: = - - - - - - - - - - - - figure lasso regression solutions compare with figure one advantage of using the lasso regularization is that the resulting optimal parameter vector often has several components that are exactly for examplein the top middle and right panels of figure the optimal solution lies exactly at corner point of the square {[ ]| | | * | * |}in this case * for statistical models with many parametersthe lasso can provide methodology for model selection namelyas the regularization parameter increases (orequivalentlyas the norm of the optimal solution decreases)the solution vector will have fewer and fewer non-zero parameters by plotting the values of the parameters for each or one obtains the so-called regularization paths (also called homotopy paths or coefficient profilesfor the variables inspection of such paths may help assess which of the model parameters are relevant to explain the variability in the observed responses {yi example (regularization pathsfigure shows the regularization paths for coefficients from multiple linear regression model yi xi ei = where for and for the error terms {ei are independent and standard normal the explanatory variables {xi were independently generated from standard normal distribution as it is clear from the figurethe estimates of the regularization paths