It has been found in the late 1990's that certain transformations of the integrand can help to increase the efficiency of quasi-Monte Carlo methods.
Prominent examples from quantitative finance are provided be the Brownian bridge construction (Moskowitz et. al., 1996})and the Principal Component Analysis construction (Acworth et. al., 1998) for sample paths of Brownian motion.
It was later observed by Papageorgiou (2002) that
1. those transformations do not increase efficiency for arbitrary problems, rather they can slow things down for some problems;
2. those transforms can be understood as orthogonal transforms of the standard normal input vector.
Imai and Tan (2007) had the idea of constructing orthogonal transforms tailored to a given (finance) problem. Their idea has been built upon by a number of authors.
Up to now, most work concentrates on making the problem ``as one-dimensional as possible'' by choosing some orthogonal transform that puts as much variance as possible onto the first input variable.
We generalize this idea and we want to know under which conditions an orthogonal transform can be found that makes QMC more efficient, and how such a transform can be constructed.