Minimal Euclidean Distance Degree of Segre-Veronese Varieties

The Euclidean Distance degree EDD(Q,X) of an algebraic variety X in a real inner product space (V, Q) counts the number of complex critical points of the distance function from a generic point in V to X. Since this invariant of X depends on Q, it is a natural problem to find or characterize inner products Q that correspond to the minimal possible EDD(Q,X). In my talk I will discuss this question for Segre-Veronese varieties, which consist of rank-1 (partially symmetric) tensors. I will show that with respect to the classical Frobenius (a.k.a. trace) inner product F(A,B)=Tr(AB), the variety X of nxm rank-1 matrices has smallest EDD(F,X)=min(n,m), whereas EDD(Q,X) with respect to a sufficiently general inner product Q on the space of nxm matrices is much higher.