\mathrm{grad}\right)$    end while Output:X