Tensor Representations for Event Time Series

Updated 19 July 2025

Tensor representations for event time series are mathematical frameworks that convert irregular event data into structured, high-dimensional arrays for analysis.
They enable efficient forecasting, anomaly detection, clustering, and relational inference by leveraging tensor factorization and deep learning techniques.
Applications span diverse fields such as neuromorphic imaging, finance, and sensor networks, highlighting their versatility and scalability.

Event time series are sequences of temporally marked discrete events that may be associated with additional structured or unstructured attributes. Tensor representations for event time series provide a mathematical and computational framework to encode, analyze, and learn from the high-dimensional, irregular, and multimodal structure inherent in such data. These representations have demonstrable advantages for tasks including forecasting, anomaly detection, relational inference, and semantic clustering. Efforts in this domain span model-driven approaches rooted in temporal point processes, latent low-rank tensor factorization, deep learning with tensorized encodings, and innovative transformations that bridge continuous-time events with fixed-size tensor arrays.

1. Principles of Tensor Construction for Event Time Series

A range of methodologies exist for mapping event time series into tensorial forms, each reflecting application-specific goals (reconstruction, similarity, interpretability, prediction):

Direct Binning and Transformation: Events are aggregated into multi-dimensional tensor grids by binning along axes such as normalized time, event attributes (e.g., energy, semantic marks), or inter-event intervals. For example, a two-dimensional E–t map bins events over normalized time and a transformed event attribute, while a three-dimensional E–t–dt cube introduces inter-event timing as a third axis (Dillmann et al., 15 Jul 2025).
Periodic and Derivative Transformation: One-dimensional event or signal time series are mapped into 2D tensors either via periodic decomposition (FFT-based reshaping to highlight dominant periods) or via derivative-based heatmaps that explicitly encode sharp changes and turning points as secondary axes (Nematirad et al., 31 Mar 2025).
Spatiotemporal Tensors: In neuromorphic event camera data, asynchronous spatial-temporal events are stacked into three-way tensors over (x, y, t) coordinates, enabling global modeling of the underlying spatio-temporal structure (Yang et al., 16 Jan 2024).
Tensorized Embeddings and Algebraic Lifts: Sequential event data are lifted to tensor algebra features using free algebra constructions or signature-based methods, capturing non-commutative, high-order dependencies (including all ordered subsequences up to a specified degree) (Toth et al., 2020).

Such tensorizations facilitate regularization, comparability, or the application of machine learning methods designed for fixed-size inputs regardless of the original sequence's irregularity or length.

2. Low-Rank and Factorized Models

Low-rank tensor factorization serves as the backbone for many event time series models, driven by the necessity of dimensionality reduction and interpretability in high-dimensional settings:

Tucker and CP Factor Models: Multimodal event time series are expressed as core tensors multiplied along each mode by loading matrices, with temporal dependence encoded in the evolution of the core tensor (Han et al., 2020). Iterative orthogonal projection algorithms (iTOPUP, iTIPUP) efficiently estimate these factors, offering convergence rates optimal with respect to both sample size and mode dimensions.
Tensor Autoregressive (TenAR) Models: Time evolution is modeled via multi-linear dynamics, with mode-specific coefficient matrices applied to lagged tensors (Li et al., 2021). This preserves the tensor structure and vastly reduces parameter count versus vectorized VAR, yielding improved interpretability.
Low-Rank Temporal Knowledge Graph Models: In temporal knowledge graph completion tasks, low-rank tensor factorization is extended by modulating entity and relation embeddings with specialized, cycle-aware temporal encodings. Model variations (e.g., Time-LowFER) support explicit separation of static and time-varying effects, with cyclic encodings capturing multi-scale temporal phenomena (Dikeoulias et al., 2022).
Mixture-based Representations: Recent frameworks use channel-wise Gaussian mixture models to represent the latent factors in tensor time series, disentangling variable-specific heterogeneity across time, location, and source channels (Deng et al., 2023). Such representations improve interpretability and generalization to new spatiotemporal conditions.

