Is this alternative function a soft-thresholding variant?

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OhMyMarkov
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Hello everyone!

The soft-thresholding function often arrises when trying to find the MAP estimate with a Laplacian model of the parameter to be estimated. It is defined as:

\[
w(y) = \left\{
\begin{array}{l l}
y+T & \text{y < -T}\\
y-T, & \text{y > T}\\
0, & \text{otherwise}\\
\end{array} \right.
\]

Now, in a different context, could this be described as a soft thresholding function?

\[
w(y) = \left\{
\begin{array}{l l}
T-y & \quad \text{if $0 < y < T$}\\
0, & \quad \text{otherwise}\\
\end{array} \right.
\]

Thanks for the help!
 
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OhMyMarkov said:
Hello everyone!

The soft-thresholding function often arrises when trying to find the MAP estimate with a Laplacian model of the parameter to be estimated. It is defined as:

\[
w(y) = \left\{
\begin{array}{l l}
y+T & \text{y < -T}\\
y-T, & \text{y > T}\\
0, & \text{otherwise}\\
\end{array} \right.
\]

Now, in a different context, could this be described as a soft thresholding function?

\[
w(y) = \left\{
\begin{array}{l l}
T-y & \quad \text{if $0 < y < T$}\\
0, & \quad \text{otherwise}\\
\end{array} \right.
\]

Thanks for the help!

Hi OhMyMarkov, :)

No, I don't think so. According to the first definition \(w(y)=0\) when \(0<y<T\). However according to the second definition \(w(y)=T-y\) when \(0<y<T\).

Kind Regards,
Sudharaka.