The proliferation and ubiquity of temporal data across many disciplines has sparked interest for similarity, classification and clustering methods specifically designed to handle time series data. A core issue when dealing with time series is determining their pairwise similarity, i.e., the degree to which a given time series resembles another. Traditional distance measures such as the Euclidean are not well-suited due to the time-dependent nature of the data. Elastic metrics such as dynamic time warping (DTW) offer a promising approach, but are limited by their computational complexity, non-differentiability and sensitivity to noise and outliers. This thesis proposes novel elastic alignment methods that use parametric \& diffeomorphic warping transformations as a means of overcoming the shortcomings of DTW-based metrics. The proposed method is differentiable \& invertible, well-suited for deep learning architectures, robust to noise and outliers, computationally efficient, and is expressive and flexible enough to capture complex patterns. Furthermore, a closed-form solution was developed for the gradient of these diffeomorphic transformations, which allows an efficient search in the parameter space, leading to better solutions at convergence. Leveraging the benefits of these closed-form diffeomorphic transformations, this thesis proposes a suite of advancements that include: (a) an enhanced temporal transformer network for time series alignment and averaging, (b) a deep-learning based time series classification model to simultaneously align and classify signals with high accuracy, (c) an incremental time series clustering algorithm that is warping-invariant, scalable and can operate under limited computational and time resources, and finally, (d) a normalizing flow model that enhances the flexibility of affine transformations in coupling and autoregressive layers.

翻译：时间数据在多个学科中的广泛普及与激增，激发了专门针对时间序列数据的相似性、分类和聚类方法的研究兴趣。处理时间序列的核心问题在于确定其两两相似性，即给定时间序列与另一时间序列的相似程度。由于数据的时间依赖性，欧氏距离等传统距离度量并不适用。动态时间弯曲（DTW）等弹性度量提供了一种有前景的方法，但其计算复杂度、不可微性以及对噪声和异常值的敏感性限制了其应用。本文提出了一种新型弹性对齐方法，该方法利用参数化微分同胚弯曲变换来克服基于DTW的度量标准的缺陷。所提出的方法可微且可逆，适用于深度学习架构，对噪声和异常值具有鲁棒性，计算高效，且具有足够的表达能力和灵活性以捕捉复杂模式。此外，我们为这些微分同胚变换的梯度开发了一种闭式解，从而能够高效搜索参数空间，并在收敛时获得更优解。利用这些闭式微分同胚变换的优势，本文提出了一系列进展，包括：（a）一种用于时间序列对齐和平均的增强型时态变换网络；（b）一种基于深度学习的时间序列分类模型，可同时以高精度对齐并分类信号；（c）一种增量式时间序列聚类算法，该算法对弯曲变换具有不变性、可扩展，并能在有限的计算和时间资源下运行；以及（d）一种归一化流模型，增强了耦合层和自回归层中仿射变换的灵活性。