{"id":"19cade3b-c92c-4cf0-ab09-40b4b5015424","arxiv_id":"2501.06247","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":0.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A broad survey of optimal transport algorithms and applications that contains mathematical errors and no original contributions.","lead":"This paper is a survey of optimal transport theory, algorithms, and applications, from Monge and Kantorovich through Sinkhorn and primal-dual methods. It offers no new results, and several technical inaccuracies limit its value as a reference.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Core algorithm restatement errors in §3.2.3 (wrong LP marginals) and §3.2.1 (impossible Sinkhorn scaling-variable dimensions) undermine the survey's reliability as a reference.","rationale":"The reader identified exactly the right load-bearing weakness: a survey's value rests on the correctness of its restatements, and the wrong LP marginals and impossible Sinkhorn variable dimensions are concrete, checkable mathematical errors in the core sections. I agree these are not cosmetic. The positive-LP reduction in Section 3.2.3 is presented as a modern reduction-based approach, and the Sinkhorn description in Section 3.2.1 is the foundational iterative method; both are central to the advertised algorithmic coverage. The errors make the paper unusable as a reference for these topics. The additional issues cited by the reader (orphaned test.png reference, citation padding) are secondary but consistent with the conclusion that the manuscript has not undergone careful technical checking. No new objection beyond the reader's is needed; the verdict of REJECT stands. I would not raise the concern to a higher severity than the reader did: the classical theory in Section 2 is largely correct, and the errors are localized rather than indicating a systematic misunderstanding of the entire field. Still, for a survey whose entire purpose is reliable exposition, a moderate-confidence reject is appropriate. Therefore my recommendation is UNCHANGED.","tokens_in":20530,"tokens_out":2318,"duration_ms":22452,"concrete_test":"Independently re-derive the positive-LP reduction in Section 3.2.3 from its source (Quanrud 2018) or from the primal OT problem (2). In particular, instantiate n=m=2 with a=(1,0), b=(0,1) and compare the feasible sets of the printed constraints {Pa=b, P^T1=1} against the correct marginals {P1=a, P^T1=b}; they should differ. Separately, verify the Sinkhorn derivation by substituting u∈R^n_+ and v∈R^m_+ into (Kv)⊙u=a and (K^T u)⊙v=b; if the paper's n×m dimensions are used, diag(u) is ill-defined. A successful check confirming both errors would settle that the survey's core algorithm descriptions are unreliable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is to provide a detailed, self-contained examination of OT algorithms. For a survey, the load-bearing assumption is that the restated algorithms are mathematically correct. That assumption fails at two concrete points. In Section 3.2.3, the reduction to a positive LP is written as min_{P∈R^{n×n}_+} {⟨C,P⟩ : Pa = b, P^T 1 = 1}. The standard Kantorovich marginals are P 1_m = a and P^T 1_n = b. The printed constraints are dimensionally inconsistent for rectangular transport plans, and even for n=m they define a different feasible set (e.g., they do not enforce the target marginal b on the columns). In Section 3.2.1, the Sinkhorn scaling variables are stated as u,v ∈ R^{n×m}_+; they should be u∈R^n_+ and v∈R^m_+. Since diag(u) is then undefined for a matrix u, the factorization P = diag(u)K diag(v) and the subsequent updates do not make sense as written. These are not typos in an incidental remark: they sit in the algorithmic core the survey claims to present. A reader relying on these sections cannot reproduce or even parse the algorithms. The presence of an orphaned 'test.png' reference and the citation padding in Section 4 corroborate a lack of careful editorial checking, but the mathematical transcription errors are the decisive flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper is a survey of optimal transport (OT) aimed at a data-science and CS audience. It covers the classical Monge and Kantorovich formulations, duality and C-transforms, entropic regularization, multimarginal and unbalanced extensions, Wasserstein barycenters, and a broad range of algorithms (network simplex, auction, Sinkhorn, Greenkhorn, primal-dual methods, and reduction-based approaches). It also discusses applications, with a focus on Optimal Transport Warping for time series. The paper's stated value is to provide a detailed, self-contained examination of OT algorithms and theory.","tokens_in":109,"tokens_out":6447,"duration_ms":124897,"significance":"If its restatements were reliable, this survey would be a useful entry point to computational OT and would help readers navigate recent algorithmic developments. The paper does cover many important modern results and makes a genuine effort to organize the literature, including the OTW application. However, a survey's contribution rests entirely on the accuracy of its exposition, and the manuscript contains several errors in load-bearing mathematical definitions and algorithmic statements. These errors currently prevent