APML: Adaptive Probabilistic Matching Loss for Robust 3D Point Cloud Reconstruction
Authors: Sasan Sharifipour
Constantino Alvarez Casado
Mohammad Sabokrou
Miguel Bordallo Lopez
Venue: NeurIPS 2025 poster ()
Keywords: point cloud loss
deep learning
one-to-one matching
optimal transport
Sinkhorn algorithm
adaptive temperature
spatio-temporal point cloud
WiFi CSI
point cloud completion
point cloud generation
shape prediction
differentiable loss
low-density regions
probabilistic assignment
Primary area: deep_learning
TL;DR: We propose APML, a differentiable and efficient loss for point cloud prediction tasks, approximating one-to-one matching using Sinkhorn iterations with adaptive temperature.

Abstract: Training deep learning models for point cloud prediction tasks such as shape completion and generation depends critically on loss functions that measure discrepancies between predicted and ground-truth point sets. Commonly used functions such as Chamfer Distance (CD), HyperCD, InfoCD and Density-aware CD rely on nearest-neighbor assignments, which often induce many-to-one correspondences, leading to point congestion in dense regions and poor coverage in sparse regions. These losses also involve non-differentiable operations due to index selection, which may affect gradient-based optimization. Earth Mover Distance (EMD) enforces one-to-one correspondences and captures structural similarity more effectively, but its cubic computational complexity limits its practical use. We propose the Adaptive Probabilistic Matching Loss (APML), a fully differentiable approximation of one-to-one matching that leverages Sinkhorn iterations on a temperature-scaled similarity matrix derived from pairwise distances. We analytically compute the temperature to guarantee a minimum assignment probability, eliminating manual tuning. APML achieves near-quadratic runtime, comparable to Chamfer-based losses, and avoids non-differentiable operations. When integrated into state-of-the-art architectures (PoinTr, PCN, FoldingNet) on ShapeNet benchmarks and on a spatio‑temporal Transformer (CSI2PC) that generates 3‑D human point clouds from WiFi‑CSI measurements, APM loss yields faster convergence, superior spatial distribution, especially in low-density regions, and improved or on-par quantitative performance without additional hyperparameter search. The code is available at:  https://github.com/apm-loss/apml.

APML: Adaptive Probabilistic Matching Loss for Robust 3D Point Cloud Reconstruction
Sasan Sharifipour', Constantino Álvarez Casado', Mohammad Sabokrou', Miguel Bordallo López'
*Center for Machine Vision and Signal Analysis (CMVS), University of Oulu (Finland)
Okinawa Institute of Science and Technology (OIST), Japan
sasan www.ssaarroiooo constantino alvarezcasado, migiieeadddmmiuuiiee miggue..bordaloeelllu www.allllolll..........o.o...l..
Ground
APML (Durs)
HyperCD
InfoCD
PCN
ShapeNet-34
ShapeNet-55
Figure 1: Impact of loss functions on point cloud completion (using FoldingNet), and generation (using CS12PC). Compared with Chamfer distance-based based losses, APML preserves structure better in sparse regions, reduces clumping, and generalizes across input modalities.
Abstract
Training deep learning models for point cloud prediction tasks such as shape completion and generation depends critically loss functions that measure dis- crepancies between predicted and ground-truth point sets. Commonly used func- tions such Chamfer Distance (CD), HyperCD, InfoCD and Density-aware CD rely on nearest-neighbor assignments, which often induce many-to-one correspon- dences, leading to point congestion in dense regions and poor coverage sparse regions. These losses also involve non-differentiable operations due to index se- lection, which may affect gradient-base optimization. Earth Mover Distance (EMD) enforces one-to-one correspondences and captures structural similarity more effectively, but its cubic computational complexity limits its practical use. We propose the Adaptive Probabilistic Matching Loss (APML), a fully differentiable approximation of one-to-one matching that leverages Sinkhorn iterations on a temperature-scaled similarity matrix derived from pairwise distances. We analyt- ically compute the temperature to guarantee minimum assignment probability, eliminating manual tuning. APML achieves near-quadratic runtime, comparable Chamfer-based losses, and avoids non-differentiable operations. When integrated into state-of-the-arr architectures (PoinTr, PCN, FoldingNet) on ShapeNet bench- marks and on a spatio-temporal Transformer (CSI2PC) that generates 3-D human point clouds from WiFi-CSI measurements, APM loss yields faster convergence, superior spatial distribution, especially in low-density regions, and improved or
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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on par quantitative performance without additional hyperparameter search. The code is available at: https://githuu: com/ aaaaaammooooo 1omm-lsss/an apm1
Introduction
Point sets are a primary representation for three-dimensional data acquired through sensors such as LiDAR, structured-light scanners, and depth cameras [13]. They are widely used in geometric learning tasks, including surface reconstruction, object generation, shape completion, and registration across different views 9 21 33 as shown in Figure LI Many of these applications involve set of output points, where supervision requires aligning predicted points with a ground truth set. This alignment is typically enforced through loss function that quantifies the similarity between point sets. Since these sets are unordered and may vary in cardinality or density, standard vector-based loss functions are not directly applicable. This has motivated the development of set-based loss functions that directly compare point sets during training 100 D.
learning to map an input representation
Widely adopted metrics such as Chamfer distance (CD) offer computational efficiency but often struggle with accurately capturing geometric details due to limitations such as sensitivity to outliers, lendency to cause point clustering, and issues arising from discrete nearest-neighbor assignments [ 17. Conversely, Earth Mover's s Distance (EMD) provides superior geometric fidelity by encouraging one-to-one correspondences but is typically too computationally expensive for direct use in large- scale deep learning [22] 77. Although several modifications to CD have been proposed to address some of its shortcomings [8 17 [6], fundamental need persists for a loss function that robustly approximates the desirable one-to-one matching properties of EMD without its prohibitive cost, while also offering smooth gradients suitable for training modern deep networks.
To address this gap, we introduce the Adaptive Probabilistic Matching Loss (APML). APML is novel, fully differentiable loss function that applies principles from optimal transport to establish soft, probabilistic correspondences between point sets. By approximating a transport plan through application of an efficient Sinkhorn-base mechanism, combined with a key innovation, a data- analytically derived temperature schedule, APML is designed to overcome the limitations of prior methods. This adaptive temperature controls the sharpness of the probabilistic assignments without requiring manual regularization tuning, thereby contributing to stable training and encouraging the generation of high-fidelity point clouds with good structural coherence and surface coverage. Our approach is designed to be broadly applicable tool for multiple point cloud prediction tasks.
The main contributions of this work are the following:
We introduce Adaptive Probabilistic Matching Loss (APML), novel loss function which enforces soft one-to-one correspondences. This approach mitigates common shortcomings of Chamfer Distance, such as point clumping, density bias, and outlier sensitivity. APML. approximates the matching quality of Earth Mover's ' Distance with near-quadratic complexity in the point count and a runtime comparable to CD-based losses.
We introduce an adaptive temperature selection mechanism where each row and column in the transport plan assigns a minimum probability mass (Pmin). This closed-form schedule removes the need for manual tuning of the Sinkhorn regularizer and adapts to the local geometric context of the point sets.
We perform quantitative and qualitative evaluations on standard point cloud completion benchmarks, including ShapeNet and PCN, using three well-established backbone models. These evaluations show that APMI. obtains performance improvements or comparable results relative to established CD-based losses when measured by metries sensitive to structural fidelity, such as EMD.
We present real-world data evaluations on the challenging MM-Fi dataset. For these, a transformer-bases architecture predicts 3D point clouds of humans in indoor spaces using WiFi Channel State Information (CSI) data as an input modality. These experiments demonstrate the ability of APML to improve training stability and the preservation of structural details and surface coverage, in a different task and domain.
APMI. is designed as a drop-in replacement for CD, requiring minimal changes system implemen- tation and introducing only one interpretable hyperparameter.

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2
Related Work
Point Cloud Prediction Tasks. Point cloud prediction tasks include completion, generation, and reconstruction from partial or transformed inputs. These tasks are relevant applications such as 3D scene understanding, autonomous systems, and human-centered sensing. Learning- based models have become the standard approach due to their ability to infer dense geometry from incomplete or abstract inputs [13]
representative method for shape completion is PCN 333, which builds on PointNet 21 and FoldingNet [3]] using an encoder-decoder design that refines a coarse output via grid deformation. This framework has been extended in models such as anowflakeNet [29], which applies progressive refinement, and Transformer-basse approaches like PoinTr (32) and SeedFormer (35), which model long-range dependencies through attention mechanisms. These models typically combine coarse- to-fine stages to first establish global shape and then improve local detail. Alongside completion from visual inputs, recent work has explored point cloud generation from RF-based signals such as WiFi-CSI. The CSI2PC model [1] applies a spatio-temporal Transformer to map CSI data into structured 3D point clouds, processing amplitude and phase information across antennas and subcarriers. It generates representations of indoor scenes with humans and furniture. Evaluation on the MM-Fi dataset [30] demonstrates generalization to unseen subjects and environments, supporting its relevance in joint communication and sensing [8. These architectures highlight the need for effective supervision in learning from unordered point sets. Since predictions are sets without fixed ordering, loss functions must provide permutation arrmutationinvarriant comparisons with accurate geometric feedback. This motivates the development of distance metrics suitable for guiding such models during training.
Point Cloud Distance Metrics and Loss Functions. Supervising deep learning models for point cloud completion and generation requires loss functions that can compare unordered sets of points. Since point clouds may vary in density and cardinality, standard vector-based losses are not directly applicable. Instead,
corennattiooniivariint set-based distances such as Chamfer Distance (CD) [10 and Earth Mover's Distance (EMD) II are commonly used. CD computes nearest-neighhor distances between the two sets in both directions and averages them. It is widely adopted due to its computational efficiency and case of implementation in frameworks used for models like PCN [33), PoinTr [32], and CSI2PC [8]. However, CD allows many-to-on mappings, which often lead to clustering in dense regions and poor coverage in sparse areas. Its reliance on discrete assignments also introduces non-differentiability, affecting gradient-based optimization [ 17. EMD, or Wasserstein-1 distance, addresses these issues by computing a one-to-one correspondence that minimizes the total transport cost between sets [10 (2). This makes it more effective in preserving global shape structure and assigning geometrically meaningful matches. Nonetheless, its cubic complexity in the number of points and requirement for equal cardinality render it impractical for training deep models at scale 77.
To improve over CD while avoiding the cost of exact EMD, several modifications have been in- troduced. Density-aware Chamfer Distance (DCD) incorporates local density weights to balance sparse and dense regions (28), though it may still amplify the influence of isolated points. Hyperbolic Chamfer Distance (HyperCD) modifies the metric space to reduce the effect of distant mismatches and sharpen local gradient behavior [17] Contrastive Chamfer Distance (InfoCD) incorporates regularization term inspired by contrastive learning to spread predicted points across the target shape, improving coverage and robustness sampling noise (16). Another recently proposed approach is the Learnable Chamfer Distance (LCD) [14), which replaces static matching rules with dynamically predicted weight distributions. LCD introduces learnable attention mechanism that adaptively em- phasizes critical reconstruction errors, improving convergence and representation learning. Through adversarial training, LCD identifies structural defects and adjusts local weightings in the distance computation, resulting in faster convergence and enhanced geometric fidelity while maintaining the computational simplicity of CD. Another alternative formulation the Sliced Wasserstein Distance (SWD) 19, which approximates the Wasserstein transport by projecting point clouds into lower- dimensional spaces and averaging transport costs across multiple random slices. SWD preserves the geometric consistency of EMD while having computational complexity comparable to CD, making it suitable for evaluating or training models where EMD would be computationally infeasible. Despite their improvements, DCD, HyperCD, InfoCD, LCD, and SWD remain based on fixed-point correspondences or approximations of transport that may still lead to partial many-to-one mappings and suboptimal gradient flow in sparse non-uniform distributions.
3

