Does Non Negativity Constrains increases the estimation error I have been working with Tensor and matrix Non negative constrained algorithms. I have never seen a non negative constrained algorithm (ex. Non Negative Tucker Decomposition NTD) with error that is comparable to the non constrained algorithm (ex.High Order SVD HOSVD). Publications normally don't compare estimation error of unconstrained algorithms with constrained algorithms instead the comparison is always between different algorithms of same type.

By Testing, I found that the result of constrained algorithms with noise is around $10^{-7}$ compared to unconstrained ones is around $10^{-15}$.

Any reference that I can start from or any explanation would be appreciated.
Are you talking about train error or test error?
Non negativity constrains make your search space of possible solutions much smaller. This means that the train error would be always higher or equal when you add non-negativity constraint. 
But higher train error can mean that you are overfitting. If the assumption of non-negativity make sense for the solution of your problem, then it is likely that this assumption would decrease the test error as you are limiting your function space and decrease the chance of overfitting.