Learning to Predict Physical Properties using Sums of Separable Functions

We present an algorithm for learning the function that maps a material structure to its value on some property, given the value of this function on several structures. We pose this problem as one of learning (regressing) a function of many variables from scattered data. Each structure is first converted to a weighted set of points by a process that removes irrelevant translations and rotations but otherwise retains full information about the structure. Then, incorporating a weighted average for each structure, we construct the multivariate regression function as a sum of separable functions, following the paradigm of separated representations. The algorithm can treat all finite and periodic structures within a common framework, and in particular does not require all structures to lie on a common lattice. We show how the algorithm simplifies when the structures do lie on a common lattice, and we present numerical results for that case.