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Probabilistic Matching and Optimal Transport. Optimal transport (OT) provides a formalism for measuring dissimilarity between distributions by computing the minimal cost of reassigning mass from one to another [20]. In the context of point cloud comparison, Earth Mover's Distance (EMD) is a special case of OT that seeks a cost-minimizing transport plan between two sets, subject to marginal constraints. EMD produces one-to-one matchings and captures global structure but requires solving linear program with cubic time complexity [22], making impractical for large-scale learning. It also assumes equal cardinality between sets, which is often not the case in real applications.
To address these limitations, Cuturi 77] introduced an entropy-regularizzed formulation of OT that smooths the transport objective by adding a negative entropy term. This modification enables a fast and differentiable approximation using the Sinkhorn- SinkhornKKoop algorithm, which iteratively normalizes the rows and columns of a cost-derived matrix. The result is doubly stochastic matrix that defines soft, probabilistic correspondences between point sets 7 27 Unlike hard nearest-neighhor matching, this approach improves gradient stability during training and allows for continuous optimization. The regularization parameter e controls the sharpness of the assignment, with lower values approaching hard matchings and higher values producing smoother distributions. This tunable behavior has made Sinkhorn-based OT a common tool in differentiable applications [12), including 3D point cloud registration and alignment under uncertainty [4 23 These formulations support learning in cases where exact matching is ambiguous or ill-defined. Building on this foundation, we propose a loss function that approximates one-to-one matching through soft, differentiable assignments while dynamically adjusting assignment sharpness using local distance structure.
3 Adaptive Probabilistic Matching Loss (APML)
Drawing from the principles of optimal transport and the computational efficiency of entropy- regularized approaches such as the Sinkhorn algorithm 77, we introduce the Adaptive Probabilistic Matching Loss (APML). APML is fully differentiable loss function that compares unordered point sets by constructing a soft, probabilistic approximation of one-to-one correspondences. The objective is to provide the geometric supervision properties of transport-based losses while avoiding the computational burden and set cardinality constraints associated with exact methods. Unlike losses, such as CD and its variants, APML does not rely on discrete index selection, which can interfere with gradient propagation. A key distinction from existing Sinkhorn- based approaches is the introduction of a data-dependent mechanism that adaptively selects the temperature paramcter controlling the sharpness of the transport distribution. This parameter is computed analytically from the pairwise distances, ensuring that each point maintains minimum level of probabilistic assignment and eliminating the need for manual tuning of the regularization.
The APML procedure begins by constructing a soft assignment matrix from the pairwise distances between predicted and ground truth point sets. Rather than computing hard matchings, the assignment distributes mass across all candidates based on cemperature-scaled similarities. Assignments are computed independently in both directions and averaged to maintain consistency. The resulting matrix is then refined using normalization steps to approximate a doubly stochastic transport plan. The adaptive control of sharpness allows the loss to adjust locally to the geometry of cach pairwise comparison, providing more stable gradients and improved point coverage.
Before defining the complete loss function, we describe the mathematical preliminaries. Let X E BBxNx xd denote the predicted point sets and X E B M xd the ground truth point sets, where B is the batch size, N and M are the number of predicted and ground truth points, respectively, and is the spatial dimensionality. For each batch element b E (1,..., B). the pairwise cost matrix C E T N is computed using the Euclidean distance:
Co.s. |o.. -Xooll:
(I)
where i E (1...., N} indexes the predicted points and j E (1,...,M indexes the ground truth points. The construction of the transport matrix, the adaptive temperature schedule, and the final loss objective are defined in the following subsections.
Adaptive Softmax. To generate soft correspondences between point sets, the APML method defines an adaptive softmax function. This function is designed to ensure that for any given point, its resulting probability distribution over potential matches assigns at least a minimum probability,

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Pmin E (0,1), 1), to its most likely match (i.e., the match with the lowest cost). This mechanism is applied independently to each row and each column the cost matrix C.
Consider generic cost vector с = (c,,02, c2...., .k. E TR representing the costs from one point to K' other points. The adaptive softmax computation for this vector proceeds as follows:
Cost Normalization: The cost vector с first normalized by subtracting its minimum value to prevent potential numerical issues with large cost values in the exponential function and to focus on relative differences. Let the normalized cost vector, defined as éj = Gj - mitl(=1..... C) for ,, K. Thus, ming Cj 0.
2. Local Gap Definition: Let G(1) and G(2) be the smallest (i.e., 0) and the second smallest different values in c, respectively. If all elements are identical (i.e., 0g = O for all j): ((2) can be considered notionally large handled as special case (see step 4). The local gap, g. is defined to ensure margin if G(2) is very close to (()) as ((2) + 6, where > 0 is a small positive constant (e.g.. 10 10-6) added for numerical stability, particularly if ((2) = 0.
3. Adaptive Temperature Calculation: To ensure that the probability assigned to the element
with the minimum cost (i.e., G(1) = 0) at least Pmin: we solve the following inequality for the
temperature T > 0, assuming K >> 1:
exp( S - 0)
1
2 Pmin
(2)
exp(-T + Σ (T(m)) 1 + (K 1)
The approximation uses the second smallest cost ((2) (via g) as a representative for other non- minimal costs to simplify the derivation T.. This leads to the adaptive temperature:
log
Pam
[K 1 Prein
T =
This expression for T is valid under the conditions K > 1 and 0 < Pmin < 1, which ensure that the logarithmic term is well-defined and strictly positive. When K 1, the assignment is trivially deterministic with probability I, and no temperature scaling is required. The constraints (K 1))m > 0 and 1 Pmin > 0 must hold to avoid numerical instability and to ensure that the denominator within the logarithm remains positive.
4. Numerical Stability for Multiple Minima:
If multiple elements in the cost vector share the same minimum value (i.c., after normalization, E(1) = c(2) = 0), the gap g becomes approximately equal to 5. A small gap leads to large temperature T, which may produce numerically unstable behavior and overly concentrated assignments. To prevent this, if ((2) < cy for small threshold cg (e.g., 10-5), we override the temperature-scaled softmax with a uniform probability distribution. Let P3 denote the assignment probability the j-th element of the vector, where then the assignment is defined as:
1 K
P5
for all j 1,..., K,
where K is the number of elements in c. This guarantees numerical stability in cases where multiple effective minima are present and avoids assigning excessively high confidence any individual element.
5. Scaled Softmax Application: If the uniform override (Step 4) is not triggered, the temperature T from Step is used to compute the soft probability distribution (PI.... Pk) over the K elements:
P3 = exp(-Té)
Σ exp(-TTg
The adaptive softmax procedure detailed above (Steps 1-5) is then applied to the overall cost matrix C. E NM (for each batch element subscript is omitted below for simplicity) to generate two initial probability matrices, Pt and P2:
5

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The matrix P1 Nx MM is obtained by applying the 5-step adaptive softmax procedure row- wise to C. For each row of C' (i.e., for the i-th predicted point X)), the input cost vector is Ci,- = (C......., C,,M) E M. In this application, K = M. Each row of Pi thus forms probability distribution: Σ==(( = for each i = I.....N.
The matrix P2 E NRM M,M is obtained by applying the 5-step adaptive softmax procedure column- wise to C. For each column j of C. (i.e., for the j-th ground truth point X,), the input cost vector is j.j (Ci.j..... CN.)) RN. In this application, K N. Each column of P2 forms probability distribution: Σ (P))as = 1 for each M.
To enforce consistency between these two directional perspectives (predicted-to-gronl truth and ground (rtth-to-redditte)) the resulting probability matrices P1 and P2 are averaged element-wise:
P (PI + P2).
(6)
This matrix P R.Nx represents the initial symmetrized soft assignment probabilities that will be further refined by Sinkhorn normalization.
Sinkhorn Normalization. The symmetrized probability matrix P E R.NXM, obtained from the adaptive softmax stage (Equation (6)). represents initial soft correspondences. However, this matrix P is not guaranteed to be doubly stochastic; that is, its row sums and column sums may not consistently adhere to the marginal constraints of transport plan (e.g., rows summing to I/N and columns to 1/M for uniform marginals, or more generally, rows and columns summing to if P is to be interpreted as a joint probability distribution between individual points).
To refine P into an approximate doubly stochastic matrix, which better reflects coherent transport plan, we apply Sinkhorm- Knopp normalization [25] This an iterative algorithm that alternates between normalizing the rows and columns of the matrix to sum to specific values (typically I in this context for each row and column, assuming we want Piş to represent the probability of matching point X, to Xj such that each point is fully "assigned"). The iterative process is performed for a fixed number of iterations, denoted as Liter (e.g.. Liter 20). In each iteration following two normalization steps are applied sequentially to the matrix P (denoting the matrix at the beginning of an iteration step as and its updated version also as P for simplicity):
1,...: Literr the
1. Column Normalization Step: Each element Pij is divided by the sum of its respective column. For all I,..., Wand] 1,..., M:
Poj
P.
Lk=1 Pkj
+
Estab
This step ensures that after its application, each column of sums approximately to I (i.e.,
Σn Ps ~ 1 for each.).
2. Row Normalization Step: Each element Pij is then divided by the sum its respective row. For all i 1, I,...,A N and ==11..... 1,..., M:
Ps
Pg + Σ Pik + Estab
(8)
This step ensures that after its application, each row of P sums approximately to (i.e., MM Piy ~ 1 for each i).
In these equations, stab is a small positive constant (e.g., 10-8) added to the denominator to prevent division by zero, ensuring numerical stability, particularly if some row or column sums happen to be zero or very close to zero during the iterations. After Liter iterations of these alternating normalizations, the resulting matrix E N X serves as the refined, approximately doubly stochastic transport plan representing the soft correspondences between the predicted point set and the ground truth point set X.
APML Objective Function. Having computed the refined, approximately doubly stochastic transport plan P, e NxM for each batch element 6 (as detailed in the Sinkhorn Normalization subsection,

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using the output of Equation 7 or 8 after Luer iterations), and utilizing the original pairwise cost matrix C2 E RN x the Adaptive Probabilistic Matching Loss (CapMi) is defined. The loss LAPMI. is computed as the expected matching cost under the learned soft assignment probabilities Po.i.j- This is averaged over all predicted points, all ground truth points (implicitly through the sum over j weighted by Pb.s.j which itself sums to 1 over j for each i), and all elements in the batch B:
M
1 LAPML
Po.. Cb...
B b=1
1=1
In this formulation, Po.s. represents the refined probability of matching the i-th predicted point to the j-th ground truth point for the 6-th element in the batch, and Cb,s. is the corresponding cost (distance) between them. The inner double summation Σ Σド= Po.s,j Cr... can be interpreted as the Frobenius inner product (Po, Cb) F, representing the total cost for the b-th pair of point sets under the soft assignment Pb-
This loss formulation encourages accurate point-to-point alignment by penalizing mismatches ac- cording to the learned soft correspondences, while the differentiability of the transport plan P (due to the differentiable nature of the adaptive softmax and Sinkhorn steps) ensures a smooth gradient flow for optimization. The adaptive control of assignment sharpness and the enforcement of bidirectional consistency, as detailed in the preceding subsections, contribute to APML providing a robust and efficient mechanism for soft matching in learning- baaseng-basee geometric tasks.
Experimental Evaluation
Experimental Setup. All experiments are conducted using Python 3. PyToorh 2.5, and CUDA 12.8. The models are trained on publicly available supercomputer, using 4 nodes, each equipped with four Nvidia Volta V100 GPUs with 32 GB of memory each and 2 Intel Xeon processors Cascade Lake, with 20 cores each running at 2,1 GHz. We use the official open-source implementations of FoldingNet, PCN, and PoinTr from 34 with their default training configurations, including learning rate, optimizer, and batch size. For point cloud generation from WiFi Channel State Information, use the CSI2PointCloud model [18], transformer-basel architecture that estimates 3D spatio- temporal point clouds from raw CSI input. The implementation is available online (2) and is used with its default training protocol. We use the official implementations of CD, HyperCD, and InfoCD for loss computation. For our proposed APML, the hyperparameters were set consistently across most experiments, unless otherwise noted: minimum assignment probability Puin 0.8. adaptive softmax gap margin 8 = 10 gap threshold for uniform override Cg = 10- number of Sinkhorn iterations Lieer = 10, and Sinkhorn stability constant Estah 10-S, Any deviations from these settings for specific experiments will be explicitly mentioned. For all experiments. we use a fixed threshold of T = 0.01 when computing the Fl Fl-score. This value corresponds to the default setting in the PoinTr evaluation framework and commonly adopted in the literature for point cloud completion tasks, as it provides reasonable balance between spatial tolerance and sensitivity to local geometric accuracy.
Evaluation Benchmark. We evaluate APML against three leading point cloud supervision objectives, CD, InfoCD, and HyperCD, across a diverse set of benchmarks encompassing both point cloud completion and cross-modal generation. Our experiments span three datasets: PCN (33], ShapeNet (SN34/SN55) (6. and MM-Fi [30], covering both synthetic and real-world modalities, including the challenging task of generating 3DD point clouds from WiFi CSI signals. In addition, we include the synthetic PCOU3D dataset (5). as shown in Figure 3 designed as a controlled environment to systematically evaluate stability and density robustness under five defined input regimes. These datasets enable evaluation in both single-category and multi-category settings. Performance is assessed using standard metrics: Chamfer Distance [TO]. Earth Mover's Distance [22] and Fl score 33 at fixed threshold. Detailed descriptions of the datasets and metric definitions are provided in Appendix [I.I] and Appendix [..2] In addition, we include two widely used alternative loss formulations in the comparison: the density- aware Chamfer variant DCD, which enhances CD by accounting for local point density, and SWD, an EMD-inspired distance based on sliced Wasserstein projections. We report DCD and SWD on configurations where they are applicable (for example, SWD requires equal cardinality between prediction and ground truth) and where we conducted additional runs; other entries are left unreported.
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Experimental Results. Table I summarizes our main validation results. For each completion dataset and backbone, we report the FI score and EMDx100; for generation, we report CD and EMDx100. In all settings, APMI. consistently achieves significantly lower EMD compared to previous losses, often by wide margins of 15-81%, %2 while maintaining comparable or slightly better F1 scores. This trend holds for both simpler models like FoldingNet and more expressive architectures like PoinTTr. Where available, the DCD and SWD rows show that APML reduces EMDx 100 relative to these surrogates while matching or improving FI (e.g., SN34 with PoinTr and FoldingNet), and on MM-Fi achieves substantially lower EMDx100 than DCD with comparable CD. The disaggregated results, additional metries (CD L1 & CD L2) for the PCN dataset, and an analysis of statistical significance are reported in Appendix
Table 1: Aggregated validation results with additional surrogates (DCD, SWD). Completion uses F1
(1) and EMDx100 (L); generation uses CD (L) EMDx100 (L). A dash (-) denotes not reported in our extended runs (c.g., due to applicability under our setup, such as SWD cqual-cardinality assumption or unstable training). Bold denotes the best result within each row.
Loss function
Datasct / Backbone
CD 10
HCD 17
InfoCD [6
DCD
28
SWD 19
APML
Point cloud completion (F1 EMDx100)
PCN
PCN
0,60 160/12.0 12.87
0.64/ 6.53
0.64 5,554
0.62 / 4.72
FoldingNet
0.43 /
0.52 22,93
0.54 / 21.24
0.56 5.34
PoinTr
0.75
8.799
0.43 9.72
0.67 5.62
ShapeNet-55
FoldingNet
0.11/26.50
0.15 23,20
0.17 / 19.55
0.20 9.07
PoinTr
0.46 7.55
0.56/ 9.02
0.35 / 10.35
0.51 6.08
ShapeNet-34
FoldingNet
0.11 27.77
16.53 0.19/ 16.53
0.18 / 22,26
0.19/11.52
0.11/
0.20 9.49
PoinTT
0,42 / 11.88
0.52 12.03
0.35 / 13.41
0,47/9.45
0.50 8.14
SN Unseen-21
FoldingNet
0.11 / 32.10
19.27
25,44 0.35 / 15.19
:
0.20 / 10.42
PoinTr
0.42 / 12.89
0,52 13:27
0.49/ 9.19
Point cloud generation from Wi-Fi FMM//MDXX0)
MM-Fi CSI2PC
04.20
34.20
0. 0.149//300 36.75
0.147 / 35.19 0.148 25.68
0.152/ 14.11
Interestingly, while F1 reflects discrete matching accuracy, it does not fully capture perceptual aligmment or structure preservation. cases where F1 scores are similar, qualitative samples (see (D and EMD values demonstrate that APML produces more geometrically faithful reconstructions. This suggests that APML acts as a regularizer toward semantically meaningful one-to-one alignments that Chamfer-basedd objectives often miss. On SN34, APML lowers EMDx100 with respect to DCD while matching or improving F1 (FoldingNet and PoinTrr) MM-Fi, APMI, achieves a CD) comparable to DCD but reduces EMDx100 by large margin (from 25,6 68 to 14.11). A comparative evaluation of visual perception against the metries for point clouds can be seen in Appendix D.I
Stability Analysis. We evaluated APML robustness to input density and batch composition variations using the synthetic PCOU3D dataset (four primitives, five regimes A-E: sparse, dense, mixed, completion). We trained FoldingNet comparing APML with CD and InfoCD. The decoder was always supervised with the 1024 poont ground truth.APML was the most stable in all regimes, with OCD = 0.0013 and FEMD 0.0025, an order of magnitude lower than CD or InfoCD On sparse inputs (regime A) CD's EMD increased 17% due to many-to-one point collapses, while APML changed < 6%, near one-to-one correspondences. This confirms that the adaptive temperature mechanism effectively regularizes matching, reducing sensitivity to density and batch composition without introducing instability. Further details are provided in Appendix C.Il
Sensitivity to Punin- APML's only interpretable hyperparameter Pmiin places lower bound on the probability mass for the closest target in the adaptive softmax. We assessed its impact on PCOU3D and MM-Fi, varying Puiin E (0.01, 0.1, 0.5, 0.8). Across sparse, dense, and completion regimes, CD and EMD vary < 5% for Pmin E [0.01, 0.8). This weak dependence stems from the analytic temperature T in Eq. 3, which scales inversely with the local distance gap g. Small gaps increase T, distributing probability: large gaps decrease T. yielding sharp assignments. Because T is recomputed row by row, changing Pmin within this broad band leaves the transport plan and gradients nearly unchanged. Quantitative results appear in Table 0 and Appendix
8

