  Max sliced Wasserstein (Max-SW) distance has been widely known as a solution
for less discriminative projections of sliced Wasserstein (SW) distance. In
applications that have various independent pairs of probability measures,
amortized projection optimization is utilized to predict the ``max" projecting
directions given two input measures instead of using projected gradient ascent
multiple times. Despite being efficient, Max-SW and its amortized version
cannot guarantee metricity property due to the sub-optimality of the projected
gradient ascent and the amortization gap. Therefore, we propose to replace
Max-SW with distributional sliced Wasserstein distance with von Mises-Fisher
(vMF) projecting distribution (v-DSW). Since v-DSW is a metric with any
non-degenerate vMF distribution, its amortized version can guarantee the
metricity when performing amortization. Furthermore, current amortized models
are not permutation invariant and symmetric. To address the issue, we design
amortized models based on self-attention architecture. In particular, we adopt
efficient self-attention architectures to make the computation linear in the
number of supports. With the two improvements, we derive self-attention
amortized distributional projection optimization and show its appealing
performance in point-cloud reconstruction and its downstream applications.
