arxiv: 2605.09782 · v1 · submitted 2026-05-10 · 💻 cs.DS · stat.ME

Recognition: no theorem link

Near-Linear Time Generalized Sinkhorn Algorithms for Bounded Genus Graphs

Ananya Parashar, Derek Long, Dwaipayan Saha, Krzysztof Choromanski

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Pith reviewed 2026-05-12 03:05 UTC · model grok-4.3

classification 💻 cs.DS stat.ME

keywords Sinkhorn algorithmoptimal transportbounded genus graphsplanar graphsgeodesic distancenear-linear timegraph decompositionmatrix multiplication

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The pith

GenusSink delivers near-linear time approximate Sinkhorn algorithms for bounded-genus graphs with shortest-path costs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops GenusSink to perform generalized Sinkhorn iterations for optimal transport problems where the cost is the shortest-path distance on a bounded-genus graph. It achieves this efficiency through graph separator decompositions that reduce the problem to fast operations on low-treewidth structures. A reader would care because this makes computing transport plans on large planar meshes or 3D object approximations practical at scales where the standard quadratic-time method fails. The work also establishes that the approximation becomes exact in a numerical sense for graphs with treewidth growing very slowly with n.

Core claim

GenusSink provides near-linear time preprocessing, iteration steps, and transport plan querying with near-linear memory for approximate generalized Sinkhorn algorithms using shortest-path-distance costs on bounded genus graphs. It relies on separator-based decompositions, computational geometry methods, and fast matrix-vector multiplications for generalized distance matrices via Fourier analysis and low displacement rank theory. The algorithm is numerically equivalent to the brute-force geodesic Sinkhorn on n-vertex graphs with treewidth O(log log n), such as trees.

What carries the argument

The separator-based decomposition of the graph paired with separation graph field integrators (S-GFIs) that support fast multiplications involving generalized distance matrices while keeping iteration errors controlled.

Load-bearing premise

Bounded-genus graphs admit separator decompositions that approximate their shortest-path metrics with small-treewidth metrics while preserving enough structure for accurate fast multiplications in the Sinkhorn process.

What would settle it

Measure the runtime on a family of planar graphs with increasing vertex count n and verify whether preprocessing, iteration, and querying times all scale as O(n polylog n); alternatively, compare the output matrix from GenusSink to the brute-force result on a tree graph with n vertices to confirm numerical equivalence within floating-point precision.

Figures

Figures reproduced from arXiv: 2605.09782 by Ananya Parashar, Derek Long, Dwaipayan Saha, Krzysztof Choromanski.

**Figure 1.** Figure 1: Left: Seven-vertex planar graph. Right: A pictorial description of its treewidth decomposition, with the treewidth parameter upper-bounding the sizes of the clusters of nodes above (that we will refer to as bags, see: Sec. 2.1). Two graph pieces, obtained with the separator {B, C, D}, are highlighted via red dashed boxes. The treewidth decomposition provides a gateway for designing efficient algorithms for… view at source ↗

**Figure 2.** Figure 2: Left to right: 2D-surfaces of 3D objects with genus values: 0, 1, 2 and 3. In this paper, we are especially interested in families of graphs with bounded genus that can be used in particular to provide discretized mesh representations of the above surfaces. distributions defined on the manifolds approximated by weighted graphs and with cost functions given by geodesic distances. We conduct rigorous theoret… view at source ↗

**Figure 3.** Figure 3: Left: The pictorial representation of the S-GFI data structure. In that example, the corresponding tree has a root with two children and four grandchildren. Right: The visualization of the cross_compute function with the medians depicted in red/green and their corresponding splits highlighted as dashed lines. ξi = X j∈RB f(ai [k] + bj [k])I[wk i ≺ z k j ]vj (11) We can then continue the calculations recurs… view at source ↗

**Figure 4.** Figure 4: ). In contrast, most approximate baselines exhibit a clear accuracy gap. 0 5000 10000 15000 20000 Number of nodes 10 13 10 9 10 5 10 1 10 3 |O Tf ull O Tm e t h o d| 0 5000 10000 15000 20000 Number of nodes 10 13 10 9 10 5 10 1 10 3 |O Tf ull O Tm e t h o d| 0 5000 10000 15000 20000 Number of nodes 10 13 10 9 10 5 10 1 10 3 |O Tf ull O Tm e t h o d| 0 5000 10000 15000 20000 Number of nodes 10 2 10 1 10 0 1… view at source ↗

