Machine learning applications on signals such as computer vision or biomedical data often face significant challenges due to the variability that exists across hardware devices or session recordings. This variability poses a Domain Adaptation (DA) problem, as training and testing data distributions often differ. In this work, we propose Spatio-Temporal Monge Alignment (STMA) to mitigate these variabilities. This Optimal Transport (OT) based method adapts the cross-power spectrum density (cross-PSD) of multivariate signals by mapping them to the Wasserstein barycenter of source domains (multi-source DA). Predictions for new domains can be done with a filtering without the need for retraining a model with source data (test-time DA). We also study and discuss two special cases of the method, Temporal Monge Alignment (TMA) and Spatial Monge Alignment (SMA). Non-asymptotic concentration bounds are derived for the mappings estimation, which reveals a bias-plus-variance error structure with a variance decay rate of $\mathcal{O}(n_\ell^{-1/2})$ with $n_\ell$ the signal length. This theoretical guarantee demonstrates the efficiency of the proposed computational schema. Numerical experiments on multivariate biosignals and image data show that STMA leads to significant and consistent performance gains between datasets acquired with very different settings. Notably, STMA is a pre-processing step complementary to state-of-the-art deep learning methods.

翻译：在计算机视觉或生物医学数据等信号处理中，机器学习应用常因硬件设备或采集会话间的差异性而面临重大挑战。这种差异性导致了领域自适应问题，因为训练数据与测试数据的分布往往存在差异。本文提出时空蒙日对齐方法以缓解此类变异性。这一基于最优传输的方法通过将多元信号的互功率谱密度映射至源域的瓦瑟斯坦重心来实现自适应。对于新领域的预测可通过滤波完成，无需利用源数据重新训练模型。本文还研究并讨论了该方法的两种特例：时间蒙日对齐与空间蒙日对齐。我们推导了映射估计的非渐近集中界，揭示了具有偏差-方差结构的误差特性，其方差衰减率为$\mathcal{O}(n_\ell^{-1/2})$。该理论保证验证了所提计算框架的有效性。在多元生物信号与图像数据上的数值实验表明，STMA能在采集设置差异巨大的数据集间实现显著且稳定的性能提升。值得注意的是，STMA可作为预处理步骤与前沿深度学习方法形成有效互补。