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Ablation of APML components. We isolated the role of each component using PCOU3D regimes A, B. and E. Removing Sinkhorn normalization markedly increases EMD, especially in dense and completion settings, due to unconstrained marginals. Using fixed temperature without adaptivity (T = 0.1) leads to unstable optimization and much higher losses, indicating the adaptive link between T and g is necessary. Dropping bidirectional symmetrization introduces directional bias and severely degrades completion. The complete APML configuration yields the lowest CD and EMD across the three regimes. Quantitative results appear in Table 7 and Appendix C.2
Next, we analyze the convergence behavior and runtime characteristics. Table 2 compare wall- clock training time and memory usage for FoldingNet on the PCN dataset using four different loss functions. Although APML introduces ~30% increase in training time per epoch relative to CD, its convergence plot, shown in Figure 2 shows substantially faster improvement in validation Fl, reaching strong performance much earlier in training. In practice, this means that APMI. could achieve competitive results in significantly fewer epochs, despite all models here being trained for 150 epochs for consistency. Other backbones show similar relative overhead (~ 15% and 4-5 RAM for FoldingNet).
Eoochs
Table 2: Wall-clock training time and peak GPU memory usage for FoldingNet trained on ShapeNer-55 (150 epochs on V100 32GB, batch size 128).
Monn by Dy hyymhn Many Junry
034
Loss
Time
Mem
433
CD
55 h
64 GB
HyperCD
57 h
< 64 GB
InfoCD
58 h
< 64 GB
Figure 2: Fl score on SN55 validation set across 150 epochs with FoldingNet. APML converges faster and achieves higher final performance.
APML
76 h
< 320 GB
Although APML's memory usage is higher due to the quadratic cost of maintaining a pairwise cost matrix, this matrix is consistently sparse with over 90% of entries falling near zero before Sinkhorn normalization. Leveraging sparsity, we can prune almost 99.9% of elements of value 10-8 with negligible impact on both qualitative and quantitative results, as seen in Appendix D.3
Discussion
Deep point-cloud supervision has long been torn between efficiency (Chamfer Distance objectives) and geometric fidelity (exact Earth Mover's Distance, EMD). Our results show that the proposed APMI. largely closes this gap: at computational cost from 15 30% higher than CD loss, APML. achieves enough one-to-one alignments to reduce EMD by ~ across three architectures and three datasets. In addition, APML reaches its peak validation EMD in fewer epochs than Chamfer-based training, further lowering the effective computational budget.
Exact EMD is useful reference for correspondence quality, but its cubic complexity and equal- cardinality constraint make full-scale training and evaluation impractical for high-resolution com- pletion and cross modal generation. We therefore include comparisons to DCD, density-aware CD variant, and SWD, an EMD-inspired sliced Wasserstein formulation. In the settings where these methods applicable and stable, APML reduces EMDx 100 and improves qualitative alignment (Table Full EMD supervision at scale is out of scope, and the appendix provides added qualitative comparisons and notes the equal-cardinality and stability constraints that shaped this evaluation.
The controlled study on PCOU3D confirms robustness to input density and batch composition: across regimes A-D the standard deviation of APML is OCD 0.0013 and FEND 0.0025, well below CD and InfoCD, and sparse-only training increases EMD by about 17% for CD but by less than 6% for APML. The analytic temperature that depends on the local distance gap maintains soft assignments when targets are ambiguous and sharp assignments when a clear match exists, which stabilizes gradients without manual tuning.
These improvements are enabled by combination of design choices APMI. First, the data-driven temperature scheme climinates the need for manually tuned Sinkhorn regulariser, as used in

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related optimal transport surrogates 77 yet remains numerically stable; rows with duplicate minima occurred in fewer than 0.6% of batches. Surprisnglyy APML sometimes increases CD-LI/L2 on the strongest backbone (PoinTrr, while still improving perceptual metrics and visual quality, as illustrated in Appendix [ This divergence reinforces recent critiques that Chamfer distance overweights dense regions and correlates poorly with human judgment [17]. The component ablations further indicate that temperature and Sinkhorn are coupled parts of a single probabilistic matching process. Fixing the temperature or removing Sinkhorn sharply degrades performance, and dropping the bidirectional symmetrization introduces directional bias, especially in completion. The full configuration consistently yields the lowest CD and EMD on PCOU3D. Second, the benefit is catptteccturr-anosttie even the weakest encoder (FoldingNet) enjoys a fivefold reduction in EMD, suggesting that APML also acts geometric regulariser when model capacity is limited. These gains persist without hyperparameter re-tuning when transferring from ShapeNet to the WiFi-CSI MM-Fi benchmark, underscoring the robustness of the adaptive temperature mechanism.
Limitations. Despite the strong empirical performance of APML,, its general applicability across architectures, and the absence of extensive tuning requirements, certain limitations remain. First, although APML eliminates the Sinkhorn regulariser E, it introduces a single hyper-parameter, the tuning was
soft-assignment threshold Pmin- We hold pmin 0.01 constant for all experiments; attempted. During training, the resulting maximum row/column probability typically increases to 0.8, but this value is outcome,, not preset. Extremely small or large Pmin values can, in principle, destabilize training, and investigating this sensitivity remains a future work. Our ablation shows that CD and EMD vary by less than 5% once Pmin E [0.01, 0.8) on PCOU3D and MM-Fi, which is consistent with adaptive temperature computed from the local gap governing the effective sharpness.
Second, the required memory still scales quadratically with point count; for 16k points, the cost matrix consumes approximately 1.8 GB, which restricts the use of larger batch sizes. Empirically, most of the similarity values fall to near-zero values after the exponential transformation, rendering the matrix effectively sparse. We currently store it densely for simplicity; exploiting sparsity or low-rank factoring [3 [5 26 could push the effective cost toward O(N log N) and is an obvious next step. Likewise, our runtime figures use plain Py Torch tensor ops; a fused CUDA kernel could further narrow the overhead when compared with CD.
Finally, our evaluation spans two synthetic completion sets and one real generation set. We have not yet measured completion in real-scan datasets such as ScanNet g or KITTI [II, nor generation beyond silhouettes. Thin structures and sensor noise may expose additional failure modes that affect all transport-based losses, although given the current empirical evidence, we find this unlikely.
Future Work and Broader Impact. Future work will explore learnable schedule-based alterna- tives to Pmin- low-rank or sliced Sinkhorn variants to reduce memory usage, and a fully optimized CUDA implementation of APML. Extending evaluation to noisy, real-world scans and -uclidean domains (c.g., surfaces or graphs) is also a priority. These steps aim to bring APML closer to practical deployment in robotics, AR, and digital twin and simulation settings where perceptual structure is more important than point-wiss precision. In contrast, the same technology might reduce the barrier to improve indoor sensing from commodity WiFi devices. APML's one-to-one regularization could enable lighter, energy energy-efficient models for edge devices, facilitating real-time 3-D feedback for low-vision navigation or affordable home robotics. At the same time, stronger completion and WiFi- based reconstruction lower the technical barrier for covert sensing and would require usage-restricted licenses and automated filters that reject models fine-tuned on non-consensual data.
6 Conclusion
We introduced Adaptive Probabilistic Matching Loss (APML), differentiable, near-quadratic surrogate for Earth Mover's Distance that brings one-to-one point-set alignment to deep point-cloud learning with only a modest (~15%) runtime overhead relative to Chamfer Distance. APML's analytically derived, data-driven temperature removes the need for manual Sinkhorn tuning, remains numerically stable across diverse inputs, and improves perceptual metrics, reducing EMD by 15-81 % on three architectures and 24 ShapeNet classes, while maintaining strong generalisation to domain- gains, minimal hyperparameter burden,
specific WiFi-CSI data. The method's is architecture-agnosse and competitive efficiency position APMI, as a drop-in replacement for Chamfer-style losses in point cloud reconstruction, completion, and generation pipelines.
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Acknowledgments and Disclosure of Funding
This research was supported by the Business Finland 6G-WISECOM project (Grant 3630/31/20241 the University of Oulu, and the Research Council of Finland (formerly Academy of Finland) through the 6G Flagship Programme (Grant 346208). Sasan Sharifipour acknowledges the funding from the Finnish Doctoral Program Network Artificial Intelligence, AI-DOC (decision number VN/3137/202-0KMM 6), supported by Finland's ' Ministry of Education and Culture and hosted by the Finnish Center for Artificial Intelligence (FCAI). The authors declare conflict of interest.
Computational resources were provided by CSC-IT Center for Science, Finland, with model training performed on the Mahti and Puhti supercomputers.
References
[1] Achlioptas, P., Diamanti, O., Mitliagkas, Guibas, L.J.: Learning representations and genera- tive models for 3d point clouds. In: International conference on machine learning. pp. 40-49, PMLR (2018)
[2] Arritmic: Csi2pointcloud:
Human point cloud generation from wifi esi. https://githur com/ Comm/Arrtmicc// wttps//iiiccsiippoocooooood (2024), accessed: 2025-05-15
[3] Blondel, M., Seguy, V., Rolet, A.: Smooth and sparse optimal transport. In: International
conference on artificial intelligence and statistics, pp. 880-889. PMLR (2018)
[4] Bringmann, K., Staals, F,, van Wordragen, G., et al.: Fine-grained complexity of earth mover's distance under translation, arXiv preprint arXiv:2403.04335 (2024)
[5] Álvarez Casado, C., Sharifipour, S., Bordallo López, M.: Pcou3d synthetic database (Oct 2025). 17390529,
17390529
[6] Chang, A.X., Funkhouser, T., Guibas, L., Hanrahan, P., Huang, Q., Li, Z., Savarese, S., Savva, M., Song, S., Su, H., et al.: Shapenet: An information-rich 3d model repository. arXiv preprint arXiv: 11Xiv:1512.03011 (2015)
[7] Cuturi, M.: Sinkhorn distances: Lightspeed computation of optimal transport. Advances in neural information processing systems 26 (2013)
[8] Dai, A.. Chang, A.X., Savva, M., Halber, M., Funkhouser, T., NieBner, M.: Scannet: Richly- annotated 3d reconstructions of indoor scenes. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 5828-588850 0828-9839-1 (2017)
[9] Dai, A., Ruizhongtai Qi, C., Nießner, M.: Shape completion using 3d-encoder-predicte cnns and shape synthesis. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 5868-5877 (2017)
[10] Fan, H., Su, H., Guibas, L.J.: A point set generation network for 3d object reconstruction from single image. In: Proceedings of the IEEE conference computer vision and pattern recognition. pp. 605-613 (2017)
[11] Geiger, A., Lenz, P., Stiller, C., Urtasun, R.: Vision meets robotics: The kitti dataset. The international journal of robotics research 32(11), 1231-1237 (2013)
[12] Genevay, A., Peyre, G., Cuturi, M.: Learning generative models with sinkhorn divergences. In: Storkey, A., Perez-Cruz, F. (eds.) Proceedings of the Twenty-First International Confer- ence on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 84, pp. 1608 1608-1617, PMLR (09-11 Apr 2018), http::://proooooiggg..... mtt.i/iss........//8//44. Senevayl8a.htm
[13] Guo, Y., Wang, H., Hu, Q. Liu, H. Liu, L., Bennamoun, M.: Deep learning for 3d point clouds: survey. IEEE transactions on pattern analysis and machine intelligence 43(12), 1338-4366 (2020)

