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arxiv: 2306.06581 · v2 · submitted 2023-06-11 · 📊 stat.ML · cs.DS· cs.LG· math.OC

Importance Sparsification for Sinkhorn Algorithm

Mengyu Li , Jun Yu , Tao Li , Cheng Meng This is my paper

Pith reviewed 2026-05-24 08:20 UTC · model grok-4.3

classification 📊 stat.ML cs.DScs.LGmath.OC
keywords Sinkhorn algorithmoptimal transportimportance samplingsparsificationunbalanced optimal transportconsistencycomputational efficiency
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The pith

Importance sparsification based on upper bounds for transport plans reduces Sinkhorn per-iteration cost from O(n^{2}) to Õ(n) while keeping estimators consistent.

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

The paper develops Spar-Sink, an importance sparsification method for the Sinkhorn algorithm applied to entropy-regularized optimal transport and unbalanced optimal transport. It constructs sampling probabilities from natural upper bounds on the unknown optimal transport plans, then builds a sparse kernel matrix that accelerates each iteration. The resulting estimators remain consistent under mild regularity conditions. On synthetic data the method improves both error and speed over competing approaches; on echocardiogram recordings it estimates cardiac cycles with accuracy matching the classical algorithm but at far lower computational cost.

Core claim

By employing natural upper bounds on optimal transport plans to define sampling probabilities and then building a sparse kernel, Spar-Sink accelerates Sinkhorn iterations to Õ(n) cost per step for sample size n, and the resulting estimators for regularized OT and UOT remain consistent under mild regularity conditions.

What carries the argument

Importance sparsification of the kernel matrix using sampling probabilities derived from upper bounds on the transport plan.

If this is right

  • Estimators for regularized OT and UOT problems are consistent under mild regularity conditions.
  • Per-iteration computational cost drops from O(n^{2}) to Õ(n).
  • On synthetic data Spar-Sink outperforms mainstream competitors in both estimation error and speed.
  • On real echocardiogram data Spar-Sink estimates cardiac cycles as accurately as classical Sinkhorn while requiring significantly less time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The sparsification strategy could be adapted to other iterative solvers that repeatedly multiply by dense kernel matrices.
  • Large-scale applications that currently avoid OT distances because of quadratic scaling might become practical once the per-iteration cost is near-linear.
  • Tighter problem-specific upper bounds would directly improve sampling efficiency and reduce variance of the resulting estimators.

Load-bearing premise

Natural upper bounds on the unknown optimal transport plans exist and can be used to construct effective sampling probabilities that preserve consistency.

What would settle it

A demonstration that the Spar-Sink estimators become inconsistent for the regularized OT cost as n grows large, or that cardiac-cycle prediction accuracy on the echocardiogram task falls below that of classical Sinkhorn.

Figures

Figures reproduced from arXiv: 2306.06581 by Cheng Meng, Jun Yu, Mengyu Li, Tao Li.

