Data-Driven Polynomial Ridge Approximation Using Variable Projection

Inexpensive surrogates are useful for reducing the cost of science and engineering studies involving large-scale, complex computational models with many input parameters. A ridge approximation is one class of surrogate that models a quantity of interest as a nonlinear function of a few linear combinations of the input parameters. When used in parameter studies (e.g., optimization or uncertainty quantification), ridge approximations allow the low-dimensional structure to be exploited, reducing the effective dimension. We introduce a new, fast algorithm for constructing a ridge approximation where the nonlinear function is a polynomial. This polynomial ridge approximation is chosen to minimize least squares mismatch between the surrogate and the quantity of interest on a given set of inputs. Naively, this would require optimizing both the polynomial coefficients and the linear combination of weights, the latter of which define a low-dimensional subspace of the input space. However, given a fixed subspace the optimal polynomial can be found by solving a linear leastsquares problem. Hence using variable projection the polynomial can be implicitly defined, leaving an optimization problem over the subspace alone. Here we develop an algorithm that finds this polynomial ridge approximation by minimizing over the Grassmann manifold of low-dimensional subspaces using a Gauss{Newton method. Our Gauss{Newton method has superior theoretical guarantees and faster convergence on our numerical examples than the alternating approach for polynomial ridge approximation earlier proposed by Constantine, Eftekhari, Hokanson, and Ward [Comput. Methods Appl. Mech. Engrg., 326 (2017), pp. 402{421] that alternates between (i) optimizing the polynomial coefficients given the subspace and (ii) optimizing the subspace given the coefficients.