Most quasi-Monte Carlo research focuses on sampling from the unit cube. Many problems, especially in computer graphics, are defined via quadrature over the unit triangle. QuasiMonte Carlo methods for the triangle have been developed by Pillards and Cools [J. Comput. Appl. Math., 174 (2005), pp. 29-42] and by Brandolini et al. ["A Koksma-Hlawka inequality for simplices," in Trends in Harmonic Analysis, Springer, 2013, pp. 33-46]. This paper presents two quasi-Monte Carlo constructions in the triangle with a vanishing discrepancy. The first is a version of the van der Corput sequence customized to the unit triangle. It is an extensible digital construction that attains a discrepancy below 12/ √ N . The second construction rotates an integer lattice through an angle whose tangent is a quadratic irrational number. It attains a discrepancy of O(log(N )/N ), which is the best possible rate. Previous work strongly indicated that such a discrepancy was possible, but no constructions were available. Scrambling the digits of the first construction improves its accuracy for integration of smooth functions. Both constructions also yield convergent estimates for integrands that are Riemann integrable on the triangle without requiring bounded variation.
Quasi-Monte Carlo (QMC) sampling has been developed for integration over [0, 1] s where it has superior accuracy to Monte Carlo (MC) for integrands of bounded variation. Scrambled net quadrature gives allows replication based error estimation for QMC with at least the same accuracy and for smooth enough integrands even better accuracy than plain QMC. Integration over triangles, spheres, disks and Cartesian products of such spaces is more difficult for QMC because the induced integrand on a unit cube may fail to have the desired regularity. In this paper, we present a construction of point sets for numerical integration over Cartesian products of s spaces of dimension d, with triangles (d = 2) being of special interest. The point sets are transformations of randomized (t, m, s)-nets using recursive geometric partitions. The resulting integral estimates are unbiased and their variance is o(1/n) for any integrand in L 2 of the product space. Under smoothness assumptions on the integrand, our randomized QMC algorithm has variance O(n −1−2/d (log n) s−1 ), for integration over s-fold Cartesian products of d-dimensional domains, compared to O(n −1 ) for ordinary Monte Carlo.
Using a multivariable Faa di Bruno formula we give conditions on transformations τ : [0, 1] m → X where X is a closed and bounded subset of R d such that f • τ is of bounded variation in the sense of Hardy and Krause for all f ∈ C d (X ). We give similar conditions for f •τ to be smooth enough for scrambled net sampling to attain O(n −3/2+ ) accuracy. Some popular symmetric transformations to the simplex and sphere are shown to satisfy neither condition. Some other transformations due to Fang and Wang (1993) satisfy the first but not the second condition. We provide transformations for the simplex that makes f • τ smooth enough to fully benefit from scrambled net sampling for all f in a class of generalized polynomials. We also find sufficient conditions for the Rosenblatt-HlawkaMück transformation in R 2 and for importance sampling to be of bounded variation in the sense of Hardy and Krause.
Many interesting problems in the Internet industry can be framed as a two-sided marketplace problem. Examples include search applications and recommender systems showing people, jobs, movies, products, restaurants, etc. Incorporating fairness while building such systems is crucial and can have deep social and economic impact (applications include job recommendations, recruiters searching for candidates, etc.). In this paper, we propose a definition and develop an end-to-end framework for achieving fairness while building such machine learning systems at scale. We extend prior work [29] to develop an optimization framework that can tackle fairness constraints from both the source and destination sides of the marketplace, as well as dynamic aspects of the problem. The framework is flexible enough to adapt to different definitions of fairness and can be implemented in very large-scale settings. We perform simulations to show the efficacy of our approach.Preprint. Under review.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.