{"id":"ed4bdccd-bd60-463f-900b-fca15d86bd44","arxiv_id":"2412.17281","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Low-tubal-rank tensors can be recovered from independent compressed measurements of each lateral slice using alternating minimization with spectral initialization, and a preconditioned variant achieves condition-number-independent iteration complexity.","lead":"This paper introduces a tensor compressed sensing model in which each lateral slice (e.g., each video frame) is measured independently, and gives two alternating-minimization algorithms with recovery guarantees for low-tubal-rank tensors. A preconditioned variant is claimed to converge very quickly, independent of the tensor condition number, which matters for streaming or distributed sensing of difficult data.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Initialization-to-contraction gap: Theorem 3 cannot provide the warm start that Lemma 4 and Theorem 5 require for large κ, so the κ-independent iteration complexity is unproven as stated.","rationale":"I agree with the reader's weakest_assumption. The paper's central advertised contribution is recovery for ill-conditioned tensors with κ-independent iteration complexity. That claim is Theorem 5 (with Corollaries 4–5) and depends directly on Lemma 4 fact (5) and the contraction proof through Eqs. (67) and (78). The only initialization guarantee (Theorem 3) degrades with κ, and the two bounds are incompatible for κ above a constant: substituting Eq. (18) into the requirement behind Eq. (67) yields r·κ^3 ≲ 10. Since no such condition is stated, the theorem's scope is larger than what the proof establishes. This is a genuine gap in the argument, not merely a disagreement with prior work. The paper does have independent value: the local-slice measurement model is a natural tensor analogue of column-wise matrix sensing, the algorithms are clearly specified, and the synthetic/video/MRI experiments show the preconditioned variant working well. Those do not, however, close the proof gap on the κ-independence claim. The exact minimization step in Eq. (23) additionally needs m_c ≥ r·n3 for H_{t,i}H_{t,i}^c to be invertible; this condition is absent from Corollaries 2–5, but it is a more easily repaired technical omission than the contraction gap. I therefore do not move the verdict: CONDITIONAL remains the right call, with the requested revisions being (i) state and prove the warm-start condition needed by Lemma 4, or add a κ restriction to Theorems 4–5, and (ii) state the m_c ≥ r·n3 assumption. If the authors can prove Lemma 4 under a weaker condition implied by Theorem 3, the paper's main claim would stand.","tokens_in":34543,"tokens_out":6878,"duration_ms":59505,"concrete_test":"Re-derive the contraction step in Theorems 4 and 5, replacing the generic condition 'Dis(U_t,U*) ≤ 0.016√r·κ^2' with the actual bound used in Eq. (67). Check whether 0.36·Dis·√r·||X*|| ≤ 0.06·σ*_min holds when Dis = 0.016√r·κ^2; for r=1 and κ=3, the left side equals 0.36·0.144·1·3·σ*_min ≈ 0.156·σ*_min, which is larger than 0.06·σ*_min, so the displayed lower bound fails. Then either (a) prove Lemma 4 fact (5) under a weaker sufficient condition that is actually implied by Theorem 3, or (b) add an explicit κ restriction (e.g., r·κ^3 ≤ 10) to Theorems 4–5 and Corollaries 2–5. If neither can be done, the κ-independent iteration complexity claim must be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline claim—κ-independent iteration complexity for Alt-ScalePGD-Min (Theorem 5, Corollary 4) and the κ^2·log(1/ε) rate for Alt-PGD-Min—rests on the contraction proofs in Theorems 4 and 5, which require the principal-angle distance Dis(U_t,U*) to be small enough that Lemma 4 fact (5) gives σ_min(V_t) ≥ 0.94 σ*_min. The proof of fact (5) needs the perturbation term 0.36·Dis·√r·||X*|| (from fact (3)) to be at most 0.06 σ*_min to reach 0.94 σ*_min; see Eq. (67). Since ||X*|| = σ_max = κ·σ_min, this requires Dis ≲ 1/(6√r·κ). The only initialization guarantee proved (Theorem 3, Eq. 18) is Dis(U0,U*) ≤ 0.016√r·κ^2. For the warm start to satisfy the contraction requirement one needs 0.016√r·κ^2 ≲ 1/(6√r·κ), i.e., r·κ^3 ≲ 10. No such κ restriction appears in Theorems 4–5 or Corollaries 2–5. For larger κ—precisely the ill-conditioned regime the preconditioned method targets—the initialization bound is vacuous (it can exceed 1) and, more importantly, the displayed lower bound in Eq. (67) does not imply σ_min(V_t) ≥ 0.94 σ*_min. The proofs of Theorems 4 and 5 then lack a valid contraction step, so the κ-independent rate is not established. This is a proof gap, not a contradiction; the experiments appear to support the method, but the theoretical claim as stated is incomplete.