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Published at December 9

cs.DB

Released Date: December 9, 2024

Authors: Leilei Du¹, Peng Cheng¹, Libin Zheng², Xiang Lian³, Lei Chen⁴, Wei Xi⁵, Wangze Ni⁶

Aff.: ¹East China Normal University; ²Sun Yat-sen University; ³Kent State University; ⁴HKUST (GZ) and HKUST; ⁵Xi'an Jiaotong University; ⁶HKUST

Notations	Description
$k$ -Dim	$k$ -dimension
$\mathcal{D}$ , $\mathcal{I}$	the input domain
$\tilde{\mathcal{D}}$ , $\mathcal{T}$	the output domain
$\mathcal{\hat{I}}$	the inferred (estimated) domain of $\mathcal{I}$ domain
$D$	an input instance
$\tilde{D},O$	an output instance instance
$\boldsymbol{v}$	a spatial data point
$\tilde{\boldsymbol{v}}$	a disturbed point of $\boldsymbol{v}$
$M_{\boldsymbol{v}}(\tilde{\boldsymbol{v}})$	the probability of randomizing $\boldsymbol{v}$ as $\tilde{\boldsymbol{v}}$
$W(\cdot)$	the wave function
$W_{k}^{p}$	a $k$ dimensional Wasserstein distance
$W_{k}^{p}$	with $p$ norm cost function
$W_{k}(\cdot)$	the $p$ -th root of $W_{k}^{p}$ (i.e., $W_{k}(\cdot)=\sqrt[p]{W_{k}^{p}}$ )
$SW_{k}^{p}$	a $k$ dimensional sliced Wasserstein distance
$SW_{k}^{p}$	with $p$ norm cost function
$b$	a high dimensional radius
$L$	the side length of an input instance
$g$	the side length of a grid cell
$d$	the number of cells along a side of grid length
$n$	the number of cells in the grid