Monte-Carlo cubature construction
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In numerical integration, cubature methods are effective, in particular when the integrands can well be approximated by known test functions, such as polynomials. However, the construction of cubature formulas has not generally been known, and existing examples are only for particular domains of integrands such as hypercube and sphere. In this study, we show that we can generally construct cubature formulas for probability measures provided that we have an i.i.d. sampler from the measure and mean values of given test functions. We also see that our method work as a means of data compression, even if we do not have sufficient prior information of the measure.
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