What is the Proximal Operator ( Prox ) of the Pseudo <mrow class="MJX-TeXAtom-ORD">

cdsommegolfzp 2022-07-04 Answered
What is the Proximal Operator ( Prox) of the Pseudo L 0 Norm?
Namely:
Prox λ 0 ( y ) = arg min x 1 2 x y 2 2 + λ x 0
Where x 0 = n n z ( x ), namely teh number of non zeros elements in the vector x.
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Answers (1)

billyfcash5n
Answered 2022-07-05 Author has 17 answers
Since both the 1 2 x y 2 2 and the λ x 0 are element wise the problem could be solved per element.

There are 2 possible solutions for the value of x i :

1. The value of y i .
2. The value 0.

For each there is a different loss hence the choice is by the higher loss:

1. Value of y i -> The loss is λ.
2. Value of 0 -> The loss is 1 2 y i 2

Then the solution is given by:
x i = { y i  if  1 2 y i 2 > λ 0  if  1 2 y i 2 λ
Mathematically, for the case 1 2 y i 2 = λ one could chose either solution.
The above operation is called Hard Threshold.
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