Title: Effective Neural Approximations for Geometric Optimization Problems
Venue: NeurIPS 2025
Accepted: yes
Keywords: geometric deep learning

computational geometry

neural networks

point clouds
Abstract: Neural networks offer a promising data-driven approach to tackle computationally challenging optimization problems. 
In this work, we introduce neural approximation frameworks for a family of geometric "extent measure" problems, including shape-fitting descriptors (e.g. minimum enclosing ball or annulus). 
Central to our approach is the \textit{alignment} of our neural model with a new variant of the classical $\varepsilon$-kernel technique from computational geometry. 
In particular, we develop a new relaxed-$\varepsilon$-kernel theory that maintains the approximation guarantees of the classical $\varepsilon$-kernels but with the crucial benefit that it can be easily implemented with \textit{bounded model complexity} (i.e, constant number of parameters) by the simple SumFormer neural network. 
This leads to a simple neural model to approximate objects such as the directional width of any input point set, and empirically shows excellent out-of-distribution generalization. 
Many geometric extent measures, such as the minimum enclosing spherical shell, cannot be directly captured by $\varepsilon$-kernels. 
To this end, we show that an encode-process-decode framework with our kernel approximating NN used as the ``process'' module can approximate such extent measures, again, with bounded model complexity where parameters scale only with the approximation error $\varepsilon$ and not the size of the input set. 
Empirical results on diverse point‐cloud datasets demonstrate the practical performance of our models.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel approach for approximating \"extent measures\" in computational geometry with neural networks. It develops a \"relaxed $\\epsilon$ theory\" which can be implemented using a neural network (SumFormer) with a bound on the number of parameters. The framework encodes input points into a latent space where they are processed using a SumFormer-based model. The model size is only dependent on the input point dimension, not on the number of points in the set, and a bound on the model parameters, depending on a user-specified approximation error, is derived.\n\nThis latent space model is then tested on both synthetic data for validation and real datasets (SQUID [a fish boundary contour dataset] and ModelNet). The experiments test multiple latent processing models and training schemes (transformer, sumformer, end-to-end, frozen processing model) with mixed results and performance on both in and out of distribution points. The appendix provides further results on varying sampling schemas and set sizes.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n- Good structure\n\n- Clear theory development and implementation thereof. Theory from the initial definition to theorems can be followed by someone not familiar with the topic.\n\n- Robost framework, which can handle point clouds of varying sizes\n\n- The latent space processing model can be retrained on new data/task by simply retraining the encoder/decoder.\n\n## Weaknesses\n\n- Derivations require \"fatness\" of the point cloud, not clear how much of a restriction this is in practice.\n\n- Missing Baselines: No deep-learning baselines and \"none-neural\" baselines.\n\nQUESTIONS:\nI'm not an expert on point cloud geometry. Reading your paper, I have the following questions:\n\n- How strong is the assumption of \"fat\" point clouds in practice? Are there important cases where this assumption doesn't hold?\n\n- Can you provide a comparison to a classical baseline (algorithms using $\\epsilon$-kernels$?), both in terms of performance and compute requirements/scaling?\n\n- Are there any deep learning baselines to compare to?\n\n- The SQUID dataset seems a bit odd (fish boundary contours?). Can this method also be applied to noisy real-world data, such as lidar scans?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a neural approach to address geometric \"extent measure\" problems. A key contribution is the introduction of the relaxed-ϵ-kernel, with a neural network employed as the fitting function. The effectiveness of the approach is demonstrated through experiments conducted on both synthetic and real datasets.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n* This work presents a neural solution for addressing geometric \"extent measure\" problems, which is an interesting attempt.\n* The paper includes some theoretical analysis.\n\nWeaknesses:\n\n* The motivation behind this work is unclear. The geometric \"extent measure\" problem is a well-studied area. It is not evident what specific advantages a neural solver brings to this problem. Furthermore, there is no discussion about the potential benefits of the proposed approach.\n* The lack of a related works section makes it difficult for readers to evaluate the distinct innovations of the proposed method compared to existing approaches.\n* There is a lack of comprehensive comparisons. The experimental section only includes results from some proposed neural variants, which does not sufficiently verify the effectiveness of the proposed method over existing works.\n* Important experimental results are missing. Specifically, there is no time analysis or visual results, which limits the readers’ ability to fully understand the effectiveness and utility of the proposed method.\n\nQUESTIONS:\n* Could the author provide a more detailed discussion of the motivation?\n* The related work section is essential, but it is currently missing.\n* It is necessary to include comparisons with neural baselines and extensive classic baselines.\n* Adding more visual comparisons would be helpful to verify that the proposed method achieves reasonable performance.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis manuscript proposes a way of approximating certain quantities about point clouds with bounded model complexity (and linear runtime complexity). It does so by introducing the concept of “relaxed epsilon-kernel”, which unlike the traditional epsilon-kernel can be built out of points in the entire convex hull, not a subset of the point cloud. This approach is evaluated in a thorough way theoretically and in practical examples.\n\nSTRENGTHS AND WEAKNESSES:\nThe main strength of the work is the theoretical contribution of introducing relaxed epsilon kernels and their relationship to commonly-used neural architectures. These seem like important contributions to the study of point clouds from the computational geometric perspective, and I can imagine they will inspire future work in this direction. Another major strength is the evaluation: all theoretical results are conclusively proven, including careful tracking of bounds and infinitesimal constants. The practical evaluation of the method in the supplemental also goes well beyond what is expected of such a theoretically heavy manuscript.