Title: PDPO: Parametric Density Path Optimization
Venue: NeurIPS 2025
Accepted: yes
Keywords: Optimal Transport

Generalized Schrodinger Bridges

Lagrantian Costs
Abstract: We introduce Parametric Density Path Optimization (PDPO), a novel method for computing action-minimizing paths between probability densities. The core idea is to represent the target probability path as the pushforward of a reference density through a parametric map, transforming the original infinite-dimensional optimization over densities to a finite-dimensional one over the parameters of the map. We derive a static formulation of the dynamic problem of action minimization and propose cubic spline interpolation of the path in parameter space to solve the static problem. Theoretically, we establish an error bound of the action under proper assumptions on the regularity of the parameter path. Empirically, we find that using 3–5 control points of the spline interpolation suffices to accurately resolve both multimodal and high-dimensional problems. We demonstrate that
PDPO can flexibly accommodate a wide range of potential terms, including those modeling obstacles, mean-field interactions, stochastic control, and higher-order dynamics. Our method outperforms existing state-of-the-art approaches in benchmark tasks, demonstrating superior computational efficiency and solution quality.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Parametric Density Path Optimization (PDPO), a framework that casts the infinite‑dimensional least‑action problem over density curves into a finite‑dimensional spline optimization in the parameter space of a push‑forward map. The authors select neural ODEs for the map, discretize the parameter curve with cubic Hermite splines, and prove that using only 3 – 5 control points suffices to keep the action error $O(h^{\\kappa-1})$.  Optimization proceeds by alternating a “path step’’ on interior control points with a “boundary step’’ on the endpoints (Algorithms 1‑3).  Experiments on obstacle‑avoidance transport, generalized momentum minimization, acceleration‑regularized dynamics, and a 1000‑d opinion‑depolarisation benchmark show 7–39 % lower action and up to 10× faster wall‑time than GSBM, NLOT and APAC‑Net in the authors’ setup.\n\nSTRENGTHS AND WEAKNESSES:\n- The paper addresses an important problem in optimal transport and provides a computationally efficient alternative to existing methods that struggle with high-dimensional settings.\n- I think the paper’s strongest point is the clean static‑coupling reformulation (Eq. 11) and its parametric version (Eq. 13) that lets one exploit off‑the‑shelf autodiff once everything lives in the parameter space. The spline error bound is convenient and, to my knowledge, new for neural push‑forward families.  \n- Empirically PDPO is much faster than GSBM on the same GPU while matching or improving action and endpoint fidelity, especially in the 1 000‑d test where runtime drops from five hours to 23 minutes.\n\nQuality‑wise, however, several issues remain:\n \n- First, the dynamic/static equivalence (Theorem 1) merely states without the assumptions on the regularity of $\\pi$. Moreover, the non‑emptiness of $\\Gamma(\\pi)$ looks restrictive but is not discussed in the main text, so I am not sure how often the theory covers the practical neural‑ODE setting. There are also no convergence guarantees for the proposed bilevel optimization procedure, which alternates between updating interior control points and boundary parameters, which I think is critical.\n\n- Second, the experimental evaluation, while showing improvements, has notable limitations. Many experiments only compare against GSBM, and the baseline comparisons may not be entirely fair due to different implementations and computational frameworks. The statistical analysis is minimal - using only three random seeds.\n\n- Third, from a technical standpoint, the method requires careful initialization of boundary parameters, and the authors acknowledge this is done \"outside of the main algorithm.\" This suggests the approach may be sensitive to initialization and require problem-specific tuning. The scalability analysis reveals concerning computational complexity of $O(N \\times M \\times (1 + d))$ for problems involving Fisher information terms, which severely limits applicability to high-dimensional problems. \n\n- Fourth, I find this statement in Section 3.3 troublesome: **\"We in general consider this step outside of the main algorithm. A parameter can be used for any functional, so once a boundary parameter has been obtained, it can be reused across multiple problems.\"** I think this makes the runtime comparison completely unfair.\n- Finally, the paper also lacks important details about convergence criteria and stopping conditions for the alternating optimization procedure. The claim that \"as long as the action is lower bounded, step 2 will converge to a local minimum\" is insufficiently rigorous without proper analysis of the optimization landscape.\n\n- Clarity-wise, the authors have done a decent job. Just one point that I am not a big fan of putting the algorithm boxes in the Appendix, especially with the split of supplementary material for NeuRIPS submission.\n\nQUESTIONS:\n- Is it possible to  provide theoretical guarantees for the convergence of the alternating optimization procedure? Under what conditions does the bilevel optimization converge to a global or local optimum? The current analysis is insufficient for a rigorous optimization method.\n- How sensitive is the method to the initialization of boundary parameters? Can you provide a systematic analysis of when the Flow Matching initialization will succeed, and what happens when it fails to find good boundary parameters?