Point Cloud Synthesis Using Inner Product Transforms
Authors: Ernst Röell
Bastian Rieck
Venue: NeurIPS 2025 poster ()
Keywords: Topological Data Analysis
TDA
Topology
Topological Deep Learning
Geometric Deep Learning
Primary area: deep_learning
TL;DR: We develop a differentiable, invertible geometrical-topological encoding of point clouds based on inner products.

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.

Point Cloud Synthesis Using Inner Product Transforms
Ernst R.231 Bastian Rieck
AIDOS Lab, University of Fribourg. Switzerland Institute of Al for Health, Helmholtz Munich, Germany Technical University Munich, Germany
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 geometricalltooollgiccl 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.
Introduction
Point clouds are a data modality of crucial relevance for numerous domains. While computer graphies is the predominant application area, where point clouds are often used as precursor to more structured representations like meshes, they also occur in higher dimensions in the form of sensor data, for instance. However, the synthesis of hitherto- unseen point clouds from a given distribution still proves to a challenging task, with numerous models aiming to address it [39]. The complexity arises because of the sparsity and 'set-like" structure of point clouds, making it hard to generalise existing machine learning architectures directly. State-of-the-ar methods thus typically require large amounts of compute, exhibiting long training and inference times. With recent work [33] demonstrating exchanging raw coordinates for Fousier-based features-can
that a change of perspective- make comparatively simple deep-learning architectures competitive in computer- vision tasks, our paper explores the question to what extent novel representations of point clouds can lead to gains in computational performance without sacrificing too much quality.
To obtain such representations, we build on multi-scale geometrical- topoloogccal descriptor [35] based on inner products of coordinates representing high-dimensional shape. Using suitable approximations, this descriptor permits us to represent a point cloud as a single 2D image. Unlike other single-view representations, however, the coordinates of the image represent a different 'domain," namely geometrical toppoooocclllolloiccall aspects. In fact, this mapping is computationally efficient and injective, making it theoretically possible to reconstruct point cloud from its descriptor (we will demonstrate that this property also holds empirically when working with approximations), We refer to this mapping as the Inner Product Transform (IPT) and use it for the generation of 3D point clouds.! Our work primarily focuses on building highly-efficient models that enable real-time inference in settings of limited computational resources while also being trainable commodity hardware.
For readers familiar with computational topology, the IPT is a special case the Euler Characteristic Transform (ECT). Our focus on point clouds permits us to formulate the IPT without any background knowledge in computational topology, while also resulting in substantially simplified proofs and more powerful statements about its theoretical properties.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Our paper is built on two core notions, the first one being that we treat point cloud generation as two- step task, with the first step being an image generation task, yielding a two-dimensional descriptor, followed by a (multi-modal) image-to-point-chee using separate machine learning models, thus substantially decreasing architectural complexity effect, we represent a point cloud as a special image that can be generated, providing a bridge to the point cloud domain. The second core notion is that the generated image is a faithful, i.c. injective, representation of the point cloud, indicating that it is possible to perfectly reconstruct point cloud from its descriptor. Our paper makes the following contributions:
reconstruction task. We realise the two steps
1. We present a novel generative image-to to po-oot---oou pipeline based on inner products that allows training and inference times to be orders of magnitude faster while retaining high quality generation.
2. We show that our representation yields a stable latent space, which permits (i) high-quality interpolation tasks, and (ii) solving different out distribbutionn tasks without the need for retraining while still maintaining high quality.
3. We demonstrate injectivity and other advantageous properties of our descriptor, allowing the generalisation of our method to point clouds of arbitrary dimensions.
Our code is available at https:/ / gi twwb.th..aaaattt. www.eioomaaaoaieeeeeeeeeeeosseee baab//nner-rree lab/inner-product transforme and is released under a BSD-3- 3-Clause license.
2
Related Work
Point clouds being a nigh-ubiquitous data modality, numerous models already exist to tackle classi- fication or generation tasks [39]. The lack of structure, as well as the requirement of permutation invariance, imposes constraints on the underlying computational architecture, typically substantially increasing model complexity [29, 41J. To solve generation tasks, many methods opt for jointly learning the generation of the shape, i.e. the surface or object the points are sampled from, well the mapping of points onto that object. This core idea drives several recent state-of-thh-as models, including Moxel CNN [21], PointFlow [40]. SoftFlow [17], Point Voxel Diffusion [44, PVD] and LION [42]. While these models exhibit high-quality results, their architectures require long training and inference times.
By contrast, our approach only needs to model a distribution of coomeetricaltooologiiio descriptors, represented as 2D images, from which we subsequently reconstruct a point cloud again. While reconstructing point clouds from images is an active field of research (see Fahim et al. [11] for a survey single-view reconstruction), such images are typically depth images snapshots taken from a specific position around the object. As such, they are not necessarily yielding a faithful, unique representation of an object. Another close analogue to our method is given by 'structure-froot woree-frromoooonn approaches [27]. which reconstruct complex geometries based on sets of images, taken from different spatial viewpoints. For our method, however, the viewpoints are represented as unit vectors on a sphere, which are used to 'probe' the point cloud from a specific direction.
Hence, crucial property of our method, the Inner Product Transform (IPT) is that it theoretically yields a faithful representation, hence permitting an injective mapping between the image domain and the shapes we aim reconstruct (or generate). We observe this property to hold empirically when working with discrete approximations. This is due to the fact that the IPT is a special case of a general ceometriialltopologicee descriptor, the Fuler Characteristic Transform (ECT), which studies shapes at multiple scales and from multiple directions, providing a unique characterisation [35]. Being a stable [10, 13] and efficient descriptor, the ECT often used to solve questions in data science, mostly in the form of "hand-crafted features for classification and regression tasks [4, 8, 22, 24, 25, 34, 37]. Recent work addressed this shortcoming and enabled the use of the ECT in machine-leariin applications in the form of a generic differentiable computational layer [31, 32] or positional encoding [3]. Notably, the ECT remains a unique characterisation even when using a finite number of directions [9, 14], meaning that, theoretically, it can be inverted to reconstruct the input data. Practically, however, inversion is presently possible only for select data modalities like planar graphs [12]. While restricted to point clouds, our method is thus the first to enable the inversion of such a deseriptor for input data of arbitrary dimensionality, making possible to use it in the context of generative models.

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Denerative mocel
IP- IP-Cndoder
VAE
VAE
IPT
Figure 1: Given a point cloud on the left, we compute its Inner Product Transform (IPT). For generative tasks, we train a generative model (middle) reconstruct and generate distribution IPT is then passed through the encoder model to obtain the
of IPTs. The (possibly-generatel)) reconstructed (or novel) point cloud. Our pipeline decoupled, permitting any generative image model to be used to generate point clouds.
3 Methods
Our method, the Inner Product Transform (IPT), can be intuitively understood as a filtering process of point clouds, employing different sets of hyperplanes, created by a of directions (i.e. normal vectors). The inner products of point cloud coordinates with direction vector are then used to parametrise curve that counts the number points below the hyperplane. By stacking curves. we obtain a 2D image representation of the point cloud. The IPT is special case of the Euler Characteristic Transform [35, ECT), but our subsequent description is self-contained, does not require any knowledge of topology, and presents simplified proofs of all properties.
3.1 Inner Product Transforms
Let S denote the unit sphere in RR. Given a point cloud X C. RR a fixed direction vector E 5 and a height R. we define the set Xe.h := far X (a, S h], where (x, {) the Euclidean dot product. The set Xg.b contains all points below the hyperplane spanned by (x.E) = h, and we denote its cardinality by x(X.,.). We then define the Inner Product Transform as
IPT(X): 5 R
(5, + x(X.,h).
point x X included in XR.ş, thus affecting XXx,,n). if and only if its height h. == (a. along& is less than h. We can thus formulate the contribution of point æ to the IPT along each direction in terms of an indicator function:
f
if (E, < h otherwise.
1=(E,h)
(2)
This enables us to rewrite Eq. (1) as
IPT(X): 5n R - N
(5, h) + Σ hg, xEX
(3)
Following ideas from Röell & Rieck [32], can replace all indicator functions with sigmoid functions, i.c.
IPT(X): 8--1 R R (E, H Σ S( -h),
REX
where À denotes scale parameter, which controls how closely the sigmoid function approximates the indicator function (see Appendix for additional ablations demonstrating the stability of this approach). In practice, we sample directions and discretise all heights with ny steps, thus
representing IPT as image of resolution n4 Rd-
For readers familiar with topology, this notation is allusion to the Euler Characteristic.

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Properties. The IPT and its approximation IPT from Eq. (4) satisfy several important properties. We first focus on the case of the IPT and note that is injective, i.e. point cloud can perfectly reconstructed from its descriptor. Unlike existing injectivity results [14, 35), which focus on geometric simplicial complexes, we prove that for point clouds in R", it sufficient to use n + affinely-independent directions for the reconstruction.
Theorem 1. Given two point clouds X, Y with X * Y, we have IPT(() # IPT(Y).
An important consequence of injectivity is that allows us to formulate a metric for the set of point clouds contained in a ball of fixed radius.
Lemma 1. Let (R( denote ball of radius R in R". For two point clouds X, C. (n( define their distance as
d(x,))
[IPT(X) (el(Y)
(5)
EEE
where E is finite set of directions and || ll2 is the -nom restricted to the interval [- R, R). The function -) satisfies the definition of a metric.
When approximating the IPT via Eq. (4) and finite number of directions in practice, the metric defined in Lemma corresponds to the pixel-wise mean squared error between discretised IPTs. Hence, given a sufficient number directions, we can formulate a loss function based on this metric, which admits a highly-efficient implementation and will turn out to lead to high-quality results. We also obtain a result that permits us calculate IPTs from disjoint unions of point clouds.
Lemma 2. Let X,Y C RR be disjoint point clouds, then
IPT((UY) = (PT(X) + IPT(Y)..
(6)
Finally, we can prove that the IPT is surjective on convex linear combinations.
Theorem 2. Given point clouds X,Y C CR", the IPT is surjective for the rational linear subspace spanned by IPT(X) and IYY,,), up to rational scaling factor. In particular, for p, 7.q E No with 0 SS p S q and q > 0 we have
LIPT(X) + 9- --PIPT(Y) = 1 (PT(
where Z Up
A direct consequence of Theorem 2 is that, along the linear interpolation between two IPTs, there are only valid IPTss, i.e. each interpolation step affords a perfect reconstruction in theory. We analyse this aspect further in Section 4.5. We note that most of these properties (except injectivity) only hold for the IPT but, as we will later demonstrate, IPT, the approximation to the IPT, retains these properties in practice.
3.2 The IP-Encoder
The existence of Theorem unfortunately does not lead to a practical algorithm for 'inverting' an IPT. We therefore suggest an approach based on neural networks and describe the IP-Encoder, which encodes IPT point cloud. Subsequently, this will enable to learn distributions of IPTs and reconstruct new point clouds in generative setting and, as we shall see, even in the setting of --ii-iiiidsititiinn data. Since our representation is permutation permutation-invariinn, our IP-Encoder model directly inherits this invariance, which substantially reduces the complexity of its architecture.
Model architecture. Given the structure of the IPT as an image, CNN architectures provide suitable base models. However, since the direction vectors E g cannot be consistently ordered along one dimension, we need to reframe the input data. Specifically. we consider an IPT, normalised to [-1, as a multi-channel, mie-dimensoonal signal. Our IP- IP-Encoder model then consists of multiple ID convolutional layers followed by fully-connected layers, resulting a conceptually simple and efficient architecture. To generate a novel IPT, we can use any generative model for images. In our experiments, we will use an architecture based on convolutional variational autoencoder, denoted by TP-VAE (see Appendix F for more architectural details).

