Variable Projection Method Sample Clauses

Variable Projection Method. The variable projection method can be used to solve the nonlinear least squares problem (3.2). The approach exploits the fact that ψ(x, y) is linear in x and takes advantage of the relatively fewer parameters contained in y, compared to x. However, rather than explicitly separating variables x and y as in coordinate descent, the variable projection approach mathematically eliminates the linear parameters x, thus obtaining a reduced cost functional that depends only on y. A Xxxxx-Xxxxxx method is then applied to the reduced cost functional. Specifically, consider ϕ(y) ≡ ψ(x(y), y) , (3.10) where x(y) is a solution of x X X(x) X x X x X 0 min ψ(x, y) = min λI x − 0 . (3.11)

Variable Projection Method. ‌ In a minimization problem, when the objective function depends on two variables, and the subproblem of minimizing with respect to just one of the two variables has an easy solution, we can use the variable projection method to reduce the number of variables. Consider the objective function f pφ, xq for each fixed φ, and define x f˜pφq min f pφ, xq, (4.5) and thus φ,x φ x