\n- Computational Complexity: The $O(N \\times M \\times (1 + d))$ scaling for Fisher information terms seems prohibitive for truly high-dimensional problems. Can you provide a more detailed computational analysis and comparison with baselines using the same computational resources and implementation quality?\n- Can you provide more rigorous statistical analysis including confidence intervals, significance tests, and fair computational comparisons? The current evaluation with only three seeds and potentially unfair runtime comparisons undermines the claimed performance improvements. Please also answer the point raised on taking initialization into runtime as I raised above.\n- What are the precise assumptions required for the two theorems , and how realistic are they in practice? When do the error bounds become meaningful, and how do they relate to the quality of the final solution?\n\nLIMITATIONS:\nYes but not enough.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe proposed method, called Parametric Density Path Optimization (PDPO), utilizes an action minimization framework incorporating additional constraints such as external potential energy and interaction and internal energy to find an action-minimizing path between probability densities. This generalizes the classical approach used in the case of Wasserstein distances which can also be seen via the action-minimizing viewpoint.\n\nThe authors work with a parameterization which allows for a finite dimensional representation making the action minimization problem computationally tractable. The authors furthermore present a theorem establishing the equivalence between a  static and dynamic representation viewpoint of their constrained action-minimization approach (Theorem 1).\n\nA cubic Hermite spline approximation of parameter space curves together with a neural ODE approach is then used to solve the optimization problem practically and an error estimate for the spline-based approximation is provided (Theorem 2).\n\nExperiments are presented that include two-dimensional settings of obstacle avoidance problems, showing improved action cost and significantly reduced computation time vs baseline methods.\nThe authors also extend their approach to incorporate an acceleration term and show 2d examples where PDPO can find solutions also in this setting. Additionally, a 1000-dimensional case of opinion depolarization problem is considered where the method again has improved computational time as compared to the baseline GSBM approach.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- overall, the paper is well-written and clearly describes the chosen problem and solution approach\n\n- significant improvements in computational complexity and improvements or matching performance in minimization of the objective function over the baseline approaches appear empirically supported by the presented experiments.\n\n- the work tackles a rather general action minimization path finding problem between probability densities of relevance to several application domains in the NeurIPS community.\n\n- the paper presents both the authors' theoretical insights and formulation (e.g. Theorem 1 and 2) as well as a practical algorithm and implementation that combines a neural ODE formulation in combination with the proposed parameterize to arrive at a practically feasible solution.\n\nWeaknesses:\n- The chosen obstacle avoidance scenarios are interesting, but may at present still be quite far from the required real-world complexity to be useful in end-applications such as in robotics, where much larger maps with significant detail would likely have to be considered to be of practical relevance. \n- To assess the practical significance of the proposed approach it may be helpful to showcase an experiment with much larger problem complexity as well.\n\nQUESTIONS:\n- The current 2D maps you considered still have a relatively simple spatial structure. Do you think your method will generalize to complex maps directly, or do you see challenges in convergence as problem complexity is increased?\n\nLIMITATIONS:\n- It would be interesting to add some discussions of potential failure cases of the proposed method as part of the limitations. An experiment where the approach could not be successfully applied yet or where other challenges of interest arise may be helpful to guide readers towards open problems within this area.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Parametric Density Path Optimization (PDPO), a novel method for calculating the optimal, action-minimizing paths between different probability densities. The core innovation is to reframe this complex, infinite-dimensional problem into a more manageable, finite-dimensional one. It achieves this by representing the probability path as the transformation (or \"pushforward\") of a simple, fixed reference density through a parametric map, which is modeled by a Neural ODE. The path is then optimized by interpolating a small number of control points (typically 3-5) in the parameter space using cubic Hermite splines, which defines the continuous evolution of the density.\n\nPDPO demonstrates superior performance compared to existing state-of-the-art methods across several challenging benchmark tasks, including obstacle avoidance, entropy-constrained transport, and high-dimensional opinion dynamics. The method is not only more computationally efficient, achieving 40-80% faster runtimes, but it also produces higher-quality solutions, with 7-39% lower action and up to 10 times better boundary accuracy. A key strength of PDPO is its flexibility, as it can be applied to a wide range of problems involving external potentials, mean-field interactions, and higher-order dynamics, such as acceleration minimization, within a single, unified framework.