where m m ∈ R k and p p ∈ R k are low-dimensional (k < M, P ), latent vector representations of material m and property p, respectively, and µ m ∈ R is a bias for material m. The materialproperty interaction term, the dot product m m p p , represents the "affinity" (if positive) or "aversion" (if negative) of material m for property p. Geometrically, the interaction term is positive (negative) if m m and p p point in roughly the same (opposite) direction. The magnitude of the interaction term depends on both the angle between m m and p p and their norms. The material bias µ m reflects variation of the values of the properties of material m independent of interactions; some materials may simply tend to have higher or lower values of the properties. See Koren et al. . From a matrix perspective, the low rank model factorizes the material-property matrix A as: