We introduce Deep Set Linearized Optimal Transport, an algorithm designed for the efficient simultaneous embedding of point clouds into an $L^2-$space. This embedding preserves specific low-dimensional structures within the Wasserstein space while constructing a classifier to distinguish between various classes of point clouds. Our approach is motivated by the observation that $L^2-$distances between optimal transport maps for distinct point clouds, originating from a shared fixed reference distribution, provide an approximation of the Wasserstein-2 distance between these point clouds, under certain assumptions. To learn approximations of these transport maps, we employ input convex neural networks (ICNNs) and establish that, under specific conditions, Euclidean distances between samples from these ICNNs closely mirror Wasserstein-2 distances between the true distributions. Additionally, we train a discriminator network that attaches weights these samples and creates a permutation invariant classifier to differentiate between different classes of point clouds. We showcase the advantages of our algorithm over the standard deep set approach through experiments on a flow cytometry dataset with a limited number of labeled point clouds.

翻译：我们提出了深度集线性化最优传输算法，该算法旨在将点云高效地同时嵌入到$L^2$空间中。这种嵌入在Wasserstein空间中保留了特定的低维结构，同时构建了一个分类器来区分不同类别的点云。我们的方法基于如下观察：在特定假设下，源于同一固定参考分布的不同点云的最优传输映射之间的$L^2$距离，近似于这些点云之间的Wasserstein-2距离。为学习这些传输映射的近似，我们采用输入凸神经网络（ICNNs），并证明了在特定条件下，这些ICNN样本之间的欧氏距离能够密切反映真实分布之间的Wasserstein-2距离。此外，我们训练了一个判别器网络，该网络为这些样本附加权重，并构建了一个置换不变分类器，以区分不同类别的点云。通过在标记点云数量有限的流式细胞术数据集上的实验，我们展示了该算法相对于标准深度集方法的优势。