On the Local Structure and Approximation Stability of Block Isotropic Gaussian Fields

Abstract: Skew-symmetric functions are a class of functions defined on a product space $M \times M$ that are antisymmetric with respect to the order of their inputs. In [13], the authors proved that non-deterministic skew-symmetric Gaussian fields cannot be stationary or isotropic and proposed an alternative notion: stationarity (isotropy) in each component space. Our work focuses on local quadratic approximations of the associated Gaussian fields. Local quadratic approximations to random fields are random polynomials parametrized by a jointly sampled gradient vector and Hessian matrix. We characterize the distribution of the corresponding random vectors and random matrices. Then, we study the error in the quadratic approximation, which is also a Gaussian field. We investigate the error induced by the quadratic approximation in three senses: the pointwise error, the maximal error over an ellipsoidal region, and the worst-case error for multivariate Gaussian inputs at a given confidence level. Next, we explore the limiting behavior of the worst-case error as the distance between an expansion point and evaluation points approaches zero and infinity. Finally, we study how, as the input dimension increases, the variance of multivariate Gaussian distributions must be restricted to keep the worst-case error bound constant.