This paper introduces a learning scheme to construct a Hilbert space (i.e., a vector space along its inner product) to address both unsupervised and semi-supervised domain adaptation problems. This is achieved by learning projections from each domain to a latent space along the Mahalanobis metric of the latent space to simultaneously minimizing a notion of domain variance while maximizing a measure of discriminatory power. In particular, we make use of the Riemannian optimization techniques to match statistical properties (e.g., first and second order statistics) between samples projected into the latent space from different domains. Upon availability of class labels, we further deem samples sharing the same label to form more compact clusters while pulling away samples coming from different classes.We extensively evaluate and contrast our proposal against state-of-the-art methods for the task of visual domain adaptation using both handcrafted and deep-net features. Our experiments show that even with a simple nearest neighbor classifier, the proposed method can outperform several state-of-the-art methods benefitting from more involved classification schemes.

本文提出了一种学习方案，用于构建希尔伯特空间，以解决无监督和半监督领域自适应问题，并通过学习每个领域到潜在空间的投影来实现，同时最小化领域方差的概念，同时最大化区分力的测量。我们利用黎曼优化技术在不同领域中进行潜在空间中的样本之间的统计属性匹配。我们进一步认为共享相同标签的样本形成更紧凑的聚类，同时拉开来自不同类别的样本。我们针对手工制作和深度网络功能使用我们的提案进行全面评估和对比，并在视觉领域自适应任务中展示了我们的实验，即使是用简单的最近邻分类器，这种方法也可以胜过几种最先进的方法。

为领域自适应学习不变的 Hilbert 空间