Figur Pairwise compute exampleTPS tw and for , right). "Learningfromuncertaincurves: The 2-Wassersteinmetricfor Gaussian processes".

Let N(µ,σ2) denote a Gaussian distribution with meanµ and variance σ2. Let χ2(n) denote a χ2 distribution withn degrees of freedom. Our analysis extensively uses the following facts about Gaussian and χ2 distributions: Definition A.1 (Gaussian and Wigner Random Matrices). We let G N(n) denote an n n randomGaussianmatrixwith i.i.d. We let W W(n)=G+GT denotean n n Wigner matrix, where G N(n). Fact A.1 (χ2 TailBound(Lemma 1of[1])).

Recently,the Hutch++ algorithm was proposed, which reduces the number of matrix-vector queries fromO(1/2) to the optimalO(1/), and the algorithm succeeds with constant probability.

The performance of any supervised learning model depends strongly on the quality of input instances.

We focus on the setting where firms are not aware of demand functions and how reference prices areformed but haveaccess to an oracle that provides a measure of consumers' responsiveness to the current posted prices.