---

[14] Huang, T., Liu, Q. Zhao, X., Chen. J., Liu, Y: Learnable chamfer dis- tance for point cloud reconstruction. Pattemn Recognition Letters 178, 43-48 (2024).
http://waaaaatatto//////o////o/p 2023. htts: http://.......tpppp/////eeeeeeee http:::://ww.ssseeeeireeee.. sciencedirect com/science/articeecc /ttcc////1///00007766666333333
[15] Li, M., Yu, Li, T., Meng, C.: Importance sparsification for sinkhorn algorithm. Journal of Machine Learning Research 24(247), 1-44 (2023)
[16] Lin, F., Yue, Y., Zhang, Z., Hou, S., Yamada, K., Kolachalama, V., Saligrama, V:: Infoed: a contrastive chamfer distance loss for point cloud completion. Advances in Neural Information Processing Systems 36, 76960-76973 (2023)
[17] Lin, M., Xu, Y., Wang. Z., Liu, Y.: Hyperbolic chamfer distance for point cloud completion. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp- 19176-19185 (2023)
[18] Määttä, T., Sharifipour, S., Bordallo López, M., Álvarez Casado, C.: Spatio-temporal
3d point clouds from wi-fi-csi data via transformer networks. In: 2025 IEEE 5th In- ternational Symposium Joint Communications & Sensing (IC&S). pp. 1-6 (2025).
http://doi.org// http://.....r/111.100///C56666 109//196466111111 2.0.C6666666.2255.... 10880665
[19] Nguyen, T., Pham, Q.H., Le, T., Pham, T., Ho, N., Hua, B.S.: Point-set distances for learning representations of 3d point clouds. In: 2021 IEEE/CVF International Conference on Computer Vision (ICCV). pp. (0458-10467 (2021). https://doii.or//1.. http:://li..rr/......1.ccvvv/// 202.000330
[20] Peyré, G., Cuturi, M., et al.: Computational optimal transport: With applications to data science. Foundations and Trends@ in Machine Learning 11(5-6), 355-607 (2019)
[21] Qi, C.R., Su, H., Mo, K., Guibas, LJ:.: Pointnet: Deep learning point sets for 3d classification and segmentation. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 652-660 (2017)
[22] Rubner, Y., Tomasi, C., Guibas, L.J.: The earth mover's distance as a metric for image retrieval. International journal of computer vision 40, 99-121 (2000)
[23] Sarlin, P.E., DeTone, D., Malisiewizz, T., Rabinovich, A.: Superglue: Learning feature matching with graph neural networks. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 4938 4938-49477 (2020)
[24] Shen, Z., Feydy, J., Liu, P., Curiale, A.H., San Jose Estepar, R., San Jose Estepar, R., Nietham- mer, M.: Accurate point cloud registration with robust optimal transport. Advances in Neural Information Processing Systems 34, 5373-5389 (2021)
[25] Sinkhorn, R., Knopp, P.: Concerning nonnegative matrices and doubly stochastic matrices. Pacific Journal of Mathematics 21(2), 343 348 (1967)
[26] Tang, X., Shavlovsky, M., Rahmanian, H., Tardini, E., Thekumparampil, K.K., Xiao, T., Ying, L.: Accelerating sinkhorn algorithm with sparse newton iterations. arXiv preprint 12223 (2024)
[27] Villani, C., al.: Optimal transport: old and new, vol. 338. Springer (2008)
[28] Wang, M., Luo, X., Li, Y.: Density-aware chamfer distance as comprehensive metric for point cloud completion. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 13014-13023 (2021)
[29] Xiang. P., Wen, X., Liu, Y.S., Cao, Y.P., Wan, P., Zheng, W., Han, Z.: Snowflakenet: Point cloud completion by snowflake point deconvolution with tkip-rr. In: Proceedings of the IEEE/CVF international conference on computer vision. pp. 5499-5509 (2021)
[30] Yang, J., Huang, H., Zhou, Y., Chen, X., Xu, Y., Yuan, S., Zou, H. Lu, C.X., Xie, L.: Mm-fi: Multi-modal non-intrusive human dataset for versatile wireless sensing. Advances in Neural Information Processing Systems 36 (2024)
12

---

[31] Yang, Y., Feng, C., Shen, Y., Tian, D.: Foldingnet: Point cloud deformation. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 206-215 (2018)
via deep grid
[32] Yu, X., Rao, Y., Wang, Z., Liu, Z., Lu, J., Zhou, J.: Pointr: Diverse point cloud completion with geometry-aware transformers. In: Proceedings the IEEE/CVF international conference on computer vision. pp. 22498-12507 (2021)
[33] Yuan, X., Chamzas, D., Mitra, N.J.: Pen: Point cloud completion network. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 774-783 (2018)
[34] Yuxumin: Pointr: Point cloud completion with transformers. https://github. com/
yuxumin/PoinTr (2021), accessed: 2025-05-15
[35] Zhou, H., Cao, Y., Chu, w., Zhu, J. Lu, T., Tai, Y., Wang, C.: Seedformer: Patch seeds based point cloud completion with upsample transformer. In: European conference on computer vision. pp. 416-432 Springer (2022)
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Question: Are new assets introduced in the paper well documented and the documentation provided alongside the assets?
Answer: [Yes]
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(https://github com/
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The answer NA means that the paper does not release new assets.
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19

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The answer NA means that the paper does not involve crowdsoureing nor research with
human subjects.
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Does the paper describe the usage of LLMs if it is important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
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20

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A Technical Appendix: Algorithm description and theoretical analysis
This appendix collects algorithm pseudocode, and a theoretical analysis and comparison against other loss functions.
A.1 APML Algorithm Summary
The computation of the Adaptive Probabilistic Matching Loss for batch of predicted point sets X and ground truth point sets X summarized in Algorithm [ This procedure includes pairwise cost computation, adaptive softmax with bidirectional matching, Sinkhorn normalization, and final loss evaluation.
Input: Predicted point sets X E BBNxx Ground truth point sets X E TR BxMxx
Input: Hyperparameters: Pmin: O, Eg (adaptive softmax); Litery Estab (Sinkhorn)
Output: Loss value LAPMI. Initialize total loss: Linall <- 0:
for b + to B do
X, + :b. X, :bb;;.];
Compute cost matrix C N where (Ct)ss = ||xo, Xo,sl2s
// Compute soft assignments from predicted to ground truth for 1 do
+ (Ch)i.s; (Pi,)i; + AdaptiveSoftmax Vec(o,, M, Puins 5, Cg);
end
// Compute soft assignments from ground truth to predicted for j M do C.j + (Co)..i (P2t)as
VelappiveSoftmaxxxcee..... N, Priin 8, Eg);
end
// Symmetrize
Pmn + (P1. + P2.0):
// Apply Sinkhorn normal ization P+ Piiti
for 1 + to Liter do M do
for j
P.j / (EN Phij Estab))
end
for i f- to
Ps: Pt.l (Ei- Pite + Esub))
end
end
// Compute loss for current batch item
Lb Σi م Pij Cb.azi
Listal Liotal + Lb:
end
CAPML f- Crotal/ B:
return LAPML:
Algorithm 1: Adaptive Probabilistic Matching Loss (APML)
A.2 Theoretical Analysis and Comparison with Other Loss Functions for Point Cloud Tasks
We provide a theoretical comparison of APML with representative loss functions commonly used point cloud prediction tasks. The comparison focuses on computational complexity, sensitivity to outliers and local density variations, and the nature of the assignment strategy. The proposed APML. is contrasted with methods based on nearest-neighhor correspondence and with loss functions inspired by optimal transport. The comparison is summarized in Table
21

---

Unlike nearest-neiie searrestneighhor-Beee based losses, which rely on discrete assignments and may suffer from clustering artifacts or instability in sparse regions, APML constructs probabilistic transport plan that is refined through iterative normalization. Its formulation avoids the high complexity of exact optimal transport by using a fixed number of Sinkhorn scaling steps, while introducing an adaptive temperature mechanism that automatically adjusts the sharpness of the assignment based on the local cost structure. This allows APML to improve alignment quality without requiring manual regularization tuning.
Table 3: Theoretical comparison of point cloud loss functions. Complexity assumes N predicted points, M ground truth points, dimensionality d, and L Sinkhorn iterations (for APML). For EMD, K max(N, N, M) (complexity often cited assuming N ~ M K).
Function
Mechanism
Complesity (Approx.)
Duneiyy
Mapping Peef-
Theorotical
Limitutions
Seusitivity
Seusitivity
ereace
tages
CD L00
Nearest (NN)
Neighbor
OUNAD)) (saive)
High
1ow
Mony-to-one
Simplicity; Relatively Outlier & density sues: Champing: Gradi
OI(N log 2d
Fasi
dy (with spotial stree-
cut quality
tures)
LMD A
Optimal Transport (Exact LP)
O(K log K)
Liw
Hligh
One-io one
High geometric
fi- Very high computa- tional cost, Cardinality constraint (std. fom)
deiity; Robusiness
DCD 33
weighty
OUNMS
Medium
Medium- High
Mont-so-une
Attempts
improved
Can boost out- tuning tuning: NN-hosed
NN
deasity
HypeCD 1b
Hapatolic
space
O(N301)
Modium
Many-so-one
Down-weights
pairs. (cutlier robust- ness)
InfoCD
Contrastive Regulrizztion
NN
O(NM6
Con-
Medium
Mediom
Many-to-cese (encourages
Mirigates clumping:
Addal setup complex- ity: Contastive tuning: Still NN-based
trastive Overhead
[via spread-
Improves cuverage
ing)
spretddnng)
APML (ours)
Sutkhors OT Ap- prss, w/ Adagiive Tomp
M(N + D)
Low- Medium
Medium-
Soft one-to-one
EMD-like progerties Differesiable; Adag- live temp. (eo globiil
Higher cast than CD, Appros, quality cvs ex- sei FMD)
High
c <tuning)
The dominant computational costs of APML arise from the pairwise distance computation, with complexity MNN and the iterative Sinkhor normalization, which requires L matrix scaling steps. This results in overall complexity of O(N M ()) For point clouds of low dimensionality (e.g., = 3) and fixed L (c.g.. L = 20), the total cost remains within a practical regime. In contrast to exact EMD solvers, APML avoids combinatorial optimization by relying on differentiable scaling operations, making it compatible with standard backpropagation pipelines.
The adaptive temperature mechanism further differentiates APML from other methods that require manual tuning of regularization parameters, such as e in entropy-regularizzed Sinkhorn distances or in HyperCD. This adaptivity provides per-instance control over the sharpness of the assignment, enabling the loss to adjust locally to the underlying geometry of the point sets. This design aims to preserve stable gradients and promote better structural alignment in both dense and sparse regions.
By integrating a transport- based formulation with adaptive regularization, APML represents a principled alternative to existing point-based losses, combining efficiency, flexibility, and improved theoretical properties for learning- based geometric matching.
B Technical Appendix: Description of datasets and metrics
This appendix describes in more detail the datasets used and the evaluation metrics chosen.
B.1 Evaluation Databases
The evaluation is conducted using three datasets, which cover standard benchmarks for point cloud completion and a modality-speciff task involving generation from WiFiibased measurements:
PCN Dataset (33 The PCN dataset is commonly used benchmark derived from ShapeNet- Core (6. It includes pairs of partial and complete 3D point clouds for a set of object categories such as airplane, cabinet, car, chair, lamp, sofa, table, and watercraft. The partial inputs typically contain 2048 points, while the ground truth shapes consist of 16384 points. These data pairs are used to assess the ability of a model to recover full geometry from incomplete observations. For the PCN evaluation we follow their standard official splits and report results on the test sets.
22

