Title: FRAM:  Frobenius-Regularized Assignment Matching with Mixed-Precision Computing
Venue: NeurIPS 2025
Accepted: yes
Keywords: graph matching

network alignment

quadratic assignment problem

linear assignment problem
Abstract: Graph matching, usually cast as a discrete Quadratic Assignment Problem (QAP), aims to identify correspondences between nodes in two graphs. Since QAP is NP-hard, many methods its discrete constraints by projecting the discrete feasible set onto its convex hull and solving the resulting continuous problem. However, these relaxations inevitably enlarge the feasible set and introduce two errors: sensitivity to numerical scales and geometric misalignment between the relaxed and original feasible domains. To address these issues, we propose a novel relaxation framework to reformulate the projection step as a Frobenius-Regularized Linear Assignment (FRA) problem. This formulation incorporates a tunable regularization term to curb the inflation of the feasible region and ensure numerical scale invariance. To solve the FRA efficiently, we introduce a scaling algorithm for doubly stochastic normalization. Leveraging its favorable computational properties, we design a theoretically grounded, accelerated mixed-precision algorithm. Building on these components, we propose Frobenius-Regularized Assignment Matching (FRAM), which approximates the QAP solution through a sequence of FRA problems. Extensive CPU experiments show that FRAM consistently outperforms all baselines. On GPUs, with mixed precision, FRAM achieves up to a 370× speedup over its FP64 CPU implementation without sacrificing accuracy.


Peer reviews:
comments: SUMMARY:
This paper proposes a novel algorithm that approximately solves the NP-hard quadratic assignment problem, a fundamental task for many AI-field applications. The method introduces a scaling factor, which can control the sharpness of relaxed matrix values and enables gradually reaching a solid permutation matrix.
In addition, it proposes a novel convergence criterion to stop the iteration at the right time. All results are theoretically supported and its superiority is experimentally proven, being approximately three times faster than the SOTA fastest method while maintaining comparable accuracy to the SOTA precise method.

STRENGTHS AND WEAKNESSES:
## Strength
1. Graph matching is a fundamental problem for the NeurIPS community.
2. The presentation is kind for many readers.
3. The method includes a theorem and its proof. 
4. The experimental results revealed that the method actually works faster, while maintaining SOTA performance.


## Weakness
### 1. Although well written, some sections are unclear.

- The reviewer is unsure if normalizing with $n$ in Eq. (12) is appropriate, given the non-linear complexity increase. Despite this, Eq. (12) claims to provide "a stable metric to assess the quality of …" (L142). A more precise justification is needed.

- “the regularization term” (L150) is not very specific. 

- In Fig. 3, the use of stacked bar charts makes it difficult to understand. This reviewer suggests using simple bar charts for clarity.

QUESTIONS:
- Could you clarify the items in Weakness 1?
- An error analysis could enhance the value of this work. Is there any possibility?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-06-27

