{"id":"3537868e-0c47-499b-a81f-8f65e64913e7","arxiv_id":"1909.01737","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every group-invariant tensor admits an invariant decomposition on a suitably enriched weighted simplicial complex, unifying translationally invariant, symmetric, and nonnegative tensor decompositions.","lead":"This paper builds a general framework for decomposing invariant tensors into sums of elementary pieces, with indices arranged on a weighted simplicial complex and symmetry enforced by a group action. It proves such decompositions always exist after enriching the complex, unifying known tensor-network and nonnegative-matrix results.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unrestricted central claim omits the connectedness hypothesis of Theorem 13; for disconnected complexes only pure tensors are representable after weight raising.","rationale":"The reader accepted the paper with the weakest assumption being the algebraic-tensor-product restriction, which the paper states explicitly and does not undermine the theorems as written. My concern is different: the central claim as stated in the abstract and in the reader's strongest claim omits the connectedness hypothesis that appears in Theorem 13. Proposition 7 preserves the support of the complex, so it cannot fix disconnectedness by weight raising alone. The counterexample with two disconnected vertices shows that arbitrary invariant tensors of tensor rank greater than one cannot be represented after any weight raising. This does not invalidate the proof of Theorem 13 under its stated hypothesis, but it means the headline existence claim is too broad as formulated. The paper should either add connectedness to the abstract and central claim or prove that the enrichment procedure can connect a disconnected complex. Because this is a scoping flaw in the central claim rather than a defect in the main proof, a conditional acceptance requiring this clarification is appropriate.","tokens_in":21773,"tokens_out":21052,"duration_ms":206897,"concrete_test":"Take the disconnected wsc Ω on {0,1} with Ω({0})=Ω({1})=1, Ω({0,1})=0, and let C2 act by swapping vertices. For each m≥1 let Ω_m be the complex obtained by multiplying every facet weight by m. Check whether the C2-invariant tensor v = e0⊗e1 + e1⊗e0 in C²⊗C² admits any (Ω_m,C2)-decomposition. Since ~F = ~F0 ⊔ ~F1 in every case, any such decomposition factorizes as a pure tensor, while v has tensor rank 2; therefore no decomposition exists. This directly settles whether the connectedness hypothesis is necessary for the unrestricted central claim.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central existence claim is stated as holding for every invariant tensor after possibly raising facet weights and refining the group action (abstract; Theorem 13 combined with Proposition 7). However, Theorem 13 explicitly assumes the weighted simplicial complex Ω is connected, and Proposition 7 only multiplies facet weights without changing the facet incidence graph, so it cannot make a disconnected complex connected. The unrestricted claim is therefore false. Concretely, let Ω be the disconnected complex on {0,1} with Ω({0})=Ω({1})=1 and Ω({0,1})=0, with C2 acting by swapping 0 and 1. For any weight raising, ~F is the disjoint union ~F0 ⊔ ~F1, so every (Ω,G)-decomposition has the form ∑_{α0,α1} v^0_{α0} ⊗ v^1_{α1} = (∑_{α0} v^0_{α0}) ⊗ (∑_{α1} v^1_{α1}), a pure tensor. The C2-invariant tensor v = e0⊗e1 + e1⊗e0 has tensor rank 2, hence admits no such decomposition. Thus connectedness is load-bearing for the claim as summarized; either Theorem 13's hypothesis must be stated in the abstract and in the strongest claim, or a proof that the allowed enrichment can connect a disconnected complex is required.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a framework for tensor decompositions whose summation indices are arranged on a weighted simplicial complex Ω and that are manifestly invariant under a group G acting on Ω. It defines (Ω,G)-decompositions, separable decompositions, purification forms, and corresponding ranks, and proves existence theorems under free actions (Theorem 13) and blending actions (Theorem 17). It establishes inequalities among the ranks and applies the framework to nonnegative tensors, recovering several known matrix and tensor decompositions (Theorem 43 and Corollary 44). The main constructive result is that, on a connected Ω, after raising facet weights and refining the group action to be free, every G-invariant tensor admits such a decomposition.","tokens_in":22000,"tokens_out":13799,"duration_ms":142980,"significance":"The framework is genuinely unifying: known cases include translation-invariant matrix product operators, symmetric tensor decompositions, and nonnegative/PSD/cp/cpsd factorizations. The proofs are mostly constructive and the rank inequalities