---

ShapeNet (SN34/SN55) [6]: The ShapeNee33 and ShapeNet55 subsets are drawn from the ShapeNetCore dataset [6], including 34 and 55 object categories, respectively. These subsets are used to test model generalization in point cloud completion tasks under more challenging conditions. for ShapeNet-34 and ShapeNet-55, we follow the hard evaluation protocol, which measures generalization to unseen categories and shapes. ShapeNet-55 includes all 55 categories for both training and evaluation. ShapeNet-34 is trained and tested 34-category subset, while ShapeNer Unseen-21 evaluates generalization by training on the same 34 categories but testing on the remaining 21. For the PCN and MM-Fi datasets, we follow their standard official splits and report results on the test sets.
MM-Fi Dataset [30]: To evaluate APML on a point cloud generation task from a different modality, we use the MM-Fi dataset 30) This dataset provides WiFi Channel State Information (CSI) measurements collected from commercial WiFi devices, along with corresponding ground truth 3D point clouds captured with LiDAR device representing human subjects performing various activities in indoor settings. This dataset is particularly relevant for assessing the robustness of loss functions in scenarios involving noisy. real- worldworld sensor data and the generation of complex, dynamic human shapes. For MM-Fi datasets, we follow their standard official splits and report results on the test sets.
PCOU3D Synthetic Dataset [5): For the additional experiments, we constructed synthetic dataset, as shown in Figure 3 as a controlled environment to evaluate the stability of APML under different input conditions. The dataset comprises four analytic primitives: cube, sphere, a pyramid, and a Gaussian blob. Each primitive is represented as a point cloud with 1024 points sampled uniformly from the interior of the shape within the interval [-0.5, 0.5] The data are partitioned into 1,600 shapes for training, 400 for validation, and 400 for testing, with all ground- truth point clouds containing exactly 1024 points. During training, the encoder receives a modified version of the ground truth, while the decoder is supervised against the complete 1024-point cloud. Five regimes considered: A) all sparse (256 points), B) all dense (1024 points), C) mixed batches with 50% sparse and 50% dense, D) random assignment with p(sparse) = 0.5, and E) completion with contiguous patch removed, leaving 512-1023 points. Regimes A-D evaluate reconstruction robustness to density variations, while regime E examines completion with missing surface regions. This setup enables systematic evaluation of stability and density sensitivity across controlled conditions.
B.2 Evaluation Metrics
To quantitatively assess and compare the performance of models trained with different loss functions for point cloud completion and generation tasks, we adopt the following standard metrics. Let X = ( } be the predicted point set and X (21,..., } be the ground truth point set.
Chamfer Distance (CD). The Chamfer Distance measures the average nearest-neighbor distance between two point sets (10). We report two common variants:
CD-L1 (Mean L2 distances): This is the sum of average Fuclidean distances between each point in one set and its closest point in the other set. Lower values are better.
(X,
Σ BEX
min 2|12 +
min 2112. Σ
(10)
N
M
FEX REX
. CD-L2 (Mean Squared T2 distances): This is the sum of average squared Euclidean distances, and is the most common CD formulation. Lower values are better.
(x
min j Σ
x11 +
1 M
Σ REX
min || 211
(11)
N
sEx
SEX
BEX
Earth Mover's Distance (EMD). EMD measures the minimum cost to transform one point set into another, reflecting overall structural similarity (22]. For point sets X and X of equal cardinality (N = =M), it is defined as the solution to an optimal assignment problem:
SEMD(X,X)
= min Σ x
| - (
(12)
8,E
23

---

Regime A
Regime
Regime
Regime
Regime
Sphere
Culbe
Pyramid
Goursiin
Figure 3: Qualitative visualization on PCOU3D. Rows correspond four analytic shapes: sphere, cube, pyramid, and Gaussian cluster. Columns correspond to the five input regimes used in our stability study: sparse input with 256 points, B dense input with 1024 points, batch level mix of sparse and dense, D random per sample mix with probability 0.5 for sparse, and E completion where a contiguous patch removed leaving between 512 and 1023 points. In each panel the full ground truth point cloud is shown in light gray and the regime specific input is overlaid in blue.
where e is a bijection. Due to computational cost and cardinality constraint (implementations often require sampling or padding if N * MM, report EMD multiplied 100 (EMDx 100). Lower values are better.
F1-Score (@r), The Fl-score assesses reconstruction accuracy by balancing precision and recall, commonly used in point cloud completion 33 Given distance threshold т, precision P(r) and recall R(T) are defined as:
P(T) = Σ
min =Ex
| - x112 <T)
(13)
N
&EX
R(T) = M
Σ
1
min SEX
|| - L112 <T) =]]2=]]
(14)
TEX
where IT--) is the indicator function. The F1-score is their harmonic mean:
P(r) P(()
FI(T) = P(T) + R(T) E
(15)
where EFi is small constant (e.g., 10 to prevent division by zero if both P(7) and R(T) are zero. our experiments, we use threshold value of = 0.01. Higher Fl-scores are better.
note that while CD-LI widely used evaluation, its suitability for accurately reflecting perceptual quality in generation tasks can be limited. As will be discussed with our qualitative results, numerically higher CD values not always correspond to visually worse point clouds, particularly when distributed soft matching is involved, and vice-versa.

---

C Technical Appendix: Disaggregated results and statistical significance
This appendix shows disaggregated results per category and complementary metrics for some of the experiments Section We evaluated APML on point cloud completion task using the PCN dataset, comparing its performance when integrated into three different backbone architectures, FoldingNet [3T, PCN [3 and PoinTr 322, against models trained with HyperCD 17 as a strong
baseline. We present the results in Table
Table 4: Performance across PCN Dataset using Metric EMD*100 100, disaggregated by categories. Best results per backbone highlighted.
PCN Object Categories
Airplane
Cabinet
Car
Chair
Lamp
Sofa
Table
Watercraft
NotdingNet CD
15.20
35.98
23.26
36.30
38.32
30.16 16
28.43
22.01
FoldingNet HCD
14.19
27.03
19.30
30.26
30.86
23.73
19.52
18.55
Not InfoCD
13.76
4.15
18.41
23.27
28.54
22.91
21.42
17.42
FoldingNet APML
3.36
5.62
3.84
6.40
7.89
5,41
5.66
4.52
PCN + CD
34.70
16.82
10.56
14.66
19.84
15.09
10.79
10.46
PCN + HCD
3,86
6.61
3,91
6,81
13.21
6.02
6.52
5.31
PCN InfoCD
3.54
5.83
3.58
5.40
10.01
5.66
5.71
4.54
PCN + APML
3.14
4.41
3,33
4.83
8.71
4.59
4.76
4.00
PoinTr CD
5.62
9.16
6.47
8.29
20.28
9.58
9.28
6.96
PoinTr HCD
4.78
9.05
6.19
7.99
18.40
9.36
8.47
6.06
PoinTr InfoCD
8.31
9.88
8.76
9.83
11.29
11.22
11.07
7.39
PoinTr APML
3.22
6.14
5.30
5.55
7.49
6.03
6.25
5.02
C.1
Stability Analysis on the PCOU3D Dataset
To assess the stability of APML. under controlled conditions, we introduce the PCOU3D synthetic dataset described in Appendix B. The dataset allows reproducible experiments across five distinct regimes: A (sparse, 256 points), (dense, 1024 points), C (batch-level mix 50% 50% B), D (random per-sample mix 0.5), and E (completion with contiguous patch removal, 512-1023 points).
We trained FoldingNet for 30 epochs using identical hyperparameters for all losses (Adam optimizer, learning rate 10 4, batch size 32). Each model was supervised with the complete 1024-point cloud. Results are reported on the test set, and lower values indicate better performance.
Table Comparison of performance using CD and EMD metrics on the PCOU3D synthetic dataset.
Lower values are better.
CD (L)
EMD (L)
Loss
A
B
D
E
C
D
APML
0.0474
0.0497
0.0502
0.0471
0.0481
0.0433
0.0459
0.0485
0.0416
0.0423
CD
0.0529
0.0413
0.0429
0.0495
0.0492
0.0706
0.0585
0.0626
0.0600
0.0583
InfoCD
0.0496
0.0557
0.0495
0.0509
0.0495
0.0578
0.0805
0.0597
0.0641
0.0639
Key observations.
Table summarizes the stability analysis across all regimes. APML achieves consistently lower losses and minimal variation between conditions. For example, the difference between the best and worst regime in EMD remains below 5%, indicating a high robustness compared to CD and InfoCD Training with sparse-only inputs (regime A) increases the EMD for CD by approximately 17% due to many-to-one assignments, while APMI shows a variation less than 6%. This confirms that APMI. maintains stable matching behavior even under strong input sparsity.
25

---

This stability originates from the adaptive temperature T, computed locally for each row of the cost matrix based on the distance gap g (see Eq. 3). When candidate targets are close (e.g., distances 0.010 and 0.012, = 0.002), T becomes large (T s 80), producing soft assignment P a (0.83, 0.......) When the best match is clear (e.g., distances 0.010 and 0.40, g = 0.39), T becomes small (T R 0.4), yielding a sharp P x (0.99, 0.006....) ). As T is recomputed for each mini-batch, ambiguous matches remain soft while confident matches stay sharp, ensuring smooth gradients regardless of density or sampling.
).
We also tested the effect of disabling the uniform fallback, which replaces the temperature scaled softmax with uniform distribution when several minima coincide (g < Eg). Removing this safeguard substantially worsened performance, raising the EMD on dense inputs from approximately 0.055 to O. 0010, and confirming its importance for numerical stability when ambiguous matches occur. These results demonstrate that the analytically derived temperature adapts automatically to local structure, avoiding the need for a manually tuned hyperparameter and maintaining stability across all sampling regimes. Qualitative reconstructions for regime A are illustrated in Fig. 4 showing that APML retains global shape integrity across all primitives while CD and InfoCD shows local collapses or flattened regions.
C.2 Ablation Studies on Pmin and APML Components
Table 6 reports an extended sweep of Pmin spanning four orders of magnitude (10-1 to 0.999) on the PCOU3D dataset, together with the corresponding results on MM-Fi for the tested values. Across all regimes (sparse input (A), dense input (B), and completion (E)), CD and EMD remain nearly invariant within the practical range Pmin E [0.01, 0.8), with variations below 5%. The same stability pattern appears on MM-Fi, confirming that APML is largely insensitive to this byperparameter.
Table 6: Ablation of Pmin on PCOU3D and MM-Fi, Lower is better. Bold values denote the best performance in each regime.
Pmin
A (Sparse)
B (Dense)
E (Completion)
MM-Fi
CD
EMD
CD
EMD
CD
EMD
CD
EMD
0.0001
0.05335
0.04722
0.05973
0.06322
0.05302
0.04809
0.01
0.05283
0.05248
0.05703
0.05559
0.05324
0.05648
0.1431
0.1686
0.1
0.05068
0.04954
0,05372
0.05907
0.05505
0.05273
0.1398
0.1661
0.25
0.05466
0.06696
0,05653
0.05586
0.05346
0.04937
0.5
0.05586
0.04945
0.05033
0.04300
0.05417
0.05093
0.1371
0.1671
0.75
0.05655
0.05121
0.06147
0.05975
0.05480
0.05094
0.8
0.05331
0.05067
0.05443
0.05191
0.05840
0.05514
0.1392
0.1637
0.9
0.05067
0.05784
0.06168
0.06910
0.05286
0.04892
0.999
0.05262
0.05040
0.05618
0.05372
0.05326
0.05200
This behaviour can be explained analytically. In APML, Pmin acts only as a lower bound on the probability assigned to the closest candidate in the adaptive softmax. The actual sharpness of the assignment is determined by the adaptive temperature T, calculated from the local distance gap g as shown Equation 3 The mapping T(Pmin) is monotonic and smooth for all < Pmin < 1. For small Pmin- T grows approximately linearly, creating softer assignments. For larger Puiin T saturates, producing sharper distributions. Because T is recomputed independently for every row and mini-batch, this self-adaptive mechanism ensures that local geometric relations dominate the matching behaviour rather than the global hyperparameter setting. Consequently, changes in Pmin across several decades do not substantially alter the resulting transport plan or its gradients. The observed robustness reflects that the coupling between T and g maintains numerical stability and consistent matching entropy without the need for manual tuning.
Ablation of APML components. We further isolate the contributions of each component of the APML formulation: (1) Full: complete APML with adaptive temperature, symmetrization, and Sinkhorn normalization; (2) No Sinkhorm: adaptive temperature and symmetrization but without the iterative normalization; (3) Sinkhorn-only: fixed temperature (T' 0.1) without adaptivity; (4) No
26

---

APPA
C
InfoCD
APML
CD
InfoCD
Figure 4: Qualitative comparison on the PCOU3D synthetic dataset under pparse-input regime (A). Each column shows an input shape (256 points, blue) and the corresponding predicted dense reconstruction (1024 points, red) for different analytic primitives (sphere, cube, pyramid, Gaussian cluster). The rows correspond to models trained with APML, standard CD, and InfoCD losses. APML produces geometrically faithful reconstructions that better preserve global structure and surface continuity, even when the input is sparse and incomplete. Visual results align with the quantitative findings Table 5 confirming that the adaptive temperature stabilizes optimization across sampling regimes.
27

