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REVIEW 4 major objections 5 minor 24 references

Modeling event intensities as non-negative B-spline combinations with neural-network-predicted coefficients yields exact likelihood evaluation, architecture flexibility, parallel training, and built-in smoothness regularization.

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2026-08-01 08:27 UTC pith:KGCBWP4H

load-bearing objection New exact-likelihood spline CIF neural TPP with real speed gains; the math is clean, but the basis is never defined beyond the training support, so the exact-NLL claim only holds on a finite interval. the 4 major comments →

arxiv 2607.21098 v1 pith:KGCBWP4H submitted 2026-07-23 cs.LG stat.ML

Smooth Neural Point Processes via B-Splines

Michele Bellomo , Riccardo Ramaschi , Alberto Dolara , Tomaso Aste This is my paper
classification cs.LG stat.ML
keywords temporal point processesconditional intensity functionB-splinesmaximum likelihood estimationneural networksmoothness regularizationexact likelihoodinter-event times
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proposes a neural temporal point process model in which the conditional intensity function is represented as a non-negative linear combination of B-spline basis functions, with the coefficients emitted by a generic neural network. Because the integrals of B-spline basis functions are known in closed form and can be precomputed, the negative log-likelihood is evaluated exactly without numerical integration or architectural constraints such as monotonicity. The representation also enables parallel evaluation of intensities across time points and makes a roughness penalty—based on the integrated squared second derivative—essentially cost-free. On synthetic and real-world benchmark datasets, the method improves time-prediction accuracy relative to a reference neural TPP baseline and achieves roughly a 12-fold training speedup.

Core claim

The central claim is that directly parameterizing the CIF as a non-negative combination of B-spline basis functions—where the coefficients are predicted by a neural network—is a practical and advantageous alternative to modeling the compensator. The paper shows that this construction guarantees a valid non-negative CIF, permits exact and efficient NLL evaluation via precomputed integrated basis functions, removes constraints on the neural architecture, and supports smoothness regularization through a quadratic roughness penalty. Empirically, the proposed model outperforms the baseline on five of seven synthetic datasets and on both real-world datasets, with substantial gains on the Hawkes an

What carries the argument

The key object is the cubic B-spline basis: the CIF is written as a sum of non-negative B-spline basis functions multiplied by non-negative coefficients predicted by the network. The basis functions have local support, and their integrals (for the compensator) and the matrix of integrated products of second derivatives (for the roughness penalty) can be precomputed once, enabling closed-form likelihood evaluation and efficient penalization.

Load-bearing premise

The model fixes a single cubic B-spline basis with 20 internal knots at quantiles of the training inter-arrival times, and the argument assumes this finite, history-independent basis can accurately represent the conditional intensity of any process in the benchmarks and beyond.

What would settle it

Run the model on a dataset generated by a Hawkes process with a very short excitation timescale (e.g., β large relative to typical inter-event times) and measure whether the learned CIF reproduces the sharp post-event peak; if the 20-knot quantile-based spline smooths the peak away and prediction error does not shrink as data increase, the expressiveness assumption is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Neural TPPs can be trained by exact maximum likelihood without numerical integration and without constraints like monotonicity on the network output.
  • Parallel evaluation of the CIF across all event times in a sequence is possible, enabling faster training, especially with attention-based architectures.
  • The smoothness penalty provides a principled, computationally cheap way to regularize the learned intensity and reduce oscillations.
  • The median-based next-event-time estimator can be computed efficiently by solving a single root-finding problem on the precomputed compensator curve.
  • The approach extends to multivariate point processes by predicting a separate set of non-negative B-spline coefficients for each event type.

Where Pith is reading between the lines

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

  • If the finite B-spline basis is expressive enough for a given process, the exact-likelihood property could be combined with more powerful architectures, such as deep state-space or graph-based models, without worrying about integration or positivity constraints.
  • A natural extension would be an adaptive knot placement mechanism—e.g., adding knots in regions of high estimated curvature—which could reduce bias for processes with sharp excitation features.
  • The roughness penalty could be interpreted as a prior over intensity functions; choosing α via smooth variational or hierarchical Bayes methods might make the model more robust and reduce tuning cost.
  • The parallel evaluation scheme could be especially valuable for streaming or online settings where the entire future intensity curve is needed quickly, but the paper does not yet report results on such tasks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a neural temporal point process model in which the conditional intensity is parametrized as a non-negative cubic B-spline in the time since the last event, with spline coefficients produced by a Transformer. Because B-splines admit closed-form integrals and roughness penalties, the negative log-likelihood and the smoothness regularizer are evaluated analytically, and the CIF can be evaluated at all event times in parallel. The experiments compare median next-event-time prediction on synthetic and real datasets with the Omi et al. (2019) baseline and report a substantial training speedup from the parallel evaluation scheme.

