I am having some difficulties to understand the difference between:

1. Basis Pursuit DeNoising (BPDN) which is often stated as:

$min \left \| x \right \|_1 s.t \left \|Ax-b \right \|_2 \leq \varepsilon$ (1)

but solutions are generally found by solving:

$min \frac{1}{2}\left \| Ax-b \right \|_{2}^{2}+\lambda \left \| x \right \|_1$ (2)

2. And LASSO, which is often stated as:

$min \left \| Ax-b \right \|_2 s.t \left \|x \right \|_1 \leq \varepsilon$ (3)

for which in Lagrangian form is also (2)

It seems to me that the natural langrangian form for BPDN would be: $min \left \| x \right \|_1 +\lambda\frac{1}{2}\left \| Ax-b \right \|_{2}^{2}$ But i never saw this in any literature. Can i conclude from this that (2) for small $\lambda$ is BPDN, and for large $\lambda$ its LASSO? or where is the difference between the two?

The main point which i do not understand about finding a BPDN solution is how to use an estimate for the measurement error or noise if this is available? If one knows for example that the noise in measurements b is for example 10%, how can i use this information about in the solution procedure? Expression 1 would allow me include this error estimate by means of epsilon.
The naive approach of putting expression 1 in a general purpose non-linear optimiser with inequality constraints and some fixed value for epsilon works however very poorly. I understand that with the value of lambda i can control the sparsity, but how can i set epsilon in the standard solution procedures, which all seem to be using the Lagrangian form (2), where the noise threshold epsilon seems absent?

Any helpful insights are really appreciated.

I am having some difficulties to understand the difference between:

1. Basis Pursuit DeNoising (BPDN) which is often stated as:

$min \left \| x \right \|_1 s.t \left \|Ax-b \right \|_2 \leq \varepsilon$ (1)

but solutions are generally found by solving:

$min \left \| Ax-b \right \|_{2}^{2}+\lambda \left \| x \right \|_1$ (2)

2. And LASSO, which is often stated as:

$min \left \| Ax-b \right \|_2 s.t \left \|x \right \|_1 \leq \varepsilon$ (3)

for which in Lagrangian form is also (2)

It seems to me that the natural langrangian form for BPDN would be: $min \left \| x \right \|_1 +\lambda\left \| Ax-b \right \|_{2}^{2}$ But i never saw this in any literature. Can i conclude from this that (2) for small $\lambda$ is BPDN, and for large $\lambda$ its LASSO? or where is the difference between the two?

The main point which i do not understand about finding a BPDN solution is how to use an estimate for the measurement error or noise if this is available? If one knows for example that the noise in measurements b is for example 10%, how can i use this information about in the solution procedure? Expression 1 would allow me include this error estimate by means of epsilon.
The naive approach of putting expression 1 in a general purpose non-linear optimiser with inequality constraints and some fixed value for epsilon works however very poorly. I understand that with the value of lambda i can control the sparsity, but how can i set epsilon in the standard solution procedures, which all seem to be using the Lagrangian form (2), where the noise threshold epsilon seems absent?

Any helpful insights are really appreciated.

I am having some difficulties to understand the difference between:

1. Basis Pursuit DeNoising (BPDN) which is often stated as:

$min \left \| x \right \|_1 s.t \left \|Ax-b \right \|_2 \leq \varepsilon$ (1)

but solutions are generally found by solving:

$min \frac{1}{2}\left \| Ax-b \right \|_{2}^{2}+\lambda \left \| x \right \|_1$ (2)

2. And LASSO, which is often stated as:

$min \left \| Ax-b \right \|_2 s.t \left \|x \right \|_1 \leq \varepsilon$ (3)

for which in Lagrangian form is also (2)

It seems to me that the natural langrangian form for BPDN would be: $min \left \| x \right \|_1 +\lambda\frac{1}{2}\left \| Ax-b \right \|_{2}^{2}$ But i never saw this in any literature. Can i conclude from this that (2) for small $\lambda$ is BPDN, and for large $\lambda$ its LASSO? or where is the difference between the two?

The main point which i do not understand about finding a BPDN solution is how to use an estimate for the noise if this is available? If one knows for example that the noise in measurements b is for example 10%, how can i use this information about in the solution procedure? Expression 1 would allow me to estimate some value of epsilon.
The naive approach of putting expression 1 in a general purpose non-linear optimiser with inequality constraints and some fixed value for epsilon works very poorly. I understand that with the value of lambda i can control the sparsity, but how can i set epsilon in the standard solution procedures, which all seem to be using the Lagrangian form (2), where the noise threshold epsilon seems lost?