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n* **Novel and Theoretically-Grounded Formulation:** The paper introduces Parametric Density Path Optimization (PDPO), a novel method that reframes a complex infinite-dimensional optimization problem into a more tractable finite-dimensional one. The core strength lies in its static formulation of the action-minimizing problem, which the authors claim is novel for handling internal and interaction energy terms, unlike previous work. This new formulation is supported by theoretical guarantees, including a theorem proving its equivalence to the classical dynamic formulation (Theorem 1) and an error bound for the cubic spline approximation (Theorem 2).\n\n* **Superior Performance and Efficiency:** The paper provides strong empirical evidence that PDPO outperforms state-of-the-art methods in benchmark tasks.\n    * **Accuracy:** It achieves 7-39% lower action costs and up to 10x better boundary accuracy compared to baselines like GSBM and NLOT.\n    * **Speed:** It demonstrates significantly faster runtimes, ranging from 40-80% on some tasks to over 10x faster in high-dimensional cases (e.g., nearly 2 hours for GSBM vs. under 10 minutes for PDPO in the GMM experiment, and over 5 hours vs. 23 minutes for the opinion dynamics problem).\n    * **Simplicity:** A key advantage is its ability to resolve complex, high-dimensional problems accurately using only 3-5 control points for the spline interpolation.\n\n* **Flexibility and Broad Applicability:** PDPO is designed as a general framework that can flexibly accommodate a wide variety of problems within a unified optimization scheme. It is shown to be effective for tasks involving:\n    * External potentials and obstacles.\n    * Stochastic Optimal Control (SOC) through a Fisher Information term.\n    * Mean-field interactions and congestion.\n    * Higher-order dynamics, such as acceleration-regularized problems, which the authors claim PDPO is the first method capable of solving in this context.\n\n### Weaknesses\n\n* **Potentially Unfair Comparison Due to Initialization:** The proposed optimization strategy begins with an efficient initialization of the boundary parameters using Flow Matching (FM). While effective, this raises a question about the fairness of the performance comparison. The significant speed-up over GSBM might be partially due to this specific initialization strategy rather than solely the core PDPO framework. The paper does not provide an ablation study or results for a GSBM baseline enhanced with a similar FM initialization, which would be necessary to isolate the true source of the efficiency gains.\n\n* **Reliance on Qualitative Visual Evidence for Key Claims:** In the obstacle avoidance experiment (Figure 2), the paper claims that competing methods violate the constraints while PDPO does not. However, visual inspection of the PDPO result (Figure 2b) suggests that some samples may still lie within the obstacle. The paper does not provide a quantitative measure of constraint violation (e.g., the percentage of probability mass inside the obstacle) to substantiate its claim of superiority, relying instead on visual interpretation which can be ambiguous.\n\n* **Acknowledged Scalability and Architectural Limitations:** The authors are transparent about certain limitations of the current method.\n    * **Scalability in SOC:** The computational cost for problems involving entropy and Fisher Information increases substantially with the problem's dimension. The authors note that the highest-dimensional SOC problem they successfully computed was 50-dimensional, implying that scalability in this domain remains a challenge.\n    * **Model Architecture:** The current implementation uses Neural ODEs with MLP architectures. This restricts its direct application to other data modalities, such as images, where methods like GSBM can use more suitable architectures (e.g., U-Nets). The authors explicitly state this as a limitation for image-based transport problems.\n\nQUESTIONS:\n### **1. On the Source of Computational Efficiency**\n\nRegarding the computational efficiency of PDPO compared to GSBM, could you clarify the contribution of the **Flow Matching (FM) initialization**? The paper notes that PDPO achieves significantly faster runtimes. Is it possible that this speed-up is largely attributable to the FM initialization step? To help disentangle the gains from the core PDPO framework versus the initialization strategy, it would be insightful to see a comparison where the GSBM baseline is also initialized using a similar FM-based approach.\n\n***\n\n### **2. Clarification on Constraint Violation in Figure 2**\n\nIn the obstacle avoidance experiment shown in Figure 2, the paper claims that both GSBM (2c) and APAC-Net (2d) violate the obstacle constraints, while PDPO effectively avoids them. However, on visual inspection, the optimized PDPO path (2b) also appears to have some samples crossing into the obstacle regions. Could you please provide a **quantitative measure of the constraint violation** (e.g., the percentage of the mass inside the obstacle) for all three methods? This would offer a more objective comparison of their performance on this task.\n\n***\n\n### **3. Request for Concrete Implementation Details**\n\nTo improve the paper's reproducibility and provide a clearer understanding of the methodology, could you offer more concrete implementation details or a working example for two key aspects of the framework?\n\n* **Parameterization of the Pushforward Map**: While the paper states that the map $T_{\\theta}$ is a Neural ODE, a more detailed description of the network architecture ($v_{\\theta}$) and how its parameters $\\theta$ are structured would be very helpful.\n* **Flow Matching Initialization**: A step-by-step walkthrough of how Flow Matching is employed to initialize the boundary parameters $\\theta_0$ and $\\theta_1$ would greatly clarify the practical application of this crucial first step.\n\nLIMITATIONS:\n### Scalability in High-Dimensional Stochastic Optimal Control (SOC)\n\nA primary limitation is the method's computational cost when dealing with problems that involve Fisher Information, which is used to solve Stochastic Optimal Control (SOC) problems.