{"id":"a357092f-a8dd-453e-9163-b54b02184048","arxiv_id":"2509.06225","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Generalized tensor completion that jointly fits a low-rank tensor and a logistic missing-not-at-random mechanism, with per-iteration error bounds and a MCAR-versus-MNAR test.","lead":"This paper proposes a generalized tensor completion method that models missingness that depends on the underlying tensor values themselves, for continuous, binary, and count data. It also provides a statistical test for whether such non-random missingness is present.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Condition 2's b0 scaling may make the missingness mask uninformative for b1, invalidating Theorems 2-3.","rationale":"The paper is a serious attempt at a difficult problem: joint modeling of tensor entries and MNAR missingness with alternating maximization, plus a testing procedure. The simulations are extensive and the BIC rank selection is useful. I nevertheless find a more fundamental issue than the initialization warm-start identified by the reader. Condition 2 imposes b0*=Θ(Rλ_min/√d3) when b1*≠0; under the paper's own scaling (d1,d2,d3=Θ(d), Condition 3), this forces b0* to grow with d. Since X entries are O(λ/d^{3/2}) by incoherence, the mask probabilities collapse to 0 or 1, leaving almost no information about b1. This contradicts the paper's claim that the conditions ensure b0 and b1 are not dominant, and it would make both the finite-sample bound for b1 and the asymptotic normality in Theorem 3 false. The fact that the simulations use b0 of order 1 (while the condition gives ~d) suggests a typographical error, but as written the theorem is not credible. Because the proof and initialization are in an unseen supplement, I cannot tell whether the proof uses a different condition. Hence I move from conditional acceptance to 'unverdictable' pending correction of Condition 2 and release of the supplement. If the condition is corrected to b0*=Θ(Rλ_min/d3^{3/2}) (or O(1)), the paper would be back to conditional acceptance.","tokens_in":16966,"tokens_out":20479,"duration_ms":223693,"concrete_test":"Set Gaussian data with d1=d2=d3=d, R=1, λ*=c d^{3/2}, b1*=1, and choose b0*=C λ*/√d as required by Condition 2. Treat X* as known and compute the oracle Fisher information I(b1)=Σ_{i,j,k} σ(1-σ) X_ijk^2 with σ=logit^{-1}(b0*+b1*X_ijk). If I(b1)→0 as d→∞ (as direct calculation shows), then b1 is not estimable from the mask, so Theorem 2's error bound for b1 and Theorem 3's normality cannot hold under the stated condition. If, alternatively, the authors intended b0*=Θ(Rλ_min/d3^{3/2}), rerun the check with that scaling: then I(b1)≍d^{3/2}, and the theorems may hold; this would confirm a typo rather than a substantive flaw.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"Condition 2 (Section 3.1) requires, when b1*≠0, that b0*=Θ(Rλ_min/√d3). With d1=d2=d3=Θ(d) and Condition 3 forcing λ_min≥d^{3/4} (up to polylog factors), this gives b0*=Ω(d^{1/4}); for the typical incoherent CP scaling λ_min≍d^{3/2}, b0*≍d. Meanwhile X_ijk=O(λ_max/d^{3/2}) by incoherence, so b0* dominates b1*X_ijk by a factor √d. Consequently P_ijk=logit^{-1}(b0*+b1*X_ijk) is exponentially close to 0 or 1 for every entry, and the mask D is asymptotically constant. The oracle logistic regression of D on X* then has Fisher information for b1 equal to Σ P(1-P)X^2, which tends to 0 super-polynomially (e.g., ≍d^{3/2}e^{-Ω(d^{1/4})} or worse). Thus b1 is not identifiable from the missingness mask, contradicting (i) the paper's rationale that 'probabilities are not dominated by either b0* or b1*X', (ii) Theorem 2's finite-sample bound on |b1-b1*|, and (iii) Theorem 3's asymptotic normality. The simulations themselves use b0∈{-1,...,2} while Rλ_min/√d3≈30 at d=50, so the numerical experiments do not satisfy the stated condition. If the intended condition was b0*=Θ(Rλ_min/d3^{3/2}) (i.e., b0=O(1)), the theorem must be restated; as written, the main claim is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17485,"tokens_out":12446,"duration_ms":138951,"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":[{"comment":"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.","section":"Section 3.1, Condition 2; Theorems 2-3"},{"comment":"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.","section":"Theorems 1-2, Algorithm 1 and Algorithm S1"}],"minor_comments":[{"comment":"Typos: 'suprior performance' should be 'superior performance'; 'Obsevation Ratio' should be 'Observation Ratio'.","section":"Section 4.1; Figures 4-5"},{"comment":"The text refers to the method as 'MTDF' but the Figure 5 legend uses 'MDTF'. Please standardize.","section":"Section 4.3 / Figure 5"},{"comment":"The displayed Hessian has an unmatched parenthesis/bracket: 'exp(-bb_A2_0 - bb_A2_1 bX_A1_ijk ))' contains an extra closing parenthesis.","section":"Theorem 3"},{"comment":"The phrase 'we exploit employ covering number arguments' should be corrected (e.g., 'we employ covering number arguments').","section":"Section 3.2"},{"comment":"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.","section":"Condition 2"}],"recommendation":"major_revision","confidential_remarks":"The scaling problem in Condition 2 may be a typo: replacing √d3 with d3^{3/2} would make b0*=O(1) for the Gaussian case and would align the condition with the simulation settings (b0∈{-1,...,2}). If that is the intended condition, the paper is potentially salvageable, but the supplement must be provided and the proofs checked for the Fisher information issue. The current written version of the theorem is not credible as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper proposes a joint likelihood for exponential-family tensor completion under MNAR, with an alternating maximization algorithm, non-asymptotic bounds, and a split-sample test of MCAR vs MNAR. The idea is good and the simulations are extensive. But there's a scaling problem in Condition 2 that I think breaks the main theorems as stated, and the proofs and initialization are in a supplement I didn't see.