ProxNest.optimisations.l2_ball_proj.sopt_fast_proj_B2(x, tau, params)

Fast projection algorithm onto the \(\ell_2\)-ball.

Compute the projection onto the \(\ell_2\) ball, i.e. solve

\[z^* = \min_{z} ||x - z||_2^2 s.t. ||y - \Phi z||_2 < \tau\]

where \(x\) is the input vector and the solution \(z^*\) is returned as sol.

Parameters

• x (np.ndarray) – A sample position \(x\) in the posterior space.

• tau (float) – Radius of likelihood \(\ell_2\)-ball.

• params (dict) – Dictionary of parameters defining the optimisation.

Returns

Optimal solution \(z^*\) of proximal projection.

Return type

np.ndarray

Notes

[1] M.J. Fadili and J-L. Starck, “Monotone operator splitting for optimization problems in sparse recovery” , IEEE ICIP, Cairo, Egypt, 2009.

[2] Amir Beck and Marc Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems”, SIAM Journal on Imaging Sciences 2 (2009), no. 1, 183–202.