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The IPT as an optimisation problem. As an alternative to the machine learning model above, we also investigate whether the inversion of an IPT can be turned into an optimisation problem.
Due to the differentiabilityy of our approximation IPT and given a known IPT, minimising a loss function, such as the one described in Lemma using backpropagation should result in suitable approximation of the unknown point cloud one wishes to recover. However, while this works in theory, it requires the input and target to be a 'correct' IPT of point cloud. In practice, this is not guaranteed: Generative models may sample from their latent spaces and output samples that are close to being an IPT of a point cloud without actually satisfying all structural constraints. To some extent, we argue that this is even the desired behaviour since generative model would otherwise just rehash its inputs [16]. Hence, any optimisation-ban reconstructions in generative setting. We nevertheless investigate such an optimisation procedure in an ablation study in the appendix (Figure S.3).
method will not necessarily result in realistic
Topological loss functions. The fact that Lemma 1 shows the IPT to be a metric naturally poses the question if its approximation IPT can be used as an effective and efficient loss term. While in principle the IPT is sufficient as loss term for training point cloud reconstruction model, training time be reduced through combination of low-resolution (64 x 64) IPT combined with the Chamfer Distance (CD). The CD is a fast-to-compute (pseudo-) metric often used in point cloud evaluation (see Appendix E for more details). The IPT loss ensures that the overall geometry and point cloud density of the object is captured, whereas the CD loss ensures that fine-grained details are accounted for. Building a joint loss, combining the IPT and CD, thus results in a density- loss term that takes the global density of a point cloud into account.
Latent space. Interpolation between samples provides valuable insight into the capacities of our generative model. Theorem 2 provides intuition for interpolation using IPTss. To this end, suppose we perform linear interpolation between point clouds X, Y. using a parameter p over the interval [0, 1], which is partitioned in q equidistant steps. Then, at step p/q. the point cloud Z Up Ug-p Y is the union of copies of X and of X is increased and the number of copies of Y is reduced. We may interpret this intuitively as transporting 'mass' from Y to X over the course of g steps. Our IP-Encoder model averages all copies during reconstruction in natural fashion, resulting in smooth transitions between point clouds.
p copies of Y. During the interpolation, the number of copies
4 Experiments
We demonstrate the effectiveness, efficiency, and overall utility of our IP-Encoder through a com- prehensive suite of experiments. Subsequently, to simplify the notation, we will only refer to the IPT, with the understanding that we are calculating an approximation of it according to Eq. (4). Throughout our experiments, the emphasis and motivating questions regard the expressivity of the IPT and its capacity to faithfully and effectively represent point clouds in practical setting. Good results with minimal architecture underpin the fact the IPT is an effective representation as shows that the data distribution is easy to learn. Among other things, we show that our [P-Encoder model can (i) effectively reconstruct shapes, (ii) create novel point clouds from generated IPTs, and (iii) effectively downsample point clouds.
All our experiments rely on a subset of the ShapeNet dataset, and we adopt the preprocessing and evaluation workflow introduced by Yang et al. [40]. Each point cloud in the dataset consists of 2048 points sampled on the surface of three shape classes (airplanes, chairs and cars). We report the minimum matching distance [2, MMD] based on the Chamfer Distance (MMD-CD) or the Earth Mover's Distance HMMD-EMD) between reconstructed point clouds (see Appendix E for brief description of these evaluation metrics). Unless otherwise mentioned, the reported Chamfer Distance scaled by 1.00 x 1 and the Earth Mover's Distance is scaled by 1.00 x 10 Our hardware consists of an NVIDIA GeForce RTX 4070 with 12GB VRAM and an 13th Gen Intel(R) Core(TM) 17-13700K with 32GB RAM, We compare our methods to several ofate-of-the-art models, namely (i) Foon [40], (ii) SoftFlow [17], (iii) ShapeGF [6], (iv) Canonical VAE [7] and (v) LION [42].
5

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Table 1:
Reconstruction results on three ShapeNet classes. The best- best-performing model is highlighted in bold, while the second best is shown in italics. Our method consistently ranks second in terms of the EMD, which is recognised as the best metric for reconstruction quality [44].
Airplane
Chair
Car
MODEL
CD (U)
EMD (L)
CD (4)
EMD (4)
CD
EMD (L)
PointFlow
1,30
5,36 0.06
10,43 2 0.02
17.54 0.16
12.93 6.94
SoftFlow
1.19
0.00
17.68 0.00
0.08
11.44 6.02
ShapeGF
1.05
0.42
5,96
12.23
9.26 5.68
Canonical VAE
0.00
0.09
8.60 0.02
5.44
6.02
LION
0.30
0.00
0.14 0.70
0.60
0.09
IP-Encoder (ours)
1.05 + 0.00 1.57 0.00
9,24
6.19 + 0.00
5.00
3.18 + 0.00
4.1
Reconstructing Point Clouds
Our first set of experiments assesses (i) the reconstruction quality of the IP-Encoder, (ii) the efficacy of the IPT as loss, and (iii) its computational efficiency.
Architecture and experimental setup. We consider the IPT as ID signal, with each direction corresponding to channel, sample 128 directions uniformly from the unit sphere, and discretise each direction into 128 steps, thus obtaining an IPT with resolution of 128 x 128. Sampling the directions randomly is motivated by the lack of canonical ordering of unit vectors in three and higher dimensions (notice that in two dimensions, the angle affords a parametrisation that results in a natural ordering). While 'pscudo-ordered" directions, for instance via spiral along the unit sphere, might potentially lead to better results, they come at the cost of generality, and we thus refrained from doing so in experiments. To improve this for future work, one could use positional encoding of the directions; at present, such an ordering is only implicitly present through the fixed ordering of columns in the image IP-Encoder consists of four 1D convolutional layers with batch normalisation, max-pooling, and SiLU activation functions (cf. Appendix F). After mapping the IPT into latent space, we apply final 3-layer MLP to predict the final point cloud. We use ReLU activation functions for the first two layers and tanh activation function for the last layer, since tanh is better suitable for bounded outputs. Subsequently, we train the IP-Encoder separately for each of the classes for 5k epochs, using CD + IPT-64 loss, denoting a weighted sum of the CD and the IPT with a resolution of 64 64.
to represent the IPT. Given such a 2D representation of an IPT, our
Table 2: Ablation study wrt. the loss function. Our joint loss, combining CD and IPT, yields balanced results without overfitting any of the two metrics.
Results. Our reconstructions (Table I and Figure 2a) are of high quality, despite a comparatively simple architec- ture. Our Encoder consistently ranks second in terms of the EMD, known to be the most suitable metric to evaluate reconstruction quality [44]. Notably. our loss term does not use the EMD. Our method also exhibits substantially reduced training times compared to other methods: The En-Encoeer model requires approximately 30.00 min on a single GPU, compared to 192.00 022.001 for PointFlow and more than of 550.00 h for LION. We additionally perform an ablation study concerning the loss term, showing that the combination CD and IPT is crucial for high reconstruction quality. As Table 2 shows, a joint loss yields the best quality. In line with prior work, CD on its own tends to adversely affect clont density [2], resulting in larger (i.e. worse) EMD scores. Conversely, training only with ECT loss shows that the reconstructed follow the density of the underlying object, but may become blurry. This is partially due to the low resolution of 64, making the combination of CD (for details) and the ECT (for global density and structure) crucial. While it would also possible to use the EMD as a loss term, we find that our IPT-64 loss function is substantially faster than the EMD (0.0006s versus 0.0291 s), making it the preferred loss function.
Airplaoe ())
Chair (1)
Car())
Loss FUNCTION
CD BMD
CD
BMD
CD
EMD
CD
1,00 8.89
10.44
32433
5.97
14.6
JPT-4
241 139
13.05
4.29
7.75
2.47
CD IT66
1,00 1,46
9.52
8.44 6.12
4.16
We observe the IPT to be an effective representation of point clouds, leading to a conceptually simple model with exceptionally fast training times.

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(a)
(b)
Figure 2: Examples of reconstructed (a) and generated (b) point clouds using our IP- IP-Encoder model for three classes in the ShapeNet dataset.
4.2 Generating Novel Point Clouds
We now view the IPT latent space, from which sample and subsequently reconstruct point clouds, demonstrating that (i) the IPT constitutes stable latent space, (ii) a distribution of shapes can be learned through the IPT, and (iii) the IP-Encoder works well in out-of-sample settings.
Architecture and experimental setup. We use IP-VAE, a CNN-style VAE [15] as our generative model, thus turning point cloud generation into image generation task. Our encoder uses four convolutional blocks with Leaky ReLU activation functions, batch normalisation, and a final linear layer for the 64-dimensional latent space embedding. We also add additional models, namely (i) Set- VAE [18], (ii) Point Voxel Diffusion [44, PVD], (iii) Point Straight Flows [38], and (iv) XCube [30]. These models lack reconstruction capabilities or the respective code, so we excluded them from the experiment in Section 4.1. 1. For training the PP-, we follow the 3- setup, using KL-divergence and MSE loss terms with B 1.00 x 10-4 We sample latent vectors from the IP-VAE and consider them to be IPTs, which subsequently map to point cloud using the IP-Encoder model. Our evaluation of generative performance follows the setup of Yang et al. [40].
Table 3: Inference time T in for all methods, measured on the same hardware. Our model is orders of magnitude faster than all others.
Results. Table 4 reports numerical results, whereas Figure 2b depicts generated samples from each class. We include different quality met- rics and note that, despite its conceptual simplicity, tently ranks among the best two models in terms of MMD-EMD, and exhibits performance
model consis-
par with more elaborate architectures. This is noteworthy since the IP-VAE is not constrained to produce exact IPTs, hence its outputs may contain artefacts. The fact that IP-Encoder model (remaining fixed throughout this experiment) can reconstruct point clouds from such out-oeesample data without additional regularisation or retraining, shows that the IPT describes a stable and expressive latent
MODEL
DEVICE
T())
PointFlow SoftFlow
0.270
0.120
ShapeGF
GPU
0.340
SetVAE
0.030
space. We observe fast training times, with the IP-VAE taking approx- imately 15.00 min, making the full training pipeline run in less than
PSF
0.040
PVD
29.900
h.00 This is in stark contrast to models like LION (550.00 or Soft-
IP-AAE
GPU CPU
0.001
Flow (z 144.00 h). Finally Table 3 shows that our inference times on commodity hardware are orders of magnitude faster than existing models.
IP-VAE
0.006
The IPT yields a suitable representation for generating high-quality point clouds, while exhibiting inference times that are orders of magnitude faster than existing models. More expressive generative models are likely to improve generative quality (at the expense of longer runtimes).
7