are explicit. The paper is careful to state that it works with algebraic tensor products, so infinite-dimensional Hilbert-space states that are not finite-rank are outside the scope. The main theorem, once the connectedness hypothesis is made explicit, is a useful contribution; the counterexample in the report shows only that the advertised unrestricted version needs amendment.","major_comments":[{"comment":"The main existence claim as stated in the abstract, in §1, and in §6 omits the connectedness hypothesis that is explicit in Theorem 13. Proposition 7 only raises facet weights and refines the action on the facet multiset; it does not change the facet incidence graph, so it cannot turn a disconnected complex into a connected one. The omission is load-bearing: let Ω be the disconnected complex on {0,1} with Ω({0})=Ω({1})=1 and Ω({0,1})=0, with C2 acting by swapping the two vertices. After any admissible weight raising, every (Ω,G)-decomposition is a pure tensor, namely (∑_α v^0_α)⊗(∑_β v^1_β), because the two facets are disjoint. The C2-invariant tensor e0⊗e1+e1⊗e0 is not pure, so it has no such decomposition. Thus the unrestricted claim is false; the abstract and introduction must either state the connectedness condition or specify an enrichment operation that can connect the complex, with a proof.","section":"Abstract; §1; Theorem 13 and Proposition 7"}],"minor_comments":[{"comment":"The step marked '∼' in the proof of Theorem 17 is compressed: after using equation (2), the sum over tuples is not literally equal to ∑_{g∈G} g·v unless one checks that every tuple satisfying {g0 0,...,gn n}=[n] gives a G-translate of v. This is true by blending and by G-invariance of v, but it should be spelled out; the notation w[g0]j in the same display is also ambiguous and should be written as w^{[g·0]}_j.","section":"Theorem 17, proof after Eq. (2)"},{"comment":"In the definition of a positive semidefinite (Ω,G)-decomposition, the condition (E[gi]_j)_{gβ,gβ'} = (E[i]_j)_{β,β'} uses the action on functions gβ defined in Section 2; adding one sentence with the functional definition would improve readability, since the matrix index convention is otherwise easy to misread.","section":"Definition 41(iii)"},{"comment":"There are a few typographical issues (for example 'Hermitain squares' in Remark 26 and inconsistent hyphenation of 'purification'); these do not affect the mathematics.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main issue is a mismatch between the abstract claim and the theorem's hypotheses; it is easily fixed by stating connectedness explicitly. No concerns about novelty or citation practice beyond the report."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper is a real contribution to the tensor-decomposition toolkit, but read Theorem 13 before trusting the abstract. The abstract claims an (Ω,G)-decomposition exists for every invariant tensor after enriching the complex. That's false for disconnected complexes. Theorem 13, the actual main result, assumes the weighted simplicial complex is connected. Proposition 7 only multiplies facet weights; it cannot connect components. Concrete counterexample: Ω with two isolated vertices, C2 swapping them. After any weight raising, the decomposition space contains only pure tensors, so the rank-two invariant tensor e0⊗e1 + e1⊗e0 has no decomposition. The conclusions section repeats the overclaim.\n\nWhat's actually new and good: the (Ω,G)-decomposition framework, existence under free action (Theorem 13) and blending action (Theorem 17), the separable and purification ranks, and the rank inequalities. Theorem 43, the correspondence for nonnegative/PSD decompositions, is clean and recovers known matrix ranks. Proofs are detailed and the main steps hold. The 'positive multiple' step in Theorem 17 is compressed, but I believe it is correct. The algebraic tensor product limitation is stated honestly.\n\nCitations are fine: self-citation to [8] is for recovery and comparison, not as premise. The paper works to show how prior results fit as special cases.\n\nWho benefits: tensor-network and quantum-information readers, and anyone working on symmetric tensor rank or nonnegative factorizations. It deserves a serious referee. After an abstract patch adding 'connected', I'd be happy to see it published. I'd bring it to reading group: the gap between abstract and theorem is a useful lesson.