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arxiv: 2604.10353 · v1 · submitted 2026-04-11 · 📊 stat.ME · math.ST· stat.TH

Uncertainty Quantification for Noisy Low-tubal-rank Tensor Completion

Jiuqian Shang , Jingyang Li , Yang Chen This is my paper

Pith reviewed 2026-05-10 15:08 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH
keywords low-tubal-rank tensortensor completionuncertainty quantificationstatistical inferencedebiasingconfidence intervalsasymptotic normalitygeophysical data
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The pith

Double-sample debiasing followed by low-rank projection yields asymptotically Gaussian estimators for tensor inference.

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

This paper develops methods for uncertainty quantification in high-dimensional tensor completion under the low-tubal-rank model. It shows how to turn noisy observations into estimators that are asymptotically normal, which in turn supports hypothesis tests and confidence intervals. The approach relies on a double-sample debiasing step before the low-rank projection. Validation comes from simulations plus an application to global total electron content data, where the resulting intervals appear reliable.

Core claim

Employing a double-sample debiasing technique followed by a low-rank projection, we construct asymptotically Gaussian estimators that yield valid statistical inference under mild assumptions. More precisely, we can perform hypothesis testing and construct confidence intervals with this result. We validate the theoretical results through extensive simulations and demonstrate the practical effectiveness of our method in completing the global total electron content data.

What carries the argument

Double-sample debiasing followed by low-rank projection, which produces estimators that are asymptotically normal and therefore usable for inference on linear forms of the tensor.

If this is right

  • Hypothesis tests become available for any linear functional of the completed tensor.
  • Entrywise confidence intervals can be formed for the imputed tensor values.
  • The framework applies directly to geophysical tensor datasets such as total electron content.
  • Numerical experiments confirm that the intervals remain robust under the stated conditions.

Where Pith is reading between the lines

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

  • The same debiasing-plus-projection idea could be tested on other tensor rank notions such as CP or Tucker rank.
  • If the mild conditions hold in practice, the method could be embedded in existing tensor completion pipelines to supply uncertainty measures automatically.
  • Adaptive sampling designs that exploit the asymptotic normality might further reduce the number of observations needed for reliable inference.

Load-bearing premise

The tensor has low-tubal-rank structure and the noise distribution plus sampling mechanism satisfy conditions that make the estimators asymptotically Gaussian.

What would settle it

Simulations or real data in which the nominal 95 percent confidence intervals achieve coverage rates far from 95 percent would show that the asymptotic Gaussianity does not hold.

Figures

Figures reproduced from arXiv: 2604.10353 by Jingyang Li, Jiuqian Shang, Yang Chen.

Figure 1
Figure 1. Figure 1: Visualizations of the example tensor used in simulation. Panels show (a) the ground truth low￾tubal-rank tensor T ∈ R 500×500×500 with rank r = 4; (b) T corrupted by i.i.d. sub-Gaussian noise with standard deviation σξ = 0.6; (c) the observed tensor with 40% of entries observed and missing entries marked in black; (d) the tensor imputed by the proposed procedure in Algorithm 1. repeated across indices, mak… view at source ↗
Figure 2
Figure 2. Figure 2: Pixel-wise performance gains of estimators across estimation stages: Init (Tinit,a, a = 1, 2), Proj (Tproj,a, a = 1, 2), and Final (Tb), and comparison to RCGD [28, Algorithm 2]. Grouped boxplots display the pixel-wise reduction in empirical absolute Bias, Variance, and MSE evaluated at 100,000 randomly sampled tensor locations (1,000 simulations). Values above the red dashed line indicate an improvement. … view at source ↗
Figure 3
Figure 3. Figure 3: Normal approximation of ⟨Tb,M⟩−⟨T ,M⟩ σbξsb √ d1d2d3/n with M = M(1) (upper left), M(2) (upper right), M(3) (bottom left), M(4) (bottom right). Each panel shows a density histogram of the standardized estimated statistics from 1000 independent simulations, and the red curve represents the p.d.f. of standard normal distributions. The simulation parameters are d1 = d2 = d3 = 500, r = 4, σξ = 0.6, and n = 0.4… view at source ↗
Figure 4
Figure 4. Figure 4: TEC reconstruction for the 24-hour storm of 2 September 2017. Panels (a)–(d) show the VISTA TEC maps (ground truth) at four representative times (00:05, 07:00, 11:50, and 20:00 UT), displayed as latitude versus local time (LT) with 12 LT centered. Panels (e)–(h) show the corresponding masked data, where each pixel is removed independently with probability 0.6. Panels (i)–(l) and (m)–(p) display two initial… view at source ↗
Figure 5
Figure 5. Figure 5: Pointwise upper and lower confidence bounds (with significance level 95%) for storm-time TEC on 8 September 2017 at four representative times (00:05, 07:00, 11:50, and 20:00 UT). Panels (a)–(d) and (e)–(h) show, respectively, the upper and lower bounds of the pointwise (1−α)–confidence intervals targeting the latent TEC map, constructed from Corollary 3.2. Panels (i)–(l) and (m)–(p) display the correspondi… view at source ↗
read the original abstract

