Proximal Soft Thresholding

Proximal operators are a key tool for solving optimization problems with non-differentiable penalty terms. The proximal soft thresholding operator corresponds to the L1 penalty, yielding exact zero coefficients (sparsity). The L2 proximal operator instead scales each coordinate by 1/(1 + lambda). An initially zero coordinate stays zero; a nonzero coordinate remains nonzero for finite supported lambda.

Experiment

Use the Proximal Soft Thresholding module to observe the difference between L1 and L2 penalties.

1. Set Input v1 to 3 and Input v2 to -1.
2. Set Penalty lambda to 1.
3. Leave Penalty as L1. Observe that x₂ becomes exactly 0 (Sparsity).
4. Change Penalty to L2. Observe the solution shifts to (1.5, -0.5), meaning no coordinate is exactly zero.
5. In Penalty = L2, set Penalty lambda to 0.5. Read solution (2, -2/3) and objective 5/3 = 1.666667; L2 results are not restricted to halves or quarters.
6. Restore L1 and set lambda to 4: both coordinates are zero and the objective is 5. At lambda 0, the input and solution coincide and objective is zero.