How to represent an operator in code

[DrDLyuan]

Hello. In the sampling process of Proposition B.3, how is the operator (I+A^{T}A)^{-1} (with some parameters omitted) implemented in code? For example, if A is an inpainting operator, how would this operation be represented in code?

[ZehaoDou-official]

Sorry for the late reply. For the inpainting operator A, you can image that A can be expressed as a diagonal matrix with its diagonal elements to be either 0 or 1. Then, for operator B = (1+tA'A)^{-1}, it is also a diagonal matrix. If A_{ii} = 0, then B_{ii} = 1. If A_{ii}=1, then B_{ii}=1/(1+t). Therefore, we know that B is a linear combination of A and Id.
This is the basic idea.

[DrDLyuan]

Sorry for the late reply. For the inpainting operator A, you can image that A can be expressed as a diagonal matrix with its diagonal elements to be either 0 or 1. Then, for operator B = (1+tA'A)^{-1}, it is also a diagonal matrix. If A_{ii} = 0, then B_{ii} = 1. If A_{ii}=1, then B_{ii}=1/(1+t). Therefore, we know that B is a linear combination of A and Id.
This is the basic idea.

Thanks. And does the following code sample the mean \mu_{k}^{FPS} derived in Proposition B.3?

if op == 'inpainting':
model_mean = model_mean_back + coef * operator.forward(Y, mask=mask)
model_mean = model_mean - (coef / (coef+1.0)) * operator.forward(model_mean, mask=mask)