
# The proximal map of the distance squared

Proximal map $\prox_{\lambda f}$ for the function $f\colon\M\to\R,\quad f(x) = d^2_{\M}(x,y)$ for some fixed value $y\in\M$. Following [1], let $\geo{x}{y}$ denote the geodesic starting in $\geo{x}{y}(0)=x$ and reaching $\geo{x}{y}(1)=y$ at time $1$. Then

If a data item $x$ contains a NaN it is set to the minimizer of the distance, i.e. initialized to the fixed item. Values can be kept unchanged by activating the optional FixedMask boolean array.

### Matlab Documentation

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 % proxDistanceSquared(M,x,f,lambda) prox of f(x) = d^2(x,y), y fixed, % and prox paramater lambda. The fixedMask can be used to fix values, i.e. % they are kept unchanged by hard reset. % % INPUT % M : a manifold % x,f : data point( sets/columns ) on the maifold M % lambda : stepsize towards f % % OPTIONAL % FixedMask : fix a certain subset of x % % OUTPUT % x : result point( sets) of the proximal map % w : indicator which terms where proxed, here: all % --- % Manifold-valued Image Restoration Toolbox 1.0 ~ R. Bergmann, 2014-10-19

### References

1. Ferreira, O P and Oliveira, P R (2002). Proximal point algorithm on Riemannian manifolds. Optimization. 51 257–70