Significance. The paper has a clean, verifiable core: the exact NLL computation in Section 3.1 is correct as a calculation, the roughness penalty is a precomputable quadratic form, and the architecture-agnostic formulation is a genuine improvement over the monotonicity-constrained compensator networks of Omi et al. The reported predictive improvements on Hawkes 1 and on the music and meme datasets are interesting and, if confirmed, would make the method practically useful. However, the central formal claim of 'exact maximum likelihood' is currently qualified by an unaddressed support issue for the B-spline basis, and the experimental support is weakened by single-run results and the use of baseline numbers quoted from a different paper rather than re-run under matched conditions. The speedup claim also needs a more careful evaluation protocol. These are fixable issues, so the paper is a plausible candidate after revision.

major comments (4)
  1. [Sections 2.4, 3.1, 4.2] The B-spline basis is never defined outside the range of the training inter-arrival times. Section 4.2 places 20 internal knots at quantiles of training inter-arrival times; with a standard clamped cubic B-spline, the basis functions vanish outside the training support. Then φ(τ) = 0 for τ beyond the largest knot, so Φ(τ) is bounded as τ→∞. Any test inter-arrival time exceeding the training maximum gives a zero intensity, hence an infinite negative log-likelihood contribution. Moreover, the median estimator in Eq. (1) has no solution whenever ∫_0^∞ φ(s) ds < log 2, and the paper gives no fallback. The 'exact NLL' claim therefore holds only on the support of the training data, not for the process on [0,∞). This needs an explicit tail extension (constant, exponential, truncated-power, or similar) and a statement of how the prediction protocol handles the no-solution case.
  2. [Table 1 and Section 4.4] The predictive accuracy results are single runs with no error bars or repeated-seed statistics. The baseline numbers are quoted from Omi et al. [14] rather than re-run under the same data splits, preprocessing, and evaluation protocol. Given the large reported differences — for example, Hawkes 1: 0.399 vs 0.848 — it is important to know whether these are stable or due to seed luck. Please report means/variance over at least several seeds and, ideally, re-run the baseline in the same framework.
  3. [Sections 4.3–4.4] The claimed 12× speedup is based on a single comparison of average epoch times on synthetic datasets, with no detailed protocol description (data loading, warm-up, sequence lengths, attention masking, GPU contention). More importantly, the parallel multi-evaluation variant is never evaluated for predictive accuracy; the Table 1 results use the 'standard' sequential setting. Since the speedup is one of the paper's central claims, the multi-evaluation training should be compared to the standard training in both accuracy and wall-clock time, including on the real datasets, with enough details to reproduce the timing.
  4. [Sections 2.4 and 4.2] The expressiveness of the fixed 20-knot basis is a load-bearing assumption but is only justified by the classical denseness of splines. No approximation bound is given, and there is no sensitivity analysis with respect to the number of knots or their placement. A sharp Hawkes excitation immediately after an event may be smoothed away by a fixed sparse knot grid, and the exact-NLL property does not protect against this bias. I am not asking for a full theory paper, but a small sensitivity study (e.g., K = 10, 20, 40 and two knot placement strategies) would make the capacity claim much more credible.
minor comments (5)
  1. [Table 1] The header 'MAE Omi et al. MAE [14]' is ambiguous. One column should be labeled 'Ours' and the other 'Omi et al.'.
  2. [Figures 1 and 2] The figures lack axis labels and a legend stating which curve corresponds to which α; please add them so the regularization effect can be assessed.
  3. [Section 2.4] The statement that cubic splines are 'the smoothest' interpolating functions should include the usual boundary/natural-spline conditions; otherwise it is imprecise.
  4. [Section 4.2] The sentence 'using the same architecture for all datasets' is followed by hyperparameters (embedding size 64, etc.); please state random seed initialization and note whether results are from a single seed.
  5. [Section 3.3] Please state explicitly what happens when the bisection search for Eq. (1) fails because Φ(τ_max) < log 2; this is related to Major Comment 1 and should be addressed in the text.

