Quantitatively, we show a variety of bounds relating the number of mistakes to well-known combinatorial dimensions.

Let A be the map from RX to RX Y that sends a function f on X to the function f(x) p P(x,y)onX Y. Similarly,let B bethemapfrom RY toRX Y thatsendsafunction gto g(y) p P(x,y). Combining these, we get exactly the stated formula. Here we elaborate further on COPT's optimization routine. As the objective Equation 3.1 is not globally convex, gradient descent can fall into local minima. But this requires anontrivialnumber (e.g.

Weintroduce COPT,anoveldistance metric between graphs defined via anoptimization routine, computing a coordinated pair of optimal transport maps simultaneously. This gives an unsupervised way to learn general-purpose graph representation, applicable tobothgraphsketching andgraphcomparison.

However, achieving such an assumption is non-trivial, leading to the following two aspects of challenges. On the one hand, it is challenging to design a model with enough capacity with limited computational resources, andexisting models areusually tailored forspecific problems which require heavytrailand-error [25,57,59].