{"id":"7d2c3440-fb51-482a-94ed-919310e6317f","arxiv_id":"2606.26688","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extends matrix adaptive randomized pivoting to tensor CUR approximations in the t-product framework, with direct bounds for one variant and alignment-conditioned bounds for the common-slice variant.","lead":"This paper extends adaptive randomized pivoting from matrices to tensors in the t-product framework by proposing ARP-T-CUR (slicewise in Fourier domain) and T-ARP (common slices across the tensor) with expected error bounds. A smart generalist might read it to see how randomized column selection techniques scale to multi-way data like images and videos.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"T-ARP bound requires frequency-alignment condition whose restrictiveness is not quantified beyond the aligned case","rationale":"The reader's weakest_assumption matches the explicit extra hypothesis needed for the T-ARP result; the full-text description confirms that the coupling difficulty is handled precisely by introducing this condition rather than removing it. No other internal inconsistency appears from the abstract-level claims.","tokens_in":1817,"tokens_out":341,"duration_ms":23593,"concrete_test":"Extract the precise statement of the frequency-alignment condition (likely in the theorem for T-ARP) and the definition of the alignment metric; then recompute the metric on the synthetic tensors used in the experiments and report its value. If the metric exceeds the regime where the bound reduces to the r+1 factor by more than a small constant, the headline T-ARP guarantee does not apply to the reported numerical cases.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim for the second construction (T-ARP) is an expected-error bound that holds only when the common tensor-level sampling indices are sufficiently close to the per-slice ARP choices; this is formalized as a frequency-alignment condition on leverage-score distributions across Fourier slices. The abstract states that the bound recovers the standard r+1 factor precisely when those distributions are aligned, but supplies no further control on the degradation when they differ. Because the coupling of indices across slices is the defining feature of T-ARP, any gap between the stated condition and the actual sampling behavior directly limits the applicability of the claimed guarantee. ARP-T-CUR avoids this issue by inheriting the matrix bound slice-wise without coupling.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper extends adaptive randomized pivoting (ARP) to tensor CUR approximations in the t-product framework. It proposes ARP-T-CUR, which applies matrix ARP independently to frontal slices in the Fourier domain and inherits expected-error bounds directly from matrix theory, and T-ARP, which enforces common lateral/horizontal slice selection across the tensor and derives an expected-error bound under an explicit frequency-alignment condition on leverage-score distributions; the bound recovers the standard r+1 factor when distributions align. The work also relates the resulting approximation to t-DEIM and reports experiments on synthetic tensors, images, and videos.","tokens_in":1966,"tokens_out":424,"duration_ms":20676,"significance":"If the central claims hold, the work supplies the first ARP-style guarantees for tensor cross approximations with shared indices, which is relevant for applications requiring consistent sampling across modes such as video compression. ARP-T-CUR inherits its bound without additional assumptions, while T-ARP's conditional bound makes the coupling explicit; the experiments provide empirical support for the practical advantage of common-index sampling.","major_comments":[{"comment":"The frequency-alignment condition is load-bearing for the T-ARP expected-error bound (stated in the abstract and developed in the T-ARP section). The bound recovers the usual r+1 factor precisely when leverage-score distributions align across Fourier slices, but the manuscript supplies no further quantitative control on the degradation factor when the distributions differ; because common-index selection is the defining feature of T-ARP, this gap limits the scope of the claimed guarantee.","section":null}],"minor_comments":[{"comment":"Notation for the t-product and Fourier slices is introduced without a self-contained summary table; adding one would improve readability for readers outside the immediate t-product literature.","section":null},{"comment":"The experimental section reports results on images and videos but does not specify how the alignment condition was monitored or estimated in those runs; a brief diagnostic would clarify whether the observed performance occurs inside or outside the regime where the bound is tight.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for highlighting the role of the frequency-alignment condition in the T-ARP analysis. We address the major comment below.","responses":[{"response":"We agree that the frequency-alignment condition is central to the T-ARP bound and that the manuscript provides no quantitative control on the degradation factor when leverage-score distributions differ across Fourier slices. A uniform bound independent of alignment would require additional structural assumptions on the tensor that do not hold in general, as common-index selection can be arbitrarily misaligned with individual slices. The explicit condition is presented precisely to make this coupling transparent and to recover the classical factor when alignment holds. In the revision we will add a dedicated remark clarifying this limitation, together with synthetic examples that illustrate the observed degradation under controlled misalignment. This will better delineate the scope of the T-ARP guarantee while preserving the paper's focus on making the cost of common indices explicit.","revision_made":"yes","referee_comment":"The frequency-alignment condition is load-bearing for the T-ARP expected-error bound (stated in the abstract and developed in the T-ARP section). The bound recovers the usual r+1 factor precisely when leverage-score distributions align across Fourier slices, but the manuscript supplies no further quantitative control on the degradation factor when the distributions differ; because common-index selection is the defining feature of T-ARP, this gap limits the scope of the claimed guarantee."}],"tokens_in":1338,"tokens_out":313,"duration_ms":28732,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper takes adaptive randomized pivoting, which was developed for matrix column selection, and moves it into the t-product tensor setting. ARP-T-CUR applies the matrix routine independently to each Fourier frontal slice and inherits the existing expected-error bound without change. T-ARP instead forces the same lateral and horizontal slices across the entire tensor, which produces a genuine tensor-level cross approximation, and the authors derive a bound that holds when the leverage-score distributions satisfy a frequency-alignment condition.