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Table 4: Evaluation metries for point cloud generation tasks, cited from their respective papers. We highlight the best result in bold and the second-best in italics. Our implementation of the EMD follows PVD [44]. Whenever available, we report coverage (COV), minimum matching distance (MMD), and I-NNA (1-nearest neighbour accuracy) for both EMD and CD.
Airplane
Cheir
Car
MMD44)
CtV
1. LNNNA())
MMD
CUV
1 L.NNA(U)
MMD
COV (7)
I.NNA())
Model
CD
EMD
EMD
CD
EMD
CD
EMD
CD
EMD
CD
EMD
EMD
EMD
EMD
PointHow
47.90
75.68
69.44
42.90
50.00
60.88
59.89
0.90
0.81
46.88
50.00
60.65
62.36
SoftPlow
0.23
0.37
46.91
47.90
70.92
69.44
2.52
1.68
41.39
47.43
39.95
63.51
1.19
0.86
42.90
44,60
62.63
64.71
Shape-GP GP
2.70
0.66
40.74
40.49
50.00
76.17
2.89
1.70
46.67
48.03
68.96
65.48
9.23
0.76
49.43
50.28
63.20
56.53
SerVAR
43.70
2.55
61.48
0.74
49.15
59.66
61.48
PVD
64.81
2.62
50.60
53.32
1.08
0.79
41.19
50.56
LION
0.22
0.37
47.16
49.63
67.41
61.23
2.64
1.55
43.94
52.11
53.70
52.34
0.91
0.75
50.00
56.53
53.41
$1.14
PSF
71.11 61.09
58.92
54.45
57.19
56.07
XCahe
52.85
53.99
48.60
57.96
54,43
VAE (Ouns)
45.67
77.28
68.40
2.34
48.34
61.33
63.07
0.70
35.23
49.12
58.52
59.23
4.3 Point Cloud Downsampling
Motivated by the promising results in terms of generative performance, we further investigate the capacity of our IP-Encoder model to upsample a downsampled point cloud.
Table 5: Reconstruction performance for consecutive down- and upsampling, with the best result shown in bold. For refer- purposes, the last row repeats results without downsampling
Architecture and experimental setup. We train a down- sampling model with the same architecture as the IP- Encoder, resulting in a point cloud with 256 points. Here, the purpose of the loss term to minimise the discrepancy of the original point cloud to its downsampled version, measured using the IPT, which we normalise because of its dependency on point cloud cardinality. We then pass the IPT of the downsampled point cloud to the IP-Encoder model to obtain an upsampled version, which, ideally, should be close to the original point cloud. Reporting both CD and EMD scores, we compare these results to uniform subsampling method.
(irplane()
Chair())
Carli)
MErron
CD
EMD
CD
EMD
CD
EMD
[P- D--Donsampler
1.18 2,58
11.70 11.42
6.42 581
Uniform
3.10 4.71
15:38
15.34
9.30 9.09
IP-Encoder
9.53
Results. Table 5 depicts numerical scores (cf. Figure S.2 in the appendix for sample point clouds). Unlike uniform subsampling, our downsampling model preserves uniform density of each object, showing that it has learned the underlying shape. We observe some loss in quality compared to reconstruction without downsampling, which is to be expected. Despite potential compounding errors, the loss in quality remains low. Due to the similar setup. we can compare results with Table and we observe that even when downsampling, we still perform on par with the best model in the reconstruction task. Notably, the IP-Enerder has not been retrained for this task, so the IPT of the subsampled point cloud is out--of-distribute This indicates that our model has captured the 'true' underlying shape characteristics.
Our IP- IP-Encoder can downsample point clouds with minimal loss of quality, demonstrating that the model effective in encoding relevant shape properties.
4.4 Practical Inversion of the IPT
While theoretical invertibility (see Theorem 1) ensures that the IPT yields an expressive summary, practical algorithms for inversion of an IPT remain an open question. However, in our setting, making use of an approximation to the IPT, i.e. IPT, we may use backpropagation to invert it without requiring an IP-Encoder. The core observation is the differentiability of the IPT with respect to the point cloud coordinates, and the fact that it constitutes a metric (see Lemma 1). The main idea is to randomly initialise a point cloud, compute its IPT, and subsequently calculate the loss between said IPT and the IPT of the target point cloud. Differentiability ensures we can compute the derivative with respect to the point cloud coordinates and update them to reduce this loss. Applying backpropagation thus ultimately results in convergence to the original point cloud. We show that
8

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Figure 3: Linear interpolation between IPTs results in smooth interpolation in the point-cloud domain. Given the IPT of two airplanes, we linearly interpolate between the pixel values and pass each step through the IP-Encoder obtain a prediction of the intermediary point cloud. Although the IP- Encoder has not been specifically trained on such data, it is able to produce meaningful reconstructions, since has learned the distribution of shapes. We provide two examples of the interpolation process, one with high pairwise similarity (top), the other with low pairwise similarity (bottom).
(i) IPTs can be inverted in practice, (ii) resolution of 128 yields high-quality reconstructions, and (iii) the IPT has stable gradients (which not clear a priori, given that are approximating an indicator function). For this experiment, we apply backpropagation for 2000 epochs with the Adam optimiser using learning rate of 0.5, which is halved in epochs 50, 100, 200, and 1000, respectively, We repeat all experiments with three resolutions and for three choices of scaling factor À in Eq. (4), which we set either be equal, half, or a quarter of selected resolution. As quality metric, we report the EMD and CD between the optimised and target point cloud.
Table 6: Inversion of the IPT with backpropagation leads to high quality reconstruction.
Results. Table and Figure S.3 in the appendix depict the numerical results and show some reconstructed samples, respec- tively. Comparing the result with Table I, we observe that back- propagation outperforms the iP-Encoder model. Table S.1 in the appendix provides the full set of results and an ablation with respect to the scale and resolution. We observe that all approxima- tions become gradually coarser as the resolution is decreased. As expected, observe problems with vanishing gradients for large values of X, since the sigmoid function approaches an indicator function in this case. This requires balancing with respect the resolution, and we empirically observed that around a quarter of the IPT resolution is sufficient for high-quality results, which is how we pick the parameter in practice. It is important to that while this method of inverting works for an IPT without any artifacts, it fails to accurately converge if the IPT is only an approximation of a true IPT, as is the case with a generated IPT. This also further motivates the use of a learned method for reconstruction.
DATASET
CD
EMD (U)
Airplane
0.52 #
0.00
Car
2.04 + 0.00
0.00
Chair
0.00
0.35
Backpropagation through IPTs directly is possible and outperforms the IP-Encoder in reconstruc- tion quality. However, this optimisation-based scheme is unable deal with noisy or generated IPTs, motivating the need of the IP-Encrder.
4.5 Interpolation in Latent Space
As our final experiment, we consider the linear interpolation between the IPT of two point clouds. In the setting of infinite directions, Theorem 2 guarantees that we only encounter valid IPTs along each interpolation. However, in practice, given our use of smaag--eeneratiee models, is not guaranteed that our input is an IPT the stricl sense. Hence, is crucial to understand the characteristic properties of the IPT in practice since not every proper shape (in other words, IPTs are not surjective). In our discretised representation, interpolating between IPTs can be achieved by interpolating per pixel. To visually assess the quality of the latent space, we reconstruct each intermediary IPT during the interpolation using the IP-Encoder. Figure depicts the resulting point clouds; observe that the intermediary point clouds still remain plausible reconstructions. Overall, this serves to highlight the utility of the IPT latent space.
of sigmoid functions is guaranteed to correspond to
The IPT permits linear interpolations, resulting in smooth interpolations between point clouds.

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5
Discussion
We propose the Inner Product Transform (IPT), which enables us to represent point clouds using 2D images by evaluating inner products. Next to showing that this representation has advanta- geous theoretical properties, we develop an modd-to-en--rrineee Notably, we simplify the point-cloud generation process by introducing an intermediate step, the generation of the IPT descriptor, for which one may use any image-generation model. Our method is (i) exceptionally fast since it uses only inner products and other simple computational building blocks, (ii) well- geol-grounded in theory (it a sufficient statistic on the space of point clouds), and, (iii) we observe that its reconstruction performance and its generative capabillitiss are, for the most part, par with more complex models. This is due to the conceptual simplicity of the model and its provably highly-stabel latent space, which facilitates performing out-of-distribution tasks without the need for retraining or additional regularisation.
pipeline for point-cloud generation.
Limitations. An unfortunate limitation arising from the disconnect between theory and practice is that our injectivity results do not guarantee that one can find suitable directions, in particular given the complex interplay between directions and resolutions of the IPT. We believe that additional guidance for selecting such parameters would be helpful. The current formulation of the IPT is also not invariant with respect rotations, which could be a desired property for some applications (or, at the very least, might make the method more robust in the regime of smaller sample sizes). However, with increasing sample size, we observe that data augmentation enables IPT- models to 'learn' equivariance despite not being intrinsically invariant (see Figure S.5 in the appendix for preliminary results). Moreover, if point clouds become nigh-dimensional, our direetion-sampling procedure becomes inefficient-this is not a problem for the shape-generation tasks we are tackling in this paper, though. A larger limitation in point-cloud generation involves the question of evaluation metrics: In the absence of ground ground-truth information, it is hard to choose the 'best' model (unlike point-cloud reconstruction, where well-defined metrics exist). In terms of other generative-mode evaluation metrics like COV and 1-NNA, our models exhibits only middling performance, ranking slightly better in terms of COV than in terms of 1-NNA. This is in stark contrast to its excellent reconstruction performance and its strong MMD scores. Taken together, this indicates that the IPT is capable of generating high-quality point clouds, but its generative part, based on simple VAEs, is not sufficiently powerful to provide diverse representations.
Future work. Hence, future research concerning more complex and expressive mmdel architec- tures and their capabilities would be intriguing and potentially lead to improved results in terms of diversity metrics, albeit at the cost of computational efficiency. We believe diffission models to be particularly suitable for generating new IPTs, given their proven track record in image generation. Another possible extension of the IP-Encoder could involve a transformer-bast due to its high capacity for tasks involving sets. This could be achieved by considering the IPT as bag of tokens with associated directions. Along those lines, the selected directions for computing the IPT would benefit from proper positional encoding, opposed to implicit (but simple) encoding, this would enable a model to better capture dependencies between individual directions. In addition, given the connection of the IPT to more involved geometrical-tope topollggal descriptors for graphs, meshes, or higher-order complexes, a natural question is to what extent our inversion results apply to such data. This would further serve to contextualise the IPT within the emerging field of topological deep learning [28] but requires extensions to handle datasets without strong geometrical signals [5], for instance. Another feasible next step would be the extension to graph generation tasks, thus building a bridge between image generation and graph generation. The caveats we raised about generative metries are all the more relevant in the context of graph learning, though [19, 26]. Finally, while we have not yet explored any single-shot or even zero-shot experiments. we believe that our work may pave the path towards novel, efficient point cloud foundation models.
architecture [36],
10

