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arxiv: 2604.07377 · v1 · submitted 2026-04-08 · 📊 stat.ME · math.ST· stat.AP· stat.TH

Poisson-response Tensor-on-Tensor Regression and Applications

Carlos Llosa-Vite , Daniel M. Dunlavy This is my paper

Pith reviewed 2026-05-10 18:36 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.APstat.TH
keywords Poisson regressiontensor-on-tensor regressioncanonical polyadic decompositioncount datamaximum likelihood estimationchange point detectiontensor covariateslongitudinal data
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The pith

Tensor-on-tensor regression models element-wise Poisson counts by constraining the coefficient tensor to a canonical polyadic form.

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

The paper develops a regression method that takes tensor-valued responses whose entries are independent Poisson counts and tensor-valued predictors. The link between predictors and Poisson rates is given by a coefficient tensor restricted to canonical polyadic low-rank structure. This restriction is used to derive maximum-likelihood algorithms that keep all estimated rates positive. The work supplies an initial error analysis for the resulting estimators and illustrates the method on three data sets: international crisis records, positron emission tomography scans, and longitudinal dyadic communication patterns.

Core claim

We introduce Poisson-response tensor-on-tensor regression (PToTR) as a framework for modeling tensor responses composed of independent Poisson counts whose rates are determined by tensor covariates through a CP-structured coefficient tensor. We develop algorithms for maximum likelihood estimation under this structure that satisfy the positivity requirement on the Poisson parameters and supply an initial theoretical error analysis for the estimators.

What carries the argument

The canonical polyadic decomposition placed on the regression coefficient tensor, which maps tensor covariates to Poisson rates while preserving positivity.

Load-bearing premise

The entries of the response tensor are independent Poisson random variables whose rates equal a positive linear function of the covariates under a fixed CP low-rank structure on the coefficient tensor.

What would settle it

A simulation in which the proposed maximum-likelihood algorithms return a negative rate estimate for at least one entry, when the true rates are known to be positive, would falsify the claim that the estimators satisfy positivity.

Figures

Figures reproduced from arXiv: 2604.07377 by Carlos Llosa-Vite, Daniel M. Dunlavy.

Figure 1
Figure 1. Figure 1: 4-way tensor covariates X(t) used in the PToTR autoregressive model for the ICEWS experiments. 2) Experimental results: We compared PToTR, Gaus￾sian ToTR [46], and OP-based ToTR [8] models involv￾ing the ICEWS database [6]. We used the subset of the database described in [8], which involves 25 countries and four quad classes as actions, leading to tensor responses Y(t) ∈ N 25×25×4 0 . The dates selected co… view at source ↗
Figure 2
Figure 2. Figure 2: BIC values from fitting PToTR, Gaussian ToTR, and [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Shepp-Logan phantom under a PET scanner, with [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: RMSE (∝ q ||B − B|| b 2) across different reconstruction methods, sample sizes, and iterations in the estimation. Results show that for increased data and parameters, PToTR(R), improves with increased iterations, while ML-EM exhibits an initial decrease in RMSE followed by a substantial increase. is significantly more parsimonious. For example, PToTR with rank R = 84 has only 63,168 parameters, making it n… view at source ↗
Figure 5
Figure 5. Figure 5: PET reconstructions for different amounts of data (4%, 16%), different number of iterations (10 and 100), and multiple [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Loglikelihoods resulting from fitting the PTANOVA [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
read the original abstract

We introduce Poisson-response tensor-on-tensor regression (PToTR), a novel regression framework designed to handle tensor responses composed element-wise of random Poisson-distributed counts. Tensors, or multi-dimensional arrays, composed of counts are common data in fields such as international relations, social networks, epidemiology, and medical imaging, where events occur across multiple dimensions like time, location, and dyads. PToTR accommodates such tensor responses alongside tensor covariates, providing a versatile tool for multi-dimensional data analysis. We propose algorithms for maximum likelihood estimation under a canonical polyadic (CP) structure on the regression coefficient tensor that satisfy the positivity of Poisson parameters and then provide an initial theoretical error analysis for PToTR estimators. We also demonstrate the utility of PToTR through three concrete applications: longitudinal data analysis of the Integrated Crisis Early Warning System database, positron emission tomography (PET) image reconstruction, and change-point detection of communication patterns in longitudinal dyadic data. These applications highlight the versatility of PToTR in addressing complex, structured count data across various domains.

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

2 major / 2 minor

Summary. The paper introduces Poisson-response Tensor-on-Tensor Regression (PToTR), a framework for regressing a tensor response whose entries are independent Poisson random variables on tensor covariates, with the coefficient tensor constrained to CP low-rank structure. It develops MLE algorithms that enforce positivity of the Poisson rates, supplies an initial theoretical error analysis for the resulting estimators, and illustrates the method on three applications: longitudinal analysis of the ICEWS crisis database, PET image reconstruction, and change-point detection in longitudinal dyadic communication data.