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the problem of solving quadratic assignment problems via the convex relaxation of replacing the set of permutation matrices by the set of doubly stochastic matrices. It focuses on a specific algorithm for solving the resulting problem by first projecting the gradient of the cost function onto the set of doubly stochastic matrices and subsequently forming a convex combination of the projected gradient with the current iterate. In particular, it investigates scaling the cost function by a constant positive factor (which alters the iterates), and considers conducting different parts of the computation in different precision. Numerical results demonstrate that the resulting scheme is faster and yields lower cost functions than the compared competitors.\n\nSTRENGTHS AND WEAKNESSES:\nWhile the overall problem is interesting and investigations on different ways of solving the relaxation are interesting, the presented work has several structural weaknesses. \n1) It does not specify any properties of the cost function. When optimizing quadratic costs over the set of permutation matrices, $||P||_F^2 = n$ is a constant, such that it can be added or subtracted without changing the minimizer. Thus, for the optimization problem over permutation matrices, it does not matter if the costs are convex, concave or neither of the two. Yet, this property heavily affects the type of problem after relaxing the permutation matrices to doubly stochastic matrices, see \n- F. Fogel, R. Jenatton, F. Bach, and A. d’Aspremont. Convex Relaxations for Permutation Problems, NeurIPS 2013.\n- N. Dym, H. Maron, and Y. Lipman. DS++ - A flexible, scalable and provably tight relaxation for matching problems, ACM Transactions on Graphics (TOG), 2017.\n- F. Bernard, C. Theobalt and M. Moeller, DS*: Tighter Lifting-Free Convex Relaxations for Quadratic Matching Problems, CVPR 2018.\n\n for exemplary works that discuss and exploit this aspect. In case the quadratic costs are (strongly) convex, the particular choice of optimization algorithm should not matter in terms of the solution quality anymore – the minimizer of the relaxed problem is unique and can be computed by a wide range of different convex optimization algorithms. For a sufficiently concave objective, the minimizer over the doubly stochastic matrices will coincide with the minimizer over the permutations, thus making the relaxation exact, but also NP-hard to minimize. These aspects are neglected in the presented paper. It remains unclear if the claim is to solve a convex problem faster, or a non-convex problem heuristically better for common problem instances. \n\n2) Independent of the properties of the cost function, the subproblem of projecting the (scaled) gradient onto the set of doubly-stochastic matrices is a convex problem. This is well-known, yet the alternating projection algorithm (Alg. 2) takes a lot of room in the paper. Results like the projection of a variable onto a 2-dimensional linear subspace (row- and column-sum equal to one) and the projection onto the non-negative orthant are presented as a theorem despite being standard results from convex optimization. Accelerations of the alternating projection method (such as Dykstra’s algorithm) or ADMM/Douglas-Rachford splitting algorithms are not discussed. In this respect, the paper significantly lacks context in the fundamentals of convex optimization. \n\n3) The aforementioned lack of context is also reflected in phrasing several very simple results from convex analysis (such as theorems 1, 2 and 4) as theorems, partially without a proper contextualization such as the considered algorithm for $\\theta=\\infty$ becoming the Frank-Wolfe Algorithm. \n\n4) In my opinion, the paper is not well-written and confusing in several aspects. For example, \n- “the numerical scale invariance of the quadratic objective, which is destroyed under the projections”  is irritating. I do not know what is meant by “numerical scale invariance of the quadratic objective”. The minimizer of any cost function remains unchanged when multiplying the costs by a positive number. If the costs are convex, then the projection does not “destroy” this property. The iterates might change, but still converge to the same point (under suitable conditions/algorithm parameters). \n- Writing that DSN finds \"the nearest doubly stochastic matrix\", and Lu et al. “adapt the DSN so that the projected gradient matrix is doubly stochastic” is irritating. What is the difference? It would be better to formalize this.\n- As discussed above, I do not think that inequality (5) is a good argument to say that some scaling invariance is destroyed. \n- $D^*$ in Theorem 3 should be defined mathematically. The current description remains a little vague. \n- Alg. (2) uses the max(X,0) for the projection onto the non-negative orthant. I recommend using the same in eq. (16). They are of course identical, but having consistency would be clearer. \n- In Theorem 6 the term “convergence rate constant” crucially needs to be defined. \n- The phrase that  the “truncation residual vanishes over iterations in SDSN” is unclear to me and should be rephrased/detailed. Similarly $\\Delta X_k$ should be related to quantities defined in the algorithm. The current version is difficult to follow.\n- The costs in eq. (18) are not identical (up to a constant) to the original costs that are minimized. Squares are missing at the Frobenius norms. Have these been included in the numerical experiments? \n\n5) Considering that one of the main technical contributions is rescaling the cost function (so far in a heuristic way/via a hyperparameter as admitted in the conclusions), I consider the contribution to be somewhat limited. The mixed precision computation is interesting, but needs to be detailed. Running competing algorithms at lower precision is – to my mind – a crucial baseline.\n\nQUESTIONS:\n- Please detail the setting of the relaxation, and put the presented work into context of the literature on such relaxations  (see 1) above).\n- Please explain the novelty in Algorithm 2, comment on why alternating projections instead of algorithms like Dykstra/ADMM with a common stopping criterion like the primal-dual gap was used (see 2) above).\n- Please revisit the presentation of the results incorporating the suggestions in 4) above.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-29"}