the paper from serving as a trustworthy reference, so the significance is contingent on substantial corrections.","major_comments":[{"comment":"The reduction of OT to a positive LP is stated with incorrect marginal constraints. The text gives min_{P∈R^{n×n}_+} {⟨C,P⟩ : Pa = b, P^T 1 = 1}. The Kantorovich marginals are P 1_m = a and P^T 1_n = b. As printed, the constraints are dimensionally inconsistent for rectangular plans, and even in the square case they define a different feasible set (for example, they do not enforce the target marginal on the columns). The same defect carries into the 'mixed packing and covering' and 'packing LP' formulations in this section, so the algorithmic content of the section is not reproducible.","section":"§3.2.3"},{"comment":"The Sinkhorn scaling variables are assigned impossible dimensions. The text states u, v ∈ R^{n×m}_+, but the correct domains are u ∈ R^n_+ and v ∈ R^m_+. With u and v as matrices, the factorization P = diag(u)K diag(v) and the elementwise update formulas u ← a/(Kv), v ← b/(K^T u) do not parse, since diag(u) of a matrix is undefined. This is not a cosmetic issue: the iteration cannot be implemented or even checked as written.","section":"§3.2.1"},{"comment":"The unbalanced optimal transport relaxation is misstated. The paper defines the relaxed problem as min_{\\tilde a,\\tilde b} L_C(a,b) + τ_1 D_φ(a‖\\tilde a) + τ_2 D_φ(b‖\\tilde b), but the transport cost term uses the fixed measures a and b rather than the optimization variables \\tilde a and \\tilde b. As written, the left-hand side is a constant plus two independent divergence terms and is not equal to the right-hand side min_{P} ⟨C,P⟩ + τ_1 D_φ(P1_m‖a) + τ_2 D_φ(P^T1_n‖b). The first term should be L_C(\\tilde a, \\tilde b). This is a definitional error in a core theoretical section.","section":"§2.6"}],"minor_comments":[{"comment":"The manuscript contains an orphaned reference to a figure 'test.png' with a URL at the end of the text; this appears to be a leftover from the production process and should be removed.","section":"End of manuscript"},{"comment":"The sentence 'For K probability distributions {a_k} defined on a shared finite domain {x_k}' is unclear, since the barycenter is supposed to live on a common target domain while the input measures may have different supports. The notation should be made precise.","section":"§2.7"},{"comment":"In the APDAGD description, the phrase 'there exists some R > 0 where ‖λ*‖_2 ≤ R < +∞ and λ* is the optimal dual solution with objective value of ‖λ*‖_2' is garbled; the clause 'with objective value of ‖λ*‖_2' should be corrected or removed.","section":"§3.2.2"},{"comment":"The applications paragraph cites several works by 'Moradi et al.' on vehicle routing, electric vehicles, and simulated annealing without explaining how these works use optimal transport. The relevance of these citations to the OT applications under discussion should be stated explicitly, or the citations should be removed.","section":"§4"}],"recommendation":"major_revision","confidential_remarks":"The citation pattern in Section 4 deserves the editor's attention. Seven consecutive citations are to works by 'Moradi et al.' by a different author (Nima Moradi) on vehicle routing and simulated annealing, and their connection to optimal transport is not explained in the text. The surname overlap with the manuscript's author (Sina Moradi) makes this a point the editor may wish to examine for citation padding. In addition, the number of mathematical transcription errors in core sections (Sections 2.6, 3.2.1, 3.2.3) suggests that the manuscript needs a careful checking pass by someone with expertise in computational OT before it can be considered for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: this is a survey with zero new technical content, and while the classical theory sections are mostly faithful to the standard references, the algorithm sections contain two concrete transcription errors that make the paper untrustworthy as a reference.\n\nWhat is actually new: nothing. It explicitly points readers to Peyré et al. for a comprehensive introduction, and every derivation is attributed to prior work. That is fine for a survey, but it means the bar is accuracy and organization. On those, the theory part does reasonably well: Monge and Kantorovich formulations, duality and C-transforms, entropic regularization, multimarginal, unbalanced OT, and Wasserstein barycenters are reproduced correctly, and the writing is clear.\n\nThe soft spot is load-bearing. Section 3.2.3 gives the reduction to a positive LP with constraints Pa = b, P^T 1 = 1. The actual Kantorovich marginals are P1 = a and P^T1 = b; as written, even in the square case the target marginal is not enforced on columns. Section 3.2.1 states the Sinkhorn scaling variables u,v ∈ R^{n×m}_+; they should be vectors of length n and m. Since diag(u) for a matrix is undefined, the factorization P = diag(u)K diag(v) and the ensuing updates do not parse. These are not side remarks; they sit in the algorithmic core the survey claims to present.