Smooth tensor decomposition with application to ambulatory blood pressure monitoring data

Irina Gaynanova; Leyuan Qian; Naresh M. Punjabi; R. Nisha Aurora

REVIEW 4 major objections 5 minor 49 references

Smooth tensor decomposition with application to ambulatory blood pressure monitoring data

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Smooth tensor model finds sleep-apnea blood-pressure link

desk verdict Solid optimization core and a genuinely useful ABPM application, but the simulation is self-referential, the final rank choice is post hoc, and the downstream regression ignores score uncertainty. read the letter →

arxiv 2507.11723 v1 pith:KQZPJR4E submitted 2025-07-15 stat.ME

Leyuan Qian , R. Nisha Aurora , Naresh M. Punjabi , Irina Gaynanova This is my paper · ORCID

classification stat.ME MSC 62H2562P1062-08

keywords tensordecompositionTuckertemporalsmoothingmissingdataambulatorybloodpressuremonitoringfunctionalanalysisSmoothHOOIwearabledevice

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that ambulatory blood pressure monitoring data, aligned as a three-way tensor of hour, measurement type, and patient, can be decomposed into smooth temporal components and subject-level scores without losing correlations among systolic blood pressure, diastolic blood pressure, and heart rate. The proposed method, SmoothHOOI, builds on the Tucker decomposition—a low-rank factorization of a multidimensional array into orthogonal factor matrices and a core tensor—and penalizes the smoothness of the reconstructed low-rank fit rather than the component matrices, a design choice the authors show is necessary because directly smoothing components under orthogonality produces unstable sinusoidal reconstructions of missing values. Simulations indicate that SmoothHOOI reconstructs smooth temporal trends from noisy, incomplete data more accurately than functional principal component analysis applied one measurement at a time. In the HYPNOS sleep-apnea application, the method detects a significant association between obstructive sleep apnea severity and overall blood pressure and heart rate levels after multiple-comparison adjustment, whereas traditional daytime and nighttime summary measures do not.

What carries the argument

The load-bearing device is the second-difference penalty $\lambda\|D L G_i R^\top\|_F^2$ applied to the reconstructed matrix $L G_i R^\top$ rather than to the component $L$; $D$ is the 24-point second-difference matrix. This placement keeps fitted curves smooth while preserving the orthogonality constraint, because penalizing $L$ directly drives the temporal components toward the sinusoidal eigenvectors of $D^\top D$. The optimization reduces to alternating top-eigenvector updates through the reparametrization $U = A^{1/2}L$ with $A = I + \lambda D^\top D$, and missing data are handled inside the loop by imputing the current low-rank fit into unobserved entries.

What would settle it

Simulate data from the same smooth low-rank model with log(ODI4) having zero true effect on the level scores, run the full pipeline (cross-validated ranks and $\lambda$, imputation, regression), and check the empirical type I error rate at the nominal 5% level; if it is materially above 5%, the reported clinical p-values are anti-conservative. In the actual HYPNOS data, a patient-level bootstrap that re-estimates the scores within each resample would show whether the association survives the added uncertainty.

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Extended reading notes

Core claim

SmoothHOOI solves a penalized Tucker decomposition in which each patient's slice is approximated by $L G_i R^\top$, with the time factor $L$ and measurement factor $R$ orthonormal, and the objective adds $\lambda\|D L G_i R^\top\|_F^2$ to the squared reconstruction error on observed entries, where $D$ is the second-difference matrix that enforces smoothness across hours. The central discovery is that smoothing the fitted signal rather than the factor $L$ avoids the instability that arises when orthogonality pushes the columns of $L$ toward the discrete sine and cosine eigenvectors of $D^\top D$. Closed-form updates for the core slices $G_i$ and for $R$ and $U = A^{1/2}L$ with $A = I + \lambda D^\top D$ convert the problem into alternating eigen-decompositions, missing entries are filled by iterative imputation inside the loop, and SVD rotations of the core tensor restore identifiability. The recovered temporal components separate overall blood pressure and heart rate level, nocturnal dipping, and individual sleep-time shifts, and regressing the leading core scores on clinical covariates reveals the sleep-apnea association that summary measures miss.