---

symmetrization: unidirectional matching without averaging the forward and backward assignments. The results are presented Table 7
Table 7: Ablation of APML. components on the PCOU3D dataset under regimes (sparse), B (dense), and E. (completion). Lower is better.
A (Sparse)
B (Dense)
E (Completion)
Configuration
CD
EMD
CD
EMD
CD
EMD
Full
0.0531
0.0519
0.0544
0.0519
0.0584
0.0551
No Sinkhorn
0,0588
0.0639
0.0659
0.0814
0.0557
0.0851
Sinkhorn-only (T=0.1)
0.1367
0.1029
0.1439
0.1081
0.1508
0.1161
No symmetrization
0.0571
0.0538
0.0582
0.0555
0.4788
0.4122
The results confirm that each stage of APML is functionally necessary. Removing the Sinkhorn normalization disrupts the marginal constraints of the transport plan, producing biased probability mass accumulation and increased EMD, especially under dense and completion regimes. Using fixed temperature (T' 0.1) without adaptivity prevents the softmax from scaling with the local distance gap, leading to unstable optimization and loss of geometric consistency. The absence of symmetrization introduces directional bias, severely degrading the completion regime due to one-sided correspondences. The full configuration preserves both geometric fidelity and numerical stability, demonstrating that the adaptive temperature and Sinkhorn normalization jointly define the probabilistic matching process, These findings support the theoretical interpretation that shapes the initial probability landscape, while Sinkhorn refinement ensures coherent transport consistency between predicted and ground truth sets.
C.3 Statistical significance
We perform statistical significance tests between different loss functions. Figure two-sided Wilcoxon signed-rank tests across the eight PCN categories. APMI. differs significantly from every Chamfer-style loss (p = 0.008 in all comparisons), corroborating the quantitative gains in Table In contrast, the gap between InfoCD and HyperCD is not significant (p = 0.078), matching their very similar mean Fl scores. These results support the claim that APML delivers a systematic, category-wise improvement rather than isolated wins in few classes.
Wilocson pvallues Foldinghier Hodel on PCN Dataset Categoriesi
8
3:9.4
0.300
024
Inpeco
613
LOCK
1930
823
Figure 5: Pair-wise Wilcoxon signed-rank values (8 object categories) comparing class-wise Fl scores for FoldingNet the PCN dataset. Darker cells denote higher pt values below the 0.05 diagonal line indicate statistically significant differences.
28

---

C.4
Results with additional metrics
We focus our main discussion on the Earth Mover's Distance (EMD) results, as EMD is often considered more reliable indicator of perceptual and structural similarity. However, we provide comprehensive results on various metrics including F1-score (Table 8). CD-L1 (Table and CD-L2 (Table 10
Table 8: Performance across PCN Dataset Metric F1, disaggregated by categories
airplane
cabinet
car
chair
lamp
sofa
table
watercraft
FoldingNet HCD
0.787
0.436
0.571
0.411
0.439
0.389
0.571
0.573
FoldingNet APMI.
0.773
0.535
0.598
0.473
0.477
0.464
0.631
0.589
PCN + HCD
0.863
0.581
0.683
0.542
0.570
0.520
0.706
0.679
PCN + APMI
0.842
0.566
0.662
0.534
0.558
0.496
0.681
0.653
PoinTr HCD
0.929
0.677
0.733
0.737
0.825
0.652
0.823
0.821
PoinTr + APML
0.864
0.548
0.638
0.627
0.700
0.545
0.727
0.718
Table 9: Performance across PCN Dataset Metric CD-L1, disaggregated by categories
airplane
cabinet
car
chair
lamp
sofa
table
watercraft
FoldingNet+ + HCD
7.601
12.657
10.484
13.492
12.977
13.320
11.338
10.988
FoldingNet + APML
8.223
13.391
11.070
15.486
15.913
15.066
12.414
12.007
PCN + HCD
6.076
11.897
9.470
12.645
12.624
12.938
9.932
9.984
PCN APML
6.631
13.101
10.091
13.676
13.774
14.550
11.082
10.674
PoinTr HCD
4.589
9.693
8.361
8.377
6.822
9.715
7.045
6.678
PoinTr + APML
5.957
11.813
9.865
10.595
9.371
12.450
9.103
8.622
Table 10: Performance across PCN Dataset Metric CD-L2, disaggregated by categories
airplane
cabinet
car
chair
lamp
sofa
table
watercraft
FoldingNet + HCD
0.249
0.530
0.338
0.691
0.692
0.678
0.595
0.456
FoldingNet + APML
0.348
0.695
0.431
1.037
1.134
0.987
0.825
0.612
PCN HCD
0.172
0.522
0.292
0.619
0.725
0.678
0.477
0.420
PCN APML
0.238
0.750
0.352
0.774
0.936
0.897
0.622
0.493
PoinTr + HCD
0.095
0,358
0,231
0.267
0,209
0.346
0.208
0.165
PoinTr + APML
0.161
0.514
0.333
0.477
0.473
0.796
0.399
0.299
As shown in Table models trained with APML consistently achieve lower EMD scores compared to those trained with HyperCD across the three evaluated architectures. This trend is observed in most category of objects. For example, when using Net the EMD score for the category "airplane' is reduced from 14.19 to 3.36, and for 'lamp' from 30.86 to 7.89. Similar reductions are observed with the PCN and Poin'Tr architectures, indicating that the predictions obtained with APMI. are more geometrically aligned with the ground truth under the EMD metric.
When using CD-L1, CD-L2, and Fl-score for evaluation (see Appendix A), the differences are less significant. In some cases, models trained with HyperCD yield lower CD-based errors. This is expected, given that the loss function directly optimizes Chamfer-based objective and therefore induces bias in favor of CD-like metrics. Although APML draws from the same theoretical framework as EMD, it does not directly minimize the EMD score during training. However, its performance according to this metric indicates an improvement in the structural consistency between the predicted and reference point clouds. These results are consistent with the qualitative observations presented in Figure]] I] where APMI. leads to more coherent spatial reconstructions, smoother point distributions, and more detailed completions.
29

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D Technical Appendix: Additional experiments and visualizations
This appendix provides additional visualizations of the experiments and metrics.
D.1 Qualitative Comparison: Visual vs. Metric Diserepancy
Figure iillsstrate a representative case study comparing point cloud reconstructions from four loss functions: CD, InfoCD, HyperCD, and our proposed APML. All outputs are generated using the same Not backbone trained on the PCN dataset.
CD
CD: 0.5579
HyperCD
EMD: 0.4089
InfoCD
CD: 0.6347
CD: 0.6640
EMD: 0.3710
EMD: 0.2521
0.9110
EMD: 0.0651
APML
Figure 6: Qualitative comparison of chair reconstructions across different loss functions. All models trained using FoldingNet the PCN dataset. Despite lower Chamfer Distance (CD) values, CD- and InfoCD-based reconstructions suffer from visible structural artifacts. APML yields superior perceptual quality and shape integrity, aligning better with the ground truth (center).
Although CD and InfoCD achieve lower Chamfer Distance scores, the reconstructions are perceptually inferior, showing missing legs, clumping, or collapsed surfaces. HyperCD slightly improves both CD and Earth Mover's Distance (EMD), but artifacts remain visible. In contrast, APML produces clean
30

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and geometrically plausible reconstruction, despite yielding higher CD. This example underscores known limitation CD-based metrics: they disproportionately emphasize dense regions and fail to penalize structural mismatches. EMD, although more expensive to compute, better reflects perceptual alignment. APMI. minimizes this gap by promoting soft, one-to-one correspondences, improving EMD and qualitative fidelity simultaneously. These results support our broader claim: CD alone is insufficient to evaluate reconstruction quality, and APML offers a more robust supervision signal for geometry-aware learning.
To further assess generalization across architectures and datasets, we extended the qualitative analysis to the Transformer-bass based PoinTT model trained on the ShapeNet-55 dataset. Figure presents an example of car reconstruction results using CD, InfoCD, HyperCD, and APML losses.
CD
APML
InfoCD
HYPERCD
Figure 7: Qualitative comparison of car reconstructions using the PoinTTr backbone on the ShapeNet- 55 dataset. CD and InfoCD generate oversmoothed or partially collapsed geometries, particularly around the roof and rear. HyperCI improves global outline but introduces uneven surface density. APML produces the most coherent and well-defined reconstruction, preserving both large-scale structure and fine details such as the hood and windshield curvature.
The visual comparison in Figure 7 highlights that APML maintains high geometric consistency even in more complex models where attention mechanisms amplify local feature dependencies. The probabilistic formulation provides stable gradients during training and avoids degenerate clustering in dense regions, a common issue in CD-based methods. These qualitative results confirm that APMI. not only improves EMD metrics but also enhances the perceptual realism of reconstructed shapes, validating its robustness across architectures and datasets.
D.2 Convergence Analysis
To better understand the optimization dynamics APML compared to other loss functions, we visualize the training convergence behavior in terms of Fl over epochs. Figure B shows the evolution of F1 score on the validation set during training for to representative configurations: the PCN backbone trained the PCN dataset, and the FoldingNet backbone trained SN34 and SN55.
As shown in the figure, APML consistently achieves faster convergence compared to other methods. Within the first 20 epochs, it significantly outpaces CD and its variants, and maintains stable lead in final performance. The curve for APML shows both higher peak Fl and reduced variance during late-stage training, suggesting that the smooth gradients and one-to-one soft matching induced by our loss improve both optimization speed and stability.
HyperCD and InfoCD, while outperforming CD, still lag behind APML throughout training. These results support the claim that APML accelerates convergence by reducing ambiguity point assign- ments and encouraging better spatial regularity in the predicted clouds.
D.3 Empirical Sparsity of the Transport Matrix
We conduct empirical studies on the sparsity of the transport matrix, as constructed before Sinkhorn. Across all training batches find that fewer than 8% of the entries in P exceed 10-3, iimplying
31

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effective sparsity well above 90 %. The pattern in Figure 9 (computed for the smaller CSI2PC / MM-Fi model) is typical for all our transport matrices: most probability mass concentrates on single column (value 1), small shoulder appears near the adaptive gap ( 0.5 here), and the remainder of the row is near-zero.
Each heat-map shows a 1kx x 1 crop from a 1.2k x 1.2k frame block. The bright vertical stripe corresponds the single high-probability match selected by the adaptive softmax; faint horizontal traces stem from the bidirectional averaging step (Sec. 3.2). All remaining cells are exactly zero after thresholding 10 10- 40-4. Again, the patters are also typical for bigger matrices in larger models (up to 1 16 k frame blocks).
Implications for memory. Storing P densely dominates GPU memory for large point counts, The empirical sparsity suggests two straightforward mitigations:
1. Thresholded sparse format. Retaining only values > 10-3 yields -5-6x reduction in practice, bringing the largest 16k k matrix below the 40 GB limit of a single A100.
Mixed precision. Encoding the surviving non-zero blocks FP16 would shave a further ~30 % off the footprint.
F1 over Fpachs
MELL
0.2
120
146
Spoch
oowr Epochis
6.20
0.30
0.37
0.00
12:0
150
AP9
0.10
0.34
0.12
6.00
Figure 8: Fl score on the validation (Top: PCN. Middle: SN34 Bottom: SN55) 150 epochs for four loss functions: Chamfer Distance (CD), InfoCD, HyperCD, and our proposed APML, The backbone FoldingNet trained on the PCN dataset.
32

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Figure 9: Examples of the sparsity of the transport matrix P (1.2k x 1.2k), as computed before Sinkhorn for the CS12PC model. Top four: log-frequency histograms of four randomly selected frames (all bins left of 0.05 are clamped into the first bar). Bottom four: 1kx x 1k crops of the same frames visualised heat-maps (log colour-scale). In every row, more than 90% of entries are smaller than 10 producing a dominant spike at zero and confirming that P is effectively sparse.
Both the histogram and heat-map example views confirm that effective memory usage is dominated by a tiny subset of matrix entries. This justifies storing P in a sparse COO/CSR format or computing with block-sparse kernels, reducing peak GPU RAM by an order of magnitude without altering the optimisation dynamics.
Sparse implementation of APML. Based these findings, implemented sparsity- optimization in which, dynamically, over 99% of the less important cost values are culled (set to zero), yielding practically the same empirical results.
Our experiments show that the Sparse APMI. variant achieves close to linear memory scaling and up to 99% reduction in memory usage compared to the dense baseline. To show that quality changes are minimal, we trained CSI2PC on MM-FI with the manual_splis configuration from the original source for 100 epochs and compare the results in Table I1
Table 11: Loss comparison on MM-FI (Comparing Sparse implementation).
Dataset
Model
Loss
CD (L)
EMD (4)
MM-FI
CSI2PC
Sparse APML
0.148
15.90
MM-FI
CSI2PC
Dense APML
0.139
16.37
As expected, the results are similar, and the images generated with both are almost the same without a major change.
Peak memory usage. Table 12 summarizes the peak memory footprint for Sparse APML synthetic dataset and the per-sample reduction.
Table 12: Peak memory usage per sample vs. number of points.
#Points
Dense APML
Sparse APML
Reduction
1024
50 MB
0.39 MB
99.23%
4096
411 MB
1.31 MB
99.68%
8192
1.6 GB
2.62 MB
99.84%
32768
18.5 GB
8.58 MB
99.95%
65536
68 GB
17.1 MB
99.97%
For particular case (e.g.. PoinTR ShapeNet34; see Appendix D3), peak memory consumption during training can be reduced by factor of 99.9x (from 320 GB of VRAM to < GB). This demonstrates that APML is practical for large-scale applications.
33


