Logarithmic gradient of an exponential squared-distance sum ($\nabla \log {\sum }_i{e}^{\Vert x-a_i{\Vert }^2}=2(x-{\sum }_i{w}_i{a}_i)$)

For fixed vectors $a_1,\dots ,a_m$ in a finite-dimensional Euclidean space, defineThe gradient and Hessian matrix of $\Phi =\log F$ areTo derive the first identity, differentiate each exponential function and divide the sum of its gradients by $F$. For the second, differentiating a normalized weight gives $\nabla w_i=2w_i(\bar{a}-a_i)$. Thus $J_{\bar{a}}=-2{\mathrm{Cov}}_w(a_i)$, where the weighted covariance matrix is the sum in the displayed Hessian matrix. Differentiating $2(x-\bar{a})$ completes the calculation.

The weighted covariance matrix is positive semidefinite, so ${\nabla }^2\Phi ⪰2I$. Also $\Phi (x)\ge \Vert x-a_1{\Vert }^2\to \infty$ as $\Vert x\Vert \to \infty$. Therefore $F$ and $\Phi$ share a unique global minimizer, characterized by $x=\bar{a}(x)$. The weights place that minimizer in the convex hull of the centres.

For stable numerical evaluation, subtract the largest exponent before computing the normalized weights. This is the log-sum-exp principle and avoids floating-point overflow without changing the gradient. A naive iteration $x_{n+1}=\bar{a}(x_n)$ need not converge: in one dimension, centres $-1,1$ give $\bar{a}(x)=-\tanh (2x)$ with derivative $-2$ at its unique fixed point. A Newton root-finding iteration or a suitable Quasi-Newton method instead uses the positive curvature of $\Phi$.