Significance. If the independence assumption and the error bounds can be rigorously justified, PToTR would provide a useful extension of tensor regression to count-valued responses that arise in international relations, imaging, and network data. The CP parameterization yields interpretable low-rank structure and the positivity-preserving algorithms address a practical constraint of Poisson models. The applications are concrete but remain illustrative; the work's impact hinges on whether the theoretical analysis can be completed and whether the model remains useful when the independence assumption is relaxed.

major comments (2)
  1. [§2] §2 (Model and likelihood): The entire MLE construction and the subsequent concentration arguments rest on the assumption that the response-tensor entries are conditionally independent Poisson random variables. In the ICEWS application (§5.1) the data are longitudinal event counts; temporal dependence across time slices is expected and would invalidate both the likelihood and any error bounds derived under independence without additional robustness analysis or a modified model.
  2. [Theoretical error analysis] Theoretical error analysis (the section following the algorithms): The analysis is labeled 'initial' and the abstract states that full derivations are not supplied. Because the central claim is that the estimators enjoy certain error bounds, the absence of complete proofs, explicit constants, or verifiable conditions on the design tensor makes it impossible to assess whether the claimed rates are attainable or merely formal.
minor comments (2)
  1. Notation for the CP factors and the inner-product operator should be introduced once and used consistently; several passages reuse the same symbol for the coefficient tensor and its unfolded version.
  2. The PET imaging application would benefit from a brief statement of how the tensor covariates are constructed from the scanner geometry; the current description leaves the link between the physical acquisition model and the regression tensor unclear.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript introducing PToTR. The comments raise important points about the model's assumptions and the completeness of the theoretical analysis. We address each major comment below and commit to revisions that will strengthen the paper.

read point-by-point responses

  1. Referee: [§2] §2 (Model and likelihood): The entire MLE construction and the subsequent concentration arguments rest on the assumption that the response-tensor entries are conditionally independent Poisson random variables. In the ICEWS application (§5.1) the data are longitudinal event counts; temporal dependence across time slices is expected and would invalidate both the likelihood and any error bounds derived under independence without additional robustness analysis or a modified model.

    Authors: We agree that the conditional independence assumption is central to the likelihood and error bounds in PToTR. In longitudinal applications such as ICEWS, temporal dependencies are indeed plausible. However, the model is intended as a first-order approximation that captures the main effects through the tensor covariates. In the revised manuscript, we will add a dedicated paragraph in Section 5.1 discussing this limitation, its potential consequences for the reported results, and outlining possible extensions (e.g., quasi-likelihood approaches or block bootstrap for inference). We will also note that the empirical performance remains informative despite the assumption. revision: partial

  2. Referee: [Theoretical error analysis] Theoretical error analysis (the section following the algorithms): The analysis is labeled 'initial' and the abstract states that full derivations are not supplied. Because the central claim is that the estimators enjoy certain error bounds, the absence of complete proofs, explicit constants, or verifiable conditions on the design tensor makes it impossible to assess whether the claimed rates are attainable or merely formal.

    Authors: The 'initial' label reflects our intent to present the core proof strategy and key technical steps concisely in the main text, with full details deferred due to space constraints. We acknowledge that this makes independent verification challenging. In the revision, we will move the complete derivations, including all constants and the precise assumptions on the design tensor (e.g., bounded norms and tensor incoherence conditions), to a supplementary appendix. This will enable full assessment of the error bounds, which we believe hold under the conditions outlined. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper defines PToTR as a tensor regression model with elementwise independent Poisson responses whose rates are given by a CP-structured coefficient tensor, then derives MLE algorithms enforcing positivity and provides initial error bounds from the resulting likelihood. These steps follow standard GLM estimation and tensor decomposition without any reduction where a claimed prediction or bound is equivalent by construction to a fitted input or self-defined quantity. No self-citation chains, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation appear as load-bearing elements. The independence assumption and CP structure are modeling choices with external statistical justification, not tautological outputs. The framework remains self-contained against standard benchmarks for Poisson tensor models.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on the assumption that tensor entries are independent Poisson counts and that a CP low-rank structure on the coefficient tensor is appropriate and estimable under positivity constraints. No free parameters or invented entities are explicitly named in the abstract.

free parameters (1)
  • CP rank
    The rank parameter of the CP decomposition must be chosen or tuned, though its specific value is not stated in the abstract.
axioms (2)
  • domain assumption Tensor response entries are independent Poisson random variables
    Invoked to justify the likelihood and the element-wise Poisson model.
  • domain assumption Coefficient tensor admits a CP decomposition that preserves positivity of rates
    Required for the proposed MLE algorithms and theoretical analysis.

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