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Acknowledgments
The authors are grateful for the productive discussions with the reviewers and their useful feedback, which has significantly strengthened work. We are indebted to them and the area chair, who, like us, believe the merits of this paper. The authors are also grateful for the thorough feedback received from Julius Rohrscheidt. This work has received funding from the Swiss State Secretariat for Education, Research, and Innovation (SERI).
Impact Statement
This paper presents work whose goal to advance the field of machine learning, specifically the problem of high-quality point cloud (or shape) generation. Given the computational efficiency of our method, we believe it to be beneficial for 'democratising' access to machine learning algorithms. There many other potential societal consequences of point cloud generation methods, none which we feel must be specifically highlighted here.
References
[1] Abramson, J., Adler, J., Dunger. J., Evans, R., Green, T., Pritzel, A., Ronneberger, O., Willmore,
L., Ballard, A. J., Bambrick, J., Bodenstein, S. W., Evans, D. A., Hung. C.-C., O'Neill, M.,
Reiman, D., Tunyasuvunakool, K., Wu, Z., Zemgulyté, A., Arvaniti, E., Beattie, C., Bertolli, O., Bridgland, A., Cherepanov, A., Congreve, M., Cowen-Rivers, A. I., Cowie, A., Figurnov, M., Fuchs, F. B., Gladman, H., Jain, R., Khan, A., Low, C. M. R., Perlin, K., Potapenko, A., Savy, P. Singh, S., Stecula, A., Thillassuddramm, A., Tong, C., Yakneen, S., Zhong, E. D., Zielinski, M., Židek, A., Bapst, V., Kohli, P., Jaderberg, M., Hassabis, D., and Jumper, J. M. Accurate structure prediction biomolecular interactions with AlphaFold Nature, 630 (8016):433-50, 2024.
[2] Achlioptas, P., Diamanti, O., Mitliagkas, I., and Guibas, L. Learning representations and generative models for 3D point clouds. In Dy. J. and Krause, A. (eds.), Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pp. 40-49. PMLR, 2018.
[3] Amboage, J., Röell, E., Schnider, P., and Rieck, B. LEAP: Local ECT-based learnable positional encodings for graphs, 2025. URL httttttss//aaxxi..orrre abs///bb/////
[4] Amézquita, E. J., Quigley. M. Y., Ophelders, T., Landis, J. B., Koenig, D., Munch, E., and Chitwood, D. H. Mcasuring hidden phenotype: Quantifying the shape barley seeds using the Euler characteristic transform. in silico Plants, 4(1):diab033, 12 2021.
[5] Ballester, R., Röell, E.,, Schmid, D. B., Alain, M., Escalera, S., Casacuberta, C., and Rieck, MANTRA: The manifold triangulations assemblage. In International Conference on Learning Representations, 2025.
[6] Cai, R., Yang, G., Averbuch-Elor, H., Hao, Z., Belongie. S., Snavely, N., and Hariharan, B. Learning gradient fields for shape generation. In Vedaldi, A., Bischof, H., Brox, T., and Frahm, Cham, Switzerland, 2020. Springer.
J.-M. (eds.), Computer Vision ECCV 2020, pp.
[7] Cheng, A.-C., Li, X., Liu, S., Sun, M., and Yang. M.-H. Autoregressive 3D shape generation via canonical mapping. Avidan, S., Brostow, G., Cissé. M., Farinella, G. M., and Hassner, (eds.), Computer Vision ECCV 2022, pp. 89-104, Cham, Switzerland, 2022. Springer.
[8] Crawford, L., Monod, A., Chen, A. X., Mukherjee, S., and Rabadán, R. Predicting clinical outcomes in Glioblastoma: An application of topological and functional data analysis. Journal of the American Statistical Association. 115(531): 139-1150. 2020.
[9] Curry, J., Mukherjee, S., and Turner, K. How many directions determine a shape and other sufficiency results for two topological transforms. Transactions of the American Mathematical Society, Series B, 1026)1000-1443 2022.

---

[10] Dlotko, P. On the shape that matters-topolopg and geometry in data science. European Mathematical Society Magazine, pp. 5-13, 2024.
[11] Fahim, G., Amin, K., and Zarif, Single-view 3D reconstruction: A survey of deep learning methods. Computers & Graphics, 94:164-190, 2021.
[12] Fasy, B. T., Micka, S., Millman, D. L., Schenfisch, A., and Williams, L. Challenges in reconstructing shapes from Euler characteristic curves, 2018. URL URUnttps:/aarxv..rrr/// 1811.11337.
[13] George, J., Osborn, O. L., Munch, E., Ridgley II, M., and Wang, E. X. On the stability of the Euler characteristic transform for perturbed embedding, 2025. URL httph::t//// //appi/////xvx/o/////// abs/2506.1990 199911
[14] Ghrist, R., Levanger, R., and Mai, H. Persistent homology and Euler integral transforms. Journal of Applied and Computational Topology, 2(1):55-66)) 60, 2018.
[15] Higgins, I,, Matthey, I.., Pal, A., Burgess, C.,, Glorot, X., Botvinick, M., Mohamed, S., and Lerchner, A. 3-VAE: Learning basic visual concepts with constrained variational framework. In International Conference on Learning Representations, 2017.
[16] Kamb, M. and Ganguli, S. An analytic theory of creativity in convolutional diffusion models. In Singh, A., Fazel, M., Hsu, D., Lacoste-Julies S., Berkenkamp. F., Maharaj. T., Wagstaff, K., and Zhu, J. (eds.), Proceedings of the 42nd International Conference on Machine Learning, volume 267 of Proceedings of Machine Learning Research, pp. 28795-28831. PMLR, 2025.
[17] Kim, H., Lee, H., Kang, W. H., Lee, J. Y., and Kim, N. S. SoftFlow: Probabilistic framework for normalizing flow on manifolds. In Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., and Lin, H. (eds.). Advances in Neural Information Processing Systems, volume 33, pp. 16388-16397. Curran Associates, Inc., 2020.
[18] Kim, J., Yoo, J., Lee, J., and Hong, S. SetVAE: Learning hierarchical composition for generative modeling of set strustrue data. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition (CVPR), pp. 15059-15068, 2021.
[19] Krimmel, M., Hartout, P., Borgwardt, K., and Chen, D. Polygraph discrepancy: A classifier- based metric for graph generation, 2025. URL https: wawpi.xxivvvoo//////// abs/ 251000612..
[20] Lee, J., Lee, Y., Kim, J., Kosiorek, A., Choi, S., and Teh, Y. W. Set transformer: framework for attention-based permutation-invarian neural networks. In Chaudhuri, K. and Salakhutdinov, R. (eds.), Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pp. 3744-3753. PMLR, 2019.
[21] Liu, Z., Tang, H., Lin, Y., and Han, S. Point-Voxed CNN for efficient 3D deep learning. In Wallach, H., Larochelle, H., Beygelzimer. A., ''Aché-Buu F., Fox, E., and Garnett, R. (eds.), Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019.
[22] Marsh, I.., Zhou, F. Y., Qin, Lu, X., Byme, H. M., and Harrington, H. A. Detecting temporal shape changes with the Euler characteristic transform. Transactions of Mathematics and its Applications, 8, 2024.
[23] Mémoli, F. and Sapiro, G. Comparing point clouds. In Proceedings the Eunographies/ACM SIGGRAPH Symposium on Geometry Processing, pp. 32-40, 2004.
[24] Munch, E. An invitation to the Euler characteristic transform. The American Mathematical Monthly, 132(():15-25, 2025.
[25] Nadimpalli, K. V., Chattopadhyay, A., and Rieck, B. Euler characteristic transform based topological loss for reconstructing 3D images from single 2D slices. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp. 571-579, 2023.
[26] O'Bray, L., Horn, M., Rieck, B., and Borgwardt, K. Exaluation metrics for graph generative models: Problems, pitfalls, and practical solutions. In International Conference on Learning Representations, 2022.
12

---

[27] Özyeşil, O.. Voroninski, V., Basri. R., and Singer, A. A survey of structure from motion. Acta Numerica, 36:305-364 2017.
[28] Papamarkou, T., Birdal, T., Bronstein, M., Carlsson. G., Curry, J., Gao, Y., Hajij, M., Kwitt, R.. Liò, P., Lorenzo, P. D., Maroulas, V., Miolane, N., Nasrin, F., Ramamurthy, K. N., Rieck, B.. Scardapane, S., Schaub, M. T., Veličković, P., Wang, B.. Wang. Y., Wei, G.-W., and Zamzmi, G. Position: Topological deep learning is the new frontier for relational learning. In Salakhutdinov, R., Kolter, Z.., Heller, K., Weller, A., Oliver, N., Scarlett, J., and Berkenkamp, F. (eds.), Proceedings of the s1 International Conference on Machine Learning, number 235 in Proceedings of Machine Learning Research, pp.
PMLR, 2024.
[29] Qi, C. R., Yi, L.,, Su, H., and Guibas, L. J. PointNet++: Deep hierarchical feature learning on point sets in metric space. In Guyon, I., Luxburg, U. V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., and Garnett, R. (eds.). Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
[30] Ren, X., Huang. J., Zeng, X., Museth, K., Fidler, S., and Williams, F. Xcube: Large-scale 3d generative modeling using sparse voxel hierarchies. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition (CVPR), pp. 4209-421111 June 2024.
[31] Rieck, B. Topology meets machine learning: An introduction using the Fuler characteristic transform. Notices of the American Mathematical Society, 72(7):719-727, 2025.
[32] Röell, E. and Rieck, B. Differentiable Fuler characteristic transforms for shape classification. In International Conference on Learning Representations, 2024.
[33] Tancik, M., Srinivasan, P., Mildenhall, B.. Fridovich-Keis Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., and Ng. R. Fourier features let networks learn high frequency functions in low dimensional domains. In Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., and Lin, H. (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 7537-7547. Curran Associates, Inc., 2020.
[34] Toscano-Duran, V., Rottach, F., and Rieck, B. Molecular machine learning using Euler charac- teristic transforms. In López Fernández, A., Rodríguez-Gonzáále, A., Leirós-Rodríguee R., Mata Miquel, C., and González Suárez, V. M. Artificial Intelligence in Biomedicine, pp. 391-405, Cham, Switzerland, 2026. Springer.
[35] Turner, K., Mukherjee, S., and Boyer, D. M. Persistent homology transform for modeling shapes and surfaces. Information and Inference: A Journal of the IMA, 314:3110-34,, 2014.
[36] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L.,, Gomez, A. N., Kaiser, I. u., and Polosukhin, I. Attention is all you need. In Guyon, I., Luxburg, U. V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., and Garnett, R. (eds.), Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
[37] von Rohrscheidt, and Rieck, B. Diss-I-ECT: Dissecting graph data with local Euler character- istic transforms. In Singh, A., Fazel, M., Hsu, D., Lacoste-Julien, S., Berkenkamp, F., Maharaj. T., Wagstaff, K., and Zhu, J. (eds.), Proceedings of the 42nd International Conference on Ma- chine Learning, volume 267 of Proceedings of Machine Learning Research, pp. 61790-61809 PMLR, 2025.
[38] Wu, L., Wang. D., Gong, C., Liu, X., Xiong, Y., Ranjan, R., Krishnamoorthi, R., Chandra, V., and Liu, Q. Fast point cloud generation with straight flows. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition (CVPR), pp. 9445 9454, 2023.
[39] Xu, Q.-C., Mu, T.-J,, and Yang, Y.-L. A survey of deep learning- based 3D shape generation. Computational Visual Media, 9())407-442, 2023.
[40] Yang, G., Huang, X., Hao, Z., Liu, M.-Y,, Belongie, S., and Hariharan, B. PointFlow: 3D point cloud generation with continuous normalizing flows. IEFF/CVE International Conference on Computer Vision (ICCV), pp. 4540-4549, 2019.
13