\n\n* The cost to compute the action functional increases substantially with the problem's dimension, `d`.\n* Calculating the Fisher Information requires solving a separate `d`-dimensional Ordinary Differential Equation (ODE).\n* The authors state that a 50-dimensional Schrödinger Bridge was the \"highest-dimensional case with Fisher Information we have successfully computed,\" highlighting a practical cap on its scalability for this class of problems.\n\n***\n\n### Architectural Restrictions\n\nThe current implementation of PDPO has architectural constraints that limit its range of application.\n\n* The framework uses Neural ODEs (NODEs) built with Multilayer Perceptron (MLP) architectures.\n* This reliance on MLPs restricts the method's direct use for certain data types, particularly images, where specialized architectures like U-Nets are more effective.\n* This is a notable limitation compared to methods like GSBM, which can be applied to image-based transport problems.\n\n***\n\n### Methodological and Evaluative Limitations\n\nThe methodology and evaluation presented in the paper have some implicit limitations.\n\n* **Dependence on Initialization:** The three-step optimization process relies on a strong initial approximation of the boundary distributions using Flow Matching (FM). While effective, this also means the final solution's quality is dependent on this prerequisite step.\n* **Potential for Unfair Comparison:** The performance gains reported over baselines like GSBM could be partially attributed to the specialized FM initialization. The paper does not provide an ablation study comparing performance when baselines are also given the benefit of a similar initialization, which makes it difficult to isolate the gains from the core PDPO algorithm alone.\n* **Convergence to Local Minima:** The optimization algorithm, being gradient-based, is guaranteed to converge to a local minimum, not necessarily a global one. The final solution is therefore sensitive to the initial path, again underscoring the importance of the initialization procedure.\n* **Qualitative Evaluation:** In some experiments, such as obstacle avoidance, the claims of superiority are based on visual inspection of plots. The paper does not provide quantitative metrics for constraint violation (e.g., the amount of probability mass inside an obstacle), relying on a qualitative assessment that can be subjective.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a method for the optimisation of paths followed by probability distributions under trajectory constraints. The proposed approach transforms the original Wasserstein-2 optimal transport problem into an equivalent dynamic reinterpretation and formulates a constrained least-action optimisation objective. This dynamic formulation more easily allows for the incorporation of additional trajectory constraints. Probability distributions and their paths and their paths are represented by parametric models. Cubic splines parameterise the paths, with endpoint densities given by flow matching models to approximate the target distributions and intermediate points approximated via neural ODEs. Theoretical results regarding the equivalence of the formulations and the approximation error are derived. Experiments on synthetic benchmarks evaluate the algorithms against the most relevant baselines in the area.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n* The paper tackles a hard problem with an interesting solution that connects different fields.\n\n* Theoretical guarantees are provided.\n\n* The discussion of experimental results seems to be solid, insightful and to go into enough depth for a conference paper.\n\n* Significant computational gains with competitive performance are demonstrated.\n\n\n### Weaknesses\n\n* The paper is quite mathematically heavy at times with notation that turns out a bit loosely defined and a few potential mistakes.\n\n    - In the introduction and subsequent restatements, the action functional equation (1) has a minimisation over two function-valued variables $\\rho$ and $v$, but they are indexed (under the $\\inf$) as scalars representing their point evaluations $\\rho(t, x)$ and $v(t, x)$, which is confusing.\n    - How can a joint probability density $\\pi$ be equated to a point tuple as $\\pi = (x, T(_\\pi(x))$? These are two objects of different nature. The same apparent mistake occurs first between lines 134 and 135 and it]s then repeated throughout the text. The set $\\tilde{\\Pi}$ they define then appears in Eq. 11, which is part of the result in Theorem 1, making the latter seem invalid.\n\n* \"Opinion\" dynamics is involved in the experiments, but the term is never explained or defined in the main text.\n\n* Quite a few of the references to the appendix are invalid in the main text, though fixed in the supplementary material.\n\nQUESTIONS:\n* The final algorithms have quite a few components and moving parts (flow matching models, neural ODEs, cubic spline parameters, Monte Carlo estimators for entropy, Fisher Information, etc.). How are hyper-parameters tuned? The appendix presents an ablation study for the number of control points, but the paper would benefit from a discussion on how sensitive the method would be to the choice of models involved and the machinery involved in the numerical integration and optimisation steps.\n\n* With the spline-based framework, the method seems flexible enough to incorporate more than 2 fixed endpoints. Have the authors considered problems where intermediate points also need to satisfy hard constraints?\n\n* Can you revise and explain the definition of $\\tilde{\\Pi}$?\n\nLIMITATIONS:\nLimitations are quite briefly mentioned in the conclusion, without much detail or discussion.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-05"}