# Degree of freedom of estimated sotuioin to the total variation problem from the ADMM algorithm

The study of the total variation problem is to solve the following problem: $$\text{minimize} ~ \frac{1}{2}||x - b||_2^2 + \lambda * \sum_i^N |x_{i+1} - x_i|$$ where $$x$$ is the unknown, $$b$$ in $$R^N$$, and N is the length of vector x.

From the ADMM method in the Standford website, the solution is like this:

x =

4.0546
4.0546
4.0546
4.0546
3.3319
3.3319
3.3319
3.3319
...


If I define the number of degree freedom in the estimator $$\hat{x}$$, $$df(\hat{x})= N - m$$, where m is the number of equal coordinates in the vector $$\hat{x}$$.

Question: Is it possible for me to obtain the value m from the above ADMM solution? The difficult part is that I am not sure whether to treat two coordinates, say 1.3015 and 1.3016 equal or not?