---

[41] Zaheer, M.. Kottur, S., Ravanbakhsh, S., Poczos, B.. Salakhutdinov, R. R., and Smola, A. J. Deep sets, In Guyon, Luxburg, U. V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., and Garnett, R. (eds.), Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
[42] Zeng, X., Vahdat, A., Williams, F., Gojeic, Z., Litany, O., Fidler, S., and Kreis, K. LION; Latent point diffusion models for 3D shape generation. In Koyejo, S., Mohamed, S., Agarwal, A., Belgrave, D., Cho, K., and Oh, A. (eds.), Advances in Neural Information Processing Systems, volume 35, pp. 10021-10039 Curran Associates, Inc., 2022.
[43] Zhang, X., Zhang, Z., Zhang. C., Tenenbaum, J., Freeman, B., and Wu, J. Learning reconstruct shapes from unseen classes. In Bengio, S., Wallach, H., Larochelle, H., Grauman, K., Cesa-Bianchi, N., and Garnett, R. (eds.), Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.
[44] Zhou, L., Du, Y., and Wu, J. 3D shape generation and completion through Voxel Diffusion. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pp.
3826-5835, 2021.
14

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Appendix
Changelog
23
Overview of the IPT
24
Additional Downsampling Figures
25
D Additional Experiments
26
Metries for Point Clouds
30
F Architectural Details
31
Properties of the Inner Product Transform
32
22

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A Changelog
We provide brief list of changes that arose during the review process, commenting shortly how we implemented the feedback received by the reviewers.
Changes in the main tex::
We fixed the typo concerning ID and 2D convolutions in Section and Section 4.
We updated the constraints Theorem 2 to enforce y to be non-zero.
We fixed the typo concerning the training times in Section 4.
We clarified all architectures in Section 3 and now discuss wansformer-basel models in
Section 5.
We clarified the ordering of directions in Section and comment on improvements (via positional encodings) in Section 5.
We added details about our hardware in Section
We updated Section 4.4 to better reflect the fact that we provide the first practical method for inverting an IPT.
Changes in the appendix:
We added new experiment on reconstructing point cloud from partial views in Table S.2.
We demonstrate outto--distribution experiments in Table S.3.
We add new ablation study with respect to the number of directions in Table S.4.
We provided new results on downsampling in Table S.5.
We provide new results reconstructing large point clouds in Table S.5 (bottom row).
We added new results for multi-class generation on the MNIST and ShapeNet13 datasets in Figure S.4 and Table S.6.
We provide a new experiment on learning equivariance via data augmentation in Figure S.5.
We wrote preliminary code for using latent diffusion models and added first visualisa- tion of the results in Figure S.6.
We prefer to address the following aspects in future work, rather than adding an inadequate discussion our work:
A discussion on how to use spherical harmonics for the IPT (we believe that this could be essential for completely new approach for calculating IPT, beyond the scope of this paper).
full comparison with models like DeepSDF (we lacked compute to finish this). A rate-distortion analysis (we rewrote the compression aspects at the suggestion of the reviewers and we believe that such an analysis should be better contextualised in follow-up work).
report of FLOPPs (we were unable to collect all this data for the larger baseline models
in time and found some reporting inconsistencies).
We very much appreciate the feedback by reviewers, which helped substantially improve our work.
23

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B Overview of the IPT
(a)
(b)
Figure S.1: An overview of the IPT calculation for 32 directions and resolution 32. For given direction vector &, we filter the point cloud with hyperplanes. The partially filtered point cloud is shown in (a) and points included in the filtration are coloured blue. For each of the 32 directions, sampled uniformly from the sphere, the respective curves along each direction are shown in (b) and the partially-cone coompeeey- curve from (a) highlighted. Note that neighbouring curves are nor necessarily related, requiring us to treat each curve as its own signal. Each of the 32 curves is discretised in 32 steps and stacked form an image representation of the point cloud (c) of size 32 32. The row corresponding to the full curve from (a) and each of the rows corresponds to the curve the same index in (b). For complex geometries such the ShapeNet data, we empirically observe that at least 128 directions are required, using resolution of 128.
24

---

C
Additional Downsampling Figures
We provide a visualisation of Table for the comparison of downsampling with the IP- Downsampler (left) and uniform subsampling (right) for all three classes.
(a) Downsampled point-clouds with the IP- Downsampler (top) and upsampled with the [P-Encader (bottom).
(b) Downsampled point clouds with uniform subsam- pling (top) and upsampled with the IP-Encoder (bot- tom).
Figure S.2: A visual comparison between down- and upsampling using our IP- IP-Downsampler (a) and uniform subsampling (b). The downsampled point clouds using the IP-Downsampler are equidistantly spread, in contrast to uniform subsampling. Compare to uniform subsampling, we achieve higher upsampling quality. Our model has bias towards equidistant point clouds and leads to a better representation of the underlying shape.
25

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D Additional Experiments
This section provides supplementary figures and tables for the experiments in the main text. Please note that some results should be considered preliminary work, pointing towards potential future applications, which we included in the interest of full transparency regarding our work.
Table S.1: Reconstruction results for rendering of the three ShapeNet classes. point cloud is randomly initialised and optimised with the IPT as loss function. We repeat this experiment for different resolutions and different scales. The ideal scale is 1/4 of the resolution (bottom row).
Airplane
Car
Chair
RESOLUTION
SCALE FACTOR
CD)
EMD (1)
CD)
EMD ())
CD
EMD (3)
128
0.29 0.00
2.97
0.00
0,65 3.00
128
64
0,75 0.00
0.18 0.00
2.15
0.000
2.10
0.41 + 0.00
32
0.00
0.00
0.00
0.39 0.00
1.00
0.35 0.00
64
2.63
0.57 0.00
0.06
0.85 0.00
5.83
1.10 0.00
32
1.88 0.01
0.40 0.00
0.01
0.70
4.29
0.78 0.00
16
0.00
0.00
0.00
0.000
0.00
0.62 0.00
32
6.21 0.01
9.85
1.092 1.000
3.54 a 0.01
4.11
0.90
8.56
12.55 1.00
2.53 0.00
=.000
0.00
0.01
0.000
9.05 0.00
1.73 0.00
Figure S.3: Examples of rendered point clouds using optimisation scheme for the three classes of ShapeNet. On the left, the point clouds are rendered a resolution 128 with a scale of 32, leading to high-qualit reconstructions. The right shows the same rendering with the scale set to 128, where we observe unstable gradients. The resulting point clouds have only partially converged, with some clear outliers.
26

---

72/04144 69
7
2/04/444 1
7
?
I
o
Figure S.4: To ensure our pipeline can capture multi-class distributions, we train the IP-Encoder and VAE on the MNIST dataset. The top row depicts the reconstructed point clouds with the ground truth in the middle row. The last row depicts point clouds generated with the IP- VAE. A latent vector is sampled from the latent distribution and decoded into IPT. The novel IPT is subsequently converted to a point cloud with the IP-Encoder. Using our evaluation we find the 1-NNA CD and EMD scores to be 70.31 and 62.50, respectively. This indicates that our pipeline can handle multi-class data distributions.
3 J
4
Figure S.5: Recent work showed that equivariance with respect to certain operations like rotations can also be achieved through data augmentation, thus obviating the need for more complex archi- tectures [1]. To assess the capabilities of the IP-Encoder in this context, we follow Qi et al. [29] and use point cloud version the MNIST dataset handwritten digits. During training, we apply random rotation to each point cloud and then compute the IPT, thus permitting the model to learn an equivariant representation of the data. As the figure shows, data augmentation is sufficient to encode rotations, resulting in an equivariant model without having to specifically add equivariance as a separate inductive bias.
Table S.2: Reconstruction of point clouds based on partial views using the GenRe [43] dataset. The point clouds in the dataset sampled from depth image in particular direction and the task consists of reconstructing the full point from the partial view. We asses the capability of the IP-Encoder to reconstruct the full point cloud from the partial view without any additional training. Thus, the model only trained on the set of complete point clouds. We use the trained IP-Encoder from Section 4.1; CD and EMD have to be multiplied by 1.00 10 and 1.00 x 102, respectively.
Chair (L)
Car (1)
MODEL
CD
EMD
CD
EMD
SoftFlow
2.79
3.30
1.85
2.79
PointFlow
2.71
3.65
1.80
2.85
PVD
3.21
2.94
1.77
2.15
IP-Encoder
1.10
1.24
0.71
0.91
27

---

Table S.3: Out-of-distribution results for the IP-Encoder model. The IP-Encoder trained on the class of airplanes is used to reconstruct the category of cars and chairs to provide an insight into true out-of-distribttion reconstruction and capacity to generalise. Results indicate that our model is not fully general and thus incapable of reconstructing arbitrary point cloud configurations. Due the simplicity of our model architecture, this is partially to be expected. We hope to improve the multi-modal capabilities of our model in future work.
AIRPLANE TO
CD
EMD
Car
22.68
9.36
Chair
203.69
52.29
Table S.4: Ablation with respect to the number of directions used in the IPT to investigate the impact on reconstruction quality. The spatial resolution is kept at 128 (the default) and we progressively reduce the number of directions. Although the reconstruction quality remains consistently high, the number of directions has, in general, positive impact reconstruction quality.
Airplane (L)
NUMBER OF DIRECTIONS
CD
EMD
1.29
2.05
1.18
1.88
16
1.09
1.79
32
1.07
1.69
64
1.56
128
1.05
1.57
Table S.5: Additional downsampling experiments. We follow the same setup as in Section 4.3 and train separate downsampling model for each cardinality. With each model, predict downsampled version of the point cloud, which is subsequently upsampled with the IP-Encoder, to obtain reconstruction of the original point cloud. We observe high quality over all cardinalities, even when the original point cloud is summarized with only 32 points (!). As a comparison, we add the original reconstruction result (ef. Table 1) without downsampling in the second to last row. The bottom part of the table demonstrates the capacity of the IP-Encoder to reconstruct larger point clouds (without downsampling). Although overall performance is still high, model has the fundamental limitation that it predicts the point clouds in single N x 3 vector. To generate large point clouds, we thus believe that different architectures, such as set transformers [20], will provide better scalability with respect to the cardinality.
Airplane (1)
Chair (L)
Car (L)
NUMBER OF POINTS
CD
EMD
EMD
CD
EMD
32
1.40
3.38
18.25
11.61
9.99
6.05
1.20
2.68
14.34
9.79
7.57
4.97
128
1.15
2.31
12.07
8.88
6.64
4.51
256
1.13
2.17
11.59
8.53
6.29
4.23
512
1.12
2.09
11.55
8.34
6.32
4.36
2048
1.05
1.57
9.24
6.19
5.82
3.18
4096
0.87
1.61
8.17
6.33
5.13
3.27
28

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Table S.6: Preliminary results for multi-class reconstruction the ShapeNet13 dataset. The dataset contains 13 classes (airplanes, cars, chairs, lamps, tables, sofas, cabinets, benches, telephones, speakers, monitors, vessels, and rifles) from the full ShapeNet dataset. We follow the experimental setup of Zhou et al. [44] and cite their results as comparison partners. Our model performs well despite its conceptual simplicity; envision more expressive architectures to perform even better.
MODEL
CD (L) EMD (4)
PVD
58.65
57.85
PointFlow
63.25
66.05
LION
51.85
48.95
IP-Encoder (Ours)
63.92
50.28
Figure S.6: Preliminary results for generating samples of the airplane class using a latent diffusion model. We train VAE on the set of IPTs of airplanes, followed by training diffusion model on the latent embeddings of the IPTs to generate new IPTs. Again, we use the IP-Encoder to map generated IPT into point cloud. This demonstrates the flexibility and generality of model, which we look forward to exploiting better in future work.
29