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a tensor compressed sensing model in which each lateral slice of a low-tubal-rank tensor is sensed by an independent local measurement matrix, and formulates recovery as a nonconvex factored problem over two tensor factors. Two alternating-minimization algorithms are proposed: Alt-PGD-Min, which alternates a projected gradient step on the orthogonal factor with an exact least-squares update on the other factor, and Alt-ScalePGD-Min, which preconditions the gradient step by an approximate Hessian. The paper proves, under Gaussian local measurements and a tensor incoherence condition, that a truncated spectral initialization provides a warm start and that both algorithms converge linearly with sample complexity stated in terms of the tensor condition number kappa; the headline claim is that Alt-ScalePGD-Min has iteration complexity O(log(1/epsilon)) independent of kappa. Numerical experiments on synthetic tensors, video, and MRI data support the practical efficacy of the proposed methods.","tokens_in":34828,"tokens_out":9515,"duration_ms":88611,"significance":"If the theoretical claims were fully established, this would be a valuable contribution: the local slice-wise sensing model is new for tensors, the sample-splitting and truncation analysis is carefully designed, and the constants in the main bounds are fixed universal numbers rather than fitted parameters. The preconditioned method's claimed kappa-independent iteration complexity would be a substantial improvement over the kappa^2 dependence of unaccelerated factoring methods and over the sub-linear rates of convex tensor nuclear norm approaches. The experimental section is reasonably extensive and includes comparisons with the closest column-wise matrix sensing method. However, the central contraction proof contains a load-bearing smallness gap: the initialization bound proved in Theorem 3 is not strong enough to enter the contraction regime required by Lemma 4 and the proofs of Theorems 4 and 5 for large kappa, which is precisely the regime the preconditioned algorithm is advertised to address. The core idea is defensible and likely repairable, but the stated kappa-independent result is not currently proven.","major_comments":[{"comment":"The displayed lower bound for sigma_min(V_t) subtracts the perturbation 0.36*Dis(U_t,U*)*sqrt(r)*||X*|| from sigma*_min*sqrt(1-Dis^2(U_t,U*)). Since ||X*|| = kappa*sigma*_min, concluding sigma_min(V_t) >= 0.94*sigma*_min requires Dis(U_t,U*) to be at most of order 1/(6*sqrt(r)*kappa). The only initialization guarantee proved is Dis(U_0,U*) <= 0.016*sqrt(r)*kappa^2 (Theorem 3, Eq. (18)), which can satisfy that smallness requirement only when r*kappa^3 is bounded by an absolute constant. No such restriction is stated in Theorem 4, Theorem 5, or Corollaries 2-5; for the ill-conditioned regime emphasized by the paper the initialization bound is vacuous, since 0.016*sqrt(r)*kappa^2 can exceed 1. Consequently the contraction steps at Eq. (79) and Eq. (88) are not justified as stated, and the kappa-independent iteration complexity of Theorem 5 and Corollary 4 is not established.","section":"Proof of Theorem 4; Lemma 4; Eq. (67)"},{"comment":"The exact V-step computes v_{t,i} = (H_{t,i}H_{t,i}^c)^{-1} H_{t,i} y_i, which requires the matrix H_{t,i}H_{t,i}^c in R^{r*n3 x r*n3} to be nonsingular. This can hold only when the number of measurements per lateral slice satisfies m_c >= r*n3. The sample-size conditions in Corollary 2, Eq. (28), and in Corollaries 3-5 only impose m_c >= max{log n2, r log n3} together with a lower bound on m_c*n2; for large n2 this permits m_c to be much smaller than r*n3. The proof of Lemma 