Optimal transport (OT) is a powerful geometric tool used to compare and align
probability measures following the least effort principle. Among many
successful applications of OT in machine learning (ML), domain adaptation (DA)
-- a field of study where the goal is to transfer a classifier from one
labelled domain to another similar, yet different unlabelled or scarcely
labelled domain -- has been historically among the most investigated ones. This
success is due to the ability of OT to provide both a meaningful discrepancy
measure to assess the similarity of two domains' distributions and a mapping
that can project source domain data onto the target one. In this paper, we
propose a principally new OT-based approach applied to DA that uses the
closed-form solution of the OT problem given by an affine mapping and learns an
embedding space for which this solution is optimal and computationally less
complex. We show that our approach works in both homogeneous and heterogeneous
DA settings and outperforms or is on par with other famous baselines based on
both traditional OT and OT in incomparable spaces. Furthermore, we show that
our proposed method vastly reduces computational complexity.

本文提出了一种基于最优传输的领域适应方法，使用仿射映射的封闭形式解决方案，并学习一个嵌入空间，使得该解决方案是最优且计算复杂性较低的。我们证明了该方法在同质和异质适应设置中均有效，并且优于或与其他基于传统 OT 和不可比空间的著名基线相当。此外，我们展示了我们的方法大大降低了计算复杂性。

超越不变表示学习：线性可对齐潜空间进行高效的解析域自适应

Beyond invariant representation learning: linearly alignable latent  spaces for efficient closed-form domain adaptation

We propose a novel problem formulation of learning a single task when the
data are provided in different feature spaces. Each such space is called an
outlook, and is assumed to contain both labeled and unlabeled data. The
objective is to take advantage of the data from all the outlooks to better
classify each of the outlooks. We devise an algorithm that computes optimal
affine mappings from different outlooks to a target outlook by matching moments
of the empirical distributions. We further derive a probabilistic
interpretation of the resulting algorithm and a sample complexity bound
indicating how many samples are needed to adequately find the mapping. We
report the results of extensive experiments on activity recognition tasks that
show the value of the proposed approach in boosting performance.

本文提出了一种新的多视角学习问题，通过匹配实际分布的矩来计算不同视角之间的最优仿射映射，进一步推导出算法的概率解释和样本复杂度界限，并在活动识别任务上进行了大量实验，证明了所提方法提高性能的价值。