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E Metrics for Point Clouds
good metric for point cloud generation balances computational speed and theoretical guarantees. Finding such metrics is a challenging task, since often computations require the consideration of all pairs of points between the two point clouds. An example the Tromov-HHusdeee which has advantageous theoretical properties, but is hard to evaluate. Two metrics are commonly used describe the distance between point clouds, the Chamfer Distance (CD) and Earth Mover's Distance (EMD), for which we present self-contained summary here. Although not metric in the mathematical sense, CD poses good balance between computational speed and quality and defined for point clouds X and Y as
distance [23].
CD(X,Y)= = Σ min ||+ - yll +
Σ
min | - yll-
(8)
DEY xEX
=EX
yEY
For the EMD, by contrast, the distance between point clouds is viewed the cost required to transport one point cloud into the other, i.e.
EMD(X,
E ((x))( d:X-Y 2EX
min
where é solves an optimal transport problem. Solving the optimal transport problem a computa- tionally intensive task that becomes prohibitive for medium to large point clouds. The properties of using the CD as loss term were investigated in Achlioptas et al. [2], revealing that reconstructions had non-uniform surface density, compared to the uniformly sampled ground truth points. A strong advantage of the CD is its computational efficiency for medium to large point cloud cardinalities.
30

---

Architectural Details
We provide the details of both the IP-Encoder and IP-VAE architecture. The IPT defined in Eq. (4) is viewed as an image of shape y4д X nhd where n, is the discretisation of the heights along each direction and n is the number of directions. the architecture, na denotes channels in the CNN; to accommodate for this, we expect the input to be of the form n4 x X R4, where ns is the batch size. our experiments we set ni n4 128 and denote a batch of IPTs in the diagram with IPT [B, 128, 128]. The internal architecture of the IP-Encoder is:
IPT [B, 128, 128]
3x
Convid
Flatten
FC-ReLU-FC
Tanh-FC
PointCloud [B,2048,3]
In the architecture above, PointCloud [B, 2048,3] denotes the final predicted batch of point clouds.
Our IP-VAE has the following architecture:
IPT 128, 128,128]
4x [oonviddlayyernooo-RRBU
Convid
(FC-MU,
Latent Space
4x SaCCory/DTaampppeeBBccccooomooop
[ConviD-Tanh]
IPT (281 128]
In the diagram above, latent embeddings denoted by FC-MU and FC-VAR, respectively.
31

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G Properties of the Inner Product Transform
This section discusses all proofs for statements from the main paper. While the first result about the injectivity is already known, provide novel, highly-accessible proof, which only requires basic concepts from linear algebra. the following, we will consider two point clouds X, C IR" to be equal if they are equal in the sense of sets; in particular, this implies that they need to have the same cardinality. We will also assume that all points are in general position.
Theorem 1. Given two point clouds X, with + Y, we have IPT(X) + IPT(Y).
Proof. Let X.Y C R" with X * Y and let C S be a set of n + affinely-indeennder directions. Given & E, write Xg
(x, x X) for the projection of X onto the one-dimensional subspace along &, with (x. {) denoting the standard Euclidean inner product. For e IR, we define Xe. {ar X (x, < I).. If for any direction & E E have Xg * Yg, we done hecause this means that we can find thresholds 4 1, E IR with 4 < l/2 such that for / E [4,, 12). the cardinality of Xg.t changes but the cardinality of Yg, does not change
vice versa. Thus, let us assume that X = YE for all directions ç. For each &, we may sort the values by magnitude and calculate differences. i.e. expressions of the form (a,, {y. We have (x. E) <), 0 for all directions E = by assumption, which we may rewrite as (æ - y. = O. Treating this as system of n + 1 linear equations, this is equivalent stating that the kernel of the corresponding linear map is the whole domain. However, since the + directions are affinely independent, all coefficients must be zero, implying that the point clouds are the same. This is a contradiction, so our initial assumption must have been wrong. Thus, there is & E such that Xe * Yg, so we have IXT(X) # IPT(Y).
Lemma 1. Lei (I)) denote ball of radius R in R For two point clouds X,y Y C (R(R),, define their distance as
d(X,Y)
[IPT(X) IPT(()
(5)
SEE
where E is a finite set of directions and ||- 1l1 is the LL-norm restricted the interval [-R, R). The function d(---) satisfies the definition of a metric.
Proof. If d(X, Y) O then |IIРT(X) IPT(Y) O and since || ||2 is metric, it follows that IPT = IPT(Y). Since the IPT is injective, we conclude that = Y. Equality in this case is seen as equality the of sets, that is up to permutation. Both the reverse implication and symmetry follow from definitions. For the triangle inequality, we verify
d(x, Y) [IPT(X)) IPT(Y)]22
[IIPT(()) IPT(Z)) (IPT(Y)) 12T
(10)
S [[IPT(X) IPT(Z)I|+ |IIP(())) IPT(2))l2
d(x, 2) + d(Z,Y).
Lemma 2. Let X,Y Y RR be disjoint point clouds, then
UPT(XUY)= IPT(X) IPT(Y).
(6)
Proof. It follows from the definition of the IPT that for all & E S and h E R we have
IPT(XUY)= Σ 1x(5,h)
=EXUY
Σ 1, + 11,(sh)
(11)
aeX
yeY
+
The second equality uses that the intersection of X and Y is empty, implying that a point is either in X or in Y.
32

---

Theorem 2. Given two point clouds X,Y Y C R", the IPT is surjective for the rational linear subspace spanned by (PT( and IPT(Y), up rational scaling factor. In particular, for p. q E No with
0 'SP P S q and q > 0 we have
P IPT((X)+ 9 - -IPP((Y)- (2)
where = U
Proof. To show surjectivity, it suffices to show that for any rational linear combination of IPTs there exists point cloud Z R" that has IPT equal to that linear combination, up to rational coefficient. This is to say that Va, B Qhy E Q such that
aIPT(X) BIPT(Y) 7IPT(Z).
(12)
Let = p/q and 3 = r/s, then
LI ( -IPT(Y)
qs
(X)PT(X) + grIPT(Y)
(13)
IPT Ugr Y).
Thus, setting = 1/qs and Z = Upp Ugr Y does the trick. We conclude that the IPT is surjective on rational linear combinations. Let 0 S p S q, statement follows from the equalities
-IPT(X)
P IPT(YY
1
IIPT(X) + (q- IPT(Y))
9
(14)
[IPT(U, Up X) + +IPT( Ug-p Y)]
IPTT( Up X KUgpP\)
q
33