3 uses the same invertibility and the spectral lower bound sigma_min(Q_{t,i}Q_{t,i}^c) >= 0.86*m_c, so the per-slice identifiability condition m_c >= r*n3 is a missing necessary condition in the stated sample-complexity guarantees.","section":"Alt-PGD-Min Algorithm; Eq. (23); Corollaries 2-5"},{"comment":"Step (d) of Eq. (79) estimates 1.41*eta*m*sqrt(r)*||X*||^2*Dis(U_t,U*) by 0.03, but substituting eta = 0.9/(m||X*||^2) and Dis <= 0.016*sqrt(r)*kappa^2 gives 1.41*0.9*0.016*r*kappa^2 = 0.0203*r*kappa^2, which is not an absolute constant. The subsequent inequality (1-x)(1+2x) >= 1 therefore requires r*kappa^2 to be small. In Theorem 5, Eq. (88), the analogous term is 1.6*0.9*sqrt(r)*Dis*kappa^2 <= 0.023*r*kappa^4, which is not bounded for large kappa. This is an independent manifestation of the same missing smallness condition and, by itself, invalidates the displayed contraction unless an explicit bound on r*kappa^2 or r*kappa^4 is imposed.","section":"Proof of Theorem 4; Eq. (79) and Eq. (88)"}],"minor_comments":[{"comment":"The main text refers to Figures 6-9 for the synthetic and video results, but the displayed figures in the manuscript are labeled Figures 1-4; the numbering should be made consistent.","section":"Figures throughout"},{"comment":"The perturbation term is printed as 0.36*Dis^2(U_t,U*)*sqrt(r)*||X*||, while Lemma 3 fact (3) gives 0.36*Dis(U_t,U*)*sqrt(r)*||X*||; the square on Dis appears to be a typo and should be corrected.","section":"Eq. (67)"},{"comment":"The step size is stated as eta = 0.8/(m||X*||), but Theorem 4 and the gradient normalization require eta = 0.8/(m||X*||^2); the missing square changes the dimensional consistency of the step size.","section":"Corollary 2"},{"comment":"The success probability is stated as 1 - 1/(n1+n3)^10, while Theorem 3 contains the term c1(n1+n2) log n3; the indices n2 and n3 should be made consistent.","section":"Corollary 1"},{"comment":"The table reports the Alt-ScalePGD-Min sample complexity with an inner factor kappa^2*r*(n1+n2), whereas Corollary 5 and the abstract use kappa^4*r*(n1+n2); one of these is a typo.","section":"Supplementary Table 1"},{"comment":"Lemma 13 cites (Kumar et al. 2022), a paper on fine-tuning pretrained features, for the standard inequality sigma_min(AB) >= sigma_min(A)*sigma_min(B); a standard linear algebra reference would be more appropriate.","section":"Lemma 13 and references"}],"recommendation":"major_revision","confidential_remarks":"The paper has a genuinely novel sensing model and the experimental work is convincing that the algorithms are practically useful. The main theoretical claim, however, is not proven as stated: the warm start obtained from Theorem 3 does not enter the contraction regime required by Lemmas 4-6 for the large-kappa case that motivates Alt-ScalePGD-Min. This is a repair that may be possible, for example by proving a stronger initialization bound with higher sample complexity or by stating an explicit bounded-kappa regime, but it is load-bearing and cannot be left as a local fix in the exposition. I also note the missing m_c >= r*n3 condition for the exact least-squares V step. I would recommend major revision rather than rejection, because the core approach is sound and the issues, while central, are identifiable and potentially fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on Wu, Sun, Fan (arXiv:2412.17281). The interesting part is the model: recovering a low-tubal-rank tensor from independent Gaussian projections of each lateral slice. That's a sensible tensor analogue of the column-wise matrix sensing work by Vaswani and Nayer, and it is genuinely new as far as the cited literature goes. The two alternating algorithms are natural extensions, the preconditioned variant is a good idea, and the experiments (synthetic, video, MRI) show the method working, with the scaled version clearly faster on ill-conditioned data.