\n\nThere are also editorial red flags: an orphaned reference to 'test.png' at the end, and the applications section leans on a string of 'Moradi et al.' papers on vehicle routing and simulated annealing. Those are by Nima Moradi, a different author, and are not optimal transport applications. The OTW discussion is a condensation of Latorre et al., which is fine but adds nothing beyond the source.\n\nSo the paper is a mixed bag: a competent restatement of standard theory undermined by avoidable errors in the parts that matter most for a methods survey. A reader who wants a quick orientation to OT theory could get value from Sections 2, but they would be better served by Peyré et al. directly. This version should not go to peer review; the math errors are basic and would need fixing before the paper is even a reliable draft. If the author corrects the algorithm sections and trims the irrelevant citations, a short survey could be salvageable, but as submitted I would desk-reject.","headline":"A readable survey with no new results whose core algorithm restatements are mathematically wrong; the theory sections are salvageable but the paper as submitted should be desk-rejected.","tokens_in":21336,"tokens_out":3194,"would_cite":false,"duration_ms":28971,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22","90C08"],"pacs":[],"model":"deepseek-v4-flash","headline":"This survey argues that optimal transport has matured from Monge's non-convex map problem into a widely used algorithmic framework, and it maps the theoretical formulations, the main approximation algorithms, and applications such as…","keywords":["optimal transport","Wasserstein distance","Kantorovich relaxation","entropic regularization","Sinkhorn algorithm","primal-dual methods","Optimal Transport Warping","time series alignment"],"falsifier":"Take the reduction of OT to a positive linear program printed in Section 3.2.3 and run it on a two-point example with $a=(0.5,0.5)$ and $b=(0.4,0.6)$: the printed constraints $Pa=b$, $P^\\top \\mathbf{1}=\\mathbf{1}$ allow plans whose column sums do not match $a$, so the computed optimum will not equal the Wasserstein distance. If the survey's claim that this is an equivalent formulation were correct, that calculation would reproduce the true OT cost.","tokens_in":20324,"feed_emoji":"📊","tokens_out":8793,"duration_ms":82838,"temperature":0.7,"pith_summary":"This paper is a survey of optimal transport (OT), the problem of moving one distribution of mass onto another at minimum cost. It aims to show that OT has grown from Monge's eighteenth-century transport-map problem and Kantorovich's plan relaxation into a practical framework for quantifying differences between probability distributions. The survey walks through the theoretical core—duality, entropic regularization, multi-marginal and unbalanced variants, and Wasserstein barycenters—then catalogues exact, approximate, and reduction-based algorithms with their complexity bounds. It closes by arguing that OT's practical reach includes machine learning, computer vision, and time-series analysis, with Optimal Transport Warping presented as a robust alternative to Dynamic Time Warping.","feed_headline":"Optimal transport survey links Monge's 1781 map to today's algorithms","feed_subtitle":"Why care: one framework now measures distributional differences across machine learning, vision, and time series.","key_machinery":"The central object is the Kantorovich transport plan $P\\in\\mathbb{R}^{n\\times m}_+$, a joint distribution whose row and column sums reproduce the source and target measures $a$ and $b$, with total cost $\\langle C,P\\rangle$. The structural workhorse is the dual pair of Kantorovich potentials $(w,z)$ and the C-transform, $w^C_j=\\min_i C_{ij}-w_i$, which converts the dual problem into a concave piecewise-affine maximization and drives auction, primal-dual, and accelerated algorithms. On the computational side, the key identity is the Sinkhorn factorization $P=\\operatorname{diag}(u)K\\operatorname{diag}(v)$ with Gibbs kernel $K_{ij}=e^{-C_{ij}/\\eta}$, which turns entropy-regularized transport into alternating row-and-column scaling updates.","core_discovery":"No new theorem is proved; the paper's contribution is an organizing synthesis. Its central claim is that optimal transport provides a unifying mathematical and computational framework for comparing distributions, and that the field's algorithmic development has reached the point where large-scale problems are tractable through entropic regularization, primal-dual methods, and reductions to linear programs. The survey presents the Kantorovich relaxation as the cornerstone that turns Monge's non-convex problem into a convex linear program, identifies the C-transform as the structural mechanism behind dual algorithms, and traces a chain of complexity improvements from Sinkhorn scaling through Greenkhorn, primal-dual accelerated methods, and reduction-based approaches. It also claims that OTW, by framing time-series alignment as an optimal transport problem with an adaptively learned cost and entropic regularization, is more flexible and robust than standard alignment methods.","pith_inferences":["A reader could test the survey's narrative by picking one cited algorithm, implementing the displayed update rules, and checking whether the stated $\\epsilon$ complexity is reproduced; discrepancies in transcribed formulas would matter more than the prose claims.","The paper's