\n\nRecommendation: send to review, but require the connectedness qualifier as a minor revision.","headline":"Solid framework and correct main theorem, but the abstract overclaims by dropping connectedness; patch that and it's a good paper.","tokens_in":22511,"tokens_out":3623,"would_cite":true,"duration_ms":35960,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A69","05E45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every tensor fixed by a group action can be rewritten so that the symmetry is explicit, after one enriches the index structure enough to make the action free.","keywords":["tensor decomposition","simplicial complex","group action","invariant tensor","tensor rank","separable decomposition","purification","nonnegative tensor"],"falsifier":"Take the 3-vertex simplex with the cyclic group acting by permuting the three tensor factors, and choose a tensor invariant under that cycle but not under all permutations. Before refinement no $(\\Sigma_2,C_3)$-decomposition exists because the decomposition would force full symmetry; after tripling the facet weight the theorem's construction must produce one. Computing both for a small concrete tensor separates the necessity of freeness from the sufficiency of the refined construction.","tokens_in":21570,"feed_emoji":"🧩","tokens_out":8667,"duration_ms":87096,"temperature":0.7,"pith_summary":"Every tensor that is fixed by a group action can be written as a sum of elementary tensors in a way that itself makes the symmetry visible, provided the indexing pattern—a weighted simplicial complex with a free action on its facets—is rich enough. The paper proves this always becomes possible after multiplying facet weights by $|G|$ and refining the action, so no invariant tensor is excluded. The same construction yields invariant separable decompositions and purification forms, together with rank inequalities showing, for instance, that imposing invariance costs at most a factor $|G|$ in rank and that the ordinary tensor rank is the largest rank in the family. For entrywise nonnegative tensors, the framework recovers the nonnegative, positive semidefinite, completely positive, and completely positive semidefinite transposed decompositions as one family.","feed_headline":"Every invariant tensor admits a symmetry-explicit decomposition","feed_subtitle":"Enriching the index network makes group symmetry explicit, unifying translationally invariant and symmetric tensor forms.","key_machinery":"The carrying object is the $(\\Omega,G)$-decomposition: a finite index set $I$ with local vectors $v^{(i)}_{\\beta}$ indexed by functions $\\beta:\\widetilde{\\mathcal{F}}_i\\to I$ from the multiset of facets incident to vertex $i$, combined as $\\sum_{\\alpha\\in I^{\\widetilde{\\mathcal{F}}}} v^{(0)}_{\\alpha|_0}\\otimes\\cdots\\otimes v^{(n)}_{\\alpha|_n}$, subject to the equivariance condition $v^{(i)}_{\\beta}=v^{(gi)}_{g\\beta}$. The proof mechanism is freeness of the action on the facet multiset: a $G$-linear map $z:\\widetilde{\\mathcal{F}}\\to G$ lets the construction average over the group so that the total sum becomes $g\\cdot v$, which equals $v$ by invariance, while freeness makes the averaging unambiguous. Weighted complexes enter because raising facet weights and re-labelling copies by group elements is what produces a free refinement of an arbitrary action.","core_discovery":"The central discovery is an existence theorem: for a connected weighted simplicial complex $\\Omega$ with a free action of the group $G$, every $G$-invariant element $v$ of the algebraic tensor product has an $(\\Omega,G)$-decomposition, meaning $v = \\sum_{\\alpha\\in I^{\\widetilde{\\mathcal{F}}}} v^{(0)}_{\\alpha|_0}\\otimes\\cdots\\otimes v^{(n)}_{\\alpha|_n}$ with local vectors satisfying $v^{(i)}_{\\beta}=v^{(gi)}_{g\\beta}$. Since any finite group action on a weighted complex can be refined to a free action by multiplying all facet weights by $|G|$ (Proposition 7), the consequence is unconditional: every invariant tensor admits an invariant decomposition after suitable enrichment, using only nonnegative multiples of the vectors of any initial tensor decomposition. The paper also proves an alternative existence criterion for blending actions (Theorem 17), establishes separable and purification analogues under the same hypotheses, and shows that the nonnegative and positive semidefinite decompositions of nonnegative tensors correspond exactly to the separable and purification decompositions of an associated diagonal state (Theorem 43).","pith_inferences":["The algebraic-tensor-product limitation suggests a natural extension: on Hilbert-space tensor products, the results should hold for the closure of finite-rank invariant states by approximation, but the paper does not address approximation error or convergence of the constructed decompositions.","The minimal facet-weight inflation needed before an invariant decomposition exists can be read as a measure of how hidden the symmetry is; comparing this quantity across complexes could yield a resource theory of symmetry-explicit representations.","Since tensor rank is the largest rank in the family, hardness results for tensor rank may transfer to the other ranks; the paper does not discuss computational complexity.","The correspondence in Theorem 43 opens a route to constructing completely positive