Title: Efficient Algorithms for Robust and Partial Semi-Discrete Optimal Transport
Venue: NeurIPS 2025
Accepted: yes
Keywords: Semi-Discrete Optimal Transport

Approximation Algorithms

Robust Optimal Transport
Abstract: The sensitivity of optimal transport (OT) to noise has motivated the study of robust variants. In this paper, we study two such formulations of semi-discrete OT in $\mathbb{R}^d$: (i) the $\alpha$-optimal partial transport, which minimizes the cost of transporting a mass of $\alpha$; and (ii) the $\lambda$-robust optimal transport, which regularizes the OT problem using the total variation (TV) distance. First, we provide a novel characterization of the optimal solutions in these settings, showing they can be represented as a restricted Laguerre diagram. Second, we exploit this characterization to establish a strong algorithmic connection between the two problems, showing that any solver for one can be adapted to solve the other with comparable precision. Third, we overcome key challenges posed in extending the cost-scaling paradigm to compute these variants of OT and present an algorithm that computes the exact solution up to $\log (1/\varepsilon)$ bits of precision in $n^{O(d)}\log (1/\varepsilon)$ time, where $n$ is the support size of the discrete distribution.  Finally, we present an $n^{1+o(1)}\varepsilon^{-O(d)}$ time approximation algorithm for the above variants of OT.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors extend the deterministic semi-discrete optimal transport problem, which involves transporting a continuous distribution to a discrete one, to a stochastic setting. Noise and outliers are considered via two variants: $\\alpha$-partial optimal transport, where only an $\\alpha$-fraction of the probability mass is transported, and $\\lambda$-robust optimal transport, where mass can be discarded at a cost of $\\lambda$ per unit.\n\nThe authors propose a novel theoretical characterization of exact solutions using restricted Voronoi cells and weight functions on the discrete distribution domain, under specific conditions. For inexact solutions, they rigorously derive a connection between the two problem variants and introduce a reduction scheme along with corresponding error bounds. To solve the $\\lambda$-robust OT problem with high accuracy, the authors generalize the combinatorial cost-scaling paradigm, incorporating all necessary modifications. The resulting algorithm achieves a complexity bound of $n^{O(d)} \\log(\\Delta/\\varepsilon)$. All results hold for arbitrary cost functions, with additional acceleration techniques developed specifically for Wasserstein distances.\n\nSTRENGTHS AND WEAKNESSES:\nOptimal Transport is an important topic, and new robust algorithms represent valuable contributions to both theory and practice.\n\nThe strongest side of the paper is its complex rigorous theoretical analysis of the proposed methods and properties of the solutions. These findings are interesting and cover all aspects of the considered setups and connections between them.\n\nDespite the amount of information, the paper’s writing is clear and well-structured making it easy to follow.\n\nFor weakness, see the Limitations section.\n\nQUESTIONS:\nHow long does it take to compute the $\\Phi$ factor, both for known distributions (e.g., normal, exponential) and unknown distributions? What methods are available for this?\n\n\n Can your methods handle distributions for which only sample data is available?\n\n\nAre the obtained convergence rates optimal with respect to their dependencies on parameters such as n (sample size), d (dimensionality) and others?\n\nLIMITATIONS:\nThe paper lacks practical validation of the proposed methods, particularly comparisons with deterministic counterparts in scenarios where robustness is essential. Even basic 2D experiments on synthetically corrupted data could have provided a proof of concept while demonstrating the methods' robustness and quality.\n\nWhile the theoretical contributions are significant, their novelty appears somewhat incremental, as they primarily combine (in a non-trivial way) existing ideas, problem formulations, and methodologies.\n\nMinor weakness: the notation $\\delta$ is ambiguous, it is used as a parameter in Section 3, and as units in Introduction.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies two formulations of semi-discrete optimal transport (OT) problem: $\\alpha$-optimal partial transport and $\\lambda$-robust OT. The authors provide theoretical investigations on the optimal solutions in these formulations, connection between the algorithms for each of the settings and present their own algorithm for each of the formulations.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths.** The paper provides a lot of theoretical results on the connection and properties of partial and robust OT formulations, their solutions and corresponding algorithms.\n\n**Weaknesses.** My main concern corresponds to the **lack of experiments** in the paper. The authors present a lot of theoretical results related to partial/robust OT formulations, propose new algorithms to solve these problems, establish their convergence properties, but **do not test them in any experiments**. I am confused by this fact, since while the paper proposes the algorithms, it should provide at least some experimental validation of their performance. I am also interested in real-world applications of the proposed algorithms, i.e., experiments showing that semi-discrete partial/robust OT formulations and corresponding algorithms are useful in practice. However, at this step, even toy experiments are missing.\n\nThe related concern corresponds to the paper **clarity**. The paper is really hard to follow, most of the text is written in a technical, mathematical style starting from the ‘Introduction’ section. While I understand that it mainly focuses on theoretical investigations, the text should be written in an accessible and understandable form:\n-  I kindly suggest the authors exclude complex explanations of their results from the ‘Introduction’ section (lines 60-94) and rather include the paper contributions in a general form trying to avoid specific mathematical notations and definitions which are then introduced one more time in main sections. \n- I also have concerns regarding the clarity of OT formulations. In the classic OT problem, the marginals of the plan should coincide with input and target measures. However, in lines 98-104, you give the definition of the OT plan replacing the equality constraint on the marginals with the inequalities. While I understand that the purpose was to give a general definition applicable for the partial/robust OT cases, the explanation might be ambiguous for the reader and should be clarified. \n- I am also wondering why the paper lacks a ‘Conclusion’/’Discussion’ section with summarization of the achieved results in a general form. \n- Another minor concern corresponds to the lines 117-118 which operate with the notions of LP formulation/duality which were not given in the text before that.