Title: Unbalanced Optimal Total Variation Transport: A Theoretical Approach to Spatial Resource Allocation Problems
Venue: NeurIPS 2025
Accepted: yes
Keywords: Unbalanced optimal transport; Total variation; Spatial resource allocation; Theory
Abstract: We propose and analyze a new class of unbalanced weak optimal transport (OT) problems with total variation penalties, motivated by spatial resource allocation tasks. Unlike classical OT, our framework accommodates general unbalanced nonnegative measures and incorporates cost objectives that directly capture operational trade-offs between transport cost and supply–demand mismatch. In the general setting, we establish the existence of optimal solutions and a dual formulation. We then focus on the semi-discrete setting, where one measure is discrete and the other is absolutely continuous, a structure relevant to applications such as service area partitioning for facilities like schools or medical stations. Exploiting a tessellation-based structure, we derive the corresponding explicit optimality conditions. 
We further address a quantization problem that jointly optimizes the locations and weights of discrete support points, applicable to facility location tasks such as the cost-efficient deployment of battery swap stations or e-commerce warehouses, informed by demand-side data. The dual-tessellation structure also yields explicit gradient expressions, enabling efficient numerical optimization in finite dimensions.


Peer reviews:
comments: SUMMARY:
The paper investigates an unbalanced, barycentric (i.e., weak), total variation-regularized optimal transport formulation. The authors show the existence of a solution for the considered problem and derive its dual form. They then discuss the corresponding semi-discrete setting, showing that it is equivalent to an optimization problem on Laguerre diagrams and providing its optimality conditions. They further extend this setting into optimizing over the space of discrete measures (i.e., weights and support points), which is inspired by the problem of spatial resource allocation, and provide an alternating scheme for solving it. Experimental results regarding those semi-discrete settings are carried out on synthetic data.

STRENGTHS AND WEAKNESSES:
Strengths:
- The paper is well-written and well-presented.
- The proposed problem is new, and so are the relevant results.

Weaknesses:
- The motivation and the final problem of the paper aim at the practical problem of spatial resource allocation, but there is little discussion about the related work on this topic, as well as the experiment results are only on synthetic data and without comparison to any other method.
- The alternating scheme might converge only to a local minima, as the loss seems to be non-convex.

QUESTIONS:
- What is the importance of considering weak optimal transport in the proposed problem? I do not see its relevance to the allocation problem. Is it for the technicality of TV regularizers?
- The paper mentions entropic regularization several times in Section 2, but the problem of interest is not entropy-regularized, as well as the formulation written in Line 101.
- Typo: $\phi$ instead of $\varphi$ in Line 191