Load-bearing premise

The load-bearing premise is that the subject-level core scores produced by the decomposition can be treated as measured outcomes in a standard regression, ignoring the uncertainty from tensor fitting, rank selection, and missing-data imputation.

Editorial extensions

If this is right

Given a tensor of temporally structured matched measurements, SmoothHOOI recovers identifiable smooth temporal components and subject-level scores without requiring a common daytime or nighttime division across patients.
The algorithm, including its missing-data imputation and cross-validation for ranks and smoothing parameter, transfers to any higher-order tensor with one temporal mode, so wearable-device streams beyond blood pressure are a direct extension.
Using the separated level, dipping, and sleep-timing scores as regression outcomes lets a study test covariate associations with each physiological process individually, which dilutes less than average-based summaries.
The HYPNOS result, if correct, implies that tensor-based scores can reveal a clinically expected effect of obstructive sleep apnea severity on blood pressure and heart rate that single-measurement summary statistics fail to detect.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Our inference: the clinical p-values would be stronger evidence if they were recomputed with a patient-level bootstrap that re-runs the full decomposition, rank selection, and imputation each time; the paper's regression treats the estimated scores as known.
Our inference: the orthogonality-versus-smoothness conflict likely extends beyond tensors, so smoothing the reconstructed fit rather than the factors is a transferable design choice for other orthogonally constrained factor models, such as sparse matrix factorization.
Our inference: a natural extension is to allow measurement-specific smoothing parameters or a cyclic boundary condition on the 24-hour clock, since the global $\lambda$ and the current second-difference matrix are simplifying choices rather than necessary ones.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, the methodological core is real: penalizing smoothness of the reconstructed fit rather than the components is a sensible twist, and the complete-data optimization derivations are clean. The reparametrization in Proposition 2 and the closed-form updates in Algorithm 1 are carefully done. Second, the empirical support is weaker than the paper claims. The simulation truth in Section 3 is built from L and R returned by SmoothHOOI on the same real data used later, so the simulation tests the method on data that already satisfies its own model. That is self-referential, not fatal, but it means the simulation cannot independently establish that the method recovers true temporal structure. The paper also reports that cross-validation selected r1=6, r2=3, lambda=12, but then the authors manually reduce to r1=3, r2=2, lambda=4 based on explained variability. That is a post hoc rank choice, and the final clinical conclusions rest on it. The stress-test concern is valid: Table 1 fits OLS regressions on g11 and g12 scores while ignoring uncertainty from tensor estimation, missing-data imputation, and rank selection. If that uncertainty is non-negligible, the reported p-values and confidence intervals are anti-conservative. The paper does not provide an errors-in-variables analysis, bootstrap, or simulation of the regression step, so the headline OSA association is not yet firmly supported.

What the paper does well: the central idea of penalizing the fit is new in the Tucker context and the Appendix A demonstration of why smoothing components directly leads to sinusoidal instability is a useful, honest empirical insight. The missing-data extension is heuristic but standard. The actigraphy validation in Figure 4 gives independent grounding for the third temporal component, and the interpretation of the extracted components as overall level, dipping, and sleep timing is convincing. The R package is a plus.

This paper deserves a serious referee. It is a useful contribution for statisticians working on wearable or functional tensor data, and the method itself is likely sound in the complete-data case. But the empirical claims need revision: the simulation should be regenerated from an independent generative model, the rank selection should be preregistered or subjected to sensitivity analysis, and the downstream regression should account for propagated uncertainty or at least be accompanied by a simulation study of the regression coefficients. I would engage with it, and I would want to see those fixes before trusting the clinical inference.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes SmoothHOOI, a Tucker/GLRAM-type decomposition for three-way tensors that penalizes second-order temporal differences of the reconstructed fit, accommodates missing values through an imputation-based iterative scheme, and selects ranks and the smoothing parameter by cross-validation. The authors derive closed-form updates for the complete-data problem, provide simulations comparing reconstruction and temporal-component recovery with FPCA, and apply the method to HYPNOS ABPM data. In the application they interpret the estimated temporal components as overall level, nocturnal dipping, and sleep timing, with an actigraphy-based validation for the third component, and they regress the estimated core scores on clinical covariates, reporting an association between log(ODI4) and an overall blood-pressure/heart-rate score that is not found with summary-statistic models.