3. Neural and Attention-based Tensorized Architectures

Deep learning models capitalize on tensor representations to extend sequence modeling capacity:

Attentional Twin RNNs: A twin recurrent architecture separately processes event streams and regular time series, fusing their outputs in a synergic layer. Attention mechanisms over event history yield interpretable infectivity matrices elucidating event-type interactions (1703.08524).
Self-Attention with Functional Time Embeddings: Time lags between events are embedded into high-dimensional tensors via explicit feature maps constructed using Bochner or Mercer expansions of translation-invariant kernels. The resulting representations are concatenated with event type embeddings, facilitating attention-based modeling of both event type and temporal proximity (Xu et al., 2019).
Task- and Anomaly-Aware Autoencoders: Hierarchical multi-scale autoencoders compress high-dimensional, irregular event time series into rich latent representations. Reconstruction losses are augmented with probabilistic (GMM-based) regularization, leading to structured latent spaces suited for unsupervised similarity learning and anomaly detection (Dou et al., 2022, Dou et al., 20 Jun 2025, Dillmann et al., 15 Jul 2025). The combination of explicit vectorization (via binning to 2D or 3D tensors), temporal normalization, and contextual reconstruction enables robust anomaly detection, semantic clustering, and similarity-based search.

4. Tensor Decomposition with Class-aware or Contrastive Objectives

A growing body of work addresses the inherent non-uniqueness and rotation invariance of tensor decomposition by incorporating class-contrastive or pseudo-graph guided penalties:

Pseudo Laplacian Contrast (PLC): Class-aware representations are extracted by integrating a pseudo-graph Laplacian penalty (constructed from cluster labels in latent space) into CP decomposition. Cross-view Laplacian contrast leverages data augmentations and an alternating least squares (ALS) solver, rotating learnt features to maximize class separability while enforcing low-rank structure (Li et al., 23 Sep 2024). Empirical results highlight significant improvements in downstream classification and cluster interpretability for event-driven data.

Recent advances facilitate multi-modal or complex event sources by developing new tensorization and aggregation strategies:

Aggregation of Multi-periodic and Derivative Signals: The Times2D approach transforms 1D time series into separate 2D tensors capturing dominant periodic patterns and higher-order derivatives. Aggregation blocks then combine these orthogonal representations for improved forecasting accuracy, especially over datasets with intricate variability and sharp transitions (Nematirad et al., 31 Mar 2025).
Networked Tensor Time Series: Multi-modal, networked event time series (e.g., in smart transportation or environmental monitoring) are modeled with tensor graph convolutional and tensor recurrent neural modules. These jointly exploit explicit topological relationships (across spatial or attribute modes) and implicit multi-way temporal dynamics, enabling robust prediction and missing value imputation (Jing et al., 2021).

6. Applications, Performance, and Practical Implications

Tensor representations for event time series underpin advancements in domains as varied as neuroimaging, high-energy astrophysics, resource and demand forecasting, networked sensor systems, and cybersecurity:

Performance Gains: Models consistently demonstrate improved anomaly detection and clustering accuracy (via autoencoder-tensor-GMM architectures), superior long- and short-term prediction (with tensor transformation and convolutional models), and robust knowledge graph completion (through low-rank, temporal-aware tensor embeddings).
Scalability and Efficiency: Low-rank projections, mixture-based latent spaces, and structured convolutional operations offer reduced parameterization, efficient training, and adaptability to large heterogeneous datasets (Toth et al., 2020, Deng et al., 2023).
Interpretability: Attention matrices, tensor contraction fields, and visualization of latent tensor spaces enhance insight into causal event dynamics, semantic grouping, and latent temporal structures.
Generalizability: Binning, normalization, and algebraic lifting of irregular event streams to fixed-size tensors allow the re-use of models across disparate scientific and industrial contexts—highlighted by success in astrophysics, finance, health informatics, and IoT security (Dillmann et al., 15 Jul 2025, Dou et al., 2022, Dou et al., 20 Jun 2025).

7. Future Directions and Methodological Challenges

The field exhibits ongoing research into:

Adaptation and Efficiency: Reductions in dynamic mixture modeling overhead, hybridizing memory modules, and efficient inference of cycle-aware or context-specific embeddings (Deng et al., 2023, Dikeoulias et al., 2022).
Broader Model Integration: Fusing tensor methods with graph structures, exogenous covariate incorporation, and advances in data augmentation and cross-modal learning.
Class-guided and Self-supervised Learning: Expanded use of pseudo-graphs, augmentative contrastive learning, and cross-view regularization to enhance class separability and generalization (Li et al., 23 Sep 2024).
Explainability: Ongoing work on mapping latent tensor factors to physical, semantic, or causal phenomena enhances the transparency and domain utility of these representation learning approaches.

Tensor representations for event time series therefore constitute a rigorous, flexible, and increasingly essential class of models, offering principled solutions to the fundamental challenges posed by complexity, heterogeneity, and irregularity in modern data streams. They support not only improved prediction and detection but also deeper scientific insight across domains.