comments: SUMMARY:
This work presents a method for graph matching by reformulating projection-based regularizations as linear assignment problems. Doing so, the authors devise an effective and elegant algorithm to control the relaxation. The method is fast without compensating for substantial accuracy. Moreover, the paper introduces a mixed-precision architecture that seems novel and is validated to contribute to the stability of the approach. Overall, this work is a well-written contribution that, although niche in its application, might be interesting to the broader field and could spur further applications beyond graph matching.

STRENGTHS AND WEAKNESSES:
Strength: 
- The paper makes a nice theoretical contribution with the formulation of a linear assignment in the projection step for graph matching. The approach is core to a graph matching algorithm by iteratively solving a regularized linear assignment. This is shown to be a fast and effective method.
- The mixed-precision scheme proposed in the paper is a further well-crafted implementation detail that is shown to be effective and well-grounded.
- The manuscript is well written and covers the body of existing work adequately, which allows the authors to situate their work in the landscape of prior work.
- The convergence proofs, although not extensive, are convincing.
- The evaluation leaves no gaps and the experiments are confirming the effectiveness of the method.

Weaknesses: 
- While the method may potentially be broadly applicable, the application is limited to graph matching in this manuscript as a rather niche paper. 
- Although this is adequately evaluated, there is cost in accuracy.

QUESTIONS:
- What are the limits of the precision? At what point does the accuracy break down dramatically?

LIMITATIONS:
None
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a novel algorithm for graph matching, reformulating the projection step as a Frobenius-regularized linear assignment (FRA) problem to address the NP-hardness of the Quadratic Assignment Problem (QAP), mitigating errors from feasible region inflation and numerical scale sensitivity via a tunable regularization term. It introduces the Scaling Doubly Stochastic Normalization (SDSN) algorithm to solve FRA efficiently. Furthermore, a theoretically grounded mixed-precision architecture that achieves acceleration on GPU with negligible accuracy loss compared to CPU. The approach outperforms baselines in accuracy and efficiency across various datasets, and provides theoretical analyses of regularization effects, convergence, and error bounds, marking the first graph matching algorithm with a theoretically sound mixed-precision design.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n\nS1. This paper improves the popular DSPFP methods by reformulating the projection step as an FRA problem, extending the selection of projections such that improve the solution. Corresponding solution, SDNS algorithm, is also proposed.\n\nS2. The paper provides a solid theoretical analysis of the proposed FRA problem and SDSN solution, including the impact of the regularization parameter, convergence properties, and error bounds. \n\nS3. Extensive experiments on real-world datasets demonstrate that FRAM outperforms baselines in both accuracy and speed. \n\nWeakness:\n\nW1. In my comprehension, the proposed FRA problem introduces a family of projection problems, which should lead to equivalent graph matching, but actually results differently due to the projection. In other words, the FRA introduces a family of alternatives in the projection step. Hence, the issue of my most concern is how to select the proper FRA problem, in other words, the parameter $\\theta$. However, such strategies are not discussed in the paper.\n\nW2. The paper only conducts the experiments with $\\lambda = 1$. It remains unclear whether the value of $\\lambda$ will influence the performance of graph matching algorithms.\n\nW3. The $O(n^2)$ space complexity leads to the worry about the scalability of the proposed FRAM method.\n\nQUESTIONS:\nQ1. It seems that each iteration in Algorithm~2 takes $O(n^2)$ time complexity, and the number of iterations is related to more parameters rather than $n$. So the complexity of Step~6 in Algorithm~1 seems not accurate, which also influences the total time complexity of Algorithm~1.\n\nQ2. What does the matching error refer to? I guess it should be the accuracy defined in Equation~(18), so please use a consistent terminology in the paper. \n\nQ3. What does $K$ refer to in Algorithm~1? I guess it should be the $F\\tilde{F}^T$. Please introduce it in Algorithm~1.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}