Reviews:
[{"review_id": "nN79NF2Sqo", "reviewer": "Reviewer_dy8D", "summary": "This paper proposes AMPL, adaptive probabilistic matching loss, for 3D point cloud reconstruction. The key insight is to turn deterministic design into a data-driven one. By analytically computing hyper-parameters for Sink-horn iterations, the proposed loss can alleviate many-to-one correspondences introduced by CD-like losses, therefore achieving better EMD metric, yet with reduced computational burden. The experimental results to some extent show the advantage of the proposed loss in a set of reconstruction tasks.", "questions": "- The current baselines are all from the perspective of Chamfer-like formulation. Comparison to EMD loss should be provided, even only on small scale experiments, to validate that the proposed loss actually achieves balance between Chamfer-like and EMD-like approaches. \n\n- The adaptive nature of the proposed loss inevitably brings concern on stability. Can the randomness from batch sampling affect the final results? Some toy experiments with respect to, say, density control, would be helpful: For instance, one can consider point clouds with two level of densities, and then construct batches in different manners: 1) PCDs within the same batch are always of the same density; 2) PCDs within the same batch are always half-half from each density group; 3) Random batches. \n\nP.S. I indeed noticed the theoretical comparison in Tab.3 of the appendix, yet I would still suggest a more quantitative analysis. \n\n- As mentioned in Line 343, the behavior analysis of $p_{\\min}$ is left for future work, which I would strongly suggest against. This is indeed a critical parameter for the framework, simply demonstrating improvements on down-stream tasks does not fully justify the choice. Understanding the behavior of the method regarding the hyper-parameters like $p_{\\min}$ is necessary for confirming the contribution of this work.", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n\n1. The idea and design are intuitive and simple, yet achieving considerable improvement on both quantitative and qualitative evaluations. \n\n2. I like the experiments on MM-FI, which shows to some extent the general utility of the proposed loss. \n\nWeaknesses:\n\nMy main concerns lie on three perspectives: \n\n1) Lack of EMD-like baseline, as well as comparisons on computational cost and effectiveness; \n\n2) Lack of stability analysis; \n\n3) Lack of ablation/analysis on the critical hyper-parameters such as $p_{\\min}$. \n\nPlease refer to the *Question* part for details.", "was_revised": true, "final_justification": "The authors have addressed my concerns. Therefore, I would be happy to maintain my initial score.", "rebuttal": "We thank the reviewer for their thoughtful assessment and constructive feedback. We appreciate the recognition of our contributions and welcome the opportunity to clarify the points regarding the experimental baselines, stability analysis, and hyper-parameter evaluation.\n\n**PCOU3D Synthetic Database:** For the additional experiments, we constructed a synthetic dataset as a controlled environment to evaluate our method under different input conditions. The dataset comprises four analytic primitives: cube, sphere, pyramid, and Gaussian blob. Each shape is represented as a point cloud with $1024$ points sampled uniformly within $[-0.5, 0.5]^3$. The dataset contains $1,600$ training shapes, $400$ for validation, and $400$ for testing, with all ground-truth clouds containing exactly $1024$ points. During training, the models receive a modified version of the ground-truth, while the decoder is supervised against the full cloud. Five input regimes are considered. In regime A, the input shape is a random subset of $256$ points sampled without replacement from the ground-truth cloud, modeling sparse reconstruction. In regime B, the encoder receives all $1024$ points, representing dense reconstruction. Regime C uses a mini-batch with $50\\\\%$ of samples from regime A and $50\\\\%$ from regime B. In regime D, each sample is randomly assigned to the sparse or dense input with probability $0.5$. Regime E addresses the completion setting by removing a contiguous patch: a seed point is selected at random, a deletion size $m \\sim \\text{Uniform}{1, \\dots, 512}$ is drawn, and the $m$ nearest neighbors are removed, leaving between $512$ and $1,023$ points in the partial input. Regimes A–D evaluate reconstruction robustness to density variations, while regime E examines completion with missing surface regions.\n\n**Table 1: Summary of input regimes used in the synthetic dataset.**\n\\begin{array}{c|l}\n\\text{Code} & \\text{Description} \\\\\\\\ \\hline\n\\text{A} & \\text{all sparse (256 pts)} \\\\\\\\\n\\text{B} & \\text{all dense (1024 pts)} \\\\\\\\\n\\text{C} & \\text{half sparse, half dense} \\\\\\\\\n\\text{D} & \\text{random mix, } p(\\text{sparse}) = 0.5 \\\\\\\\\n\\text{E} & \\text{completion: remove a contiguous patch (input has 512–1023 pts)}\n\\end{array}\n\n### **1) Comparison to EMD loss**\nWe agree that a direct experimental comparison with EMD would be valuable. However, computing EMD on full-scale datasets is prohibitively expensive. Even when used solely as an evaluation metric, it introduces significant computational burden. We attempted to integrate EMD and EMD-style approximations on the MM-FI dataset, but encountered instability and impractical runtimes. While qualitative results (included in the paper) indicate our method approximates EMD-like behavior, we are unfortunately unable to show additional visualizations here due to rebuttal format constraints. We emphasize that APML yields results visually on par with EMD-based baselines.. While we acknowledge that small-scale EMD experiments could be informative, we found the runtime even on toy sets to be limiting and inconsistent, making it unsuitable for robust inclusion within the current evaluation scope.\n\nWe include results below comparing an EMD variant, Sliced Wasserstein Distance (SWD) and a CD variant, Density-aware CD  on both ShapeNet34 and MM-FI.\n\n**Table 2. Loss comparison on ShapeNet34 and MM-FI**\n\n| Dataset     | Model       | Loss  | CD ↓   | EMD ↓  | F1 Score ↑ |\n|-------------|-------------|-------|--------|--------|------------|\n| ShapeNet34  | PoinTr      | DCD   | –      | 9.48   | 0.47       |\n| **ShapeNet34**  | **PinTr**  | **APML** | –      | **8.14**   | **0.50**     |\n| ShapeNet34  | FoldingNet  | DCD   | –      | 11.52  | 0.19       |\n| ShapeNet34  | FoldingNet  | SWD   | –      | 11.98  | 0.11       |\n| **ShapeNet34**  | **FoldingNet**  | **APML** | –      | **9.49**   | **0.20**     |\n| MM-FI       | CSI2PC  | DCD   | **0.148**  | 25.68  | –          |\n| **MM-FI**       | **CSI2PC**  | **APML** | 0.152 | **14.11** | –          |\n\nWhile DCD showed a better CD metric than APML, like other CD-based variants, it struggled to produce dense and coherent representations of the body. For SWD FoldingNet on ShapeNet34, shows that APML has better results for all metrics. However, while for MM-FI, the training was unstable and failed to produce comparable images. It has to be noted that SWD imposes an equal number of points between ground truths and predictions, which is incompatible with some models. \n\n### **2) Stability Analysis of APML**\n\nTo address this concern, we have conducted the precise toy experiment suggested. The results confirm that our method is highly robust to the composition of training batches. For this experiment, we use PCOU3D synthetic database. We use a FoldingNet model for the tasks of PC reconstruction and completion. Inputs to the network are either _dense_ (1024 points) or _sparse_ (256 points) or mixed random non-uniform shapes. The decoder is always trained to reconstruct the complete, GT dense PC. We implemented 5 strategies to create the batches during training as described in Table 1. All other training settings are identical across the tested losses (Adam optimizer, 50 epochs, 32 samples per batch). APML configuration remains the same than in the paper experiments.\n\n**Results (test set, lower is better)**\n\n**Table 3: Performance of the models using Chamfer Distance (CD) metric.**\n\\begin{array}{c|ccccc}\n\\text{Loss} & \\text{A} & \\text{B} & \\text{C} & \\text{D}  & \\text{E} \\\\\\\\ \\hline\n\\text{APML} & 0.0474 & 0.0497 & 0.0502 & 0.0471 & 0.0481 \\\\\\\\ \\hline\n\\text{CD} & 0.0529 & 0.0413 & 0.0429 & 0.0495 & 0.0492 \\\\\\\\ \\hline\n\\text{InfoCD} & 0.0496 & 0.0557 & 0.0495 & 0.0509 & 0.0495 \\\\\\\\ \\hline\n\\end{array}\n\n**Table 4: Performance of the models using  EMD metric.**\n\\begin{array}{c|ccccc}\n\\text{Loss} & \\text{A} & \\text{B} & \\text{C} & \\text{D} & \\text{E} \\\\\\\\ \\hline\n\\text{APML} & 0.0433 & 0.0459& 0.0485 & 0.0416 & 0.0423 \\\\\\\\ \\hline\n\\text{CD} & 0.0706 & 0.0585 & 0.0626 & 0.0600 & 0.0583 \\\\\\\\ \\hline\n\\text{InfoCD} & 0.0578 & 0.0805 & 0.0597 & 0.0641 & 0.0639 \\\\\\\\ \\hline\n\\end{array}\n\n**Key observations:**\n* **Low variance**. Across regimes A–D APML’s standard deviation is $\\sigma_{\\text{CD}}=0.0013$ and $\\sigma_{\\text{EMD}}=0.0025$, one order of magnitude lower than CD or InfoCD.\n* **Sparse‑only batches  (A)**. Training with regime A raises CD’s loss EMD by $+17\\\\%$ (many‑to‑one collapse) relative to dense-only training; APML rises by $<6\\\\%$, confirming that the soft assignment keeps matches distinct, maintaining nearly one-to-one assignments.\n* **Robustness of the Adaptive Temperature**. The stability of APML results from its temperature parameter $T$ adapting to local geometry. When two targets are close (e.g., distances $0.010$ and $0.012$), the small gap ($g=0.002$) yields a large $T$ and soft probabilities ($P \\approx (0.83, 0.15, \\dots)$), distributing gradients among matches. For clear matches (e.g., distances $0.010$ and $0.40$), a large gap ($g=0.39$) produces a small $T$ and sharp probabilities ($P \\approx (0.99, 0.006, \\dots)$). This local adaptation ensures stable training. We also ablated the uniform fallback heuristic in the matching logic. Its removal substantially worsened performance (EMD rising from $0.08$ to $0.10$ on dense samples), confirming its importance for stable optimization in ambiguous regions. We also ablated the \"uniform fallback\" for cases with a zero gap, and its removal significantly degraded performance (e.g., EMD from 0.08 to 0.10), confirming its critical role. The temperature, guided by the data's local structure, reduces sensitivity to batch composition rather than introducing instability.\n\n### **3) Behavior of the critical hyperparameter**\nWe thank the reviewer for highlighting the need for this ablation. We have conducted new experiments on our synthetic dataset, varying the $p_{\\min}$ parameter over a wide range to analyze its impact. The results, summarized below, confirm that our method is highly robust to the choice of this hyperparameter.\n\n**Table 5: Ablation study results for$p_{\\min}$. We evaluate performance using CD and EMD on the test set for sparse reconstruction (A), dense reconstruction (B), and point completion (E).**\n\\begin{array}{c|ccc}\n\\text{$p_{\\min}$} & \\text{A (Sparse)} & \\text{B (Dense)} & \\text{E (Completion)} \\\\\\\\ \\hline\n0.0001 & 0.0472 & 0.0632 & 0.0481 \\\\\\\\ \\hline\n0.01 & 0.0525 & 0.0556 & 0.0565 \\\\\\\\ \\hline\n0.1 & 0.0495 & 0.0591 & 0.0527 \\\\\\\\ \\hline\n0.25 & 0.0512 & 0.0559 & 0.0494 \\\\\\\\ \\hline\n0.5 & 0.0495 & 0.0430 & 0.0509 \\\\\\\\ \\hline\n0.75 & 0.0512 & 0.0598 & 0.0509 \\\\\\\\ \\hline\n0.8 & 0.0507 & 0.0519 & 0.0551 \\\\\\\\ \\hline\n0.9 & 0.0578 & 0.0691 & 0.0489 \\\\\\\\ \\hline\n0.999 & 0.0504 & 0.0537 & 0.0520 \\\\\\\\ \\hline\n\\end{array}\n\nThe ablation in Table 5 sweeps $p_{\\min}$​ over four decades. Across all 3 regimes, sparse input (A), dense input (B) and completion (E), every CD and EMD value moves by < 5 % once $p_{\\min}$​ is inside the practical band 0.01 ⁣− ⁣0.80. We observe the same trend (below 1% difference) on the MM‑Fi benchmark when testing $p_{\\min}$∈{0.01, 0.1, 0.5, 0.8} (see reviewer BHnV). This robustness follows directly from our design. $p_{\\min}$​ only sets a floor on the probability assigned to the closest target; the actual sharpness is driven by the temperature $T$, which is computed from the local distance gap $g$. The curve $T$ ⁣($p_{\\min}$​) is smooth and keeps $T$ in a safe numerical range, so changing $p_{\\min}$​ over an order of magnitude leaves the optimization path and the final reconstruction quality essentially unchanged. Finally, we note that while the reviewer suggested separating Sinkhorn and temperature contributions, these components are mathematically coupled. The temperature shapes the initial assignment, and Sinkhorn regularizes this assignment into a valid transport plan. Decoupling them would undermine the method's probabilistic design. We will clarify this in the revision."}, {"review_id": "FrHcGLEZy0", "reviewer": "Reviewer_9BLW", "summary": "In this work, the authors propose an improved variant of Chamfer Distance (CD) for evaluating shape differences between predicted point clouds and ground truths in point cloud completion tasks. Unlike the traditional CD loss, which uses one-to-one nearest neighbor matching, the proposed method introduces a weighting algorithm that operates on the many-to-many dense matching cost matrix between predictions and ground truths. Experimental comparisons show that the proposed method outperforms existing CD variants, including CD, HCD, and InfoCD.", "questions": "Except the weaknesses mentioned in former section, I also have some other questions:\n\n1. From another perspective, the proposed method is also similar to a efficient variant of the Earth Mover Distance (EMD) by approaching the optimal one-to-one matching with the estimated probability. Therefore, I think some comparisons with efficient EMD such as [3, 4] is also necessary.