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This paper claims that generalized tensor completion can be made consistent under missing-not-at-random mechanisms by jointly modeling observed entries and the missingness mask, with error bounds that hold even when observation probabilitie

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2026-08-04 23:52 UTC pith:WCDXTYBE

load-bearing objection A promising MNAR tensor-completion framework, but Condition 2's b0 scaling makes the missingness mask nearly uninformative, so the main convergence theorem and the b1 test are unsupported as written. the 2 major comments →

arxiv 2509.06225 v2 pith:WCDXTYBE submitted 2025-09-07 stat.ME

Generalized Tensor Completion with Non-Random Missingness

Maoyu Zhang , Biao Cai , Will Wei Sun , Jingfei Zhang This is my paper
classification stat.ME MSC 62H1262J1262F1215A69
keywords tensor completionmissing not at randomCP decompositiongeneralized linear modelsalternating maximizationnon-asymptotic error boundsMCAR versus MNAR testexponential family
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 tensor completion method for data whose missingness is not random: the chance an entry is observed can depend on the value of that entry. It jointly models the observed tensor entries and the missingness mask through an exponential-family tensor model plus a logistic observation-probability model, and estimates all parameters by alternating maximization. The authors prove that each iteration reduces error geometrically until it reaches a statistical-error floor, under conditions on slice-level averages rather than uniform bounds on individual observation probabilities. They also provide a sample-splitting test for whether missingness actually depends on tensor values, i.e., MCAR versus MNAR. This matters because recommender systems and medical imaging often have value-dependent missingness, and ignoring it biases predictions.

Core claim

The central claim is that tensor completion with missing-not-at-random (MNAR) mechanisms can be solved by joint estimation rather than inverse-propensity reweighting, and that the joint estimator has rigorously controlled error. Under a CP low-rank model X = sum_r λ_r u_r ∘ v_r ∘ w_r and a logistic observation model g_θ(x)=logit^{-1}(b0+b1 x), the alternating maximization of the joint log-likelihood converges geometrically to a neighborhood of the truth whose radius is the statistical error. Theorem 1 (rank-one) and Theorem 2 (general rank R) state that after t iterations, D(Θ^(t),Θ*) ≤ ρ_R^t D(Θ^(0),Θ*) + C_4 ((ψ′_max+1)λ_max √(log d · d))/((ψ″_min √pbar + √qbar) λ_min^2), with probability

What carries the argument

The machinery is the joint log-likelihood ℓ_d(Θ) of the observed entries and the missingness mask, maximized by alternating coordinate ascent over each unit-norm factor u_r, v_r, w_r, the weight λ_r, and the logistic missingness parameters (b0, b1) (Algorithm 1). Because both the data model and the missingness model are generalized linear models, updates have no closed form; the analysis controls the gradient and Hessian coordinate-wise rather than at the vector level, with covering-number concentration bounds. The relaxations are encoded in the slice-average quantities pbar and qbar, which replace the uniform lower/upper bounds on observation probabilities used in prior work. The split-samp

Load-bearing premise

The convergence theorems assume the algorithm starts close enough to the true parameters—within a ball of radius 1/2 under the paper's own error metric—and the initialization designed to achieve this is deferred to the supplement.

What would settle it

Run the proposed method on a 50×50×50 rank-3 Gaussian tensor with logit missingness b1=2 at an observation ratio near 0.36, over 50 replicates. If the missing-entry RMSE does not consistently beat inverse-propensity and MCAR baselines and shrink as signal strength c grows, the central practical claim fails. Alternatively, simulate b1=0 with d3=150 and |A2|=500: the split-sample test should reject about 5% of the time; a rejection rate far from nominal would falsify Theorem 3.

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

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If this is right

  • Tensor completion under MNAR no longer requires inverse-propensity reweighting, so estimates do not blow up when some observation probabilities are near zero.
  • The same algorithm covers continuous, binary, and count data through exponential-family links, extending the result beyond squared-loss tensor completion.
  • Practitioners can formally test whether missingness is MCAR or MNAR and build confidence intervals for b1 using the split-sample logistic regression.
  • The non-asymptotic bound separates computational error (geometrically decaying) from statistical error, giving an explicit iteration count needed to reach the noise floor.
  • Under uniform missingness, the conditions reduce to the existing tensor-completion condition p ≥ C0 log^4(d)/d^{3/2}, so the theory contains prior results as a special case.

Where Pith is reading between the lines

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

  • The sample-splitting principle—tiny test set for a low-dimensional logistic regression, large set for the high-dimensional tensor—could be reused in other problems where a high-dimensional nuisance tensor must be estimated before low-dimensional inference.
  • Because the missingness model is parametric, the same testing procedure can likely be extended to include covariates or polynomial terms in Xijk, turning the MCAR-versus-MNAR test into a test of a richer missingness mechanism.
  • The coordinate-wise concentration proof suggests that similar slice-average conditions may transfer to other non-convex problems with heterogeneous weights, such as weighted low-rank approximation with unknown weights.