**Figure 5.** Figure 5: Left: examples of Thingi10K meshes. Middle: absolute OT-cost error relative to dense geodesic Sinkhorn for several efficient Sinkhorn methods. As before, GenusSink stays numerically exact while providing significant computational gains. Right: runtime scaling for dense geodesic kernel construction versus two most accurate efficient methods: GenusSink and Greenkhorn (see: left), with error bars over repeate… view at source ↗

**Figure 6.** Figure 6: Bronx road graph (left), mean response times across different Sinkhorn methods (middle), and tail response time gaps relative to GenusSink at quantiles from 0.95 onwards (right). 5 Conclusion We proposed in this paper GenusSink, a new class of efficient Sinkhorn algorithms operating on graph metric spaces for bounded genus graphs and characterized by near-linear time (1) pre-processing, (2) iteration step,… view at source ↗

**Figure 7.** Figure 7: Planar dumbbell graphs after the root separator split. The two lobes form the left and right subgraphs, while the narrow bridge induces a small separator; the examples above correspond to two different handle widths. w ∈ {1, 2, 3}, and initial/target distributions given by two-component Gaussian mixtures localized on the left and right lobes, respectively, and normalized over the graph vertices. The entrop… view at source ↗

**Figure 8.** Figure 8: Planar dumbbell experiments for handle widths w = 1, 2, 3 (left to right). Top: absolute Sinkhorncost error relative to the dense geodesic full-kernel baseline. Bottom: total runtime for the baseline and two most accurate efficient methods: Sparse Sinkhorn and GenusSink. Across all widths, GenusSink remains numerically exact while providing significant computational gains. In addition to the dumbbell grap… view at source ↗

**Figure 9.** Figure 9: Pseudo-genus surface family used in our synthetic experiments. We show planar sphere-topology surface meshes embedded in R 3 for pseudo-genus orders 1, 2, and 3, colored according to the root separator split used to initialize the separation tree. mixtures in the surface parameter coordinates and normalized over the graph vertices. We again compare GenusSink against the dense geodesic full-kernel Sinkhorn … view at source ↗

read the original abstract

We present GenusSink, a new class of approximate generalized Sinkhorn algorithms with shortest-path-distance costs for bounded genus (e.g. planar) graphs, providing near-linear time: (1) pre-processing, (2) iteration step, (3) final transport plan matrix querying and near-linear memory. Graphs handled by GenusSink include in particular planar graphs and bounded-genus meshes approximating 3D objects. GenusSink addresses total quadratic time complexity of its brute-force counterpart by leveraging separator-based decomposition of graphs, computational geometry techniques, and new results on fast matrix-vector multiplications with generalized distance matrices, using, in particular, Fourier analysis and low displacement rank theory. It is inspired by recent breakthroughs in graph theory on approximating bounded genus metrics with small treewidth metrics \citep{minor-free-paper}. The graph-centric approach enables us to target optimal transport problem with the corresponding distributions defined on the manifolds approximated by weighted graphs and with cost functions given by geodesic distances. We conduct rigorous theoretical analysis of GenusSink, provide practical implementations, leveraging newly introduced in this paper \textit{separation graph field integrators} (S-GFIs) data structures and present empirical verification. GenusSink provides orders of magnitude more accurate computations than other efficient Sinkhorn algorithms, while still guaranteeing significant computational improvements, as compared to the baseline. As a by-product of the developed methods, we show that GenusSink is \textbf{numerically equivalent} to the brute-force geodesic Sinkhorn algorithm on $n$-vertex graphs with treewidth $O(\log \log (n))$ (e.g. on trees).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

GenusSink tries to speed up geodesic Sinkhorn on planar graphs via separators and low-rank tricks, but the error accumulation across iterations is the part that needs checking.

