Section: New Results

Minkowski sums and Hadamard products of algebraic varieties

Participant : Alessandro Oneto.

In [26], we study two particular geometric constructions. Given two affine algebraic varieties, we define their Minkowski sum as the (Zariski) closure of the set of coefficient-wise sums of pairs of points in the two varieties. Given two projective varieties, we define their Hadamard product as the (Zariski) closure of the set of coefficient-wise multiplications of pairs of points in the two varieties. In particular, we focus on computing their dimensions and degrees in terms of the ones of the original varieties. Hadamard products are of particular interests as they can be used to parametrize particular families of tensors which rise naturally by studying Restricted Boltzmann Machines, which are particular structures used in Statistics and Machine Learning.

This is a joint work with N. Friedenberg, and R. Williams.