Significance. If the claims hold, the paper would make a useful methodological contribution: penalizing the reconstructed fit rather than the component matrices is a principled response to the orthogonality-smoothness conflict, and the complete-data optimization steps are derived cleanly with closed-form updates. The paper also provides an automatic tuning procedure, an R package, and an independent actigraphy check on one temporal component, which are concrete strengths. However, the simulation truth is generated from the method's own output on the same real data, and the downstream regression treats the estimated tensor scores as noise-free outcomes while ranks are chosen after inspecting the cross-validation results. These issues bear directly on the two headline claims, reliable reconstruction in simulations and detection of a clinically meaningful association in the application, so the current evidence is not yet fully convincing.

major comments (4)

[3, Simulations] The simulation truth is generated from SmoothHOOI itself: the fixed L and R used to construct Msmooth are obtained by applying SmoothHOOI to the real HYPNOS data, and the subject-specific core matrices are sampled from the empirical distribution of the fitted core. This makes the simulation a self-consistency check rather than a test against a plausible alternative data-generating process, and it biases the comparison with FPCA in favor of SmoothHOOI, since the true model is exactly the SmoothHOOI model while FPCA is not designed for matched multivariate structure. The conclusion that SmoothHOOI 'reliably reconstructs smooth temporal trends from noisy, incomplete data' would be substantially strengthened by an independent simulation design, for example generating smooth trajectories from splines or a mixed-effects model and then comparing both methods, or by a real-data validation benchmark.
[4.3, Eq. (8) and Table 1] The outcomes g11 and g12 in the downstream linear regressions are estimated quantities whose uncertainty stems from missing-data imputation (15.9% missingness), the tensor decomposition itself, and the selection of ranks and lambda. The reported standard errors and p-values are ordinary least-squares quantities computed as if the scores were observed without error. No errors-in-variables correction, bootstrap, or simulation of the score distribution is provided. Since the manuscript's headline application claim is the detection of an ODI4 association that summary-based approaches miss, this inference gap is load-bearing; the confidence intervals in Table 1 may be anti-conservative. I request a sensitivity analysis or a simulation of the regression step, such as resampling imputations and refitting the decomposition, or a permutation test on the raw tensor that does not rely on the estimated scores being exact.
[4.3, Rank selection after cross-validation] The final model configuration is not the configuration selected by the proposed cross-validation: five-fold CV yields (r1, r2, lambda) = (6, 3, 12), and the authors then adopt (3, 2, 4) after inspecting explained variability and noting that CV may overestimate ranks in the presence of missingness. This post hoc rank reduction introduces an additional selection step whose uncertainty is not incorporated in any downstream p-value or confidence interval. At minimum, the manuscript should report the regression results under the CV-selected ranks as a sensitivity check, or pre-specify a rank-selection rule before component variability is examined.
[2.3, Missing-data algorithm] The convergence argument based on a non-increasing objective is established for the complete-data case only; the outer imputation loop, in which missing entries are replaced by the current fit LGiR^T, is heuristic, and no fixed-point property or convergence guarantee is given for the overall missing-data procedure. Proposition 1 also assumes a complete index set. The simulations assess reconstruction performance but do not isolate the effect of the imputation step. The manuscript should either provide a convergence statement for the missing-data algorithm or explicitly describe it as heuristic and examine its sensitivity in the simulations.

minor comments (5)

[1, Introduction] In the paragraph discussing existing smoothing approaches, 'regularizaiton' should be 'regularization'.
[2.2, Eq. (4)] The displayed difference matrix D has nonzero entries in the corners, indicating circular boundary conditions; this should be stated explicitly, since the usual second-difference matrix for 24 hourly points without wrap-around would have different boundary rows.
[2.5, Identifiability] The SVD-based rotation fixes the representation only when the relevant singular values are distinct; the manuscript should state this condition and explain how near-ties or sign ambiguities are handled in practice.
[4.3, Figure 4] The vertical axis label in Figure 4 reads 'G score'; it should be 'g31 score' to match the notation used in the text.
[Appendix D.5] The standard error formula for the effect size uses sqrt(var(beta1) + var(beta2)), which assumes independence between the coefficient estimates from the two score regressions; if this independence is intended, it should be stated, and if not, the covariance term should be included.

