92 research outputs found
Quasi-Monte Carlo for 3D Sliced Wasserstein
Monte Carlo (MC) integration has been employed as the standard approximation
method for the Sliced Wasserstein (SW) distance, whose analytical expression
involves an intractable expectation. However, MC integration is not optimal in
terms of absolute approximation error. To provide a better class of empirical
SW, we propose quasi-sliced Wasserstein (QSW) approximations that rely on
Quasi-Monte Carlo (QMC) methods. For a comprehensive investigation of QMC for
SW, we focus on the 3D setting, specifically computing the SW between
probability measures in three dimensions. In greater detail, we empirically
evaluate various methods to construct QMC point sets on the 3D
unit-hypersphere, including the Gaussian-based and equal area mappings,
generalized spiral points, and optimizing discrepancy energies. Furthermore, to
obtain an unbiased estimator for stochastic optimization, we extend QSW to
Randomized Quasi-Sliced Wasserstein (RQSW) by introducing randomness in the
discussed point sets. Theoretically, we prove the asymptotic convergence of QSW
and the unbiasedness of RQSW. Finally, we conduct experiments on various 3D
tasks, such as point-cloud comparison, point-cloud interpolation, image style
transfer, and training deep point-cloud autoencoders, to demonstrate the
favorable performance of the proposed QSW and RQSW variants.Comment: Accepted to ICLR 2024 (Spotlight), 25 pages, 13 figures, 6 table
An Algorithm for the Machine Calculation of Minimal Paths
Problems involving the minimization of functionals date back to antiquity. The mathematics of the calculus of variations has provided a framework for the analytical solution of a limited class of such problems. This paper describes a numerical approximation technique for obtaining machine solutions to minimal path problems. It is shown that this technique is applicable not only to the common case of finding geodesics on parameterized surfaces in R3, but also to the general case of finding minimal functionals on hypersurfaces in Rn associated with an arbitrary metric
Representations for the Three-Body T-Matrix, Scattering Matrices and Resolvent in Unphysical Energy Sheets
Explicit representations for the Faddeev components of the three-body
T-matrix continued analytically into unphysical sheets of the energy Riemann
surface are formulated. According to the representations, the T-matrix in
unphysical sheets is explicitly expressed in terms of its components taken in
the physical sheet only. The representations for the T-matrix are then used to
construct similar representations for the analytic continuation of the
three-body scattering matrices and the resolvent. Domains on unphysical sheets
are described where the representations obtained can be applied. A method for
finding three-body resonances based on the Faddeev differential equations is
proposed.Comment: LaTeX file of the published version of the pape
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