Douglas–Rachford

Douglas–Rachford Algorithm

The (Parallel) Douglas–Rachford ((P)DR) Algorithm was generalized to Hadamard manifolds in [Bergmann, Persch, Steidl, 2016].

The aim is to minimize the sum

\[F(x) = f(x) + g(x)\]

on a manifold, where the two summands have proximal maps $\operatorname{prox}_{\lambda f}, \operatorname{prox}_{\lambda g}$ that are easy to evaluate (maybe in closed form or not too costly to approximate). Further define the Reflection operator at the proximal map as

\[\operatorname{refl}_{\lambda f}(x) = \exp_{\operatorname{prox}_{\lambda f}(x)} \bigl( -\log_{\operatorname{prox}_{\lambda f}(x)} x \bigr)\]

.

Let $\alpha_k\in [0,1]$ with $\sum_{k\in\mathbb N} \alpha_k(1-\alpha_k) = \infty$ and $\lambda > 0$ which might depend on iteration $k$ as well) be given.

Then the (P)DRA algorithm for initial data $x_0\in\mathcal H$ as

Initialization

Initialize $t_0 = x_0$ and $k=0$

Iteration

Repeat until a convergence criterion is reached

  1. Compute $s_k = \operatorname{refl}_{\lambda f}\operatorname{refl}_{\lambda g}(t_k)$
  2. within that operation store $x_{k+1} = \operatorname{prox}_{\lambda g}(t_k)$ which is the prox the inner reflection reflects at.
  3. Compute $t_{k+1} = g(\alpha_k; t_k, s_k)$
  4. Set $k = k+1$

Result

The result is given by the last computed $x_K$.

For the parallel version, the first proximal map is a vectorial version, where in each component one prox is applied to the corresponding copy of $t_k$ and the second proximal map corresponds to the indicator function of the set, where all copies are equal (in $\mathcal H^n$, where $n$ is the number of copies), leading to the second prox being the Riemannian mean.

Interface

 DouglasRachford(M, F, proxMaps, x)

Computes the Douglas-Rachford algorithm on the manifold $\mathcal M$, initial data $x_0$ and the (two) proximal maps proxMaps.

For $k>2$ proximal maps the problem is reformulated using the parallelDouglasRachford: a vectorial proximal map on the power manifold $\mathcal M^k$ and the proximal map of the set that identifies all entries again, i.e. the Karcher mean. This henve also boild down to two proximal maps, though each evauates proximal maps in parallel, i.e. component wise in a vector.

Input

  • M – a Riemannian Manifold $\mathcal M$
  • F – a cost function consisting of a sum of cost functions
  • proxes – functions of the form (λ,x)->... performing a proximal map, where ⁠λ denotes the proximal parameter, for each of the summands of F.
  • x0 – initial data $x_0\in\mathcal M$

Optional values

the default parameter is given in brackets

  • λ – ((iter) -> 1.0) function to provide the value for the proximal parameter during the calls
  • α – ((iter) -> 0.9) relaxation of the step from old to new iterate, i.e. $t_{k+1} = g(α_k; t_k, s_k)$, where $s_k$ is the result of the double reflection involved in the DR algorithm
  • R – (reflection) method employed in the iteration to perform the reflection of x at the prox p.
  • stoppingCriterion – (stopWhenAny(stopAfterIteration(200),stopWhenChangeLess(10.0^-5))) a StoppingCriterion.
  • parallel – (false) clarify that we are doing a parallel DR, i.e. on a Power manifold with two proxes. This can be used to trigger parallel Douglas–Rachford if you enter with two proxes. Keep in mind, that a parallel Douglas–Rachford implicitly works on a Power manifold and its first argument is the result then (assuming all are equal after the second prox.

and the ones that are passed to decorateOptions for the decorators.

Output

  • xOpt – the resulting point of the Douglas Rachford algorithm
  • record - if activated (using the record key, see RecordOptions an array containing the recorded values.
source

Options

DouglasRachfordOptions <: Options

Store all options required for the DouglasRachford algorithm,

Fields

  • x - the current iterate (result) For the parallel Douglas-Rachford, this is not a value from the Power manifold but the mean.
  • s – the last result of the double reflection at the proxes relaxed by α.
  • λ – ((iter)->1.0) function to provide the value for the proximal parameter during the calls
  • α – ((iter)->0.9) relaxation of the step from old to new iterate, i.e. $x^{(k+1)} = g(α(k); x^{(k)}, t^{(k)})$, where $t^{(k)}$ is the result of the double reflection involved in the DR algorithm
  • R – (reflection) method employed in the iteration to perform the reflection of x at the prox p.
  • stop – (stopAfterIteration(300)) a StoppingCriterion
  • parallel – (false) inducate whether we are running a pallel Douglas-Rachford or not.
source

For specific DebugActions and RecordActions see also Cyclic Proximal Point.

Literature