# The Soft-thresholding Function

#### OhMyMarkov

##### Member
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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#### Sudharaka

##### Well-known member
MHB Math Helper
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.