\n\nThe proofs are serious and mostly self-contained, and the paper does a good job adapting the matrix proof skeleton to the t-product algebra. But there is a load-bearing gap in the theory as stated. Theorem 3 guarantees Dis(U0,U*) ≤ 0.016√r κ². Lemma 4's fact (5) — the lower bound on σ_min(V_t) — needs the perturbation term 0.36 Dis √r ||X*|| to be below about 0.06 σ*_min. Since ||X*|| = κ σ*_min, that requires Dis ≲ 1/(6√r κ). The initialization bound only satisfies this when r κ³ ≲ 10, and no such restriction appears in Theorems 4–5 or the corollaries. So the claimed κ-independent iteration complexity for Alt-ScalePGD-Min and the κ² log(1/ε) rate for Alt-PGD-Min are not established as stated for the ill-conditioned regime the paper advertises. This is a proof gap, not a contradiction; the experiments look consistent with the claims, but the theorem statements need a κ restriction or a sharper initialization.\n\nThere is also a second, smaller gap: the exact V-minimization step inverts an (r n3)×(r n3) matrix, which requires m_c ≥ r n3 per slice, but the corollaries' sample conditions don't include this. The supplementary complexity table has a mismatch in the κ power for Alt-ScalePGD-Min (κ^2 inside vs κ^4 in the main text), one lemma is attributed to a machine learning fine-tuning paper rather than a matrix analysis source, and no code is released. These are minor and fixable.\n\nOverall: the local TCS model is a real contribution, the algorithm design is sensible, and the experiments give genuine support. The proof gap in the central theorem is the main issue. I'd send this to peer review, with the expectation of major revision. The authors should either prove a κ-robust initialization, add the missing κ restriction to all theorem statements, or weaken the claims. For a reader in tensor sensing or non-convex recovery, it's worth engaging with. I would not cite the κ-independent rate as proven until the gap is closed; I would cite the measurement model.","headline":"Useful new local tensor sensing model, but the headline κ-independent convergence is not proven as stated because the Theorem 3 warm start is too weak for the contraction lemma unless r κ³ is bounded.","tokens_in":35470,"tokens_out":6293,"would_cite":true,"duration_ms":52899,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A69","90C26","94A12"],"pacs":[],"model":"deepseek-v4-flash","headline":"A low-tubal-rank tensor can be recovered from independent compressive measurements of each lateral slice, and a preconditioned alternating-minimization method reaches epsilon-accurate recovery in O(log 1/epsilon) iterations independent of…","keywords":["tensor compressed sensing","low tubal rank","local measurements","non-convex optimization","alternating minimization","preconditioned gradient descent","sample complexity","t-SVD"],"falsifier":"Take a synthetic tensor with $r=4$ and $\\kappa=10$ and initialize by the proved spectral bound: $\\mathrm{Dis}(U_0,U_\\star)\\le 0.016\\sqrt{4}\\,10^2=3.2$, which is larger than the small constant required by the contraction step (67); running Alt-ScalePGD-Min on such a tensor and checking whether the $O(\\log 1/\\epsilon)$ iteration bound actually holds would settle whether the condition-number-free claim is valid as stated.","tokens_in":34140,"feed_emoji":"🎥","tokens_out":13930,"duration_ms":115477,"temperature":0.7,"pith_summary":"The paper sets out to prove that a low-tubal-rank tensor can be recovered from local measurements, meaning each lateral slice is sensed by its own independent random matrix rather than the whole tensor being projected as one unit. It formulates recovery as a nonconvex factored problem $\\mathcal{X}^\\star = \\mathcal{U} * \\mathcal{V}$ and gives two alternating algorithms: Alt-PGD-Min and a preconditioned version Alt-ScalePGD-Min. The headline result is that Alt-ScalePGD-Min reaches $\\epsilon$-accurate recovery in $O(\\log 1/\\epsilon)$ iterations regardless of the tensor condition number $\\kappa$, with quantified per-slice sample complexity. This matters for distributed, streaming, video, and MRI settings, where the data arrive slice by slice and whole-tensor sensing is infeasible.","feed_headline":"Slice-wise sensing plus preconditioning recovers low-rank tensors fast","feed_subtitle":"Condition-number-free iteration counts make ill-conditioned video and MRI recovery practical.","key_machinery":"The load-bearing machinery is the low-tubal-rank factorization $X = U * V$ with $U^c * U = I_r$ under the t-product, in which tensor