Public defence: Christos Chatzis

Multi-way datasets (also referred to as higher-order tensors) are commonly analysed using unsupervised matrix and tensor factorizations to reveal the underlying patterns. When one of the ways that the data evolves across is time, the objective of such analyses often becomes the identification and tracking of the innate temporal dynamics.

Consider a dataset arranged as a three-way entity x features x time array, which can correspond to, for example, medical measurements of different subjects during a longitudinal study or word usage of certain users on a social media platform over a given time window.

Understanding more about the complex underlying dynamics may then call for (a) uncovering how features evolve over time at the population or group level or (b) recovering entity-specific temporal profiles that describe how individual entities express the underlying patterns over time.

Existing matrix and tensor factorization techniques for setting (a), including PARAFAC2, a tensor factorization that allows structural variation across one mode, often ignore the inherently sequential nature of the time mode.

Conversely, methods that explicitly model temporal evolution frequently come with weak or unexplored uniqueness guarantees or overly restrictive structural assumptions. This thesis addresses that gap by proposing two time-aware factorization methods that incorporate temporal ordering while remaining sufficiently flexible to capture evolving patterns, namely

• t(emporal)PARAFAC2, which promotes smooth changes across the evolving factors of PARAFAC2 and,
• dynamical coupled matrix factorization (dCMF), which enforces a linear dynamical system (LDS) structure on component trajectories.

We demonstrate the effectiveness of tPARAFAC2 in terms of pattern recovery on both synthetic and real data, showing improved robustness to noise and stronger performance in low-signal settings, particularly when the underlying patterns evolve slowly.

However, if the input does not conform to PARAFAC2 structure, our experiments indicate dCMF might be the better choice due to the lack of such specific structural requirements. Furthermore, we show that dCMF can incorporate prior information about the temporal evolution achieving highly accurate recovery, even in very noisy settings.

Finally, we examine the uniqueness behavior of dCMF and observe empirically unique solutions across all the experimental settings considered.

Real-world temporal data frequently contains missing or corrupted entries. In order to enable fitting regularized (possibly in all modes) decomposition methods, such as the proposed tPARAFAC2 and dCMF, on partially observed data we propose two approaches by extending alternating optimization alternating direction method of multipliers (AO-ADMM)-based algorithms: an expectation-maximization-based (EM) method and a weighted-optimization-based method.

Extensive synthetic experiments show that the two approaches achieve comparable recovery accuracy, while the EM-based method is faster. In two real-data settings, we further demonstrate the practical value of enabling regularization through our approaches: in a chemometrics application, non-negativity eliminates infeasible values and improves factor recovery, while in a metabolomics application, temporal regularization improves recovery at high levels of missingness.

Most existing works focus on group-level temporal patterns and therefore provide only a coarse description of temporal variation across subjects. Since subject-level dynamics may be crucial to understanding the relevant complex systems, in this thesis, we also address setting (b) by appropriately arranging the input data and using coupled matrix factorization (CMF) and PARAFAC2 to recover subject-specific temporal trajectories across multiple domains. Comparing with the state-of-the-art, we demonstrate that the proposed approaches can capture more detailed individual temporal profiles, reflecting the heterogeneity of subjects.

In a microbiome dataset, the additional flexibility of the proposed methods reveals differences in temporal trajectories related to delivery mode, while in a sensitization dataset, earlier sensitization of subjects born with csection was made visible. In a metabolomics dataset, the proposed approaches revealed body mass index and insulin resistance related differences in temporal trajectories.

A further contribution is that the recovered factors are systematically assessed in terms of replicability and interpretability: we introduce a principled way to evaluate replicability of the temporal trajectories.