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.

介绍了一种名为深度集合线性化最优输运的算法，用于将点云有效地嵌入到 L^2 空间中，并在构建一个分类器来区分不同类别的点云的同时，保留 Wasserstein 空间内特定的低维结构。通过使用输入凸神经网络 (ICNNs) 学习这些输运映射的近似，我们发现在特定条件下，从这些 ICNNs 中采样的欧氏距离与真实分布的 Wasserstein-2 距离非常接近。此外，我们训练了一个判别器网络，对这些样本进行加权，并创建一个排列不变的分类器，以区分不同类别的点云。通过在仅拥有有限标记点云的流式细胞术数据集上进行实验，展示了我们算法相对于标准的深度集合方法的优势。