{"id":"f297ca96-ce24-45dd-af9e-2bfd6189b44d","arxiv_id":"2606.30173","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces a new quotient manifold and compatible retractions for TT-format tensors, then applies Riemannian GD and CG to the completion problem with a claimed reduction in projection cost.","lead":"This paper constructs a quotient manifold from the left-orthogonal TT decomposition and develops Riemannian gradient descent and conjugate gradient algorithms for low-rank tensor completion. A generalist might read it for potential efficiency gains when completing missing entries in high-dimensional data arrays used in machine learning and imaging.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Compatibility of recursive polar/QR retractions with quotient structure is required for the claimed reduction in horizontal projection unknowns","rationale":"The reader's weakest_assumption directly identifies the prerequisite for the strongest_claim. Full-text inspection reveals no internal contradiction or missing step that would invalidate the construction, so the load-bearing point remains the same and the UNVERDICTED status is unaffected.","tokens_in":1660,"tokens_out":298,"duration_ms":28271,"concrete_test":"In the section deriving the horizontal projection (likely the Riemannian gradient or CG step), extract the linear system solved for the projection coefficients on a single TT core; count its dimension and confirm it is O(r) rather than O(r^2) per block when the left-orthogonal gauge and new retraction are used.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim asserts that the left-orthogonal TT construction yields a quotient manifold whose horizontal projection reduces unknowns per block from quadratic in the TT-ranks to near-half scaling. This reduction is possible only if the family of Riemannian metrics and the two novel retractions (recursive polar and recursive QR) are compatible with the quotient, i.e., the retractions map horizontal vectors to horizontal vectors and the vertical space remains the tangent to the fiber under the chosen gauge. The reader's weakest assumption isolates exactly this compatibility; without it the projection step reverts to a dense quadratic system and the efficiency claim fails.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper constructs a quotient manifold from left-orthogonal TT cores, equips it with a family of Riemannian metrics, defines two novel retractions (recursive polar and recursive QR) claimed to be compatible with the quotient structure, and applies Riemannian gradient descent and conjugate gradient to the tensor completion problem. It asserts that this framework reduces the number of unknowns in the horizontal projection step from quadratic dependence on the TT-ranks to near-half scaling, while numerical tests show reconstruction accuracy comparable to existing TT geometric methods.","tokens_in":1789,"tokens_out":505,"duration_ms":29275,"significance":"If the retraction compatibility and resulting projection reduction are rigorously established, the work could improve the per-iteration cost of quotient-based TT optimization for high-order tensors. The approach builds directly on standard TT orthogonalization and Riemannian quotient geometry without introducing free parameters or circular derivations.","major_comments":[{"comment":"§4 (Retractions): The central efficiency claim requires that the recursive polar and QR retractions map horizontal vectors to horizontal vectors and preserve the vertical space as the tangent to the fiber. The manuscript states compatibility but supplies no explicit verification (e.g., no lemma showing that the output of the recursive polar step lies in the horizontal space defined by the left-orthogonal gauge). Without this, the horizontal projection reverts to a dense quadratic system and the claimed reduction in unknowns per block cannot hold.","section":"§4"},{"comment":"§3.2 (Horizontal projection): The abstract asserts a reduction 'from a quadratic dependence on the TT-ranks to a near-half scaling.' The derivation of the new projection operator (or its complexity count) must be shown explicitly; the current text does not display the before-and-after flop counts or the block-wise linear system that is being solved.","section":"§3.2"}],"minor_comments":[{"comment":"Notation for the family of metrics is introduced without an explicit formula for the inner product on the horizontal space; adding the expression would clarify how the metrics differ from the standard quotient metric.","section":"§3.1"},{"comment":"The numerical section reports accuracy but does not tabulate iteration counts or wall-clock times against the baseline quotient methods, making the efficiency claim difficult to assess quantitatively.","section":"§5"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the thorough review and constructive feedback on our work constructing a quotient manifold for left-orthogonal TT tensors and applying it to completion via Riemannian optimization. We address each major comment below and will incorporate revisions to strengthen the explicit verification and derivations as requested.","responses":[{"response":"We agree that an explicit verification of compatibility is needed to rigorously support the efficiency claim. The recursive polar and QR retractions are constructed to respect the left-orthogonal gauge by design (via the recursive orthogonalization structure of TT), but the manuscript presents this implicitly through the definitions rather than via a dedicated lemma. In the revision we will add Lemma 4.3 (or equivalent) proving that the output of each recursive step lies in the horizontal space: specifically, that the updated cores satisfy the defining orthogonality conditions for the horizontal complement to the vertical (fiber) tangent space. This will confirm that the subsequent horizontal projection operates only on the reduced block structure.","revision_made":"yes","referee_comment":"[§4] §4 (Retractions): The central efficiency claim requires that the recursive polar and QR retractions map horizontal vectors to horizontal vectors and preserve the vertical space as the tangent to the fiber. The manuscript states compatibility but supplies no explicit verification (e.g., no lemma showing that the output of the recursive polar step lies in the horizontal space defined by the left-orthogonal gauge). Without this, the horizontal projection reverts to a dense quadratic system and the claimed reduction in unknowns per block cannot hold."},{"response":"We accept that the complexity reduction must be derived explicitly rather than asserted. The current text describes the reduced system size per block but omits the full before-and-after flop-count comparison and the explicit block-wise linear system. In the revision we will expand §3.2 (and add a short appendix if needed) to display: (i) the standard quotient projection as a dense quadratic system in the TT-ranks, (ii) our gauge-reduced block-wise system whose unknowns scale as approximately half the quadratic term, and (iii) the resulting flop counts for each block. This will make the claimed near-half scaling fully transparent.","revision_made":"yes","referee_comment":"[§3.2] §3.2 (Horizontal projection): The abstract asserts a reduction 'from a quadratic dependence on the TT-ranks to a near-half scaling.' The derivation of the new projection operator (or its complexity count) must be shown explicitly; the current text does not display the before-and-after flop counts or the block-wise linear system that is being solved."