I understand that you are expecting to see equivalent estimates when calling the "least squares" function and the "lasso" function with the regularization value or lambda set to 0.
Assuming that you are using MATLAB's Least square functions in the optimization toolbox, if you refer the documentation of the functions "lsqlin", "lsqnonneg" and "lasso", you will find that the "lsqlin" and "lsqnonneg" functions use a different least square formula when compared to the "lasso" function.
You can refer the following link for more information on the "lsqlin" function:
You can refer the following link for more information on the "lsqnonneg" function:
You can refer the following link for more information on the "lasso" function:
The user community might be able to help you better if you could provide the exact "lasso regularization" function and the "least square" function that you are trying to compare. It would also be helpful if you could also provide the data that you are using to compare the above functions.