Circularity Check

0 steps flagged

No significant circularity: the spline CIF derivation is self-contained and benchmark evaluation is genuine out-of-sample prediction.

full rationale

The core derivation is self-contained: the CIF is defined as a non-negative B-spline combination with coefficients output by a neural network, the NLL is evaluated with precomputed integrated B-spline basis functions, and the roughness penalty is a fixed quadratic form in the spline coefficients. The median-based next-event estimator is solved from the model's own compensator and compared against true held-out inter-arrival times, so it is a genuine prediction rather than a refit. The regularization parameter alpha is selected on a validation set, and the reported test MAE is not used to fit any parameter. The only self-citations, refs [8] and [9], appear in introductory examples (finance and Poisson processes) and are not load-bearing for the method. The finite-knot/tail-domain concern about clamped B-splines outside the training inter-arrival range is a model-validity and expressivity issue, not a circularity: it does not reduce the stated derivation to its own inputs. No step in the paper's derivation chain is equivalent by construction to a fitted parameter or an imported self-citation, so no circular step is identified.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

No new physical entities. The method relies on standard TPP theory, spline machinery, and a finite-dimensional approximation assumption.

free parameters (4)
  • Regularization coefficient α = selected per dataset from {10^-3,...,10^-10, 0} via validation
    Controls smoothness and generalization; chosen by validation search, not derived.
  • Number of internal knots K = 20
    Fixed for all experiments; determines spline capacity.
  • Context length n = 20 (last 20 events)
    Number of past events used as input; chosen by hand.
  • Spline degree p = 3
    Cubic splines; standard choice.
axioms (5)
  • domain assumption Temporal point process likelihood L = (∏ λ(T_i)) exp(-∫_0^T λ(s) ds)
    Standard theorem in point process theory cited via [1].
  • domain assumption Random Time Change Theorem: Φ(T_i) - Φ(T_{i-1}) ~ Exp(1), so the median-based estimator solves Φ(τ̂)=log(2)
    Relied on for the goodness-of-fit metric (Eq. 1, Section 2.3).
  • standard math B-spline basis functions are non-negative, have local support, and integrate to precomputed exact functions I_k(τ)
    Standard spline theory; used for exact NLL.
  • standard math Cubic splines minimize roughness P(s)=∫(s'')² dx and the penalty is w^T R w
    Standard result from nonparametric regression (Eq. 2-4).
  • ad hoc to paper The fixed 20-knot cubic B-spline basis is flexible enough to approximate the true CIFs in the benchmarks
    No error bound or adaptive knot selection is provided; this is the central capacity assumption.

pith-pipeline@v1.3.0-alltime-deepseek · 7282 in / 13912 out tokens · 116128 ms · 2026-08-01T08:27:38.050972+00:00 · methodology

0 comments
read the original abstract

Temporal point processes (TPPs) provide a general and flexible framework for modeling sequences of events in continuous time. Neural networks have been successfully employed to model TPPs in a highly expressive and data-driven way. Neural TPPs are typically trained via Maximum Likelihood Estimation (MLE) by minimizing the negative log-likelihood (NLL), which depends on both the conditional intensity function (CIF) and its integral over time, the compensator. Recent neural TPP approaches enable exact evaluation of the NLL without numerical integration. However, these methods typically model the compensator rather than the CIF directly, impose constraints on the neural network architecture, and are computationally expensive during training, as event contributions to the NLL are evaluated sequentially rather than in parallel. In this work, we propose a novel neural TPP model that directly parametrizes the CIF as a non-negative combination of B-spline basis functions, whose coefficients are predicted by a neural network. This formulation enables exact evaluation of the NLL, preserves full flexibility in the neural architecture, allows efficient parallelization during training, and naturally supports CIF smoothness regularization through the integrated squared second derivative. Experiments on both synthetic and real-world datasets show improved computational efficiency and predictive accuracy compared to the reference neural TPP baseline.

Figures

Figures reproduced from arXiv: 2607.21098 by Alberto Dolara, Michele Bellomo, Riccardo Ramaschi, Tomaso Aste.

Figure 1
Figure 1. Figure 1: Regularization effect of different smoothing parameters [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Regularization effect of different smoothing parameters [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

24 extracted references

  1. [1]

    Springer, 2003

    Daryl J Daley, David Vere-Jones, et al.An introduction to the theory of point processes: volume I: elementary theory and methods. Springer, 2003

    2003

  2. [2]

    Seismicity analysis through point-process modeling: A review.Seismicity patterns, their statistical significance and physical meaning, pages 471–507, 1999

    Yosihiko Ogata. Seismicity analysis through point-process modeling: A review.Seismicity patterns, their statistical significance and physical meaning, pages 471–507, 1999

    1999

  3. [3]

    Wilson Truccolo, Uri T Eden, Matthew R Fellows, John P Donoghue, and Emery N Brown. A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects.Journal of neurophysiology, 93(2):1074–1089, 2005

    2005

  4. [4]

    CRC press, 2013

    Peter J Diggle.Statistical analysis of spatial and spatio-temporal point patterns. CRC press, 2013. 8

    2013

  5. [5]

    Self-exciting point process modeling of crime.Journal of the american statistical association, 106(493):100–108, 2011