treatment of time-series warping suggests a concrete extension: applying the same transport-warping formulation to multivariate, unevenly sampled, or noisy time series, where Dynamic Time Warping's alignment assumptions are known to be brittle.","Because the survey presents complexity bounds drawn from different sources under different assumptions, an implicit takeaway is that these bounds should not be compared across sections without first checking assumptions like $m=n$, bounded costs, or entropic regularization.","If the Gibbs-kernel/Sinkhorn machinery is as central as the survey suggests, algorithmic improvements for entropy-regularized OT should transfer directly to time-series warping, since OTW uses the same regularized transport structure."],"forward_implications":["If the survey's account is accurate, a reader can treat OT as a practical tool for high-dimensional distribution comparison, with epsilon-approximation algorithms reported at roughly $O(n^2/\\epsilon)$ or $O(n^{2.5}/\\epsilon)$ arithmetic operations in the balanced square case.","The Sinkhorn and Greenkhorn family makes entropy-regularized OT scalable, with each Sinkhorn iteration costing $O(mn)$ and with explicit dependence on the desired accuracy $\\epsilon$ in the iteration bounds.","The reduction-based perspective connects OT to packing and covering linear programs and to bipartite matching, implying that faster-than-$O(n^2/\\epsilon)$ OT algorithms would carry consequences for classical matching problems.","OTW gives a transport-based alternative to Dynamic Time Warping for time-series alignment, with an adaptively learned cost and an entropy term that balances temporal alignment against probabilistic structure.","The survey's organisation implies that Wasserstein barycenters, unbalanced OT, and multi-marginal OT are the natural frontiers where theoretical guarantees and scalable algorithms remain incomplete."],"supporting_citations":[{"why":"supplies the book-length formulation of OT and the algorithmic toolbox that the survey's theory and algorithm sections restate","marker":"[Peyré et al., 2019]"},{"why":"underlies the entropic-regularized scaling iteration that Section 3.2.1 is built on","marker":"[Sinkhorn, 1964]"},{"why":"gives the near-linear Sinkhorn iteration bound and introduces Greenkhorn","marker":"[Altschuler et al., 2017]"},{"why":"provides the APDAGD primal-dual accelerated framework and the refined worst-case Sinkhorn iteration bound","marker":"[Dvurechensky et al., 2018]"},{"why":"supplies the dual-extrapolation primal-dual method with the $O(n^2/\\epsilon)$ arithmetic-operation target","marker":"[Jambulapati et al., 2019]"},{"why":"is the cited source for reductions to positive, packing, and mixed packing-covering linear programs in Section 3.2.3","marker":"[Quanrud, 2018]"},{"why":"establishes the reduction from OT to packing LPs and bipartite matching that motivates the reduction-based complexity discussion","marker":"[Blanchet et al., 2024]"},{"why":"introduces OTW, the time-series warping method highlighted as the main application","marker":"[Latorre et al., 2023]"},{"why":"is the cited source for the multi-marginal, unbalanced, and Wasserstein barycenter formulations in Sections 2.5 through 2.7","marker":"[Lindheim, 2023]"}],"fun_headline_variants":["Optimal transport survey maps algorithm progress from Monge to Sinkhorn","OT algorithms: from Kantorovich to scalable Sinkhorn, a survey","Survey recaps optimal transport's rise from theory to ML and time series","From Monge to Sinkhorn: a survey of optimal transport algorithms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The survey's value as a reference stands or falls on the assumption that every displayed formula and complexity bound faithfully restates the cited algorithm, because a survey supplies no new derivations to absorb transcription errors.","fun_headline_variants_meta":{"raw":{"variants":["Optimal transport survey maps algorithm progress from Monge to Sinkhorn","OT algorithms: from Kantorovich to scalable Sinkhorn, a survey","Survey recaps optimal transport's rise from theory to ML and time series","From Monge to Sinkhorn: a survey of optimal transport algorithms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000509,"raw_usage":{"total_tokens":2457,"prompt_tokens":903,"completion_tokens":1554,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":1474}},"tokens_in":519,"tokens_out":1554,"duration_ms":11906,"temperature":1.0,"reasoning_tokens":1474,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:27:53.197276+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the reduction of OT to a positive linear program printed in Section 3.2.3 and run it on a two-point example with $a=(0.5,0.5)$ and $b=(0.4,0.6)$: the printed constraints $Pa=b$, $P^\\top \\mathbf{1}=\\mathbf{1}$ allow plans whose column sums do not match $a$, so the computed optimum will not equal the Wasserstein distance. If the survey's claim that this is an equivalent formulation were correct, that calculation would reproduce the true OT cost.","supporting_citations":[{"cited_title":"Otw: Optimal transport warping for time series","cited_arxiv_id":null,"evidence_quote":"introduces OTW, the time-series warping method highlighted as the main application"}],"review_version":1}