Transportation of samples across different domains is a central task in several machine learning problems. A sensible requirement for domain transfer tasks in computer vision and language domains is the sparsity of the transportation map, i.e., the transfer algorithm aims to modify the least number of input features while transporting samples across the source and target domains. In this work, we propose Elastic Net Optimal Transport (ENOT) to address the sparse distribution transfer problem. The ENOT framework utilizes the $L_1$-norm and $L_2$-norm regularization mechanisms to find a sparse and stable transportation map between the source and target domains. To compute the ENOT transport map, we consider the dual formulation of the ENOT optimization task and prove that the sparsified gradient of the optimal potential function in the ENOT's dual representation provides the ENOT transport map. Furthermore, we demonstrate the application of the ENOT framework to perform feature selection for sparse domain transfer. We present the numerical results of applying ENOT to several domain transfer problems for synthetic Gaussian mixtures and real image and text data. Our empirical results indicate the success of the ENOT framework in identifying a sparse domain transport map.

翻译：不同域之间的样本迁移是多个机器学习问题中的核心任务。在计算机视觉和语言领域的域迁移任务中，一个合理的要求是迁移图的稀疏性，即迁移算法旨在修改最少数量的输入特征，同时将样本从源域迁移到目标域。在这项工作中，我们提出弹性网络最优传输（ENOT）来解决稀疏分布迁移问题。ENOT框架利用$L_1$-范数和$L_2$-范数正则化机制，在源域和目标域之间找到稀疏且稳定的迁移图。为了计算ENOT迁移图，我们考虑了ENOT优化任务的对偶形式，并证明了ENOT对偶表示中最优势函数的稀疏梯度提供了ENOT迁移图。此外，我们展示了ENOT框架在稀疏域迁移中进行特征选择的应用。我们给出了将ENOT应用于合成高斯混合数据以及真实图像和文本数据的多个域迁移问题的数值结果。我们的实证结果表明，ENOT框架在识别稀疏域迁移图方面取得了成功。