Figure 1
Figure 1. Figure 1: An illustration of the Sinkhorn algorithm (Left panel) and our Spar-Sink method (Right panel). The non-zero elements of each matrix are labeled with colors. we focus on the general scenario where the ground cost between supports is bounded, i.e., Cij ≤ c0 for some constant c0 > 0. Therefore, we have the upper bound (T ∗ ε ) ijCij ≤ c0 p aibj . Such an inequality motivates us to use the sampling probability… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of subsampling-based methods w.r.t. RMAE(OT) versus increasing s (in log-log scale). Each row represents a different data generation pattern (C1— C3), and each column represents a different ε. Different methods are marked by different colors, respectively, and each line type represents a different dimension d. Vertical bars are the standard errors. For the UOT problem shown in (5), we set the to… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of subsampling-based methods w.r.t. RMAE(UOT) versus increasing s (in log-log scale). Each row represents a different data generation pattern (C1— C3), and each column represents a different sparsity ratio (R1—R3). Different methods are marked by different colors, respectively, and each line type represents a different dimension d. Vertical bars are the standard errors. different s are shown in … view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of different methods w.r.t. RMAE(OT) versus log10(n) under C1. Each subfigure represents a different ε, and each color marks a specific method. Vertical bars are the standard errors. (a) CPU time (in seconds) for estimating the Wasserstein distance under C1. (b) CPU time (in seconds) for estimat￾ing the WFR distance under C1, R2 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of different methods w.r.t. computational time. Different methods are marked by different colors. Each line type represents a different value of ε. 5.2 Computational Cost and CPU Time Consider the computational cost of Algorithm 3. Constructing the sketch Ke requires O(n 2 ) time, and such a step can be naturally paralleled. The matrix Ke contains at most s non-zero elements, and thus calculatin… view at source ↗
Figure 6
Figure 6. Figure 6: Echocardiogram videos data set visualization. Two basic periods, diastole and systole, form a cardiac cycle. for Spar-Sink. These observations indicate that the proposed algorithms are suitable for dealing with large-scale OT and UOT problems. 6 Echocardiogram Analysis Echocardiography has been widely used to visualize myocardial motion due to its fast image acquisition, relatively low cost, and no side ef… view at source ↗
Figure 7
Figure 7. Figure 7: (From left to right) Each column is associated with an individual corresponding to a specific state of cardiac function, i.e., health, heart failure, and arrhythmia, respectively. (Top row) Input echocardiogram videos. (Middle row) Normalized WFR distance matrices computed by the Spar-Sink algorithm. (Bottom row) MDS in 2D: each point corresponds to a frame and is colored by the corresponding time iteratio… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of subsampling-based methods w.r.t. RMAE(UOT) versus increasing s (in log-log scale). Each row represents a different sparsity ratio (R1—R3), and each column represents a different λ. Different methods are marked by different colors, respectively, and each line type represents a different dimension d. Vertical bars are the standard errors. C.2 Asymptotic Convergence To demonstrate the asymptotic… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of different methods w.r.t. RMAE(OT) versus increasing n (in log-log scale). Each column represents a different data generation pattern (C1—C3). Vertical bars are the standard errors [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of different methods w.r.t. RMAE(UOT) versus increasing n (in log￾log scale). Each column represents a different sparsity ratio (R1—R3). Vertical bars are the standard errors. IBP, which are direct extensions of Nys-Sink and Rand-Sink to approximate Wasserstein barycenters, respectively. The input probability measures b1, b2, b3 ∈ ∆n−1 are generated as: • b1 is an empirical Gaussian distributio… view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of different methods w.r.t. log10(Error) versus increasing s/s0(n) under different levels of ε. Each color marks a specific method, and each line type represents a different dimension d. Vertical bars are the standard errors [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: (Top row) For each digit, 8 out of the 15 rescaled and translated images are randomly chosen for illustration. (Middle row) Barycenters approximated by the IBP method. (Bottom row) Barycenters approximated by the Spar-IBP method. method outperforms competitors in most circumstances, with its advantage becoming more prominent as the value of ε decreases. The comparison of CPU time has similar pattern to [… view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of different methods on color transfer. Subfigure (b): (Top row) Transferred images. (Bottom row) The corresponding point clouds in RGB space. The color of the dot represents each point’s RGB value. 340.29s. Such results demonstrate the effectiveness and efficiency of our Spar-IBP method for approximating Wasserstein barycenters. Appendix D. Applications Following the recent work of Le et al. (… view at source ↗
Figure 14
Figure 14. Figure 14: The performance of SSAE on digit interpolation and reconstruction tasks. In the subfigure (b), odd rows correspond to real images. References Dimitris Achlioptas and Frank Mcsherry. Fast computation of low-rank matrix approxima￾tions. Journal of the Association for Computing Machinery, 54(2):1–19, 2007. Dimitris Achlioptas, Zohar S Karnin, and Edo Liberty. Near-optimal entrywise sampling for data matrices… view at source ↗
read the original abstract

Sinkhorn algorithm has been used pervasively to approximate the solution to optimal transport (OT) and unbalanced optimal transport (UOT) problems. However, its practical application is limited due to the high computational complexity. To alleviate the computational burden, we propose a novel importance sparsification method, called Spar-Sink, to efficiently approximate entropy-regularized OT and UOT solutions. Specifically, our method employs natural upper bounds for unknown optimal transport plans to establish effective sampling probabilities, and constructs a sparse kernel matrix to accelerate Sinkhorn iterations, reducing the computational cost of each iteration from $O(n^2)$ to $\widetilde{O}(n)$ for a sample of size $n$. Theoretically, we show the proposed estimators for the regularized OT and UOT problems are consistent under mild regularity conditions. Experiments on various synthetic data demonstrate Spar-Sink outperforms mainstream competitors in terms of both estimation error and speed. A real-world echocardiogram data analysis shows Spar-Sink can effectively estimate and visualize cardiac cycles, from which one can identify heart failure and arrhythmia. To evaluate the numerical accuracy of cardiac cycle prediction, we consider the task of predicting the end-systole time point using the end-diastole one. Results show Spar-Sink performs as well as the classical Sinkhorn algorithm, requiring significantly less computational time.

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.

Referee Report

2 major / 2 minor

Summary. The paper proposes Spar-Sink, an importance sparsification method for the Sinkhorn algorithm to approximate entropy-regularized OT and UOT. It derives sampling probabilities from natural upper bounds on the unknown optimal transport plan to build a sparse kernel, reducing per-iteration cost from O(n²) to Õ(n). The authors claim the resulting estimators are consistent under mild regularity conditions, outperform competitors on synthetic data in error and speed, and match classical Sinkhorn on echocardiogram data for cardiac cycle estimation while being faster.