Circularity Check

1 steps flagged · score 4.0 of 10

The simulation benchmark is self-referential: its ground-truth L and R are SmoothHOOI's own fits to the real data, so favorable recovery is partly assured by construction; the core optimization and actigraphy validation remain independent.

self definitional [Section 3 Simulations, first paragraph (data-generating model for Msmooth)]
"Each signal slice Mi ∈ R24×3 in Msmooth is generated as Mi = LGiR⊤, with fixed matrices L ∈ R24×r1 and R ∈ R3×r2 obtained from SmoothHOOI applied to real data (Section 4) with r1 = 3, r2 = 2 and λ = 4. The subject-specific core matrices Gi for i = 1,...,n are generated by sampling (g11,g12,g21,g22,g31,g32) i.i.d from N(μ̂,Σ̂), where μ̂ and Σ̂ are the empirical mean and covariance estimated from the estimated core tensor Ĝ."

The simulation's 'true' temporal factors L and R are exactly the estimates returned by SmoothHOOI on the real HYPNOS tensor, and the 'true' core-score distribution is the empirical distribution of SmoothHOOI's estimated core. Thus the data-generating process is, by construction, the same model class (5) that SmoothHOOI is designed to fit. The simulation therefore measures how well the algorithm recovers its own fitted basis from noisy and subsampled copies of that basis, rather than testing whether the low-rank smoothness assumption actually matches an independent physiological process.

full rationale

The derivation chain for the method itself is otherwise self-contained: Propositions 1–3 are proved in Appendix B, the identifiability rotation relies on the standard external MLSVD framework of De Lathauwer et al. (2000a), and the third temporal component is checked against actigraphy-estimated sleep periods (Figure 4), an independent variable not used in the decomposition. I found no load-bearing self-citation: prior work on smoothing Tucker decompositions (Timmerman and Kiers 2002) is external and used only for contrast. The downstream regression analysis (Section 4.3, Table 1) treats the estimated core scores as noise-free outcomes and ignores uncertainty from missing-data imputation, rank selection, and score estimation; this is a genuine errors-in-variables and multiple-testing concern, but it is not a by-construction circularity because the reported ODI4 coefficient is not mathematically forced by the scores' definition. The moderate score reflects the self-referential simulation benchmark, while acknowledging that the optimization machinery, the comparison with FPCA, and the actigraphy-based validation give the paper substantial independent content.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The method introduces two free parameters (ranks and smoothing) that are tuned on the data; no new physical entities are introduced. The main unstated assumptions are the ignorability of missingness and the low-rank smooth model.

free parameters (3)

lambda (smoothing parameter) = 4 (final application; 12 from CV; grid [1,50] in simulations)
Controls the strength of the temporal smoothing penalty in (5); selected by cross-validation, but the final application value was chosen after manual rank reduction, making it dependent on the data.
r1 (temporal rank) = 3 (CV gave 6)
Number of temporal components; selected by cross-validation then manually reduced to 3 in Section 4.3 based on explained variability.
r2 (measurement rank) = 2 (CV gave 3)
Number of measurement components; reduced manually to 2 in Section 4.3; affects the interpretability of R components.

assumptions (5)

domain assumption Temporal smoothness: the true reconstructed signal is smooth over the 24-hour grid, encoded via second-order differences in D.
Justified by the nature of circadian blood pressure, but the circular boundary in (4) assumes smoothness across midnight and the grid is aligned to clock time, which may misalign individual sleep schedules.
domain assumption Low-rank Tucker structure: the ABPM tensor is well approximated by a Tucker decomposition with small r1 and r2, with subject mode uncompressed.
Central to the method; if the data are not approximately low-rank, the decomposition and downstream scores are not valid.
domain assumption Missing data are ignorable: missingness does not depend on unobserved values.
The iterative imputation in Section 2.3 imputes missing entries using the low-rank model without modeling missingness; the paper does not assess the missingness mechanism, and missing rates vary widely across patients.
standard math Distinct singular values for identifiability: the SVD-based rotation in Section 2.5 yields identifiable components only if singular values are distinct.
Standard property of SVD; the paper notes identifiability 'by conditions of SVD as long as singular values are distinct'.
ad hoc to paper Manual rank reduction is valid: the final ranks r1=3, r2=2 are chosen by explained variability, not by an independent validation.
Section 4.3 states the authors overrule the cross-validation choice because it may overestimate ranks; this post hoc selection affects the regression results.

