Plug-and-Play Regularization

A group of regularization terms that can not be directly written down as function are learned plug-and-play (PnP) priors. These are terms based on deep neural networks, which are trainted to implement the proximal map corresponding to the regularization term. Such a PnP prior can be used in the same way as any other regularization term.

The following example shows how to use a PnP prior in the context of the Kaczmarz solver.

using RegularizedLeastSquares
A = randn(32, 16)
x = randn(16)
b = A*x;

For the documentation we will just use the identity function as a placeholder for the PnP prior.

model = identity

identity (generic function with 1 method)

In practice, you would replace this with a neural network:

using Flux
model = Flux.loadmodel!(model, ...)

The model can then be used together with the PnPRegularization term:

reg = PnPRegularization(1.0; model = model, shape = [16]);

Since models often expect a specific input range, we can use the MinMaxTransform to normalize the input:

reg = PnPRegularization(1.0; model = model, shape = [16], input_transform = RegularizedLeastSquares.MinMaxTransform);

Custom input transforms can be implemented by passing something callable as the input_transform keyword argument. For more details see the PnPRegularization documentation.

The regularization term can then be used in the solver:

solver = createLinearSolver(Kaczmarz, A; reg = reg, iterations = 32)
x_approx = solve!(solver, b)

16-element Vector{Float64}:
 -1.802812124670742
 -1.3265763654937361
 -1.321146834924962
 -1.8944412955785452
 -0.6503702164653191
  0.49369186164588763
  2.1679453127766344
  0.7703030421106773
  1.3525979387733256
 -2.4125038915760952
 -2.481275429019287
 -0.5115922311697247
 -0.08718171558058296
  1.0089307960521232
 -0.4110415181252849
  0.09615221796411699

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