Optimization#

bartorch.optim provides BART’s iterative algorithms for linear inverse problems. A solver is configured by its constructor and called as solver(y, A, x0=None), with y the data and A a LinearOperator; it reproduces BART’s iteration, including its step sizes, penalty updates and stopping rules. An iteration block is one step of the same iteration as a torch.nn.Module. A leading axis of y beyond the codomain of A is a batch of independent problems. Inverse problems and their solvers states the problems and algorithms, and Differentiation through reconstruction the backward pass of each solver.

Solver	Problem	Regularization	Backward pass
`CG`	\(\min_x \lVert Ax-y\rVert^2 + \lambda\lVert x\rVert^2 + \sum_i w_i \lVert G_i x - b_i\rVert^2\)	`Tikhonov` terms	Implicit: one further conjugate-gradient solve
`IST`	\(\min_x \tfrac12\lVert Ax-y\rVert^2 + g(x)\)	One term with \(G = I\)	Unrolled
`FISTA`	As IST, with momentum	One term with \(G = I\)	Unrolled
`ADMM`	\(\min_x \tfrac12\lVert Ax-y\rVert^2 + \sum_j g_j(G_j x)\)	Any number of terms, any \(G_j\)	Unrolled; each x-update implicit
`PRIDU`	As ADMM	Any number of terms, any \(G_j\)	Unrolled
`POCS`	Feasibility: repeated projections	Projections and terms at unit weight	Unrolled

Terms are the objects of bartorch.priors; an ImplicitPrior is accepted wherever a regularization term is.

Linear least squares#

Object	Description
`CG`	Conjugate gradients on the normal equations, with Tikhonov terms
`Tikhonov`	Quadratic penalty \(w \lVert G x - b\rVert^2\) for `CG`

Regularized least squares#

Object	Description
`IST`	Iterative soft thresholding (proximal gradient)
`FISTA`	Fast iterative soft thresholding (accelerated proximal gradient)
`ADMM`	Alternating direction method of multipliers
`PRIDU`	Chambolle-Pock primal-dual iteration
`maxeigen`	Power-iteration estimate of the largest eigenvalue of \(A^H A\)

Projection methods#

Object	Description
`POCS`	Projection onto convex sets, repeated sweeps
`POCSBlock`	One sweep of the projections

Functional interface#

Each function constructs the corresponding solver and calls it: optim.fista(y, A, term, maxiter=30) is optim.FISTA(term, maxiter=30)(y, A).

Object	Description
`cg`	`CG` in one call
`ist`	`IST` in one call
`fista`	`FISTA` in one call
`admm`	`ADMM` in one call
`pridu`	`PRIDU` in one call
`pocs`	`POCS` in one call, with the projections in place of an encoding

Iteration blocks#

state = block.start(y, A, x0) initializes a run, state = block(state, A) takes one step and block.output(state, A) returns the image. Step sizes and penalty weights are torch.nn.Parameter objects, frozen until requires_grad_() is called on them.

Object	Description
`ISTBlock`	One iterative soft-thresholding step
`FISTABlock`	One fast iterative soft-thresholding step
`ADMMBlock`	One ADMM step
`PRIDUBlock`	One primal-dual step

Fixed-point methods#

Object	Description
`FixedPoint`	An iteration block iterated to its fixed point and differentiated implicitly there (deep equilibrium)

Data scaling#

Object	Description
`data_scaling`	BART’s estimate of the data scale by which `pics` divides the data before it iterates

Operators and solvers assembles a BART reconstruction from an operator, a term and a solver.

Optimization

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