\n\nAnother restriction of the established theory corresponds to the usage of the $p$-Wasserstein distance in the partial/robust OT formulations. This point should be stated as a limitation of this work since general OT formulations might cover many other costs.\n\n**In summary,** I have doubts about the relevance of this paper for the current conference. The paper focuses on the theoretical results related to the  partial/robust OT formulations and does not include proper experimental validation of the suggested algorithms  (no experiments at all). Theoretical results might represent an interest for the ML community, but their motivation, ideas and originality should be clearly explained in the paper’s main text while now the exposition is very technical and hard to follow.  However, currently the exposition of the paper seems to be more suitable for a mathematical venue where the significance of the established theory (nearly 30 pages in Appendix) could be carefully checked.\n\nQUESTIONS:\n- Could you conduct some experiments justifying your theoretical conclusions? \n- Could you please provide a comment regarding the originality of your approach which follows the ideas of the one suggested in (Agarwal et al., 2024)?\n\nLIMITATIONS:\nThe limitations of the approach are stated only in the NeurIPS paper checklist. I kindly suggest the authors create a separate ‘Limitations’ section in the main text or Appendix of their paper and list the limitations there. One limitation is missing here - the paper establishes results only for the $p$-Wasserstein distance as the OT cost.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nPaper studies two robust formulations of optimal transport (OT) : $\\alpha$-optimal partial transport ($\\alpha$-OPT), which transports only a fraction of the mass of the two input distributions, and $\\lambda$-robust optimal transport ($\\lambda$-ROT), which penalizes untransported mass via a total variation regularization. While both variants have been well-studied in fully discrete settings, e.g. by Caffarelli and MacCann (ref [9] of the paper), this work proposes to extend them to semi-discrete OT. The authors introduce novel characterizations of optimal solutions using restricted weighted Voronoi diagrams, which captures the points of the Voronoi cells that are within a given distance.\n\nAuthors show that any solver for $\\lambda$-ROT can be leveraged to solve $\\alpha$-OPT, thus unifying the algorithmic treatment of these problems. They propose two algorithmic contributions: (1) a high-precision combinatorial algorithm for semi-discrete $\\lambda$-ROT using a refined cost-scaling approach (2) a near-linear-time approximation algorithm. Theoretical analysis includes correctness proofs, runtime bounds, and approximation guarantees even when only a fraction of the mass is transported.\n\nThe paper does not include empirical experiments as it is primarily theoretical in nature. Authors clearly articulate limitations, such as the exponential dependence on dimension.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper addresses an important problem: robust and partial OT in semi-discrete settings, for which no/few prior algorithms exists.\n- The characterizations via restricted Voronoi diagrams are very elegant and provide useful geometric insight.\n- The paper is exceptionally well-crafted, presenting the ideas clearly and accessibly despite the complexity of the subject matter.\n- Although I didn't check all the proofs, I think the algorithm and results are reasonable.\n\nWeaknesses:\n- While the contribution is theoretical in nature, numerical evaluations are not provided.\n- the connection between this work and Bourne et al., 2018, are not discussed\n\nBourne, D. P., Schmitzer, B., & Wirth, B. (2018). Semi-discrete unbalanced optimal transport and quantization. arXiv preprint arXiv:1808.01962.\n\nQUESTIONS:\n- Could you provide numerical experiments (even in simple cases) that could validate experimentally your algorithm and show its practical utility?\n- how does this work relates to Bourne et al., 2018?\n- the connection between $\\alpha$-OPT and $\\lambda$-ROT has been thoroughly examined in the work of Caffarelli and MacCann (ref [9] of the paper). Could you clarify how your results relate to their duality findings?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: SUMMARY:
This paper presents two categories of semi-discrete OT which are the α-partial transport problem and TV-regularized optimal transport formulation with regularization parameter λ. The two categories can be demonstrated in one common formulation. The theoretical analysis reveals a fundamental connection between these problems, demonstrating that their optimal solutions can be characterized as restricted Laguerre diagrams - a geometric insight that enables us to establish their algorithmic equivalence. Building on this theoretical foundation, novel computational algorithm are developed. A rigorous analysis of the algorithm's precision and computational complexity is presented.

STRENGTHS AND WEAKNESSES:
Strengths:
1. Provides a novel characterization of optimal solutions as restricted Laguerre diagrams, deepening the geometric understanding of partial semi optimal transportation with TV regularization. The theory is mathematically rigorous.
2. Establishes a strong equivalence between α-partial and λ-robust OT, enabling solvers for one problem to be adapted to the other.
3. The proposed method achieves exact solutions with guaranteed precision, while also providing an efficient approximation scheme, with their respective computational complexities rigorously analyzed. 
Weaknesses:
The theoretical findings lack empirical validation and practical substantiation.

QUESTIONS:
1. Could numerical experiments be incorporated to validate the algorithm's efficacy?
2. These two robust formulations of semi-discrete optimal transport provide mathematically grounded mechanisms. It will enhance noise resistance through distinct regularization paradigms. Could this theoretical claim be empirically validated through some experiments?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-03