\n\nI will preface this by saying that I am not an expert on computational geometric problems like this one, so there is definitely a chance I missed a mistake in a proof or some other major limitation. I will carefully read other reviewers’ comments as well as the authors’ response.\n\nNonetheless, I could not find any major issue in this work. If anything, the sole reason I would not rank this manuscript higher is the didactic exposition of the method. For something as visual as point cloud processing, there is a single didactic figure (Fig. 3), and I cannot understand exactly what it is showing. What is u? Where is the origin, so I can understand p as a vector? Also, the font is way too small. I would really much prefer if all of sections 2 and 3 were followed by a study of the same didactic 2D point cloud: it would make an intuitive understanding of the concepts significantly easier.\n\nI would also encourage the work to be double-checked for typos (e.g., “computatoin”) and missing articles.\n\nQUESTIONS:\nWhat theoretical effects does changing the L1 normalization for a softmax have? Which results remain valid, and which are not? As a reader, I was a bit perplexed by this choice, and while I am convinced by the results that it is in practice better, I was left wondering how much of the prior 3 sections are actually theoretically justified when one performs this change.\n\nWhat are some examples of unfaithful extent measures that one can keep in mind when reading the text? What are some examples of linearizable and non-linearizable ones? Adding such examples in the text will make Sec. 4 feel much more grounded.\n\nTo clarify: once trained, the same trained network can be used to process point clouds of different sizes, correct?\n\nWhat does the “hyperparameter search” entail?\n\nWhat is the (input) noise sensitivity of this entire pipeline?\n\nLIMITATIONS:\nIn my opinion, the main limitation of the work is its applicability: it is relevant only for a small number of “simple” point cloud operations (as opposed to, e.g., reconstruction or triangulation), and likely has diminishing returns as the dimension of the data approaches the number of data points. That being said, these limitations are discussed accurately and openly, and I do not penalize this work for it.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper defines the relaxed-$\\varepsilon$-kernel as a kind of universal coreset for approximating faithful extent measures. The paper then generalizes this approach to unfaithful extent measures via a polynomial lifting/linearization trick. It is shown that a sumformer can approximate an algorithm to compute a relaxed kernel. This motivates the design of an architecture to approximate kernels, which, when composed with a learned encoder and decoder, can be used to approximate extent measrues. The experimental results compare a few architectures and training protocols on the tasks of learning to approximate kernels and extent measures.\n\nSTRENGTHS AND WEAKNESSES:\nIt seems the paper's most important contribution is in defining and proving results about relaxed $\\varepsilon$-kernels. However, this is outside my usual domain of expertise, and I am not sure how much of a contribution this is relative to previous work in computational geometry. This paper might receive more careful and incisive peer review at a computational geometry conference rather than NeurIPS.\n\nI am skeptical of the learning aspect of this work, and in particular of the whole \"algorithmic alignment\" idea. It seems the argument is that because a particular architecture X (in this case based on the sumformer) is capable of enacting a particular algorithm Y, that therefore X will learn to approximate Y during training. I don't see why this follows. Since no theoretical justification for this claim is provided (nor would I expect it would be possible to prove something like this), it seems like more rigorous empirical evaluation of the learning claim is merited.\n\nIf it were the case that the kernel approximation network were really learning to approximate Algorithm 1 to compute relaxed-$\\varepsilon$-kernels, I'd expect it to have nearly perfect generalization to arbitrary input point clouds. However, in Figures 9 and 10 in the appendix, test error appears to get worse during training for the two OOD datasets. This suggests that the model is not actually learning a universal algorithm for computing relaxed $\\varepsilon$-kernels, but rather is learning some heuristics that work on a particular dataset.\n\nOn a more philosophical note, given that Algorithm 1 is simple and efficient and that it comes with hard theoretical guarantees, what is the advantage of learning to approximate Algorithm 1? Even if it were guaranteed that the kernel approximation network would learn to approximately enact Algorithm 1, it would still be more efficient and reliable just to use Algorithm 1 directly.\n\nQUESTIONS:\n- If I understand correctly, the complexity of Algorithm 1 is bounded in the number of input points, but scales exponentially with the dimension $d$. Isn't this the more pernicious scaling? Standard algorithms for convex hull etc. work just fine on a large number of points in low dimension (2 or 3). In what context do you envision your method or model being used. In other words, in what range of dimensions, approximation quality $\\varepsilon$, etc. is the approximation/time tradeoff reasonable?\n- The fatness of the point set is also an important scaling parameter. Does the fatness not scale with the number of points in the worst case?\n- What exactly do you mean by the \"alignment\" or \"resemblance\" of a model with an algorithm? I can see how you can design very specific model weights to enact the algorithm, as you do to prove Theorem 3.4. But this is just rewriting the algorithm in a different language. Does the fact that you can do this by hand somehow imply that the model class should generically learn to enact the approximation algorithm you want? Or at least that it will learn to approximate the algorithmic result better than any other model architecture would?\n- Does the alignment happen in practice? I.e., can you see if the learned weights agree with the ones you would impose by hand to encode the approximation algorithm?\n- In Algorithm 1, the required $\\delta$-net size still scales exponentially with $d$. Doesn't this pose a problem for the bounded model complexity?\n- Did you try any experiments in dimension higher than 5? In high dimension is presumably where the scaling performance would really start to matter.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}