Non-Negative Matrix Factorization for Event Data

Technical Summary: Non-Negative Matrix Factorization for Event Data (EventNMF)

Problem Statement
Continuous-time event data, where entities emit instantaneous events over time (e.g., neuronal spikes, seismic events, social interactions), is ubiquitous across domains. While Non-Negative Matrix Factorization (NMF) is a standard tool for uncovering interpretable structure in such data, existing applications typically require preprocessing the raw event times via binning or smoothing to generate temporal signals. This discretization introduces significant drawbacks: it risks erasing entity-level heterogeneities and fine-grained temporal features, alters the statistical properties of the underlying point process, and introduces a difficult-to-control bias-variance tradeoff regarding bin width or kernel bandwidth selection.

Methodology: EventNMF
The paper introduces EventNMF, a continuous-time non-negative factorization model that operates directly on event times without binning. The core methodology is built upon the following components:

• Generative Model: Each entity $i$ is modeled as an inhomogeneous Poisson process with an intensity function $\lambda_i(t)$. The intensity is assumed to admit a low-rank factorization:
  $$\lambda_i(t) = \sum_{r=1}^R u_{ir} f_r(t)$$
  where $u_{ir} \geq 0$ are non-negative loadings specific to the entity, and $f_r(t) \geq 0$ are shared latent temporal factors.
• Parametric Basis: To make the model parametric, each latent factor $f_r(t)$ is expanded over a fixed non-negative B-spline basis $\{\phi_b\}_{b=1}^B$:
  $$f_r(t) = \sum_{b=1}^B \gamma_{rb} \phi_b(t)$$
  where $\gamma_{rb} \geq 0$. This allows the model to approximate a wide range of intensity shapes while maintaining non-negativity.
• Objective Function: The parameters (loading matrix $U$ and coefficient matrix $\Gamma$) are estimated by minimizing the negative log-likelihood of the observed counting measures under the Poisson process assumption. The loss function for entity $i$ is:
  $$l_i(U, \Gamma) = u_i^\top \Gamma I(\Phi) - \sum_{\tau \in E_i} \log(u_i^\top \Gamma \Phi(\tau))$$
  where $I(\Phi)$ is the vector of integrals of the basis functions and $E_i$ is the set of event times for entity $i$.
• Optimization: The authors derive efficient multiplicative updates for $U$ and $\Gamma$ that guarantee non-negativity throughout the optimization process. The computational complexity per iteration is $O(RBE)$, linear in the number of events, making the method scalable.
• Theoretical Connection: The paper demonstrates that when the B-spline degree is zero, EventNMF reduces exactly to standard Poisson-NMF applied to binned counts. Thus, EventNMF generalizes existing binned approaches to the continuous domain.

Key Contributions

1. Direct Continuous-Time Modeling: EventNMF eliminates the need for binning or smoothing, operating directly on raw event times to preserve fine-grained temporal features and statistical properties.
2. Unified Framework: It connects dimensionality reduction (NMF) with point process intensity estimation within a single framework using Poisson likelihoods.
3. Efficient Algorithm: The derivation of multiplicative updates provides a simple, mathematically principled, and computationally efficient estimation procedure.
4. Empirical Validation: The method is validated on synthetic data and three real-world applications:
   • Seismology: Analysis of California-Nevada earthquake data (2014–2024), successfully separating distinct seismic patterns (e.g., the 2019 Ridgecrest mainshock vs. persistent Geysers activity) into interpretable temporal and spatial factors.
   • Neuroscience: Single-trial analysis of the Allen Visual Coding Neuropixels dataset, recovering distinct population-level response motifs (onset transients, sustained rhythms, suppression) and mapping them to specific brain areas.
   • Social Networks: Analysis of face-to-face contacts in a primary school, unsupervised recovery of class-specific interaction patterns, and differentiation between lesson-time (steady) and free-period (peaked) interactions.

Results and Performance

• Synthetic Experiments: On synthetic data with known latent factors, EventNMF outperforms binned NMF baselines (NARFD) and functional PCA methods (PPCA). The study highlights a bias-variance tradeoff: higher-degree splines capture smooth latent processes better than degree-0 (binned) approaches, while excessive basis functions can lead to overfitting. The optimal configuration was found at spline degree $p=3$.
• Sparsity Robustness: The method demonstrates robustness in sparse regimes; having more entities significantly improves recovery when events per entity are low (<1), while performance stabilizes in dense regimes.
• Comparison: Unlike PPCA, which yields signed components and struggles with monotone cumulatives, EventNMF provides non-negative loadings and rate templates that align with the physical nature of event rates. Unlike NARFD, EventNMF avoids the sensitivity to bin width parameters.

Significance and Claims
The paper claims that EventNMF provides a "mathematically principled, easy to implement, and computationally efficient" alternative to current state-of-the-art methods for event data. Its primary significance lies in removing the preprocessing bottleneck of binning, thereby allowing for the discovery of interpretable temporal patterns that might otherwise be smoothed out or lost. The authors position the work as a bridge between functional data analysis and point process modeling.

The authors remain modest regarding future directions, noting that while the current method handles Poisson processes effectively, extending the framework to self-exciting processes (such as coupled multivariate Hawkes processes) remains an open theoretical and empirical challenge. They also suggest that scalability to datasets with billions of events could be further improved via random subsampling.