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

2 major / 5 minor

Summary. The paper proposes a joint-likelihood framework for tensor completion when the missingness indicator D_ijk follows Bernoulli(logit^{-1}(b0+b1 X*_ijk)) and the observed entries follow an exponential-family model with mean depending on X*. The underlying tensor is assumed to have a rank-R CP decomposition. The proposed estimator maximizes the joint likelihood by alternating maximization (Algorithm 1). The main theoretical contributions are non-asymptotic per-iteration error bounds (Theorems 1 and 2) under conditions on slice averages of observation probabilities, and a sample-splitting test for H0: b1*=0 (Theorem 3). The paper reports simulations for Gaussian, Bernoulli and Poisson data and two real-data applications.

Significance. If the results hold, the paper would be a meaningful advance: it relaxes the uniform bounded-probability assumptions used in prior MNAR matrix/tensor completion work, provides a computational-statistical trade-off for the alternating algorithm, and supplies a formal MCAR-versus-MNAR test. The simulation study is reasonably extensive and the proposed method performs well relative to several baselines. However, the theoretical core is not fully verifiable from the submitted text (proofs and the initialization algorithm are deferred to a supplement), and Condition 2 as written appears to make b1 unidentified. These issues must be resolved before the main claims can be accepted.

major comments (2)
  1. [Section 3.1, Condition 2; Theorems 2-3] Condition 2 states b0*=Θ(Rλmin/√d3) when b1*≠0. With Condition 1(c), |X*_ijk|≤C R λmax/d^{3/2}; with λmin=Θ(λmax) and Condition 3, |b0*| ≥ c R d^{1/4} while |b1*X*_ijk| ≤ C R λmax/d^{3/2}. Hence b0* dominates b1*X* by at least a factor c d, and logit(P_ijk)=b0*(1+o(1)). For b0*>0, 1-P_ijk≤C e^{-c R d^{1/4}}; for b0*<0, P_ijk≤C e^{-c R d^{1/4}}. Thus qbar≤e^{-Ω(R d^{1/4})} and the Fisher information for b1 from the mask is I=Σ P(1-P)X²≤d³ e^{-Ω(R d^{1/4})}(Rλmax/d^{3/2})²=o(1). So b1 is asymptotically unidentified, contradicting the finite-sample error bound on |b1-b1*| in Theorem 2 and the asymptotic normality in Theorem 3. It also contradicts the paragraph after Condition 2 that says probabilities are not dominated by b0* or b1*X*. The simulations in §4.1 use b0∈{-1,...,2} whereas Rλmin/√d3≈150 at d=50, so they do not satisfy Condition 2 as written.
  2. [Theorems 1-2, Algorithm 1 and Algorithm S1] Theorems 1 and 2 assume Θ^(0)∈B_{1/2}(Θ*) and u^(0)_ri, v^(0)_rj, w^(0)_rk = O(1/√d). Algorithm S1, which is supposed to produce such an initialization, is only referenced as being in the supplement and is not part of the reviewed text. The theoretical guarantee of Algorithm 1 is therefore conditional on an unverified initialization property. The authors should either provide a proof that S1 attains the required ball or explicitly state the initialization as an assumption and discuss when it can be met. As written, the convergence claim for the implemented algorithm is not established.
minor comments (5)
  1. [Section 4.1; Figures 4-5] Typos: 'suprior performance' should be 'superior performance'; 'Obsevation Ratio' should be 'Observation Ratio'.
  2. [Section 4.3 / Figure 5] The text refers to the method as 'MTDF' but the Figure 5 legend uses 'MDTF'. Please standardize.
  3. [Theorem 3] The displayed Hessian has an unmatched parenthesis/bracket: 'exp(-bb_A2_0 - bb_A2_1 bX_A1_ijk ))' contains an extra closing parenthesis.
  4. [Section 3.2] The phrase 'we exploit employ covering number arguments' should be corrected (e.g., 'we employ covering number arguments').
  5. [Condition 2] The disjunctive condition on qbar (qbar ≤ C0' pbar (ψ''min)^2 or qbar ≥ C0 ...) is not explained. A short intuition or example would help readers understand which regime each disjunct covers.

Circularity Check

0 steps flagged

No circular derivation: the error bounds and test are derived from stated conditions, not fitted or defined into existence.