semidefinite decompositions with controlled rank by choosing a weighted complex whose free action is adapted to the symmetry of the tensor, rather than working on a single edge."],"forward_implications":["Translationally invariant matrix product operator forms and symmetric tensor decompositions are recovered as instantiations, so the existence theorems apply directly to them.","The cost of explicit invariance is controlled: for a free action of a finite group, $\\operatorname{rank}_{(\\Omega,G)}(v)\\le |G|\\operatorname{rank}_{\\Omega}(v)$, and the same bound holds for the separable rank; for a normal subgroup $H$, $\\operatorname{rank}_{(\\Omega,G)}(v)\\le |G/H|\\operatorname{rank}_{(\\Omega,H)}(v)$.","The ordinary tensor rank is the largest of all ranks considered: on any connected complex, $\\operatorname{rank}_{\\Omega}(v)\\le\\operatorname{rank}_{\\Sigma_n}(v)$, and the same transfer holds for separable and purification ranks.","For every entrywise nonnegative multipartite tensor, its nonnegative $(\\Omega,G)$-rank equals the separable rank of an associated diagonal positive semidefinite matrix, and its psd $(\\Omega,G)$-rank equals a purification rank, so all the rank inequalities carry over.","If the group action is blending, no enrichment is needed: every invariant element has an $(\\Omega,G)$-decomposition, including fully symmetric tensors on the $n$-simplex, even when the local spaces are infinite-dimensional."],"supporting_citations":[{"why":"Supplies the definition of weighted simplicial complexes on which the whole framework is built.","marker":"[7]"},{"why":"Provides the finite-dimensional symmetric tensor decomposition that Theorem 17 uses to handle blending actions.","marker":"[6]"},{"why":"Gives the algorithmic symmetric tensor decomposition that makes the construction in Theorem 17 effective.","marker":"[5]"},{"why":"Is the special-case theory this paper generalizes: translationally invariant matrix product operators, separable decompositions, purification forms, and the nonnegative-matrix correspondence that Theorem 43 extends.","marker":"[8]"}],"fun_headline_variants":["Invariant tensor decompositions exist after complex enrichment","Group symmetry made explicit in tensor decompositions","Enriching simplicial complexes yields invariant tensor decompositions","All invariant tensors decompose with explicit symmetry","New framework unifies invariant tensor decompositions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proofs start from a finite sum of elementary tensors, so the framework applies only to tensors in the algebraic tensor product; in infinite dimensions this excludes states that are not finite-rank, and the existence theorems do not by themselves cover them.","fun_headline_variants_meta":{"raw":{"variants":["Invariant tensor decompositions exist after complex enrichment","Group symmetry made explicit in tensor decompositions","Enriching simplicial complexes yields invariant tensor decompositions","All invariant tensors decompose with explicit symmetry","New framework unifies invariant tensor decompositions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000626,"raw_usage":{"total_tokens":2906,"prompt_tokens":964,"completion_tokens":1942,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":1871}},"tokens_in":580,"tokens_out":1942,"duration_ms":12678,"temperature":1.0,"reasoning_tokens":1871,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:08:45.196877+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the 3-vertex simplex with the cyclic group acting by permuting the three tensor factors, and choose a tensor invariant under that cycle but not under all permutations. Before refinement no $(\\Sigma_2,C_3)$-decomposition exists because the decomposition would force full symmetry; after tripling the facet weight the theorem's construction must produce one. Computing both for a small concrete tensor separates the necessity of freeness from the sufficiency of the refined construction.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the definition of weighted simplicial complexes on which the whole framework is built."},{"cited_title":"Comon, G","cited_arxiv_id":null,"evidence_quote":"Provides the finite-dimensional symmetric tensor decomposition that Theorem 17 uses to handle blending actions."},{"cited_title":"Brachat, P","cited_arxiv_id":null,"evidence_quote":"Gives the algorithmic symmetric tensor decomposition that makes the construction in Theorem 17 effective."},{"cited_title":"Mixed states in one spatial dimension: decompositions and correspondence with nonnegative matrices","cited_arxiv_id":"1907.03664","evidence_quote":"Is the special-case theory this paper generalizes: translationally invariant matrix product operators, separable decompositions, purification forms, and the nonnegative-matrix correspondence that Theorem 43 extends."}],"review_version":1}