Reviews:
[{"review_id": "rLfoqE5fU1", "reviewer": "Reviewer_7RQM", "summary": "This 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.", "questions": "1. 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.", "limitations": "yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The 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.", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their **thorough positive review** and appreciate their extensive deliberations, which show a great attention to detail! Indeed we propose an effective representation of point clouds that allows a simplification of the generative pipeline. \nIn some sense our method is comparable to latent diffusion. Where latent diffusion aims to represent a large image with a smaller one, we propose to represent a point cloud with an image. Our paper aims to show that this representation is expressive, flexible and allows for a much simplified architecture. This perspective opens up a plethora of opportunities for other data modalities as well.\nSince we are the very first to attempt such an approach, we had to carefully scope our experiment section to provide strong evidence that our method is indeed an expressive and suitable representation. We agree that there are many opportunities for future research and extensions to the capabilities of our method, which we find rather exciting! \n\n---\n\n## Summary\n\n---\n\nHere's a brief summary of our changes:\n\n**New experiments (with preliminary results):**\n- **Efficiency:** Reported FLOPS for IPT and baselines; will also add timing comparisons with Point Straight Flows, SPVD, and hardware details.  \n- **Scalability:** Added 4K-point generation experiments showing favorable scaling; extend discussion on hierarchical extensions (e.g., Point-E).  \n- **Generalization:** Conduct ShapeNet-13 and 3D MNIST multi-category experiments; preliminary results suggest competitive performance.  \n- **Cross-domain:** Preliminary reconstruction experiments across categories (Airplane→Car/Chair); add discussion of limitations; challenging tasks for **all models**.  \n- **Approximate collisions:** Add more interpolation/collision examples to support injectivity claims.  \n\n**Clarifications and revisions:**\n- Clarify **motivation** for minimal convolutional architecture and note potential transformer-based improvements.  \n- Correct **typo** regarding 1D vs. 2D convolutions.  \n- Emphasize **simplicity + flexibility** of IPT and discuss broader opportunities.  \n- Extend discussion of **limitations** (scaling, cross-category generation) and future directions.  \n\n---\n\n## Details\n\n---\n\nPlease find our **detailed responses** below. \n\n> The key insight is that this representation is provably injective (under certain conditions), and simple CNNs can effectively learn to invert it\n\nVery much so! Our motivation to use a convolutional architecture is to show the IPT to be an expressive representation. That said, there are opportunities to improve our model. For instance, our encoder model only _implicitly_ encodes the directions, and a transformer based architecture viewing the IPT as a bag of tokens could further improve model performance. The current model is therefore not the best model per se, but the most _minimal_ model that can invert the IPT effectively for distributions of shapes.\n\n> I also appreciate the clarity of presentation and the attention to efficiency … Lastly, while the method is efficient, the timing comparisons (Table 3) lack consistent hardware reporting …recent method focused on efficient generation (e.g., Point Straight Flows, SPVD)\n\nThanks for your positive feedback! We were not only excited by the simplicity of our method, but are also very pleased by its computational performance. We thus very much appreciate the suggestion to report FLOPS and find that  the FLOPS required for the forward pass of the IPT-Encoder model are $571080704$, compared to $3628146688$ FLOPS for a standard ResNet-18 image classifier, which is approximately 6 times more. For our revised manuscript we will also **include the FLOPS** for PointFlow and other methods, if available, as a comparison.\n\nIn terms of _timing and hardware_, Point Straight Flows reports a sampling time of $0.04$ seconds (on an Nvidia RTX 3090 GPU) which would be on par with SetVAE ($0.03$ seconds) (on an GTX 1080ti GPU). We used an NVIDIA GeForce RTX 4070 with 12GB VRAM and a 13th Gen Intel(R) Core(TM) i7-13700K with 32GB RAM. Where possible **we will also report the hardware used by other methods**.\n\nAs for the _comparison to SPVD_, we find that the generation of ~600 “car” point clouds takes between 15–45 minutes, whereas our model only takes a **couple of seconds** on the **same hardware** (see above). Our method can additionally handle larger batch sizes, leading to further performance gains. We will **add this comparison to the revised manuscript**.\n\n> there’s no indication the method scales to higher-resolution shapes (e.g., 4K or 128K points) \n\nThanks for this suggestion! To show that our method also scales to larger point clouds, we have retrained our Encoder to generate 4K points, observing that computational performance is not significantly impacted, whereas generation quality was **slightly improved** (cf. Table 1 in our manuscript; this is not surprising given the fact that we are using more points, but it points towards **favorable overall scalability**).\n\nPlease find the encoding quality numbers for 4K points below:\n\n| Dataset  |   MMD-CD |   MMD-EMD | \n|:---------|:----------|:---------------------------|\n| Airplane | 0.867208 |   1.60767 |\n|Chair |  8.16931  |   6.32711 |\n| Car | 5.13266  |   3.27385 |\n\nScaling this to ultra-large point clouds is going to require additional changes in the architecture (which we kept simple on purpose for this paper), and we aim to address this in future work. We will **extend the discussions and limitations accordingly**.  The standard approach in the literature is to use a hierarchical approach with an upsampler model as is done with Point-E (Point-E: A System for Generating 3D Point Clouds from Complex Prompts, arXiv:2212.08751), for instance; we believe this to be a **feasible strategy for the IPT as well, which we are excited to pursue in the future**.\n\n(Please also refer to the response for reviewer `Ff9T`; effectively, we ran additional experiments that demonstrate that our method also preserves relevant features as we further _decrease_ the cardinality of the respective point clouds.)\n\n> there’s no real out-of-distribution or cross-category generalization shown / unseen shape categories\n\nWe agree that true cross-category generation is a very relevant (but challenging) task; we are preparing out-of-distribution experiments and will add them to the revised manuscript. In the meantime, to show that our model can handle more challenging shape distributions we have conducted a **new experiment** and trained our generation pipeline on 13 classes of ShapeNet (following the setup of LION). **Preliminary results** indicate that our method performs on a par on a subset of the test set (see table below). Since the evaluation over the full test set takes over a week to complete, we have included the evaluation results on a subset of the data for now; we will add the final results in our revision.\n\n| Model | 1-NNA CD | 1-NNA EMD | \n|:------|:--------|:-----------|\n| PVD |   58.65 | 57.85| \n| PointFlow | 63.25 | 66.05 |\n| LION      | 51.85 | 48.95 |\n| IPT (ours, 176 samples) |  54.95|  48.96|\n| IPT (ours, 352 samples) |  63.92|  50.28|\n\nWe will also add an additional experiment for the generation of 3D MNIST point clouds to show that our method works well in a multi-category setting. For a comparison to other methods in a multi-category setting, we will evaluate our performance on the full ShapeNet-13 dataset, as proposed by LION (LION: Latent Point Diffusion Models for 3D Shape Generation, arXiv:2210.06978).\n\nFinally, as an experiment of the performance with respect to unseen shapes (that is, cross-domain reconstruction), we ran preliminary experiments using an IPT-Encoder model on the \"Airplanes\" dataset and applying it to reconstruct the IPT of \"Cars\" and \"Chairs\" shapes, respectively. Since we are considering different datasets, this task is extremely challenging for any model and unsurprisingly impacts performance considerably. On the other hand, training the IPT-Encoder on the joint dataset, performance very much improves again. The latent space for a trained encoder is still stable with respect to modifications of the model (i.e. modifications within the shape class), but the shape changes from an Airplane (with prominent features like wings/tail/...) to a car/chair are too complicated to be captured without having access to additional training samples. We will expand on this behaviour in the revised manuscript.\n\n| Airplane to   |   MMD-CD |   MMD-EMD |\n|:----------|---------------------:|----------------------:|\n| Car       |     22.676         |       9.363         |\n| Chair     |      203.688     |   52.287            |\n\n\n\n> Why 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\nWe experimented with both variants and found the 1D variant to work only slightly better. This was a **typo** from our side and it will be rectified in the updated manuscript. Thanks for the diligent read!\n\n> Some study of approximate collisions would support your injectivity claims in practice\n\nThis is a great point! In the interpolation experiment, we used two rather similar airplanes, where only the engines and tail were different. However, to our knowledge, our method handles such cases well, and **we will add more examples of this kind to the manuscript**.\n\nWe thank the reviewer again for their time and engagement with our work. We hope that our clarifications and proposed revisions address your questions and would kindly ask you to consider increasing your score if they do."}, {"review_id": "PaWLbyxUi9", "reviewer": "Reviewer_Qui9", "summary": "The 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.", "questions": "1. 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?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength:\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", "was_revised": true, "final_justification": "The authors addressed most of my concerns in the rebuttal. This work is significant to the academic community as it advances point cloud processing by offering a more efficient and accessible approach, potentially broadening the use of machine learning algorithms for point cloud tasks. I raise my rating to Borderline accept.", "rebuttal": "We thank the reviewer for their **thoughtful review, which recognizes our core contributions**. The faithfulness of the IPT has a significant premise, namely that the generation of point clouds can be viewed as a pure image generation task. Moreover, we found there to be a significant amount of future avenues of research. We indeed believe that the computational efficiency and stability are strong a advantage that make our methods very suitable in cases where hardware is restricted. Moreover, we consider similar approaches to be an excellent starting point for graph generation or higher-order data generation.\n\n---\n\n## Summary\n\n---\n\nHere's a summary of our changes:\n\n**New experiments and additions:**\n- **Rotation invariance:** Add existing results with random rotation (data augmentation); clarify IPT equivariance properties; discuss possible extensions via spherical harmonics.  \n- **Injectivity:** Clarify that injectivity also holds for a finite number of directions (Appendix F, Theorem 1); highlight backpropagation experiments (Sec. 4.4, Table C.1) confirming practical invertibility.  \n- **Generative model choice:** Expand discussion on limitations of VAEs; motivate future directions with diffusion models, transformers, or general token-based formulations.  \n\n**Clarifications and revisions:**\n- **Ordering of directions:** Clarify use of randomly sampled but fixed directions; explain treatment as unordered multi-channel 1D signal; mention positional encodings as possible improvement.  \n- **Main contribution framing:** Strengthen emphasis that IPT is a **natural and expressive representation** for point clouds, even with lightweight architectures.  \n- **Future work:** Explicitly outline potential extensions (diffusion, transformer-based IPT, invariance mechanisms).  \n\n---\n\n## Details\n\n---\n\nPlease find our **detailed responses** below.\n\n> The experimental results, while respectable, do not demonstrate a significant improvement over existing methods in terms of reconstruction and generation metrics.\n\nWhile we agree that if we had trained a large diffusion model with superior results our claims would be somewhat “easier” to sell, in doing so we would have overlooked the subtle but critical claim we aim to make in this paper: That is, we find the IPT to be a **natural and suitable representation for point clouds**! Had we trained a large model, the additional capacity could compensate for the lack of expressivity. With our tiny models that is simply not an option, therefore our inductive biases have to be task-adequate, supporting our claim, making it possible to translate tasks in the point cloud domain to task in the image domain.\n\nWe would also like to point out that our architecture was **deliberately chosen to be lightweight**; for image classification tasks, for instance, SOTA architectures would nowadays also have to be larger. To **further improve reconstruction quality**, we are considering to exchange the VAE with a (conditional) diffusion model; we believe that, in general, inspirations from latent diffusion can lead to substantial improvements to our generative and the IPT-Encoder.\n\nThe **primary focus** of this work is to demonstrate the general suitability of the IPT framework for such tasks; we believe that we have accomplished this goal. For our **revised version**, we are **scaling up the framework to include diffusion models**, but within the restricted time of the rebuttal, we do not yet have any finished experiments to show for.\n\n> It is not very clear how the authors organize the IPT as an image. How do they order the direction vectors? How do you order the direction vectors? Does different ordering affect recosntruction and generation results?\n\nThanks for raising this great point! For two dimensions, the IPT admits a natural parametrisation of the directions with the angle $\\theta$. However, in 3D (and higher dimensions) this is no longer possible, since we would need at least two parameters ($\\theta$ and $\\phi$) for a parametrisation of the unit sphere. Instead we use a set of **randomly sampled directions** and stack them in an unordered but _fixed_ fashion. Therefore, the only spatial structure in the IPT image is along the columns and forms the motivation to view the signal as a multi-channel 1D signal. Our approach implicitly encodes the directions and an explicit encoding of the directions through positional encoding could further improve the model. We will **clarify these aspects in our revision**.\n\n> The current formulation of IPT is not invariant to rotations.\n\nThis is a great observation! Equivariance and invariance are indeed natural properties for us to consider. We already performed extensive experiments in 2D and 3D to see how well the model can handle such cases and found that simple **data augmentation** in the form of random rotations enables the model to learn the equivariance despite not being intrinsically equivariant. We will **add this experiment to the appendix**. In addition, we will **investigate improved formulations of the IPT**, drawing, for instance, upon **spherical harmonics** to obtain \"builtin\" invariance. Notice that the IPT is **equivariant** with respect to the orthogonal group (or _covariant_ in the strictest sense); we will **clarify this in our revision**.\n\n> The injectivity results are only valid in the case of an infinite number of directions\n\nWe appreciate the reviewer for raising this point and hope to **clarify this**. From a theoretical perspective, the IPT is injective for an infinite number of directions, but this result also holds for a **finite number of directions as we prove in Appendix F, Theorem 1**. While this shows injectivity in theory, it does not show if an IPT can actually be inverted in practice. To show that one can actually invert it in a practical and computationally efficient manner, we perform the Backpropagation experiment in Section 4.4 and Table C1 in the Appendix. This shows that both in theory and in practice the IPT can be inverted, and we are the **first paper to do so for general data**.