Any helpful insights are really appreciated.

I am having some difficulties to understand the difference between:

1. Basis Pursuit DeNoising (BPDN) which is often stated as:

$min \left \| x \right \|_1 s.t \left \|Ax-b \right \|_2 \leq \varepsilon$ (1)

but solutions are generally found by solving:

$min \frac{1}{2}\left \| Ax-b \right \|_{2}^{2}+\lambda \left \| x \right \|_1$ (2)

2. And LASSO, which is often stated as:

$min \left \| Ax-b \right \|_2 s.t \left \|x \right \|_1 \leq \varepsilon$ (3)

for which in Lagrangian form is also (2)

It seems to me that the natural langrangian form for BPDN would be: $min \left \| x \right \|_1 +\lambda\frac{1}{2}\left \| Ax-b \right \|_{2}^{2}$ But i never saw this in any literature. Can i conclude from this that (2) for small $\lambda$ is BPDN, and for large $\lambda$ its LASSO? or where is the difference between the two?

The main point which i do not understand about finding a BPDN solution is how to use an estimate for the measurement error or noise if this is available? If one knows for example that the noise in measurements b is for example 10%, how can i use this information about in the solution procedure? Expression 1 would allow me include this error estimate by means of epsilon.
The naive approach of putting expression 1 in a general purpose non-linear optimiser with inequality constraints and some fixed value for epsilon works however very poorly. I understand that with the value of lambda i can control the sparsity, but how can i set epsilon in the standard solution procedures, which all seem to be using the Lagrangian form (2), where the noise threshold epsilon seems absent?

Any helpful insights are really appreciated.

I am having some difficulties to understand the difference between:

1. Basis Pursuit DeNoising (BPDN) which is often stated as:

$min \left \| x \right \|_1 s.t \left \|Ax-b \right \|_2 \leq \varepsilon$ (1)

but solutions are generally found by solving:

$min \frac{1}{2}\left \| Ax-b \right \|_{2}^{2}+\lambda \left \| x \right \|_1$ (2)

2. And LASSO, which is often stated as:

$min \left \| Ax-b \right \|_2 s.t \left \|x \right \|_1 \leq \varepsilon$

for which in Lagrangian form is also

$min \frac{1}{2}\left \| Ax-b \right \|_{2}^{2}+\lambda \left \| x \right \|_1$ (3)

The main point which i do not understand about finding a BPDN solution is how to use an estimate for the noise if this is available? If one knows for example that the noise in measurements b is for example 10%, how can i use this information about in the solution procedure? Expression 1 would allow me to estimate some value of epsilon.
The naive approach of putting expression 1 in a general purpose non-linear optimiser with inequality constraints and some fixed value for epsilon works very poorly. I understand that with the value of lambda i can control the sparsity, but how can i set epsilon in the standard solution procedures, which all seem to be using the Lagrangian form (2), where the noise threshold epsilon seems lost?

Any helpful insights are really appreciated.

I am having some difficulties to understand the difference between:

1. Basis Pursuit DeNoising (BPDN) which is often stated as:

$min \left \| x \right \|_1 s.t \left \|Ax-b \right \|_2 \leq \varepsilon$ (1)

but solutions are generally found by solving:

$min \frac{1}{2}\left \| Ax-b \right \|_{2}^{2}+\lambda \left \| x \right \|_1$ (2)

2. And LASSO, which is often stated as:

$min \left \| Ax-b \right \|_2 s.t \left \|x \right \|_1 \leq \varepsilon$ (3)

for which in Lagrangian form is also (2)

It seems to me that the natural langrangian form for BPDN would be: $min \left \| x \right \|_1 +\lambda\frac{1}{2}\left \| Ax-b \right \|_{2}^{2}$ But i never saw this in any literature. Can i conclude from this that (2) for small $\lambda$ is BPDN, and for large $\lambda$ its LASSO? or where is the difference between the two?

The main point which i do not understand about finding a BPDN solution is how to use an estimate for the noise if this is available? If one knows for example that the noise in measurements b is for example 10%, how can i use this information about in the solution procedure? Expression 1 would allow me to estimate some value of epsilon.
The naive approach of putting expression 1 in a general purpose non-linear optimiser with inequality constraints and some fixed value for epsilon works very poorly. I understand that with the value of lambda i can control the sparsity, but how can i set epsilon in the standard solution procedures, which all seem to be using the Lagrangian form (2), where the noise threshold epsilon seems lost?

Any helpful insights are really appreciated.