Title: Point Cloud Synthesis Using Inner Product Transforms
Venue: NeurIPS 2025
Accepted: yes
Keywords: Topological Data Analysis

TDA

Topology

Topological Deep Learning

Geometric Deep Learning
Abstract: Point cloud synthesis, i.e. the generation of novel point clouds from
 an input distribution, remains a challenging task, for which numerous
 complex machine learning models have been devised. 
 We develop a novel method that encodes geometrical-topological
 characteristics of point clouds using inner products, leading to a
 highly-efficient point cloud representation with provable expressivity
 properties.
 Integrated into deep learning models, our encoding exhibits high
 quality in typical tasks like reconstruction, generation, and
 interpolation, with inference times orders of magnitude faster than
 existing methods.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe work proposes Inner Product Transforms (IPT) to represent point-clouds from images. Basically, by combining a sufficient number of cleverly positioned views of point-cloud data, one may reconstruct the full point-cloud using the inner products of coordinates. The work proves that building n+1 independent directions (planes), one may reconstruct points clouds in R^n. The results indicate that the IPT is comparable to current state-of-the-art methods regarding reconstruction in quantitative and qualitative aspects. Additionally, the authors claim that the method is orders of magnitude faster for training and inference. Finally, IPT may be used as in conjuntion with generative methods to create novel point clouds from a pretrained latent-space encoder-decoder model.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\nThe work is well grounded and well motivated. The essential concepts necessary to understand IPT are presented in a simple and effective manner, so that even unfamiliar readers will understand the main contributions and how they were achieved using only the main text. Additional details on the appendix help readers looking for a deeper dive into the subject.\n\nAdditionally, I enjoyed the limitations discussion, where the authors were frank regarding the disconnect between theory and practice regarding the number of planes needed to reconstruct a point cloud. Such good discussions lead to more openings for future works and I appreciate this.\n\nFinally, I liked that the authors submitted source code in addition to the manuscript.\n\n# Weaknesses\nI found no description of hardware configuration used for training the models, limiting the assessment regarding performane gains compared to other models. The authors mention that the model is trained on \"commodity hardware\" (Pg. 1, L. 32), although the actual hardware used is not mentioned anywhere in the text.\n\nFurthermore, the authors mention the compression capabilities of IPT without arguably performing *compression* experiments. Upsampling after downsampling is not a compression experiment. I would suggest removing the term \"compression\" in this case, since no metrics, such as Rate-Distortion (RD) curves, were presented. Additionally, this experiment was performed using a single configuration downsizing the point clouds to 256 points and upsampling it back to the original size. More extreme configurations (16, 32, 64, ...) points would make the authors' argument stronger.\n\n# Minor comments\n- Define the 1-NNA acronym in the text.\n- L. 270 \"As a comparison partner, we use...\". the term \"partner\" is odd here.\n\n# References\n[1] Park, J. J., Florence, P., Straub, J., Newcombe, R., & Lovegrove, S. (2019). DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation. 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 165–174. https://doi.org/10.1109/CVPR.2019.00025\n\nQUESTIONS:\nIs the method capable of performing point-cloud completion from partial data? An example of this can be seen in DeepSDF [1] (see the references above).\n\nWhy is Table 3 is mentioned after Table 4 in the text? Additionally, in Table 3, the authors only insert the device for ShapeGF and IPT-VAE. What is the device used for the other models?\n\nLIMITATIONS:\nYes, the authors have adequately addressed the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel framework for point cloud synthesis based on a two-stage process: (1) generating a compact and injective inner-product-based representation (IPT) of the point cloud, and (2) reconstructing the full 3D point cloud from the IPT using a learned encoder. The IPT encodes multi-directional height filtrations as 2D images, inspired by the Euler Characteristic Transform but simplified and adapted to point clouds. The authors prove injectivity of the representation in theory, and show that its approximation retains strong empirical properties. This framework enables extremely fast and lightweight generation and reconstruction pipelines, using simple convolutional architectures and supporting general tasks such as reconstruction, generation, interpolation, and upsampling.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\nThe core idea of using inner product filtrations to encode point clouds as images is both elegant and theoretically grounded, leading to a highly compressed and expressive representation. The two-step pipeline—separating image generation and reconstruction—reduces architectural complexity and training cost, while maintaining high generation quality. The paper further demonstrates strong generalization in out-of-distribution settings, thanks to the stability and structure of the IPT latent space.\n\nWeaknesses\n\n1. The paper proves that the Inner Product Transform (IPT) is injective in the continuous case, but the practical implementation relies on a finite number of directions. Could the authors provide a brief empirical analysis or visualization showing how reconstruction quality varies with the number of sampled directions? This would help clarify how well the theoretical guarantee holds under practical constraints.