Alternating gradient descent

alternating_gradient_descent(M::ProductManifold, f, grad_f, p=rand(M))
alternating_gradient_descent(M::ProductManifold, ago::ManifoldAlternatingGradientObjective, p)

perform an alternating gradient descent

Input

• M: the product manifold $\mathcal M = \mathcal M_1 × \mathcal M_2 × ⋯ ×\mathcal M_n$
• f: the objective function (cost) defined on M.
• grad_f: a gradient, that can be of two cases
  • is a single function returning an ArrayPartition or
  • is a vector functions each returning a component part of the whole gradient
• p: an initial value $p_0 ∈ \mathcal M$

Optional

• evaluation: (AllocatingEvaluation) specify whether the gradients work by allocation (default) form gradF(M, x) or InplaceEvaluation in place of the form gradF!(M, X, x) (elementwise).
• evaluation_order: (:Linear) whether to use a randomly permuted sequence (:FixedRandom), a per cycle permuted sequence (:Random) or the default :Linear one.
• inner_iterations: (5) how many gradient steps to take in a component before alternating to the next
• stopping_criterion: (StopAfterIteration(1000)) a StoppingCriterion
• stepsize: (ArmijoLinesearch()) a Stepsize
• order: ([1:n]) the initial permutation, where n is the number of gradients in gradF.
• retraction_method: (default_retraction_method(M, typeof(p))) a retraction(M, p, X) to use.

Output

usually the obtained (approximate) minimizer, see get_solver_return for details

alternating_gradient_descent!(M::ProductManifold, f, grad_f, p)
alternating_gradient_descent!(M::ProductManifold, ago::ManifoldAlternatingGradientObjective, p)

perform a alternating gradient descent in place of p.

Input

• M: a product manifold $\mathcal M$
• f: the objective functioN (cost)
• grad_f: a gradient function, that either returns a vector of the subgradients or is a vector of gradients
• p: an initial value $p_0 ∈ \mathcal M$

you can also pass a ManifoldAlternatingGradientObjectiveago containing f and grad_f instead.

for all optional parameters, see alternating_gradient_descent.

AlternatingGradientDescentState <: AbstractGradientDescentSolverState

Store the fields for an alternating gradient descent algorithm, see also alternating_gradient_descent.

Fields

• direction: (AlternatingGradient(zero_vector(M, x)) a DirectionUpdateRule
• evaluation_order: (:Linear) whether to use a randomly permuted sequence (:FixedRandom), a per cycle newly permuted sequence (:Random) or the default :Linear evaluation order.
• inner_iterations: (5) how many gradient steps to take in a component before alternating to the next
• order the current permutation
• retraction_method: (default_retraction_method(M, typeof(p))) a retraction(M,x,ξ) to use.
• stepsize: (ConstantStepsize(M)) a Stepsize
• stopping_criterion: (StopAfterIteration(1000)) a StoppingCriterion
• p: the current iterate
• X: (zero_vector(M,p)) the current gradient tangent vector
• k, ì`: internal counters for the outer and inner iterations, respectively.

Constructors

AlternatingGradientDescentState(M, p; kwargs...)

Generate the options for point p and where inner_iterations, order_type, order, retraction_method, stopping_criterion, and stepsize` are keyword arguments

AlternatingGradient <: DirectionUpdateRule

The default gradient processor, which just evaluates the (alternating) gradient on one of the components