pyproximal.optimization.cls_primal.TwIST

class pyproximal.optimization.cls_primal.TwIST(callbacks: Callbacks = None)

Two-step Iterative Shrinkage/Threshold

Solves the following minimization problem using Two-step Iterative Shrinkage/Threshold:

\[\mathbf{x} = \argmin_\mathbf{x} \frac{1}{2} ||\mathbf{b} - \mathbf{Ax}||_2^2 + g(\mathbf{x})\]

where \(\mathbf{A}\) is a linear operator and \(g(\mathbf{x})\) is any convex function that has a known proximal operator.

Notes

The TwIST algorithm can be expressed by the following recursion:

\[\mathbf{x}^{k+1} = (1-\alpha) \mathbf{x}^{k-1} + (\alpha-\beta) \mathbf{x}^k + \beta \prox_{g} (\mathbf{x}^k + \mathbf{A}^H (\mathbf{b} - \mathbf{A}\mathbf{x}^k)).\]

where \(\mathbf{x}^{1} = \prox_{g} (\mathbf{x}^0 + \mathbf{A}^T (\mathbf{b} - \mathbf{A}\mathbf{x}^0))\).

The optimal weighting parameters \(\alpha\) and \(\beta\) are linked to the smallest and largest eigenvalues of \(\mathbf{A}^H\mathbf{A}\) as follows:

\[\begin{split}\alpha = 1 + \rho^2 \\ \beta =\frac{2 \alpha}{\Lambda_{max} + \lambda_{min}}\end{split}\]

where \(\rho=\frac{1-\sqrt{k}}{1+\sqrt{k}}\) with \(k=\frac{\lambda_{min}}{\Lambda_{max}}\) and \(\Lambda_{max}=max(1, \lambda_{max})\).

Experimentally, it has been observed that TwIST is robust to the choice of such parameters. Finally, note that in the case of \(\alpha=1\) and \(\beta=1\), TwIST is identical to IST.

Methods

__init__([callbacks])
callback(x, *args, **kwargs)	Callback routine
finalize([nbar, show])	Finalize solver
memory_usage()	Compute memory usage of the solver
run(x, xold[, niter, show, itershow])	Run solver
setup(proxg, A, b, x0[, alpha, beta, eigs, ...])	Setup solver
solve(proxg, A, b, x0[, alpha, beta, eigs, ...])	Run entire solver
step(x, xold[, show])	Run one step of solver