\n\nThe explicit statement of that coupling and the recovery of the usual r+1 factor under perfect alignment are the concrete additions. The link to t-DEIM is noted, and the experiments on synthetic tensors, images, and videos give a basic check that common-index sampling can outperform standard tensor baselines in those cases.\n\nThe soft spot is the alignment condition itself. The abstract defines it as the distance between the common tensor sampling rule and the per-slice ARP choices, yet supplies no further analysis of how the error factor behaves when the distributions differ across frequencies. Because the whole point of T-ARP is the shared indices, any gap between the stated condition and real data directly limits how often the claimed guarantee applies. ARP-T-CUR sidesteps the issue by staying slicewise.\n\nThe work sits in numerical linear algebra for tensors. Readers who already use CUR or randomized column selection on tensors will see the two constructions and the experiments as useful reference points. The derivations are straightforward lifts from the matrix case plus one new condition, so the technical foundation is solid enough to warrant referee time.\n\nSend it to review. The extension is honest and the experiments are relevant; a referee can check the condition's restrictiveness and the practical tightness of the bounds.","headline":"Extends matrix ARP to t-product tensors via slicewise ARP-T-CUR and coupled T-ARP, but the latter's bound rests on a frequency-alignment condition whose practical range is not quantified.","tokens_in":2493,"tokens_out":433,"would_cite":false,"duration_ms":26639,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Adaptive randomized pivoting extends to tensors with two constructions: slicewise ARP-T-CUR inheriting matrix bounds and coupled T-ARP under a frequency-alignment condition.","keywords":["tensor CUR approximation","adaptive randomized pivoting","t-product","Fourier domain sampling","cross approximation","expected error bound","frequency alignment condition"],"falsifier":"Construct a tensor whose leverage-score distributions differ markedly across Fourier frequencies, run T-ARP, and check whether the observed approximation error exceeds the bound by more than the factor predicted by the measured alignment distance.","tokens_in":2704,"feed_emoji":"📐","tokens_out":751,"duration_ms":31291,"temperature":0.7,"pith_summary":"The paper extends adaptive randomized pivoting from matrices to tensors in the t-product framework. ARP-T-CUR applies the matrix method independently to each Fourier frontal slice, yielding a slicewise CUR with an expected-error bound taken directly from matrix results. T-ARP instead selects the same lateral and horizontal slices for the entire tensor, producing a true tensor cross approximation, but requires an explicit frequency-alignment condition on the sampling rules to obtain a comparable error bound. When leverage-score distributions match across frequencies, the bound recovers the familiar r+1 factor; numerical tests on synthetic data, images, and videos compare the methods to standard tensor cross baselines.","feed_headline":"Tensor ARP couples slices across frequencies for cross approximation","feed_subtitle":"T-ARP enforces common indices and bounds error by how aligned the per-frequency leverage scores remain.","key_machinery":"The frequency-alignment condition measuring deviation between a common tensor-level sampling rule and the individual Fourier-slice ARP rules, which controls the error bound for the coupled T-ARP construction.","core_discovery":"Adaptive randomized pivoting extends to the tensor setting by either applying it independently per Fourier slice (ARP-T-CUR) or by enforcing common slice indices across all slices (T-ARP); the latter requires a frequency-alignment condition that quantifies how far the shared sampling rule deviates from the per-slice rules, and the condition yields an expected-error bound that reduces to the standard matrix factor when the leverage scores are aligned.","pith_inferences":["If the alignment condition holds for many real-world tensors such as video data, T-ARP would give structurally coherent approximations at modest extra cost.","The common-index requirement could be relaxed by allowing limited frequency-dependent adjustments while still preserving a tensor-level CUR form.","The same alignment idea might apply to other tensor factorizations that mix Fourier and spatial sampling.","Direct comparison of storage and runtime between ARP-T-CUR and T-ARP on large tensors would quantify the practical price of enforcing common indices."],"forward_implications":["ARP-T-CUR delivers a Fourier-slicewise CUR whose expected error is bounded exactly as in the matrix ARP theory.","T-ARP produces a genuine tensor cross approximation using the same lateral and horizontal slices throughout the tensor.","The expected-error bound for T-ARP holds once the frequency-alignment condition is satisfied and recovers the r+1 factor when leverage scores align.","The resulting tensor cross approximation connects directly to t-DEIM.","Experiments on images and videos indicate that common-index sampling improves performance over independent-slice baselines."],"fun_headline_variants":["T-ARP samples common indices across tensor frequencies","ARP-T-CUR pivots each Fourier slice independently","Tensor cross approx error bounded by frequency leverage alignment","Adaptive randomized pivoting extends to t-product tensors","Frequency alignment condition for T-ARP error bounds"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The frequency-alignment condition must hold so that the common sampling rule stays close enough to the slice-wise rules; if the distributions of leverage scores differ strongly across frequencies the stated T-ARP bound no longer applies.","fun_headline_variants_meta":{"raw":{"variants":["T-ARP samples common indices across tensor frequencies","ARP-T-CUR pivots each Fourier slice independently","Tensor cross approx error bounded by frequency leverage alignment","Adaptive randomized pivoting extends to t-product tensors","Frequency alignment condition for T-ARP error bounds"]},"model":"grok-4.3","cost_usd":0.005055,"raw_usage":{"total_tokens":2461,"prompt_tokens":663,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":50549500,"prompt_tokens_details":{"text_tokens":663,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1728,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":663,"tokens_out":70,"duration_ms":17593,"temperature":1.0,"reasoning_tokens":1728,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T04:37:17.243555+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Construct a tensor whose leverage-score distributions differ markedly across Fourier frequencies, run T-ARP, and check whether the observed approximation error exceeds the bound by more than the factor predicted by the measured alignment distance.","supporting_citations":[],"review_version":1}