    George O Mohler, Martin B Short, P Jeffrey Brantingham, Frederic Paik Schoenberg, and George E Tita. Self-exciting point process modeling of crime.Journal of the american statistical association, 106(493):100–108, 2011

    2011

  6. [6]

    Hawkes processes in finance

    Emmanuel Bacry, Iacopo Mastromatteo, and Jean-François Muzy. Hawkes processes in finance. Market Microstructure and Liquidity, 1(01):1550005, 2015

    2015

  7. [7]

    Hawkes processes and their applications to finance: a review.Quantitative Finance, 18(2):193–198, 2018

    Alan G Hawkes. Hawkes processes and their applications to finance: a review.Quantitative Finance, 18(2):193–198, 2018

    2018

  8. [8]

    Hawkes processes in finance: An application in modeling transactions in eua futures.CONTEMPORARY MATHEMATICS, 6(5):5590–5611, 2025

    Michele Bellomo, Francesco Grimaccia, et al. Hawkes processes in finance: An application in modeling transactions in eua futures.CONTEMPORARY MATHEMATICS, 6(5):5590–5611, 2025

    2025

  9. [9]

    Can we do without the hairdresser? a mathematical solution.The College Mathematics Journal, pages 1–6, 2025

    Michele Bellomo. Can we do without the hairdresser? a mathematical solution.The College Mathematics Journal, pages 1–6, 2025

    2025

  10. [10]

    Spectra of some self-exciting and mutually exciting point processes

    Alan G Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1):83–90, 1971

    1971

  11. [11]

    Springer, 2021

    Patrick J Laub, Young Lee, and Thomas Taimre.The elements of Hawkes processes. Springer, 2021

    2021

  12. [12]

    Neural survival recommender

    How Jing and Alexander J Smola. Neural survival recommender. InProceedings of the Tenth ACM International Conference on Web Search and Data Mining, pages 515–524, 2017

    2017

  13. [13]

    The neural hawkes process: A neurally self-modulating multivariate point process.Advances in neural information processing systems, 30, 2017

    Hongyuan Mei and Jason M Eisner. The neural hawkes process: A neurally self-modulating multivariate point process.Advances in neural information processing systems, 30, 2017

    2017

  14. [14]

    Fully neural network based model for general temporal point processes.Advances in neural information processing systems, 32, 2019

    Takahiro Omi, Kazuyuki Aihara, et al. Fully neural network based model for general temporal point processes.Advances in neural information processing systems, 32, 2019

    2019

  15. [15]

    Automatic differentiation in machine learning: a survey.Journal of machine learning research, 18(153):1–43, 2018

    Atilim Gunes Baydin, Barak A Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. Automatic differentiation in machine learning: a survey.Journal of machine learning research, 18(153):1–43, 2018

    2018

  16. [16]

    Monotonic networks.Advances in neural information processing systems, 10, 1997

    Joseph Sill. Monotonic networks.Advances in neural information processing systems, 10, 1997

    1997

  17. [17]

    Neural likelihoods via cumulative distribution functions

    Pawel Chilinski and Ricardo Silva. Neural likelihoods via cumulative distribution functions. In Conference on uncertainty in artificial intelligence, pages 420–429. PMLR, 2020

    2020

  18. [18]

    chapman and hall/CRC, 2017

    Simon N Wood.Generalized additive models: an introduction with R. chapman and hall/CRC, 2017

    2017

  19. [19]

    Green and Bernard W

    Peter J. Green and Bernard W. Silverman.Nonparametric Regression and Generalized Linear Models. Springer US, Boston, MA, 1994

    1994

  20. [20]

    Last.fm dataset

    Oscar Celma. Last.fm dataset. http://ocelma.net/MusicRecommendationDataset/ lastfm-360K.html

  21. [21]

    Meme-tracking and the dynamics of the news cycle

    Jure Leskovec, Lars Backstrom, and Jon Kleinberg. Meme-tracking and the dynamics of the news cycle. InProceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD ’09), pages 497–506. ACM, 2009. https: //snap.stanford.edu/data/memetracker9.html

    2009

  22. [22]

    Attention is all you need.Advances in neural information processing systems, 30, 2017

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need.Advances in neural information processing systems, 30, 2017

    2017

  23. [23]

    Keras.https://keras.io, 2015

    François Chollet et al. Keras.https://keras.io, 2015

    2015

  24. [24]

    Kerashub

    Matthew Watson, François Chollet, Divyashree Sreepathihalli, Samaneh Saadat, Ramesh Sam- path, Gabriel Rasskin, Scott Zhu, Varun Singh, Luke Wood, Zhenyu Tan, Ian Stenbit, Chen Qian, Jonathan Bischof, et al. Kerashub. https://github.com/keras-team/keras-hub , 2024. 9

    2024