High-dimensional tensor data often exhibit strong temporal correlations that appear as low-dimensional structures in the frequency domain. While the low-tubal-rank tensor model effectively captures these spectral features, making it potentially suitable for geophysical data, existing methods primarily focus on point estimation. Uncertainty quantification (UQ) of imputed values and rigorous statistical inference for these models remain largely unexplored. In this work, we propose a flexible inference framework for linear forms of high-dimensional tensors. Employing a double-sample debiasing technique followed by a low-rank projection, we construct asymptotically Gaussian estimators that yield valid statistical inference under mild assumptions. More precisely, we can perform hypothesis testing and construct confidence intervals with this result. We validate the theoretical results through extensive simulations and demonstrate the practical effectiveness of our method in completing the global total electron content data. We demonstrate, using those numerical results, that our entrywise confidence intervals are robust and reliable, yielding informative uncertainty quantification that captures underlying variability.

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

1 major / 1 minor

Summary. The paper develops an inference framework for linear functionals of high-dimensional tensors under the noisy low-tubal-rank model. It uses double-sample debiasing followed by a low-rank projection (via t-SVD) to produce asymptotically Gaussian estimators, enabling hypothesis tests and entrywise confidence intervals under mild assumptions on the tensor structure, noise, and sampling. The claims are supported by simulations and an application to global total electron content (TEC) data, where the resulting intervals are reported to be robust.

Significance. If the asymptotic normality and coverage guarantees hold under realistic conditions, the work would fill an important gap by moving low-tubal-rank tensor completion from point estimation to valid statistical inference, which is particularly relevant for geophysical applications where uncertainty quantification is needed. The double-sample debiasing approach and the demonstration on real TEC data are concrete strengths that could make the method practically useful once the supporting conditions are clarified.

major comments (1)
  1. [Abstract] Abstract: The central claim of asymptotically Gaussian estimators and valid CIs after debiasing plus low-tubal-rank projection rests on 'mild assumptions,' but these assumptions do not explicitly require or verify a spectral gap of sufficient size (relative to the noise level after sampling) on the Fourier-domain singular values. Without such a gap the projection step can reintroduce bias of the same order as the debiasing correction, invalidating the Gaussian limit and the subsequent inference; this condition is load-bearing for the geophysical application where the low-tubal-rank model is typically only approximate.
minor comments (1)
  1. [Abstract] The abstract repeats the phrase 'we demonstrate' in consecutive sentences; a single concise statement of the numerical validation would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our work. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim of asymptotically Gaussian estimators and valid CIs after debiasing plus low-tubal-rank projection rests on 'mild assumptions,' but these assumptions do not explicitly require or verify a spectral gap of sufficient size (relative to the noise level after sampling) on the Fourier-domain singular values. Without such a gap the projection step can reintroduce bias of the same order as the debiasing correction, invalidating the Gaussian limit and the subsequent inference; this condition is load-bearing for the geophysical application where the low-tubal-rank model is typically only approximate.

    Authors: We appreciate the referee for identifying this important technical point. The low-rank projection step does require that the Fourier-domain singular values maintain a sufficient gap relative to the post-sampling noise level; otherwise the projection can indeed reintroduce bias comparable to the debiasing term and invalidate the asymptotic normality. Our analysis incorporates this requirement through the stated conditions on the tensor structure and noise (which bound the singular values away from the noise floor), but we agree that the abstract and the discussion of assumptions would benefit from an explicit statement of the spectral-gap condition. We will revise the abstract and Section 3 accordingly to make the requirement clear. For the TEC application, the low-tubal-rank model is approximate, yet our numerical results indicate that the estimated singular values exhibit a clear gap and that the resulting intervals remain reliable; we will add a brief remark on this empirical observation in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation

full rationale

The paper constructs asymptotically Gaussian estimators for linear functionals of low-tubal-rank tensors by applying a double-sample debiasing step followed by projection onto an estimated low-tubal-rank subspace via t-SVD. This follows standard statistical debiasing practices to separate estimation from inference without any quoted reduction of the target Gaussian limit or variance estimator to a fitted parameter or self-referential definition. No load-bearing self-citations, uniqueness theorems imported from the same authors, or ansatz smuggling appear in the abstract or described chain; the mild assumptions are external to the construction itself. The derivation is therefore self-contained against the stated inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim depends on the low-tubal-rank structure of the tensor and on noise conditions sufficient for a central-limit theorem to apply after debiasing and projection; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The observed tensor admits an approximate low-tubal-rank structure
    Required for the projection step to recover the signal and for the asymptotic analysis to hold.
  • domain assumption Noise terms satisfy regularity conditions allowing asymptotic normality after debiasing
    Needed for the constructed estimators to be asymptotically Gaussian.

pith-pipeline@v0.9.0 · 5458 in / 1408 out tokens · 102427 ms · 2026-05-10T15:08:47.996393+00:00 · methodology

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Reference graph

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