Statistical learning of materials properties or functions so far starts with a largely silent, nonchallenged step: the choice of the set of descriptive parameters (termed descriptor). However, when the scientific connection between the descriptor and the actuating mechanisms is unclear, the causality of the learned descriptor-property relation is uncertain. Thus, a trustful prediction of new promising materials, identification of anomalies, and scientific advancement are doubtful. We analyze this issue and define requirements for a suitable descriptor. For a classic example, the energy difference of zinc blende or wurtzite and rocksalt semiconductors, we demonstrate how a meaningful descriptor can be found systematically.

Let us assume that f is a continuous function defined on the unit ball of R d , of the form f (x) = g(Ax), where A is a k × d matrix and g is a function of k variables for k d. We are given a budget m ∈ N of possible point evaluations f (x i ), i = 1, . . . , m, of f , which we are allowed to query in order to construct a uniform approximating function. Under certain smoothness and variation assumptions on the function g, and an arbitrary choice of the matrix A, we present in this paper 1. a sampling choice of the points {x i } drawn at random for each function approximation; 2. algorithms (Algorithm 1 and Algorithm 2) for computing the approximating function, whose complexity is at most polynomial in the dimension d and in the number m of points.Dedicated to Ronald A. DeVore on his 70th birthday.Communicated by Emmanuel Candès. M. Fornasier ( )Due to the arbitrariness of A, the sampling points will be chosen according to suitable random distributions, and our results hold with overwhelming probability. Our approach uses tools taken from the compressed sensing framework, recent Chernoff bounds for sums of positive semidefinite matrices, and classical stability bounds for invariant subspaces of singular value decompositions.Keywords High-dimensional function approximation · Compressed sensing · Chernoff bounds for sums of positive semidefinite matrices · Stability bounds for invariant subspaces of singular value decompositions Mathematics Subject Classification (2010) 65D15 · 03D32 · 68Q30 · 60B20 · 60G50

The availability of big data in materials science offers new routes for analyzing materials properties and functions and achieving scientific understanding. Finding structure in these data that is not directly visible by standard tools and exploitation of the scientific information requires new and dedicated methodology based on approaches from statistical learning, compressed sensing, and other recent methods from applied mathematics, computer science, statistics, signal processing, and information science. In this paper, we explain and demonstrate a compressed-sensing based methodology for feature selection, specifically for discovering physical descriptors, i.e., physical parameters that describe the material and its properties of interest, and associated equations that explicitly and quantitatively describe those relevant properties. As showcase application and proof of concept, we describe how to build a physical model for the quantitative prediction of the crystal structure of binary compound semiconductors.

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