\n\nWhat's new: the framework genuinely extends the matrix MNAR work (Ma and Chen, TenIPS) to generalized CP tensors, allows observation probabilities with slice-wise averages rather than uniform bounds, and the hypothesis test is a useful addition. The coordinate-wise concavity analysis is a plausible technical contribution.\n\nThe soft spot: Condition 2 says when b1* != 0, b0* = Theta(R lambda_min / sqrt(d3)). With the incoherence assumption, |b1 X*_ijk| = O(R lambda_max / d^{3/2}). So b0* dominates b1 X* by a factor of order sqrt(d). That makes the logit b0 + b1 X approximately b0, so the observation probabilities are all near 0 or 1, and the mask D carries almost no information about b1. The Fisher information for b1 in the D model is then vanishingly small. Theorem 2's error bound for |b1 - b1*| and Theorem 3's asymptotic normality can't hold in that regime. The paper even says the condition is meant to keep b0 and b1 X from dominating each other, which is exactly what the stated scaling fails to do. The simulations use b0 in {-1,...,2}, while the condition would demand something like d^{1/4} or larger, so the numerics don't satisfy the theory either. This looks like a typo—maybe they meant d3^{3/2} in the denominator—but as written it's load-bearing.\n\nThe secondary issue is the initialization: Theorems 1-2 require starting in a 1/2-ball with coordinate-wise O(1/sqrt(d)) errors, and Algorithm S1 is only in the supplement. So the algorithmic convergence claim is conditional on an unverifiable warm start.\n\nWho should read it: statisticians working on tensor completion, missing data, and non-convex optimization. The idea has value, but the current version is not publishable as is. I'd send it to a serious referee with instructions that the main theorem needs to be fixed and the supplement made available.\n\nSummary: if the b0 scaling is corrected and the initialization is properly analyzed, this could be a useful paper. Right now I wouldn't rely on the theory.","headline":"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.","tokens_in":17815,"tokens_out":6223,"would_cite":false,"duration_ms":68187,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H12","62J12","62F12","15A69"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["tensor completion","missing not at random","CP decomposition","generalized linear models","alternating maximization","non-asymptotic error bounds","MCAR versus MNAR test","exponential family"],"falsifier":"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.","tokens_in":16909,"feed_emoji":"🧩","tokens_out":9943,"duration_ms":103278,"temperature":0.7,"pith_summary":"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.","feed_headline":"Tensor completion survives missing-not-at-random data","feed_subtitle":"Joint estimation recovers tensors even near observation probabilities 0 or 1; new test detects MNAR.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the CP low-rank decomposition (4) that the true tensor is assumed to satisfy.","marker":"Kolda and Bader (2009)"},{"why":"Baseline nonconvex tensor completion theory whose uniform-missing statistical error bound the paper's bound matches when P_ijk is constant.","marker":"Cai et al. (2022)"},{"why":"Prior MNAR matrix completion via inverse propensity weighting; its requirement that observation probabilities be bounded away from 0 and 1 motivates the relaxed condition.","marker":"Ma and Chen (2019)"},{"why":"Extends the IPS approach to tensors under MNAR; comparison method whose low-rank-propensity and noiseless assumptions the paper avoids.","marker":"Yang et al. (2021)"},{"why":"Matrix completion with data-dependent missingness probabilities; represents the bounded-probability assumption the paper relaxes to slice averages.","marker":"Bhattacharya and Chatterjee (2022)"},{"why":"Standard statistically optimal noisy tensor completion results used as benchmark for conditions and rates.","marker":"Xia et al. (2021)"},{"why":"Generalized CP decomposition used as the GCP baseline in simulations.","marker":"Hong et al. (2020)"},{"why":"BIC criterion used to select tensor rank in the implementation.","marker":"Schwarz (1978)"}],"fun_headline_variants":["Joint estimation + MNAR test for robust tensor completion","Tensor completion without reweighting: works even for MNAR","New tensor completion handles missing-not-at-random data","Provably accurate tensor completion for non-random missingness","One framework: complete tensors and test for MNAR"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Joint estimation + MNAR test for robust tensor completion","Tensor completion without reweighting: works even for MNAR","New tensor completion handles missing-not-at-random data","Provably accurate tensor completion for non-random missingness","One framework: complete tensors and test for MNAR"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001529,"raw_usage":{"total_tokens":5969,"prompt_tokens":768,"completion_tokens":5201,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":5123}},"tokens_in":512,"tokens_out":5201,"duration_ms":49857,"temperature":1.0,"reasoning_tokens":5123,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T23:52:07.252047+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the CP low-rank decomposition (4) that the true tensor is assumed to satisfy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Baseline nonconvex tensor completion theory whose uniform-missing statistical error bound the paper's bound matches when P_ijk is constant."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the IPS approach to tensors under MNAR; comparison method whose low-rank-propensity and noiseless assumptions the paper avoids."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Matrix completion with data-dependent missingness probabilities; represents the bounded-probability assumption the paper relaxes to slice averages."},{"cited_title":"Yuan, and C.-H","cited_arxiv_id":null,"evidence_quote":"Standard statistically optimal noisy tensor completion results used as benchmark for conditions and rates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Generalized CP decomposition used as the GCP baseline in simulations."}],"review_version":1}