multiplication is block-circulant convolution and becomes elementwise matrix products in the Fourier domain. The algorithm alternates an exact least-squares solve for each $V$ slice (a closed form via $H_{t,i}H_{t,i}^c$ inversion) with a projected gradient step on $U$ followed by QR projection onto the orthogonal-tensor constraint. Alt-ScalePGD-Min changes only the $U$ update to $\\hat U_{t+1}=U_t-\\eta T_t*V_t^c*(V_t*V_t^c)^{-1}$; the extra inverse is an $r\\times r\\times n_3$ tensor and is cheap to compute. The proof tracks the principal-angle distance $\\mathrm{Dis}(U_t,U_\\star)$ through a Wedin-type sin $\\theta$ bound, and uses sample splitting so each iteration sees fresh independent measurements.","core_discovery":"The paper's central claim is Theorem 5 with Corollaries 4 and 5: under Assumptions 1 and 2 (tubal rank $r$, tensor incoherence, i.i.d. Gaussian local sensing matrices), the truncated spectral initialization followed by Alt-ScalePGD-Min recovers an $\\epsilon$-accurate estimate in $O(\\log 1/\\epsilon)$ iterations, with total per-slice sample complexity $O(\\kappa^4 \\mu^2 r n_3 \\log n_3 (\\kappa^4 r (n_1+n_2) + n_1 \\log 1/\\epsilon))$ with high probability. The companion result for Alt-PGD-Min gives iteration complexity $O(\\kappa^2 \\log 1/\\epsilon)$ and establishes that the local-sensing model itself is tractable: exact $V$-updates decouple across slices and a projected gradient on $U$ converges linearly once $U_0$ is near $U^\\star$. Experiments on synthetic tensors, video, and MRI report that Alt-ScalePGD-Min keeps its convergence rate when $\\kappa$ grows, while the un-preconditioned version slows down as $\\kappa^2$ predicts.","pith_inferences":["The same preconditioning idea should transfer to other tensor formats: replacing $(V*V^c)^{-1}$ by the inverse Gram tensor is cheap whenever the core size is small, so Tucker or tensor-train factorizations might inherit condition-number-free convergence.","The paper's fixed step size of 0.8 for Alt-ScalePGD-Min across all video and MRI experiments suggests the preconditioned update may be insensitive to $\\kappa$ in practice, which would make it a plug-and-play solver for low-tubal-rank inverse problems beyond the measured setting.","Removing sample splitting, which the conclusion names as future work, would eliminate the $\\log 1/\\epsilon$ term from the sample complexity; a leave-one-out analysis is the natural route, as in matrix completion.","The proof of Theorem 5 relies on a small-basin contraction condition (inequality (67)) while the proved initialization bound is $\\mathrm{Dis}(U_0,U_\\star)\\le 0.016\\sqrt{r}\\,\\kappa^2$; for large $\\kappa$ this gap needs a sharper initialization or a larger basin, so the advertised regime is most rigorous for bounded $\\kappa$."],"forward_implications":["Video and dynamic-MRI volumes can be reconstructed from independent compressed measurements of individual frames, with no need to sense or hold the whole tensor at once.","Because Alt-ScalePGD-Min's iteration complexity is $O(\\log 1/\\epsilon)$ uniformly in the condition number, ill-conditioned low-tubal-rank tensors do not require a longer run just because they are harder to condition.","The per-slice sample-complexity formula contains a shared-factor term that decreases with the number of lateral slices $n_2$, so adding more frames or slices amortizes the cost of learning the common tensor factor $U$.","The exact minimization step for $V$ is decoupled across lateral slices, so each iteration can be parallelized across slices, matching distributed or streaming sensing architectures.","When the number of lateral slices is large, the analysis shows the total per-slice sample complexity drops to roughly $O(\\kappa^4\\mu^2 r n_1 n_3\\log n_3 \\log 1/\\epsilon)$, which is close to the information-theoretic degrees of freedom up to small factors."],"supporting_citations":[{"why":"Defines the t-product, t-SVD, identity and orthogonal tensors on which the factorization $X=U*V$ and all algorithms rest.","marker":"(Kilmer and Martin 2011)"},{"why":"Provides the column-wise local-measurement recovery framework, the truncated spectral initialization fact used in Theorem 3, and the LRcCS baseline in the