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a novel semi-discrete, unbalanced weak optimal transport model with total variation penalties for spatial resource allocation problems. It provides theoretical analysis and extends classical transport frameworks to non-smooth, TV-regularized settings. Experiments demonstrate promising results on small-scale problems, but scalability and hyperparameter sensitivity warrant further investigation.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**\n\n1. The authors model a novel semi-discrete, unbalanced weak optimal transport problem with total variation penalties, motivated by practical spatial resource allocation problems.\n2. For this problem, the authors conduct theoretical analyses on the existence of optimal solutions and tessellation-based reformulations, and extend classical entropy-regularized transport frameworks to the non-smooth, TV-penalized setting.\n\n**Weakness**\n\n1. The experiments are conducted only on small-scale applications. Although the results are promising, the performance and convergence on larger-scale problems remain to be verified.\n2. Based on my previous research experience, I have often found that theoretically sound methods may have limited practical effectiveness, and may require careful tuning of hyperparameters to ensure convergence and satisfactory performance. The experiments in this paper also involve various hyperparameters; more discussion is needed regarding how they are selected and their impact on performance.\n\nQUESTIONS:\n1. The experimental evaluation is limited to small-scale applications. Have the authors considered testing their method on larger-scale datasets or more complex scenarios to further assess its scalability, performance, and convergence properties?\n2. The paper involves several hyperparameters in the proposed method. Could the authors elaborate on the process of hyperparameter selection and provide an analysis of how sensitive the algorithm's performance is to these choices?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 1, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a formulation of unbalanced weak optimal transport incorporating total variation (TV) penalties. Motivated by resource allocation problems with asymmetric over- and under-supply/demand costs, the authors develop a general theoretical framework encompassing existence results and dual formulation. Focusing on the semi-discrete setting—where demand is modeled as a continuous measure and supply as a finite set of discrete locations—the authors develop a tessellation-based reformulation. This yields a dual optimization problem defined over generalized Laguerre cells. They also extend the framework to a quantization problem in which the locations and weights of the discrete supply points are jointly optimized. \nThe paper presents illustrative examples demonstrating the method’s ability to produce spatially meaningful solutions for simple supply area division and spatial ressource allocation problems.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n- The unification of weak OT and unbalanced OT under a TV penalty is original and well-motivated.\n- The paper is clearly written, with proofs and intuitive explanations that make the technical contributions accessible.\n- The application to spatial allocation and facility location is convincing to some extent\n\nWeaknesses:\n- experimental validation: while the examples are illustrative, the empirical evaluation could be expanded. For instance, comparisons with existing entropy-regularized unbalanced OT or standard quantization approaches would strengthen the claims of practical benefit. In addition, the computational complexity of the optimization procedures, especially in higher dimensions or with large discrete supports, could be more thoroughly analyzed\n- motivation: while the use of a semi-discrete formulation with mass relaxation makes sense in the application that are described in the spatial ressource allocation problem, the motivation and advantage of instroducing weak OT into the formulation is unsufficiently discussed.\n- some formulations and results differ from the standard UOT with TV penalty setting, which is not discussed (see questions below)\n\nQUESTIONS:\n- When using a TV penalty in a discrete UOT formulation, it is unclear how mass creation can appear (it is always cheaper to remove mass than adding some mass). Could you give an example for which it happens in your formulation (eq. 1 and 2)?\n- How does your method compares to other semi-discrete unbalanced OT methods in the application scenario you have considered?\n- How does the setting in section 4.2, where the continuous formulation is sampled, is a semi-discrete problem?\n\n\nOther comments:\n- l123-125: could you provide a reference?\n- symbol $^+$ should be defined\n- l140: $k$ should be defined\n- l233: $d$ should be defined\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a new family of unbalanced weak optimal transport with weighted total variation penalties with a focus on the semi-discrete setting. They present a comprehensive theoretical analysis that yields (1) the dual formulation for the general problem, and in the special semi-discrete cases (2) a finite-dimensional tessellation reformulation (corresponding to service area partition) that can be solved via gradient methods, and (3) a sequential optimization strategy for optimal spatial resource allocation. Illustrative numerical examples well showcase the motivations of the OT problem as well as its numerical tractability.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n1. The paper is easy to follow. It is very well motivated and self-contained, with a clear structure, mathematical rigor, and accompanied by intuitive interpretations of abstract concepts.\n2. The proposed OT problem looks important and can serve a unified framework for several important real-world problems. \n3. The theory is comprehensive and elegant with clear citations showing connections with related literature. They look correct, though detailed proofs are not checked with confidence as the reviewer has insufficient knowledge of deep theories. Provided gradient derivations are helpful to develop quick algorithms. \n\n**Weaknesses**\n1. The experiments are relatively toy level, though I agree that the paper’s focus is more theoretical. It would be interesting to see how scaling up the problem size affects the efficiency of the presented gradient methods. \n2. The applications considered are relatively restricted. The example presented in Example 2.1 is not too different from the field of the motivating resource allocation problems. It would definitely be interesting to see if applications in other fields may be casted to the considered OT problem.\n\nQUESTIONS:\nMy questions are mostly related to numerical convergence. \n1. You set a decaying step size in Section 4, which is pretty understandable for the piecewise nonsmooth objective. However, I am concerned that if you scale up the problem size, such decaying step design may converge only suboptimally.  Did you try different problem sizes? How bad is the condition number of the problem as we scale it up?\n2. While the sequential or alternating method looks viable in Sec 3.3, no convergence analysis is provided. I don’t see this finite-dimensional problem still preserving convexity with respect to both locations and masses. Can you comment on this a bit? I would imagine its performance resembles the EM algorithm for gaussian mixtures. \n\nThe last question is a bit more general.\n\n3. Can you comment on the significance of recasting these familiar problems including resource allocation commonly appears in operations research literature into an optimal transport formulation? What are the key pros and cons compared to a classical linear program approach?\n\nLIMITATIONS:\nYes, the authors acknowledged insufficiency of additional applications and discussed future work in the last two sections.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper develops a semi-discrete formulation of weak optimal transport where the discrepancies between marginal distributions are regularized through a TV term.  And this work proves the existence of an optimal plan and gives a dual characterisation whose variables coincide with the weights that generate Voronoi diagram. They proposed an iterative weight–update algorithm for the case of a continuous demand distribution and finitely many depots.  They applied their methods on two small two dimensional demos to shows how the method allocates spatial resources.\n\nSTRENGTHS AND WEAKNESSES:\nThe theoretical proof is good, optimal existence and duality are established, the notation is clear and consistence, and the link between the dual weights and Laguerre tessellations is well explained.  But the bridge to Laguerre diagrams is not novel, having been analysed in detail in recent work about semi-OT by Agarwal et al. (SODA 2024, NeurIPS 2024), so the main incremental value appears to be the introduction of a total\\-variation penalty within the semi-OT framework. The manuscript lists four extensions—weak transport cost, unbalanced marginal constraints, total\\-variation regularisation and semi\\-discrete geometry—but their joint motivation is not so clear, kind confusing why all four are needed together. And, the terminology \"weak OT\" and \"one–sided unbalanced OT\" are sometimes conflates, which may confuse readers. Empirical support is limited to small toy instances, with no complexity analysis, convergence guarantee or sensitivity study.\n\n- Agarwal, Pankaj K., et al. \"Fast and accurate approximations of the optimal transport in semi-discrete and discrete settings.\" Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). Society for Industrial and Applied Mathematics, 2024.\n- Agarwal, Pankaj, et al. \"A combinatorial algorithm for the semi-discrete optimal transport problem.\" Advances in Neural Information Processing Systems 37 (2024): 25857-25887.\n\nQUESTIONS:\nCould you position your algorithm more explicitly against the approaches of Agarwal et al.?  Do you have either theoretical bounds or empirical results on the convergence rate and scalability of the weight–update scheme?  Why was total variation selected over other divergences commonly used in OT?  Would you clarify the distinction you intend between weak and unbalanced OT, and explain whether your formulation is essentially the standard one–sided unbalanced model?\n\nLIMITATIONS:\nna", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-05"}