}],"tokens_in":1341,"tokens_out":556,"duration_ms":18355,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core new piece is the quotient manifold built from the left-orthogonal TT cores, together with a family of admissible metrics and two retractions (recursive polar and recursive QR) that are meant to stay inside the quotient structure. That is a concrete step beyond standard TT Riemannian work.\n\nThe construction itself is laid out clearly enough in the abstract and the stress-test note isolates the exact point that matters: the horizontal projection simplifies only if the retractions send horizontal vectors to horizontal vectors and leave the vertical space as the tangent to the fiber. If that holds, the reduction from quadratic to near-half unknowns per block is real; if not, the projection step stays as expensive as before. The paper states the claim but supplies no derivation or explicit projection formula in the text we have, so the efficiency gain cannot be verified yet.\n\nThe numerical section is described only as “comparable accuracy,” with no tables or setup details visible. That is a minor gap for an algorithmic paper but still leaves the practical payoff untested.\n\nThe work sits squarely inside the existing literature on Riemannian tensor completion. Readers already running TT-based geometric methods would be the natural audience; they could check the retractions and run their own timing tests. The framework is coherent on its own terms and engages the right prior results, so it deserves a serious referee even if the projection claim needs tightening.","headline":"The paper gives a new quotient manifold for left-orthogonal TT plus recursive polar/QR retractions, but the claimed halving of projection cost rests on unshown compatibility between those retractions and the horizontal space.","tokens_in":2248,"tokens_out":363,"would_cite":false,"duration_ms":21259,"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":"A quotient manifold from left-orthogonal tensor train cores yields Riemannian methods that cut the number of unknowns in horizontal projections from quadratic to near-linear scaling in the ranks.","keywords":["tensor train decomposition","low-rank tensor completion","Riemannian optimization","quotient manifold","horizontal projection","polar decomposition","QR decomposition"],"falsifier":"A side-by-side count of the number of free parameters appearing in the horizontal projection step for the new quotient construction versus a standard quotient construction, performed on identical TT ranks and tensor order.","tokens_in":2581,"feed_emoji":"","tokens_out":678,"duration_ms":19171,"temperature":0.7,"pith_summary":"The paper develops Riemannian optimization algorithms for completing low-rank tensors represented in tensor train format. It builds a quotient manifold from the left-orthogonal TT cores, equips it with a family of Riemannian metrics, and constructs retractions via recursive polar and QR factorizations that preserve the quotient structure. Gradient descent and conjugate gradient iterations are then derived on this geometry. The resulting horizontal projections require only near-half the unknowns per block compared with earlier quotient approaches, while numerical tests show reconstruction accuracy remains comparable to existing TT geometric solvers.","feed_headline":"Quotient manifold halves unknowns in TT tensor completion","feed_subtitle":"Left-orthogonal cores define a geometry whose projections scale linearly rather than quadratically with rank, yielding faster Riemannian sol","key_machinery":"The quotient manifold induced by the left-orthogonal TT cores, together with retractions obtained from recursive polar and QR decompositions that respect the recursive orthogonalization of the TT format.","core_discovery":"Leveraging the left-orthogonal property of the TT cores produces a quotient manifold that admits admissible Riemannian metrics together with retractions realized by recursive polar and QR decompositions; the associated Riemannian gradient descent and conjugate gradient algorithms therefore streamline the horizontal projection step by reducing the number of unknowns per block from a quadratic dependence on the TT-ranks to a near-half scaling.","pith_inferences":["The same left-orthogonal quotient construction could be reused for other inverse problems on TT tensors, such as denoising or regression, without redesigning the manifold.","If the efficiency gain scales with tensor order, the method may become practical for tensors whose order exceeds the range tested in the paper.","The recursive polar and QR retractions might admit closed-form expressions for their differentials, which would further accelerate higher-order optimization schemes."],"forward_implications":["Riemannian gradient descent and conjugate gradient algorithms become available on the new geometry and inherit the reduced projection cost.","The computational cost per iteration drops because each block of the horizontal projection now depends linearly rather than quadratically on the TT ranks.","Reconstruction accuracy stays comparable to existing state-of-the-art TT-based geometric completion methods.","The same retraction and metric constructions apply to any tensor whose TT cores satisfy the left-orthogonal condition."],"fun_headline_variants":["TT quotient manifold halves unknowns in tensor completion","Left-orthogonal cores yield TT quotient Riemannian geometry","Riemannian descent on TT quotient simplifies projections","Quotient retractions via polar QR for TT completion"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The left-orthogonal property of the TT cores can be used to define a quotient manifold that admits a family of Riemannian metrics together with retractions realized by recursive polar and QR decompositions that remain compatible with the quotient structure.","fun_headline_variants_meta":{"raw":{"variants":["TT quotient manifold halves unknowns in tensor completion","Left-orthogonal cores yield TT quotient Riemannian geometry","Riemannian descent on TT quotient simplifies projections","Quotient retractions via polar QR for TT completion"]},"model":"grok-4.3","cost_usd":0.006295,"raw_usage":{"total_tokens":2852,"prompt_tokens":614,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":62953000,"prompt_tokens_details":{"text_tokens":614,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2180,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":614,"tokens_out":58,"duration_ms":20640,"temperature":1.0,"reasoning_tokens":2180,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T05:14:58.132046+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A side-by-side count of the number of free parameters appearing in the horizontal projection step for the new quotient construction versus a standard quotient construction, performed on identical TT ranks and tensor order.","supporting_citations":[],"review_version":1}