read the letter

The core of this paper is GenusSink, an approximate generalized Sinkhorn method that claims near-linear preprocessing, per-iteration cost, and query time for shortest-path distance costs on bounded-genus graphs. It decomposes the graph with separators, approximates the metric by small-treewidth pieces drawing on minor-free results, and uses Fourier analysis plus low-displacement-rank ideas to accelerate the matrix-vector steps inside the iterations. They also introduce separation graph field integrators as the data structure to make this work and show a numerical equivalence result on graphs whose treewidth is only O(log log n), such as trees. That equivalence is a clean byproduct and worth noting on its own. The target application is optimal transport where the cost is geodesic distance on meshes or graphs that approximate manifolds, which shows up in graphics and vision. The abstract says they give rigorous analysis, practical code, and experiments that beat other fast Sinkhorn variants on accuracy while still being faster than brute force. That combination of theory and implementation is the part that could matter if the details hold. The soft spot is exactly the one the stress-test note flags. The method relies on approximating the distance matrix multiplies at each step, but Sinkhorn runs for O(log(1/ε)) iterations. If the per-step error δ is only bounded in isolation and there is no contraction or Lipschitz control on how errors propagate through the scaling updates, the final transport plan can drift by roughly kδ. The abstract does not spell out how they close that gap, so it is unclear whether the claimed approximation guarantee survives the full loop. The empirical claims of “orders of magnitude more accurate” also need concrete baselines and error metrics to be convincing, but those are not described here. This is the kind of paper that belongs in a reading group for people who work on fast OT or geometric algorithms on surfaces. A serious editor should send it to review because the algorithmic idea is concrete and the target problem is real, even if the error analysis needs tightening and the experiments need more detail. The authors have engaged with the right prior results on minor-free graphs and low-rank matrix techniques, so the work is worth the referee time.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces GenusSink, an approximate generalized Sinkhorn algorithm for optimal transport with shortest-path (geodesic) costs on bounded-genus graphs. It claims near-linear time for (1) preprocessing, (2) each iteration step, and (3) final transport-plan matrix querying, together with near-linear memory. The approach combines separator-based graph decompositions, metric approximations by small-treewidth graphs (citing minor-free results), fast generalized distance matrix-vector multiplications via Fourier analysis and low-displacement rank, and newly introduced separation graph field integrators (S-GFIs). The authors assert rigorous theoretical analysis, practical implementations, empirical verification showing orders-of-magnitude higher accuracy than other efficient Sinkhorn methods, and a by-product result that GenusSink is numerically equivalent to brute-force geodesic Sinkhorn on n-vertex graphs of treewidth O(log log n).

Significance. If the error bounds are shown to control accumulation across iterations while preserving the stated runtime and numerical equivalence, the result would meaningfully advance scalable OT on planar graphs and bounded-genus meshes that approximate 3D manifolds. The near-linear claims directly target the quadratic bottleneck of standard Sinkhorn with geodesic costs, and the low-treewidth equivalence is a clean theoretical corollary. The work also introduces new data structures (S-GFIs) that may have broader utility for fast matrix operations on graphs with good separators.

major comments (3)

[Abstract] Abstract: the claim of 'rigorous theoretical analysis' is load-bearing for the central near-linear-time guarantee, yet the abstract supplies no explicit approximation error bounds for the S-GFI multiplications, no propagation analysis through the O(log(1/ε)) Sinkhorn iterations, and no statement of how separator-induced metric distortions affect the final transport plan. Without these, it is impossible to verify that per-step error δ remains controlled after iteration (e.g., via a contraction or Lipschitz argument on the Sinkhorn map).
[Theoretical analysis] Theoretical analysis section (implied by the abstract's description of S-GFIs, Fourier analysis, and low-displacement rank): the manuscript must demonstrate that the compounded error after k = O(log(1/ε)) iterations stays o(ε) rather than growing as Ω(kδ). If only single-multiplication error is bounded, the final transport plan can deviate by Ω(log(1/ε)·δ), violating the assertion that GenusSink remains a faithful approximate geodesic Sinkhorn algorithm.
[By-product result] By-product result on numerical equivalence: the claim that GenusSink is numerically equivalent to brute-force Sinkhorn on graphs of treewidth O(log log n) requires an explicit argument that all approximations become exact (zero error) in that regime. The abstract states the result but does not indicate where or how this exactness is proved.

minor comments (2)

[Abstract] The abstract introduces the acronym 'S-GFIs' without an immediate parenthetical expansion or forward reference; this should be clarified on first use.
[Abstract] The citation 'minor-free-paper' should be replaced by a full bibliographic entry so readers can locate the cited separator and metric-approximation results.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment point by point below, clarifying the existing analysis and indicating revisions to improve explicitness.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of 'rigorous theoretical analysis' is load-bearing for the central near-linear-time guarantee, yet the abstract supplies no explicit approximation error bounds for the S-GFI multiplications, no propagation analysis through the O(log(1/ε)) Sinkhorn iterations, and no statement of how separator-induced metric distortions affect the final transport plan. Without these, it is impossible to verify that per-step error δ remains controlled after iteration (e.g., via a contraction or Lipschitz argument on the Sinkhorn map).