\n\n2. The method claims to support tasks like interpolation and upsampling, but these parts are not described in much detail. It would be helpful to include a few more examples or clarifications explaining how interpolation is implemented in the latent space, and what mechanisms ensure geometric plausibility in the interpolated outputs.\n\n3. There are minor writing issues that slightly affect readability, such as “exhibits performance on a par with” or “we found the IPT space to be smooth”. These could be rephrased more formally or precisely to improve presentation.\n\nQUESTIONS:\nsee weaknesses\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents the Inner Product Transform (IPT), a novel method for point cloud representation using inner products, converting point clouds into 2D images that capture their geometrical and topological features. The core contributions include a new image-to-point-cloud pipeline that reduces training and inference times while preserving generation quality, and the demonstration of IPT's injective property allowing point cloud reconstruction from its descriptor. The authors also highlight the stable latent space provided by IPT, which supports high-quality interpolation and out-of-distribution tasks without additional retraining. This work advances point cloud processing by offering a more efficient and accessible approach, potentially broadening the use of machine learning algorithms for point cloud tasks. The theoretical foundation and versatile applications of IPT, including reconstruction, generation, compression, and upsampling of point clouds in arbitrary dimensions, further enhance its value.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\nIts main advantage is the introduction of a novel and efficient point cloud representation method. The IPT shows promise in improving computational efficiency for point cloud tasks. The authors provide a solid theoretical foundation for their method, including proofs of key properties like injectivity, which adds credibility to their approach. \n\nWeaknesses:\n1. The experimental results, while respectable, do not demonstrate a significant improvement over existing methods in terms of reconstruction and generation metrics. \n2. It is not very clear how the authors organize the IPT as an image. How do they order the direction vectors? \n3. The current formulation of IPT is not invariant to rotations.\n4. The injectivity results are only valid in the case of an infinite number of directions\n\nQUESTIONS:\n1. How do you order the direction vectors? Does different ordering affect recosntruction and generation results?\n2. Why do you use VAE as the generative model? Have you tried more advanced diffusion models?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel representation for point clouds, called the Inner Product Transform (IPT), which encodes a point cloud as a multi-directional height histogram (essentially a 2D “image”). The key insight is that this representation is provably injective (under certain conditions), and simple CNNs can effectively learn to invert it. The authors use a lightweight CNN-based encoder and VAE to generate point clouds by first generating IPTs and then inverting them. The method achieves competitive reconstruction and generation results on ShapeNet while being significantly faster than flow- and diffusion-based baselines.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper presents a simple and elegant idea, backed by solid theory and clean implementation. The injectivity of IPT is well-motivated, and the authors convincingly demonstrate that their learned inverse can effectively reconstruct or upsample shapes from low-resolution IPTs back to 2048-point clouds. I also appreciate the clarity of presentation and the attention to efficiency; this is one of the few papers in the space that seriously benchmarks speed.\n\nThat said, there are some essential experimental gaps. First, everything is trained and evaluated on 2048-point clouds — and there’s no indication the method scales to higher-resolution shapes (e.g., 4K or 128K points). This is particularly limiting because the value of efficient generation becomes most critical in real-world applications that involve extensive or dense point clouds, such as LiDAR scans or detailed shape modeling. Second, although they highlight the stability of the latent space, there’s no real out-of-distribution or cross-category generalization shown. The one case they call “out-of-sample” is stemming from resolution downsampling, which is not the same thing. Lastly, while the method is efficient, the timing comparisons (Table 3) lack consistent hardware reporting, which makes it hard to judge the absolute magnitude of the speedup. They also don’t compare against at least one relevant recent method focused on efficient generation (e.g., Point Straight Flows, SPVD).\n\nOverall, I like the direction and believe the IPT idea could have a broader impact, but I’d like to see how the authors respond to the experimental limitations.\n\nQUESTIONS:\n1. Can your method scale to higher-resolution point clouds? A reconstruction or generation experiment at 4096 or more points would really help here.\n\t2.\tYou claim IPT encodes a stable latent space — but did you try testing it on unseen shape categories (e.g., train on chairs/planes, test on cars)? \n\t3.\tFor your speed claims (Table 3), could you confirm what GPU each baseline was measured on? Or alternatively, report FLOPs to standardize the comparison?\n\t4.\tWhy use 1D convs for the encoder but 2D convs for the VAE? This asymmetry isn’t clearly explained, and it seems like IPTs could support both.\n\t5.\tFinally, have you observed how IPT behaves when two very similar shapes are encoded at low resolution? Some study of approximate collisions would support your injectivity claims in practice.\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-08"}