Multivariate Interpolation on Unisolvent Nodes -- Lifting the Curse of Dimensionality

Abstract

We present generalizations of the classic Newton and Lagrange interpolation schemes to arbitrary dimensions. The core contribution that enables this new method is the notion of unisolvent nodes, i.e., nodes on which the multivariate polynomial interpolant of a function is unique. We prove that by choosing these nodes in a proper way, the resulting interpolation schemes become generic, while approximating all continuous Sobolev functions. If in addition the function is analytical in the Trefethen domain then, by validation, we achieve the optimal exponential approximation rate given by the upper bound in Trefethen's Theorem. The number of interpolation nodes required for computing the optimal interpolant depends sub-exponentially on the dimension, hence resisting the curse of dimensionality. Based on this, we propose an algorithm that can efficiently and numerically stably solve arbitrarydimensional interpolation problems, and approximate non-analytical functions, with at most quadratic runtime and linear memory requirement.

Adversarial attacks for machine learning denoisers and how to resisit them

“…Based on our prior work [18][19][20] on polynomial interpolation and regression we, here, propose a method for generating polynomial noise distributions. Consider the multi-index set A m,n,p = {α ∈ N m : α p ≤ n}, where…”

“…In fact these nodes maintain the approximation quality of the 1D-version, 20 i.e, if f : [−1, 1] m −→ R is a regular function and Q Am,n,p,f denotes its interpolant in P Am,n,p , then lim n→∞ Q Am,n,p,f = f uniformly and (exponentially) fast. 20,66 In particular, choosing Euclidean degree p = 2 results in an "optimal" choice. 20,66 We incorporate these insights into the following definitions.…”

“…20,66 In particular, choosing Euclidean degree p = 2 results in an "optimal" choice. 20,66 We incorporate these insights into the following definitions.…”

“…In fact the Lagrange polynomials are a basis of Π A , can be expressed analytically in (Newton-form) and evaluated in any argument x 0 ∈ R 2 efficiently and numerically stable. 20 Documentation of implementation and further details are provided in. 22 The notion allows to set up numerically stable regression schemes, 20 which we introduce in a simplified version matching our needs herein.…”

“…ii) Representation of general (non-uniform) noise patterns based on polynomial approximations [18][19][20][21][22] iii) An efficient computational scheme generating adversarial attacks.…”