MHB Is This a Valid Soft-Thresholding Function?

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The discussion centers on the validity of a proposed soft-thresholding function in relation to its definition. The original soft-thresholding function is defined to output zero when the input is between zero and a threshold T. In contrast, the alternative function suggests a non-zero output of T minus y for inputs within the same range. A participant argues that this discrepancy in outputs disqualifies the second function from being considered a valid soft-thresholding function. The conversation highlights the importance of precise definitions in mathematical contexts.
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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.
 
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