Tractability of tensor product problems in the average case setting

• Authors: Anargyros Papageorgiou;Iasonas Petras
• Venue: Journal of Complexity
• Year: 2011

It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem S={S"d} in the average case setting to be weakly tractable but not polynomially tractable. As a result of the tensor product structure, the eigenvalues of the covariance operator of the induced measure in the one-dimensional problem characterize the complexity of approximating S"d, d=1, with accuracy @e. If @?"j"="1^~@l"j0, we know that S is not polynomially tractable iff lim sup"j"-"~@l"jj^p=~ for all p1. Thus we settle the open problem by showing that S is weakly tractable iff @?"j""n@l"j=o(ln^-^2n). In particular, assume that @?=limj-~@l"jjln^3(j+1), exists. Then S is weakly tractable iff @?=0.