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}:
 -0.20469813638386514
  0.27138214054054677
 -0.5699509061542583
 -0.5010975045421957
  2.8464698745150443
 -2.6791786559345336
 -0.7610130301119158
  1.06671094220032
 -0.6141284680064087
  0.5365406478382755
 -0.5199082023497104
 -0.3901790888422161
 -0.7754631207268199
  0.2561462755371098
  0.12662423159885172
  0.013478946132349012

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