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Max sliced Wasserstein (Max-SW) distance has been widely known as a solutionfor less discriminative projections of sliced Wasserstein (SW) distance. Inapplications that have various independent pairs of probability measures,amortized projection optimization is utilized to predict the ``max" projectingdirections given two input measures instead of using projected gradient ascentmultiple times. Despite being efficient, Max-SW and its amortized versioncannot guarantee metricity property due to the sub-optimality of the projectedgradient ascent and the amortization gap. Therefore, we propose to replaceMax-SW with distributional sliced Wasserstein distance with von Mises-Fisher(vMF) projecting distribution (v-DSW). Since v-DSW is a metric with anynon-degenerate vMF distribution, its amortized version can guarantee themetricity when performing amortization. Furthermore, current amortized modelsare not permutation invariant and symmetric. To address the issue, we designamortized models based on self-attention architecture. In particular, we adoptefficient self-attention architectures to make the computation linear in thenumber of supports. With the two improvements, we derive self-attentionamortized distributional projection optimization and show its appealingperformance in point-cloud reconstruction and its downstream applications.

Self-Attention Amortized Distributional Projection Optimization for Sliced Wasserstein Point-Cloud Reconstruction