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Cite this review

Pith. "Pith review of Smooth tensor decomposition with application to ambulatory blood pressure monitoring data." pith.science (2026). https://pith.science/paper/KQZPJR4E

@misc{pith2026250711723,
  author       = {Pith},
  title        = {Pith review of: Smooth tensor decomposition with application to ambulatory blood pressure monitoring data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQZPJR4E}},
  note         = {Machine review of arXiv:2507.11723}
}

read the original abstract

Ambulatory blood pressure monitoring (ABPM) enables continuous measurement of blood pressure and heart rate over 24 hours and is increasingly used in clinical studies. However, ABPM data are often reduced to summary statistics, such as means or medians, which obscure temporal features like nocturnal dipping and individual chronotypes. Functional data analysis methods better capture these temporal dynamics but typically treat each ABPM measurement separately, limiting their ability to leverage correlations among matched measurements. In this work, we observe that aligning ABPM data along measurement type, time, and patient ID lends itself to a tensor representation--a multidimensional array. Although tensor learning has shown great potential in other fields, it has not been applied to ABPM data. Existing tensor learning approaches often lack temporal smoothing constraints, assume no missing data, and can be computationally demanding. To address these limitations, we propose a novel smooth tensor decomposition method that incorporates a temporal smoothing penalty and accommodates missing data. We also develop an automatic procedure for selecting the optimal smoothing parameter and tensor ranks. Simulation studies demonstrate that our method reliably reconstructs smooth temporal trends from noisy, incomplete data. Application to ABPM data from patients with concurrent obstructive sleep apnea and type 2 diabetes uncovers clinically relevant associations between patient characteristics and ABPM measurements, which are missed by summary-based approaches.

Figures

Figures reproduced from arXiv: 2507.11723 by the authors.

**Figure 1.** Tensor representation of ABPM data with the low-rank approximation, where L captures temporal components, R represents measurement components, and I denotes the identity matrix. The core tensor G contains individual-level scores for all subjects along the temporal and measurement directions. There are several roadblocks to the adoption of (2) to ABPM data: (i) the compression does not account for time ordering, tha… view at source ↗

**Figure 2.** Boxplots for losses for M (a) and L (b) across 100 replications. Case 1 corresponds to fixed ranks, and Case 2 to flexible ranks. Loss of L is only computed in Case 1. Oracle refers to SmoothHOOI with tuning parameters selected to minimize LM. Case 2, we select the number of components for FPCA to explain at least 99% of the variance (default setting). Since the number of selected components can differ across SBP, D… view at source ↗

**Figure 3.** (a): Coefficients for each estimated time component Lk from baseline HYPNOS ABPM data, k = 1, 2, 3. (b)–(d): Effect of temporal components Lk on DBP (b), SBP (c), and HR (d). The solid black line represents the sample mean of DBP, SBP, and HR over the 24-hour period with colored lines representing positive (blue curve) and negative (red curve) shifts with respect to L components. To validate the interpretation of th… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Relationship between g31 scores and actigraphy-estimated sleep periods. Blue bars represent the time of sleep periods for each subject. Red dots denote the average midpoint of sleep for each of 10 equal-width intervals spanning the range of g31 scores (e.g., [−4.37, −3…

**Figure 5.** Figure 5: (a): The estimated effect size of log-transformed ODI4 on DBP, SBP, and HR; (b): Estimated DBP, SBP, and HR profiles when ODI4 is set to its first quartile (Q1), median, and third quartile (Q3) values (unit: events/hour), with all other covariates held at their means. …

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