Automatically extracting robust representations from large and complex time series data is becoming imperative for several real-world applications. Unfortunately, the potential of common neural network architectures in capturing invariant properties of time series remains relatively underexplored. For instance, convolutional layers often fail to capture underlying patterns in time series inputs that encompass strong deformations, such as trends. Indeed, invariances to some deformations may be critical for solving complex time series tasks, such as classification, while guaranteeing good generalization performance. To address these challenges, we mathematically formulate and technically design efficient and hard-coded *invariant convolutions* for specific group actions applicable to the case of time series. We construct these convolutions by considering specific sets of deformations commonly observed in time series, including *scaling*, *offset shift*, and *trend*. We further combine the proposed invariant convolutions with standard convolutions in single embedding layers, and we showcase the layer capacity to capture complex invariant time series properties in several scenarios.

Motif Discovery involves identifying recurring patterns and locating their occurrences within a time series without prior knowledge about their shape or location. In practice, Motif Discovery faces several data-related challenges, leading to various definitions of the problem and multiple algorithms addressing these challenges to different extents. However, there has been no systematic evaluation and comparison of these diverse approaches. Consequently, this paper presents a comprehensive literature review covering data-related challenges, motif definitions, and algorithms. We also analyze the strengths and limitations of algorithms carefully chosen to represent the literature diversity. The analysis is structured around key research questions identified from our review. Our experimental findings provide practical guidelines for selecting Motif Discovery algorithms suitable for a given task and suggest directions for future research.

Analyzing inter-individual variability of physiological functions is particularly appealing in medical and biological contexts to describe or quantify health conditions. Such analysis can be done by comparing individuals to a reference one with time series as biomedical data. This paper introduces an unsupervised representation learning (URL) algorithm for time series tailored to inter-individual studies. The idea is to represent time series as deformations of a reference time series. The deformations are diffeomorphisms parameterized and learned by our method called TS-LDDMM. Once the deformations and the reference time series are learned, the vector representations of individual time series are given by the parametrization of their corresponding deformation. At the crossroads between URL for time series and shape analysis, the proposed algorithm handles irregularly sampled multivariate time series of variable lengths and provides shape-based representations of temporal data. In this work, we establish a representation theorem for the graph of a time series and derive its consequences on the LDDMM framework. We showcase the advantages of our representation compared to existing methods using synthetic data and real-world examples motivated by biomedical applications.

Motif Discovery consists of finding repeated patterns and locating their occurrences in a time series without prior knowledge about their shape or location. Most state-of-the-art algorithms rely on three core parameters: the number of motifs to discover, the length of the motifs, and a similarity threshold between motif occurrences. Setting these parameters is difficult in practice and often results from a trial-and-error strategy. In this paper, we propose a new algorithm that discovers motifs of variable length given a single motif length and without requiring a similarity threshold. At its core, the algorithm maps a time series onto a graph, summarizes it with persistent homology - a tool from topological data analysis - and identifies the most relevant motifs from the graph summary. We propose two versions of the algorithm, one requiring the number of motifs to discover and another, adaptive, that infers the number of motifs from the graph summary. Empirical evaluation on 9 labeled datasets, including 6 real-world datasets, shows that both algorithm versions significantly outperform state-of-the-art algorithms.