\n\n> Why do you use VAE as the generative model? Have you tried more advanced diffusion models?\n\nConsidering other types of models for generative modeling is something that is **certainly possible** and something we wish to explore in **future work**. Expanding on the previous point, one could also consider the IPT as a bag of tokens with an additional direction coordinate. This could potentially lead to transformer architectures for the reconstruction (and generation) of IPTs and point clouds and is (most likely) a very fruitful future line of work. For added context, we also wish to mention that our work is the **very first to provide a practical method to invert a topological transform** and show its efficacy as an expressive representation for point clouds. We are excited about the **abundance of future directions** and are already working on followup work to  explore some of these perspectives. Including all of the directions (some pun intended!) into our work may be beyond the scope of a single paper, making it harder to understand the primary claims. We will, however, investigate which additional models can be shown as part of this work without losing the focus on representational efficiency.\n\nWe thank the reviewer again for their time and engagement with our work. We hope that our clarifications and proposed revisions address your questions and would kindly ask you to consider increasing your score if they do."}, {"review_id": "kDNl8v19Zf", "reviewer": "Reviewer_UrXB", "summary": "This 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.", "questions": "see weaknesses", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\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.", "was_revised": false, "final_justification": "", "rebuttal": "We really appreciate your **thoughtful and positive** review, which summarizes our contributions **most admirably in the way we intended**! Our paradigm of separating the generative pipeline into _generation_ and _reconstruction_ is similar to latent diffusion, where we do point-cloud-to-image autoencoding allowing an image generative model to generate point cloud representations. \nThe **crucial difference** is that our pipeline is cross-domain; we hope that this idea will be applied to different data modalities as well, the main premise being that by transforming a “difficult” data modality into an image, we simplify the generation pipeline.\n\n---\n\n## Summary\n\n---\n\nHere's a brief summary of our changes.\n\n**New experiments and additions:**\n- **Number of directions (see below for preliminary results):** Provide empirical analysis and visualizations showing reconstruction quality vs. number of directions; clarify in the main text (Table 1, Appendix C.1).  \n- **Interpolation:** Add extended examples and varied results; clarify pixel-wise linear interpolation of IPTs; expand intuitive discussion with theoretical justification (Theorem 2).  \n\n**Clarifications and revisions:**\n- Clarify how **backpropagation experiments** confirm expressivity and reconstruction quality.  \n- Improve **writing precision** (phrasing such as “on a par with”, “smooth IPT space”).  \n- Strengthen discussion of **geometric plausibility** in interpolated outputs.  \n\n---\n\n## Details\n\n--- \n\nPlease find our **detailed responses** below.\n\n> 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?\n\nOften a mathematical proof of invertibility indeed does _not_ provide any practical guidance as to how one would have to do it in practice or if it can perform well, as correctly pointed out by the reviewer. Thus, in order to assess the ability to invert the IPT we **performed backpropagation experiments** to check that the gradients can, in fact, be correctly backpropagated. This shows that the IPT is **expressive and can reconstruct** the point cloud with high quality, given sufficient directions. The current version of the paper already shows some of these results in Section 4.4 and Table C.1 of the Appendix. Since the table is admittedly somewhat dense we will (a) **further clarify this in the main text** in our revision, while (b) also adding **additional visualizations** and tables that show the behavior of our method as the number of directions is varied. We appreciate this suggestion! \n\nAs a **preliminary experiment and preview**, please refer to this table, showing the _reconstruction quality_  for the \"Airplane\" class as a function of the number of directions (cf. Table 1 in the main text). Our method is **very stable and maintains high quality** even for a small number of directions. In the future, we aim to further analyze and elucidate on this point. Thanks for this great suggestion!\n\n|   Number of directions |         MMD-CD |          MMD-EMD |\n|:-----------|-------------------------:|--------------------------:|\n|        4 |                  1.287 |                   2.047 |\n|        8 |                  1.177 |                   1.882 |\n|        16 |                  1.090 |                   1.789 |\n|        32 |                  1.070 |                   1.685 |\n|        64 |                  1.033 |                   1.559 |\n\n> 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\nThanks for this suggestion! We will **extend the examples** and their **discussion** in our revision, while also adding a more varied set of interpolation results that clarify how the interpolation is performed. The **gist of interpolation** is that we do a pixel-wise linear interpolation of the IPT, viewed strictly as an image. For two IPTs the interpolant will be $\\mathrm{IPT}_\\alpha = \\alpha * \\mathrm{IPT}_1 + (1-\\alpha)* \\mathrm{IPT}_2$ for $\\alpha \\in [0,1]$. To approximate the interpolated point cloud at a particular level of $\\alpha$, we use our encoder model.\n\nThe **theoretical justification** why such a construction works is that the IPT is _linear_ with respect to adding or removing points (we prove this in the paper in Theorem 2). That is to say that computing the pixelwise sum of two IPTs is the _same_ as computing the IPT of the two point clouds simultaneously. Heuristically speaking, the interpolant $\\mathrm{IPT}_\\alpha$ is thus a kind of \"weighted virtual point cloud\" the model aims to reconstruct. Theorem 2 makes this precise; and to further elucidate this construction, we will **extend the manuscript with a more intuitive discussion**.\n\n> 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”. \n\nThanks for your diligent read! We will **rectify these formulations in our revision** and go over the paper again to ensure precise formulations.\n\nWe would like to thank the reviewer again for their engagement and positive review! If you believe we have adequately addressed your considerations we would like to kindly ask you to consider increasing your score and we would more than happy to respond to any further questions or remarks."}, {"review_id": "Rzzqw97UmX", "reviewer": "Reviewer_Ff9T", "summary": "The 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.", "questions": "Is 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?", "limitations": "Yes, the authors have adequately addressed the limitations.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# 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", "was_revised": true, "final_justification": "The authors answered my concerns, and, from my point of view, the questions posed by my fellow reviewers. Given that, I don't see a reason to reject the work, so I'm increasing my score.", "rebuttal": "We thank the reviewer for their positive view concerning our work! The fact that we perform **comparable to state-of-the-art methods** is indeed striking due to the simplicity of our architecture. Representing point clouds as images has allowed us to simplify the pipeline for generation considerably. For future work, we believe that this allows state-of-the-art image generative models to be used for the generation or segmentation of point clouds.\n\nWe also appreciate that the reviewer likes our discussion on the **practical considerations**. The theoretical results typically provide no insight into the practical feasibility. Hence, to show that our method works in practice, we perform the backpropagation experiment in Section 4.4 and Table C.1 in the appendix. The table evaluates how well a point cloud can be reconstructed if the resolution of the IPT is progressively reduced. We will emphasize this perspective more.\n\n---\n\n## Summary\n\n---\n\nHere's a brief summary of our changes:\n\n**New experiments (with preliminary results):**\n- **Varying point cloud sizes:** Already retrained for 32–512 points; quantitative results included, qualitative visuals will be added. Reconstruction quality remains high even in the low-cardinality regime.\n- **Point cloud completion:** Preliminary results for cars and chairs show strong performance; will extend to more categories and compare to DeepSDF.  \n- **Compression:** We will work on providing a rate–distortion analysis to Section 4.4.  \n\n**Clarifications and revisions:**\n- Add full **hardware configuration** (RTX 4070, i7-13700K, 32GB RAM) and training times across tables.  \n- Highlight the **practical perspective** of the backpropagation experiments.  \n- Define **1-NNA** explicitly; replace “comparison partner.”  \n- Fix **table order** (Table 3/4) and unify device reporting.  \n- Adjust or remove the **Impact Statement** as per formatting rules. \n\n---\n\n## Details\n\n---\n\nPlease find **detailed** responses to your additional questions and remarks below.\n\n>  the authors claim that the method is orders of magnitude faster for training and inference … found no description of hardware configuration used for training the models, limiting the assessment regarding performance gains compared to other models.\n\nThis was indeed an oversight of our part, which we will **rectify in our revision**. Our hardware consists of an NVIDIA GeForce RTX 4070 with 12GB VRAM and an 13th Gen Intel(R) Core(TM) i7-13700K with 32GB RAM. Due to our model being small, training our Encoder and VAE takes only approximately **45 minutes** combined. In contrast, generating the samples with PVD **alone** takes _at least_ an hour and training takes much longer leading to a major advantage of our method. \n\n> Furthermore, the authors mention the compression capabilities of the IPT without arguably performing compression experiments. \n\nThe authors were not familiar with the metric mentioned by the reviewer and are very thankful for having pointed that out. Next to **rephrasing** our manuscript, we believe that it may be possible to **add an RD** curve to our backpropagation experiments (Section 4.4), which assesses the quality versus resolution. This is a great suggestion and will provide an additional quantification of our experiment!\n\n> Additionally, this experiment was performed using a single configuration downsizing the point clouds to 256 points and upsampling it back to the original size.\n\nThis is a great point and we have already **retrained our** model to include **various point cloud sizes**, ranging from 32 to 512 (please see the table below). Visually, the smaller point clouds look good and capture the key characteristics from the shape. Unfortunately, rebuttal restrictions prevent us from showing these images, but we observe that the low-cardinality point clouds still **capture the core features** and we will **add such visualizations in our revised manuscript**.  Notice that the table demonstrates that our model can upsample the shape to a **high-quality reconstruction, even when using a small number of key points**.\n\n| Number of points | Airplane, MMD-CD | Airplane, MMD-EMD | Car, MMD-CD  | Car, MMD-EMD | Chair, MMD-CD | Chair, MMD-EMD |\n|---------|-------------|--------------|---------|---------|----------|-----------|\n| 32      | 1.398       | 3.376        | 9.993   | 6.047   | 18.251   | 11.615    |\n| 64      | 1.197       | 2.683        | 7.572   | 4.969   | 14.336   | 9.788     |\n| 128     | 1.146       | 2.312        | 6.644   | 4.510   | 12.070   | 8.878     |\n| 256     | 1.129       | 2.169        | 6.289   | 4.233   | 11.591   | 8.530     |\n| 512     | 1.124       | 2.091        | 6.316   | 4.361   | 11.548   | 8.336     |\n\n> Is 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\nThanks for this suggestion! We attempted this experiment for the car and chair category and our **preliminary experiments point towards its feasibility**. Specifically, we used point cloud samples from depth maps and applied our Encoder to complete the point cloud given the IPT of the _partial_ point cloud. We did not train our model on the partial point clouds but only on the “full” point clouds, making the task arguably more challenging. We are currently working on additional experiments and aim to **extend the manuscript accordingly**.\n\nAs the **preliminary results** show (other methods/numbers are quoted from \"Point Voxel Diffusion, arxiv:2104.03670\"), our method appears to perform very well in this setting! We will **extend these experiments** in more challenging settings and compare to DeepSDF as well. We appreciate this suggestion by the reviewer!\n\n| Category | Model         | MMD-CD      | MMD-EMD    |\n|----------|---------------|---------|--------|\n| Chair    | SoftFlow | 2.786   | 3.295  |\n|          | PointFlow | 2.707   | 3.649  |\n|          | DPF-Net | 2.763   | 3.320  |\n|          | PVD    | 3.211   | 2.939  |\n|          | IPT (ours)    | **1.103** | **1.238** |\n| Car      | SoftFlow | 1.850   | 2.789  |\n|          | PointFlow | 1.803   | 2.851  |\n|          | DPF-Net  | 1.396   | 2.318  |\n|          | PVD    | 1.774   | 2.146  |\n|          | IPT (ours)    | **0.707** | **0.912** |\n\n> Define the 1-NNA acronym in the text and Line 270 \"As a comparison partner, we use...\". the term \"partner\" is odd here.\n\nThanks for these suggestions! The 1-NNA metric refers to the 1-Nearest-Neighbour-Accuracy, first defined by Lopez-Paz and Oquab [1]. We will **clarify this terminology** and update our use of “partner”.\n\n[1] Revisiting Classifier Two-Sample Tests, Lopez-Paz and Oquab, ICLR 2017\n\n> Why 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\nThanks for spotting this! This seems to be due to LaTeX placing tables differently; we will fix this and apologize for the confusion. For Table 3, we aimed to use as little repetition as possible and used the GPU for all models except when explicitly mentioned (second to last row). In response to your comments and those by other reviewers, we will **add a more extensive description** of the hardware used. For the table you mentioned, our hardware consists of an NVIDIA GeForce RTX 4070 with 12GB VRAM and an 13th Gen Intel(R) Core(TM) i7-13700K with 32GB RAM.\n\n> The \"Impact Statement\" section is outside of the 9 page limit. However, it is not a numbered section, as in the text body, nor an appendix. Therefore, I'm not sure if it should count towards the 9 page limit or not.\n\nThanks for your diligence! While the paper FAQ states that such a statement is not required, we decided to add it to point out some potential benefits of our method. To our knowledge, it does **not count towards the limit,** but we are happy to remove it for a revision.\n\nWe hope to have addressed your concerns and questions and look forward to updating the manuscript accordingly. Please let us know if you have any additional issues, concerns, or remarks!"}]

Author rebuttal:

Meta-review: The final ratings are 3 borderline accepts, 1 accept. The AC have read the reviews and rebuttal, and discussed the submission with the reviewers. The reviewers raised a number of points during the review phase including limited experiments and analysis, clarity of methodology, substantiating claims of compression. The authors were able to address these points during the rebuttal and discussion phases, and the reviewers reached a positive consensus. The AC recommends the authors to incorporate the feedback and suggestions provided by the reviewers, and the materials presented in the rebuttal, which would improve the next revision of the manuscript.
Decision: Accept (poster)