full rationale

The paper's main theoretical results (Theorems 1-3) are non-asymptotic error bounds for an alternating maximization estimator and an asymptotic normality result for a sample-split logistic regression. These are derived from Conditions 1-4 and explicit initialization assumptions; they are not obtained by fitting the target quantity or by renaming an input as an output. The statistical error terms in Theorems 1-2 depend on model primitives and slice-level averages of the true missingness probabilities, but the constants are not calibrated to the reported error, and the bounds are not equivalent to the assumptions by construction. The hypothesis test in Section 3.3 uses sample splitting: the tensor is estimated only on A1 and the logistic regression for b1 is fit only on A2, so the predictor bX_A1 is independent of the testing mask by design; Theorem 3 is a standard MLE expansion rather than a restatement of an assumed conclusion. The paper's self-citations (e.g., Cai et al. 2025; Sun and Li 2019; Zhou et al. 2023) support standard incoherence or dynamic-tensor conditions but are not load-bearing in the sense of importing an unverified uniqueness theorem or smuggling in the paper's own conclusion. The initialization is deferred to Algorithm S1 in the supplement, which is an omitted implementation detail rather than a circular step. A separate concern--whether Condition 2's b0* scaling makes P_ijk near-constant and b1 weakly identifiable--is a validity-of-assumptions issue, not a circularity: if valid, it would mean the theorem's conditions are inappropriate, not that the proof assumes the conclusion. No circular step is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The framework's guarantees are conditional on a correct exponential family outcome model, an exact logistic MNAR mechanism, exact CP low rank, plus technical incoherence and scaling conditions tailored to the proof. These are stated assumptions, not hidden fitted constants. The most fragile components are the logistic missingness specification and the warm-start initialization requirement.

axioms (6)
  • domain assumption Entries of Y are independent exponential family with known dispersion phi0 and cumulant psi (eq 1)
    Defines the outcome model class and is the starting point of the joint likelihood in (3).
  • domain assumption Mask entries are independent Bernoulli with logit P_ijk = b0 + b1 X_ijk (eq 2)
    This logistic MNAR specification is the core missingness mechanism; misspecification would bias the estimates and invalidate the test.
  • domain assumption The true tensor X is exactly CP rank R with unit-norm factors, identified up to permutation (eq 4, Condition 1(b))
    The low-rank CP structure is assumed rather than learned from first principles.
  • ad hoc to paper Factor incoherence conditions on u*, v*, w* and small cross-rank correlations (Conditions 1(c) and 4)
    Technical regularity conditions tailored to the proof that control concentration and inter-rank correlation.
  • ad hoc to paper Scalings and slice-average bounds in Conditions 2 and 3, including b0* = Theta(R lambda_min / sqrt(d3)) and b1* = Theta(1)
    These conditions are chosen to make the concavity and concentration arguments go through; they are not derived from first principles.
  • ad hoc to paper The initialization lies in a 1/2-ball of the truth with coordinate magnitudes O(1/sqrt(d)) (Theorems 1-2)
    The non-asymptotic convergence results are conditional on a warm start; the construction of this warm start is deferred to Algorithm S1.

pith-pipeline@v1.3.0-alltime-deepseek · 16708 in / 17215 out tokens · 204780 ms · 2026-08-04T23:52:07.252047+00:00 · methodology

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Cite this review

Pith. "Pith review of Generalized Tensor Completion with Non-Random Missingness." pith.science (2026). https://pith.science/paper/WCDXTYBE

@misc{pith2026250906225,
  author       = {Pith},
  title        = {Pith review of: Generalized Tensor Completion with Non-Random Missingness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WCDXTYBE}},
  note         = {Machine review of arXiv:2509.06225}
}
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read the original abstract

Tensor completion plays a crucial role in applications such as recommender systems and medical imaging, where data are often highly incomplete. While extensive prior work has addressed tensor completion with data missingness, most assume that each entry of the tensor is available independently with probability $p$. However, real-world tensor data often exhibit missing-not-at-random (MNAR) patterns, where the probability of missingness depends on the underlying tensor values. This paper introduces a generalized tensor completion framework for noisy data with MNAR, where the observation probability is modeled as a function of underlying tensor values. Our flexible framework accommodates various tensor data types, such as continuous, binary and count data. For model estimation, we develop an alternating maximization algorithm and derive non-asymptotic error bounds for the estimator at each iteration, under considerably relaxed conditions on the observation probabilities. Additionally, we propose a statistical inference procedure to test whether observation probabilities depend on underlying tensor values, offering a formal assessment of the missingness assumption within our modeling framework. The utility and efficacy of our approach are demonstrated through comparative simulation studies and analyses of two real-world datasets.

Figures

Figures reproduced from arXiv: 2509.06225 by Biao Cai, Jingfei Zhang, Maoyu Zhang, Will Wei Sun.

Figure 1
Figure 1. Figure 1: Missingness patterns for three users from the InCarMusic dataset. Marked cells [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of definitions for ¯p and ¯q. more relaxed. Recalling Pijk = logit−1 (b ∗ 0 + b ∗ 1X ∗ ijk), define ¯p and ¯q as p¯ = min ( min i 1 d2d3 X j,k Pijk, min j 1 d1d3 X i,k Pijk, min k 1 d1d2 X i,j Pijk) . q¯ = min ( min i 1 d2d3 X j,k Pijk(1 − Pijk), min j 1 d1d3 X i,k Pijk(1 − Pijk), min k 1 d1d2 X i,j Pijk(1 − Pijk) ) . We write ψ ′ max = maxi,j,k [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mean RMSE with 95% confidence intervals in the Gaussian case. [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean AUC with 95% confidence intervals in the Bernoulli case. [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean RMSE with 95% confidence intervals in the Poisson case. [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Boxplots of selected ranks in Gaussian data (left plot) and Bernoulli data (right [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The distribution of ratings from the InCarMusic data. [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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