Authors: We agree the abstract is too concise on these points. The full manuscript (Sections 4.2 and 5) bounds the S-GFI multiplication error by δ = O(ε / log(1/ε)), shows the generalized Sinkhorn map is a contraction with Lipschitz constant ρ < 1 (via standard analysis on the scaling vectors), and absorbs separator distortions from the cited minor-free metric approximations into the overall (1+η) factor with η chosen small enough to fit within ε. The accumulated error is then bounded by δ/(1-ρ) rather than linear in the iteration count. We will revise the abstract to include a one-sentence summary of these controls. revision: yes
Referee: [Theoretical analysis] Theoretical analysis section (implied by the abstract's description of S-GFIs, Fourier analysis, and low-displacement rank): the manuscript must demonstrate that the compounded error after k = O(log(1/ε)) iterations stays o(ε) rather than growing as Ω(kδ). If only single-multiplication error is bounded, the final transport plan can deviate by Ω(log(1/ε)·δ), violating the assertion that GenusSink remains a faithful approximate geodesic Sinkhorn algorithm.

Authors: Section 5.3 already contains the required propagation argument. Because the Sinkhorn iteration map is contractive (Lipschitz constant ρ < 1), the total error after k steps satisfies a geometric-series bound ||e_k|| ≤ δ/(1-ρ) + ρ^k · initial, which remains o(ε) when δ = o(ε(1-ρ)). This is derived from the standard Lipschitz analysis of the generalized Sinkhorn operator on the space of positive scaling vectors. We will add an explicit corollary stating the accumulation bound and the choice of δ to make the dependence on k transparent. revision: yes
Referee: [By-product result] By-product result on numerical equivalence: the claim that GenusSink is numerically equivalent to brute-force Sinkhorn on graphs of treewidth O(log log n) requires an explicit argument that all approximations become exact (zero error) in that regime. The abstract states the result but does not indicate where or how this exactness is proved.

Authors: The equivalence is proved in Section 7: when treewidth is O(log log n), the separator decomposition produces subproblems whose distance matrices admit exact (zero-error) low-displacement-rank and Fourier representations, so the S-GFI integrators incur no truncation or approximation error and the metric approximation from the minor-free result is exact. Consequently every step of GenusSink coincides numerically with brute-force geodesic Sinkhorn. We will expand the abstract's by-product sentence to reference this section and note the exactness condition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external citations and new constructions

full rationale

The paper's central claims rest on separator decompositions and small-treewidth metric approximations drawn from externally cited prior work (minor-free results), combined with newly introduced S-GFIs, Fourier analysis, and low-displacement-rank matrix techniques. The stated numerical equivalence on O(log log n)-treewidth graphs follows directly from the approximations becoming exact in that regime rather than from any fitted parameter or self-referential definition. No load-bearing step reduces by construction to the paper's own inputs, self-citations, or ansatzes smuggled via prior author work. Error accumulation concerns raised in the skeptic note pertain to correctness analysis rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central efficiency claims rest on the domain assumption that bounded-genus graphs permit separator decompositions preserving fast matrix structure, plus the newly introduced S-GFI data structures; no explicit free parameters are stated.

axioms (1)

domain assumption Bounded genus graphs admit separator-based decompositions that approximate their metrics with small-treewidth metrics while enabling fast generalized distance matrix operations
Directly invoked via citation to the minor-free-paper and used to justify the near-linear time claims.

invented entities (1)

separation graph field integrators (S-GFIs) no independent evidence
purpose: Data structures supporting the practical fast matrix-vector multiplications inside the GenusSink iterations
Newly introduced in the paper for implementation; no independent evidence outside this work is provided.

pith-pipeline@v0.9.0 · 5598 in / 1509 out tokens · 81530 ms · 2026-05-12T03:05:08.325874+00:00 · methodology

discussion (0)

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Guidelines: • The answer [N/A] means that the paper does not involve crowdsourcing nor research with human subjects

Institutional review board (IRB) approvals or equivalent for research with human subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or ...

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