Significance. If the consistency result holds and the sampling preserves the fixed-point property of Sinkhorn iterates, the work would offer a scalable approach to regularized OT with statistical guarantees, which is valuable for large-scale applications such as the demonstrated cardiac imaging task. The explicit use of upper bounds for sampling (rather than post-hoc fitting) and the Õ(n) complexity reduction are potential strengths if the technical conditions on the bounds are verified.

major comments (2)
  1. [Abstract] Abstract: The consistency of the Spar-Sink estimators is stated to hold under mild regularity conditions after importance sampling with probabilities derived from 'natural upper bounds for unknown optimal transport plans.' No explicit construction is given showing these bounds are computable from the cost matrix and marginals alone in Õ(n) time, nor that the resulting p_{ij} dominate the true plan entries sufficiently (e.g., p_{ij} ≥ c · π_{ij} / n^α with c, α independent of n) to ensure the bias term vanishes in the consistency proof. This assumption is load-bearing for all subsequent claims on complexity, consistency, and empirical performance.
  2. [Theoretical results] Theoretical results (implied by abstract claims): The fixed-point property of the sparsified Sinkhorn iterates must be preserved for consistency to carry over from the dense case. If the upper bounds are only loose envelopes (e.g., from row/column sums of exp(−C/ε)), the effective support after sparsification may miss mass concentrating on a vanishing fraction of entries, violating the conditions needed for the bias to vanish. A concrete counter-example or domination lemma is needed to confirm this does not occur under the stated mild conditions.
minor comments (2)
  1. [Abstract] Abstract: The real-data experiment reports that Spar-Sink 'performs as well as' classical Sinkhorn for end-systole prediction but provides no quantitative metric (e.g., MSE or correlation) or error-bar information to support the parity claim.
  2. [Experiments] Experiments: Synthetic data results claim outperformance in estimation error and speed, but the description lacks details on the number of Monte Carlo repetitions, variance estimates, or how sampling probabilities were tuned, making robustness difficult to assess.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our work. We address each major comment point-by-point below and will revise the manuscript to incorporate clarifications that strengthen the theoretical presentation.

read point-by-point responses

  1. Referee: Abstract claims consistency of Spar-Sink estimators under mild conditions after importance sampling with probabilities from 'natural upper bounds,' but provides no explicit construction showing these bounds are computable from the cost matrix and marginals alone in Õ(n) time, nor that resulting p_ij dominate true plan entries sufficiently (e.g., p_ij ≥ c · π_ij / n^α with c, α independent of n) to ensure bias vanishes.

    Authors: We agree an explicit algorithmic construction would improve clarity. The sampling probabilities are formed from row- and column-wise upper bounds on the transport plan that depend only on the marginal vectors a, b and the cost matrix C (via the kernel K = exp(−C/ε)); these can be evaluated with O(n) vector operations per sampled row without materializing the full n×n matrix. Under the paper’s mild regularity conditions (bounded costs, strictly positive marginals), the resulting p_ij satisfy the required domination with α = 0 and c independent of n, which is used to control the bias term in the consistency proof. We will add a dedicated subsection with the precise algorithm and the domination argument. revision: yes

  2. Referee: Fixed-point property of sparsified Sinkhorn iterates must be preserved for consistency to carry over. If upper bounds are only loose envelopes, effective support after sparsification may miss mass concentrating on a vanishing fraction of entries, violating conditions for bias to vanish. A concrete counter-example or domination lemma is needed.

    Authors: The consistency argument (Theorem 3) is established directly for the output of the sparsified procedure and does not require the sparsified iterates to share the identical fixed point with the dense algorithm; the proof controls the Monte-Carlo error of the importance-sampled kernel and shows that this error vanishes in probability under the stated assumptions. The chosen upper bounds are proportional to the kernel entries and therefore cannot systematically omit mass on sets whose measure does not vanish. We will insert a short domination lemma (new Lemma 3.1) that formalizes this coverage property from the regularity conditions, rendering a counter-example unnecessary. revision: yes

Circularity Check

0 steps flagged

No circularity; consistency rests on external assumptions about bounds, not self-definition or fitted predictions

full rationale

The paper introduces Spar-Sink by constructing sampling probabilities from stated natural upper bounds on the unknown plan, then proves consistency of the resulting estimators under mild regularity conditions. No step reduces a claimed prediction or consistency result to a fitted parameter by construction, nor does any load-bearing premise collapse to a self-citation or ansatz smuggled from prior author work. The derivation chain therefore remains self-contained against the stated assumptions and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of usable upper bounds for sampling and on mild regularity conditions whose precise form is not detailed in the abstract; no free parameters or invented entities are explicitly introduced in the provided text.

axioms (1)
  • domain assumption Mild regularity conditions suffice for consistency of the sparsified estimators
    Invoked in the abstract to support the theoretical consistency claim for regularized OT and UOT.

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    employs natural upper bounds for unknown optimal transport plans to establish effective sampling probabilities, and constructs a sparse kernel matrix to accelerate Sinkhorn iterations, reducing the computational cost of each iteration from O(n²) to Õ(n)

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Reference graph

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