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arxiv: 2606.06205 · v1 · pith:I5U6I22Qnew · submitted 2026-06-04 · 💻 cs.LG

Non-Negative Matrix Factorization for Event Data

Rapha\"el Romero This is my paper

Pith reviewed 2026-06-28 02:50 UTC · model grok-4.3

classification 💻 cs.LG
keywords non-negative matrix factorizationevent dataPoisson processB-splinescontinuous-time modelingtemporal templatespoint processes
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The pith

EventNMF models entity event times as Poisson processes whose intensities factorize through non-negative B-splines to recover shared temporal templates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents EventNMF as a way to uncover structure in continuous-time event data without first binning the timestamps into counts. Each entity's events are treated as draws from a Poisson process whose rate function is expressed as a non-negative linear combination of B-spline basis functions that are shared across entities. A straightforward estimation routine then extracts the common temporal templates. The resulting procedure is shown to be mathematically consistent, simple to code, and computationally light while including ordinary binned NMF as the special case of constant (degree-zero) splines.

Core claim

EventNMF is a continuous-time non-negative factorization model that operates directly on event times: each entity's events are modeled as a Poisson process whose intensity factorizes through a non-negative B-spline basis, and a simple estimation procedure recovers interpretable temporal templates shared across entities.

What carries the argument

Non-negative B-spline basis factorization of the intensity function inside a Poisson point process model for event times.

If this is right

  • Binned-count NMF is recovered exactly when the spline degree is set to zero.
  • Bias-variance trade-offs can be explored by varying spline degree and knot placement without changing the overall estimation procedure.
  • The method recovers ground-truth factors on synthetic data drawn from a latent-factor Poisson process.
  • It produces usable templates on real event streams drawn from neuroscience, seismology, and social-network settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same spline-Poisson construction could be paired with other point-process likelihoods such as Hawkes processes.
  • Domains that record events at sub-second resolution may see larger gains from avoiding binning than domains with coarser natural timescales.
  • One could test whether the recovered templates remain stable when the number of observed entities is reduced while holding the total event count fixed.

Load-bearing premise

The true event intensities admit an accurate low-rank representation as non-negative combinations of B-spline basis functions under the Poisson process model.

What would settle it

Generate synthetic event data from a known low-rank B-spline Poisson model, run EventNMF, and check whether the recovered templates match the generating factors up to permutation and scaling.

Figures

Figures reproduced from arXiv: 2606.06205 by Rapha\"el Romero.

Figure 1
Figure 1. Figure 1: Illustration of our EventNMF method on a synthetic example with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Train NLL (left), Test NLL (centre), and NFISE (right) vs. number of basis functions [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sparsity robustness across population size N and event density. (a): normalized NMSE between learned and true factor shapes, after permutation matching. (b): nor￾malized MSE between true and learned entity-level intensi￾ties. Shaded bands show ±1 standard deviation over 3 seeds. Effect of data sparsity. To study how EventNMF performs under data spar￾sity, we vary the average number of events per entity fro… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison with PPCA in terms of recovered factor cumulatives. True (black), EventNMF [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison with NARFD. The error bars and shaded band show ±1 std over 5 seeds. NARFD [1] models longitudinal count data as Yi(tj ) ∼ Poisson(µij ) with µij = u ⊤ i ΓΦ(tj ), where Φ(t) is a B￾spline basis. For event data, NARFD discretizes the time axis into J bins, making the input observations depend on the choice of bin width. EventNMF can be viewed as a continuous-time counterpart of NARFD: both per￾fo… view at source ↗
Figure 6
Figure 6. Figure 6: Two factors recovered by EventNMF. (a) Temporal intensity profiles of the Geysers factor (green) and the Ridge￾crest factor (red); the dotted line marks the M7.1 earthquake (July 2019). (b) Geographic map of cell loadings; each cell is coloured by a blend of its Geysers loading (green) and Ridgecrest loading (red–red). We apply EventNMF to ten years of seismic data (January 2014–January 2024) from the USGS… view at source ↗
Figure 7
Figure 7. Figure 7: EventNMF analysis of a single trial from the Allen Visual Coding Neuropixels dataset. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: EventNMF on the primary school dataset ( [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Ground truth temporal factors for the synthetic data experiment. [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Wall-clock time per multiplicative-update iteration as a function of the number of entities [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
read the original abstract

Continuous-time event data, in which entities emit instantaneous events over time, arises naturally across many domains such as neuroscience, seismology, and social networks. Non-negative matrix factorization (NMF) is a natural tool to uncover interpretable structure in such data, but it has so far only been applied after binning or smoothing the entity-level counting measures. This preprocessing step comes with the risk of erasing entity-level heterogeneities and fine-grained temporal features. In this paper, we introduce EventNMF, a continuous-time non-negative factorization model that operates directly on event times: each entity's events are modeled as a Poisson process whose intensity factorizes through a non-negative B-spline basis, and a simple estimation procedure recovers interpretable temporal templates shared across entities. The resulting method is mathematically principled, easy to implement, and computationally efficient. We further show that standard binned-count approaches arise as the special case of degree-zero splines, explore bias-variance tradeoffs and compare against existing methods on a synthetic latent factor model, and demonstrate the effectiveness of EventNMF on several real-world applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces EventNMF, a continuous-time NMF model for event data in which each entity's events are modeled as an inhomogeneous Poisson process whose intensity is factorized non-negatively over a shared B-spline basis. The method operates directly on event times without binning, recovers interpretable temporal templates, reduces to standard binned NMF when using degree-zero splines, includes bias-variance analysis and synthetic recovery experiments, and is demonstrated on real-world applications in neuroscience, seismology, and social networks.

Significance. If the estimation procedure and recovery guarantees hold, the work supplies a mathematically principled, computationally efficient alternative to binned NMF that preserves fine-grained temporal structure and entity-level heterogeneity. The explicit reduction to the binned case and the provision of bias-variance trade-offs are concrete strengths that could make the approach immediately usable in domains that already rely on NMF for count data.

minor comments (3)
  1. The abstract and introduction state that the estimation procedure is 'simple' and 'computationally efficient,' but the precise optimization objective, initialization strategy, and convergence criteria should be stated explicitly in §3 or §4 with pseudocode or a short algorithm box.
  2. In the synthetic experiments, the recovery metric (e.g., template correlation or reconstruction error) and the range of spline degrees and knot placements tested should be reported in a table or figure caption so that the bias-variance claims can be directly verified.
  3. Notation for the B-spline basis functions and the non-negativity constraints on the factor matrices should be introduced once in §2 and used consistently; a small table summarizing symbols would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of EventNMF and the recommendation of minor revision. The report correctly identifies the core contributions: direct operation on event times via non-negative B-spline factorization of Poisson intensities, the explicit reduction to binned NMF, and the bias-variance analysis. No specific major comments were raised that require point-by-point rebuttal.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines EventNMF directly via a Poisson process intensity that factorizes over a shared non-negative B-spline basis, then presents an estimation procedure whose output is the recovered templates. The abstract and description show the degree-zero spline case recovering binned NMF as a special case, with bias-variance analysis and synthetic recovery experiments. No quoted step reduces a claimed prediction to a fitted input by construction, no self-citation is load-bearing for the central claim, and no uniqueness theorem or ansatz is smuggled in. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies insufficient detail to enumerate specific free parameters, axioms, or invented entities; the model appears to rest on standard Poisson-process and B-spline assumptions whose precise parameterization is not stated.

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