experiments.","marker":"(Nayer and Vaswani 2022)"},{"why":"Supplies the covering-number and sub-exponential Bernstein inequalities that control spectral norm concentration in the proofs.","marker":"(Vershynin 2018)"},{"why":"Provides the sin theta theorem used to convert spectral closeness of the initialization into a bound on the principal-angle distance.","marker":"(Wedin 1972)"},{"why":"Introduces the tensor incoherence condition, adapted as Assumption 1, that makes slice-wise low-tubal-rank recovery identifiable.","marker":"(Zhang and Aeron 2016)"},{"why":"Establishes exact low-tubal-rank tensor recovery from global Gaussian measurements and defines the tensor spectral norm; used as a convex baseline in the comparison table.","marker":"(Lu et al. 2018)"},{"why":"Supplies the sample-splitting technique that keeps measurement batches independent across iterations in the convergence analysis.","marker":"(Hardt and Wootters 2014)"},{"why":"Motivates the truncated spectral initialization with thresholding that produces the warm start in Stage I.","marker":"(Chen and Candes 2015)"}],"fun_headline_variants":["Slice-wise measurements meet preconditioned tensor recovery","Condition-number-independent tensor recovery from local data","Preconditioning accelerates non-convex tensor recovery","Local sensing plus scaling: fast tensor recovery"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The guarantees assume the spectral initialization lands inside a small contraction basin around the true factor space, and the initialization bound proved in the paper only guarantees that closeness when the condition number is small; for the large-$\\kappa$ regime the advertised condition-number-free result relies on a warm start that is not proved.","fun_headline_variants_meta":{"raw":{"variants":["Slice-wise measurements meet preconditioned tensor recovery","Condition-number-independent tensor recovery from local data","Preconditioning accelerates non-convex tensor recovery","Local sensing plus scaling: fast tensor recovery"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000434,"raw_usage":{"total_tokens":2316,"prompt_tokens":1154,"completion_tokens":1162,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":770,"completion_tokens_details":{"reasoning_tokens":1106}},"tokens_in":770,"tokens_out":1162,"duration_ms":10596,"temperature":1.0,"reasoning_tokens":1106,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:39:56.600175+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a synthetic tensor with $r=4$ and $\\kappa=10$ and initialize by the proved spectral bound: $\\mathrm{Dis}(U_0,U_\\star)\\le 0.016\\sqrt{4}\\,10^2=3.2$, which is larger than the small constant required by the contraction step (67); running Alt-ScalePGD-Min on such a tensor and checking whether the $O(\\log 1/\\epsilon)$ iteration bound actually holds would settle whether the condition-number-free claim is valid as stated.","supporting_citations":[{"cited_title":"E.; and Martin, C","cited_arxiv_id":null,"evidence_quote":"Defines the t-product, t-SVD, identity and orthogonal tensors on which the factorization $X=U*V$ and all algorithms rest."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the column-wise local-measurement recovery framework, the truncated spectral initialization fact used in Theorem 3, and the LRcCS baseline in the experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the covering-number and sub-exponential Bernstein inequalities that control spectral norm concentration in the proofs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the sin theta theorem used to convert spectral closeness of the initialization into a bound on the principal-angle distance."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the tensor incoherence condition, adapted as Assumption 1, that makes slice-wise low-tubal-rank recovery identifiable."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the sample-splitting technique that keeps measurement batches independent across iterations in the convergence analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Motivates the truncated spectral initialization with thresholding that produces the warm start in Stage I."}],"review_version":1}