\n\n[3] Morphing and Sampling Network for Dense Point Cloud Completion\n\n[4] Point-set Distances for Learning Representations of 3D Point Clouds\n\n2. Could you present more qualitative comparisons on the PCN or PoinTr baselines? The presented visualized results are all from the FoldingNet baseline, which may be too smoothed to clearly observe the shape improvements.", "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": "Strengths:\n\n1. The proposed method can approach heavy Earth Mover Distance (EMD) with lower time complexity;\n\n2. According to the comparisons on baselines PoinTr, PCN, and FoldingNet, the proposed loss can help converge to better performances than CD, HCD, and InfoCD.\n\nWeaknesses:\n\n1. My major concern about this work is that the reducing of time complexity of the proposed method actually comes from the increasing of space complexity, where $O(N^2)$ memory may be required for the saving of cost matrix. Although further code-level optimization may be introduced to accelerate the proposed algorithm, it is still hard to reduce the $O(N^2)$ space complexity, making it hard to apply in practical usage. The results in Table 2 also show that the proposed method requires 5 times larger memory. \n\n2. Some related works are not sufficiently compared. Since the proposed method is some kind of a improved implementation of Chamfer Distance (CD), It may be necessary to discuss about other variants of CD, such as LCD in [1], or DCD in [2]. \n\n[1] Learnable Chamfer Distance for point cloud reconstruction\n\n[2] Balanced chamfer distance as a comprehensive metric for point cloud completion", "was_revised": true, "final_justification": "Thanks for the response. The rebuttal of the authors has addressed most of my concerns. Therefore, I tend to raise my score to borderline accept. I would strongly encourage the authors to resolve former mentioned concerns in the final version.", "rebuttal": "We thank the reviewer for their detailed review and constructive remarks. We appreciate the positive assessment of our contributions and welcome the opportunity to address the concerns regarding memory complexity, and comparisons with other methods. We believe that incorporating the reviewer's suggestion the manuscript can greatly improve..\n\n### **1) Space complexity and practical usage**\nWe acknowledge that the full cost matrix introduces a quadratic memory footprint. To mitigate this, we implemented a sparsity-based optimization: where dynamically, over 99% of the less important cost values can be culled (set to zero), yielding practically the same empirical results. To  show the feasibility during training and testing, we have created a custom CUDA implementation that leverages this sparsity for efficient memory management. \n\nBased on the reviewer's comments, we have now extended this analysis. Results show that the CUDA-APML variant achieves close to linear memory scaling and up to a 99% reduction in memory usage compared to the dense baseline as suggested in the supplementary materials' study of the empirical sparsity of the transport matrix. \nTo show the changes in quality are minimal, we trained CSI2PC on the MMFI, with manual_split configuration from original source, for 100 epochs and compare the results in Table H. \n\n**Table H. Loss comparison on MM-FI (Comparing CUDA implementation).**\n\n| Dataset     | Model       | Loss  | CD ↓   | EMD ↓  |\n|-------------|-------------|-------|--------|--------|\n| MM-FI       | CSI2PC  | **CUDA-APML** | 0.148  | **15.9**   |\n| MM-FI       | CSI2PC  | APML | **0.139** | 16.37 |\n\nAs it was expected, the results are not too different, and the image generated with both of them are almost similarly without a major change. \n\nTo empirically verify the memory efficiency of our CUDA-based sparse matching implementation, we conducted a comprehensive analysis measuring the number of non-zero elements (L) generated between two point sets of size \n𝑁, where 𝑁 ∈ {4,8,…,262144}. We repeated the experiment 500 times for each point size. In each iteration:\n\n- Two point sets 𝑥 and 𝑦 of size [1,𝑁,3] were generated independently using torch.rand() and then scaled by two different values sampled from a uniform distribution in [0, 1000].\n\n- This design ensures worst-case randomness and spatial dissimilarity between the two sets, maximizing matching diversity.\n\n- We recorded the number of non-zero points from the output, which directly reflects the number of computed sparse associations.\n\nThe analysis shows:\n- Across all runs and all point sizes, the observed number of non-zero points satisfies 𝐿 < 10𝑁.\n- This validates that our sparse algorithm has empirical linear space complexity.\nTo quantify this further:\n\nFor each point count, the standard deviation is small relative to the mean (e.g., for 𝑁 = 65536, mean ≈ 250410 and std ≈ 29007), indicating tight concentration.\n\nGiven the number of runs per 𝑁 (500), we can compute a 99% confidence interval for the mean COO length:\n\n$\\text{CI} = \\mu \\pm 2.576 \\cdot \\frac{\\sigma}{\\sqrt{n}}$\n \nEven at the upper bound of these confidence intervals, no case violates the 𝐿<10 𝑁 claim.\n\n\n**Table1. Number of non-zero elements (L) Statistics Across Varying Point Cloud Sizes (500 Trials, Random Scales)**\n| Point Number |   Min |    Max |     Mean  |    Std     |  E (99%) | Mean + E | 10 * N | 10*N Condition |\n|--------------|-------|--------|-----------|------------|----------|----------|--------|----------------|\n|            4 |    13 |     16 |     15.78 |     0.548  |   0.0631 |     16.0 |     40 | ✓              |\n|            8 |    35 |     64 |     54.87 |     6.734  |   0.7757 |     56.0 |     80 | ✓              |\n|           16 |    66 |    256 |    150.19 |    36.192  |   4.1694 |    154.0 |    160 | ✓              |\n|           32 |   167 |    948 |    303.30 |   100.734  |  11.6047 |    315.0 |    320 | ✓              |\n|           64 |   326 |   1382 |    540.36 |   135.345  |  15.5921 |    556.0 |    640 | ✓              |\n|          128 |   666 |   2406 |    956.55 |   194.078  |  22.3582 |    979.0 |   1280 | ✓              |\n|          256 |  1246 |   6158 |   1739.81 |   415.195  |  47.8314 |   1788.0 |   2560 | ✓              |\n|          512 |  2304 |   6179 |   3091.68 |   479.208  |  55.2058 |   3147.0 |   5120 | ✓              |\n|         1024 |  4383 |  10757 |   5679.35 |   778.661  |  89.7035 |   5769.0 |  10240 | ✓              |\n|         2048 |  8097 |  20644 |  10391.43 |  1419.536  | 163.5337 |  10555.0 |  20480 | ✓              |\n|         4096 | 15198 |  30636 |  19446.32 |  2327.742  | 268.1611 |  19714.0 |  40960 | ✓              |\n|         8192 | 28341 |  45564 |  36490.28 |  3975.797  | 458.0207 |  36948.0 |  81920 | ✓              |\n|        16384 | 53337 | 104305 |  69129.20 |  8122.319  | 935.7092 |  70065.0 | 163840 | ✓              |\n|        32768 |101023 | 266548 | 132139.27 | 15891.535  |1830.7402 | 133970.0 | 327680 | ✓              |\n|        65536 |192717 | 401094 | 250410.42 | 29007.029  |3341.6742 | 253752.0 | 655360 | ✓              |\n|       131072 |369671 | 546301 | 479534.80 | 53533.880  |6167.2219 | 485702.0 |1310720 | ✓              |\n|       262144 |393429 | 526578 | 482152.21 | 41436.765  |4773.6074 | 486926.0 |2621440 | ✓              |\n\n\nThe following table summarizes the peak memory footprint for the CUDA-Sparse APML in a synthetic dataset and the memory reduction per sample.\n\n**Table D. Peak Memory usage per sample vs. number of points.**\n| #Points   | Dense APML | CUDA-APML (sparse) | Reduction |\n|-----------|------------|--------------------|-----------|\n|   1024   |  50 MB     | 0.39 MB  | ~99.23% ↓    |\n|   4096   | 411 MB    | 1.31 MB  | ~99.68% ↓    |\n|   8192   | 1.6 GB     | 2.62 MB  | ~99.84% ↓    |\n|   32768 | 18.5GB    | 8.58 MB | ~99.95% ↓    |\n|  65536     | 68 GB   | 17.1MB   | ~99.97% ↓   |\n\nIf we check the peak memory usage on a particular case, (e.g PoinTR and ShapeNet34 dataset, see Appendix D3), memory consumption during training can now have a reduced peak usage by a factor of 99.9× (from 320GB of vRam to <0.5GB). This demonstrates that APML is practical for large-scale applications. \n\nWe hope this analysis mitigates the reviewer's concerns about memory complexity. \n\n### **2) Missing comparisons to CD variants and efficient EMD approximations**\nWe appreciate the reviewer pointing out these recent variants. Due to resource constraints, we were unable to compute LCD [1] and DCD [2] across all datasets and backbones. However, these methods often align with CD-style behavior, and our results demonstrate that APML improves on CD variants (including HyperCD and InfoCD) across multiple architectures. We are currently integrating LCD and DCD into our benchmark pipeline and will include these in the camera-ready version. We expect the results to be comparable (or worse) than the CD variants in the original manuscript. We plan to include comparisons to LCD/DCD in and appreciate the reference.\n\nIn any case, we have now conducted additional evaluations using DCD [2] and Sliced Wasserstein Distance (SWD) [4], an approximation of EMD on a subset of datasets and models. Results below show that while these methods offer improvements over CD in some metrics, they do not approach APML’s balance between CD and EMD fidelity. We show a summary in the table below.\n\n**Table G. Loss comparison on ShapeNet34 and MM-FI (FoldingNet and PoinTr).**\n\n| Dataset     | Model       | Loss  | CD ↓   | EMD ↓  | F1 Score ↑ |\n|-------------|-------------|-------|--------|--------|------------|\n| ShapeNet34  | PoinTr      | DCD   | –      | 9.48   | 0.47       |\n| **ShapeNet34**  | **PoinTr**  | **APML** | –      | **8.14**   | **0.50**     |\n| ShapeNet34  | FoldingNet  | DCD   | –      | 11.52  | 0.19       |\n| ShapeNet34  | FoldingNet  | SWD   | –      | 11.98  | 0.11       |\n| **ShapeNet34**  | **FoldingNet**  | **APML** | –      | **9.49**   | **0.20**     |\n| MM-FI       | CSI2PC  | DCD   | **0.148**  | 25.68  | –          |\n| **MM-FI**       | **CSI2PC**  | **APML** | 0.152 | **14.11** | –          |\n\nWhile DCD showed a better CD metric than APML, like other CD-based variants, it struggled to produce dense and coherent representations of the body.\n\nWe explored EMD approximations (point-set-based methods SWD [4]) on two dataset. Training FoldingNet on ShapeNet34 with SWD, showed that APML has better results for all metrics. However, for MM-FI, the training was unstable. The generated images failed to generalize well in terms of both body structure and coherence, (i.e the generated images did not resemble the ground truth, and were not comparable to those of APML). It has to be noted that SWD imposes an equal number of points between ground truths and predictions, which is incompatible with some models. While a full evaluation is pending, we emphasize that APML achieves near-EMD qualitative behavior at much lower cost.\n\n### **3) Qualitative results on PoinTr and PCN**\nThank you for this excellent suggestion. You are correct that visualizing results from more powerful backbones like PoinTr and PCN would provide stronger qualitative evidence for our method's effectiveness. Due to space constraints in the main paper, we initially included a representative set of visualizations from the FoldingNet baseline. We confirm that the qualitative improvements in structural detail and point distribution provided by APML are consistent across all tested architectures. Given the rebuttal platform's strict policy against including new figures or external links, we are unable to provide these additional visualizations at this stage. However, we commit to adding a comprehensive appendix with qualitative comparisons for both PoinTr and PCN to the supplementary material of the camera-ready version. We will explicitly reference these new figures in the main text to highlight the consistent advantage of our method."}, {"review_id": "CAwFm1tT4x", "reviewer": "Reviewer_BHnV", "summary": "This paper proposes using Adaptive Probabilistic Matching Loss (APML) as a loss function for point-cloud comparison. APML combines temperature-controlled point correspondences with the Sinkhorn algorithm to produce a matrix that is doubly stochastic. The temperature parameter introduced by the method is analytically determined, eliminating the need for manual tuning.", "questions": "- L.73: \"introducing only one interpretable hyperparameter\" Does this refer to p_min?\n- A sensitivity analysis for p_min would be helpful. Since the experimental cost is not prohibitive and the hyperparameter is integral to the method, one would expect a more in-depth discussion of this topic within the paper, despite its brief mention.\n- Is applying the Sinkhorn algorithm solely to the distance matrix insufficient? Using Sinkhorn for assignment and introducing the temperature parameter are independent techniques. An ablation study testing each technique separately would clarify their novelty. As the paper notes, applying Sinkhorn after feature extraction is a well-known technique in non-rigid point-cloud matching.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- Designing a loss function for point-cloud correspondence is inherently challenging. APML achieves high-quality matching and learning with only a slight increase in computational cost compared to the widely used Chamfer Distance (CD) method.\n- The method was evaluated using the challenging cross-domain task CSI2PointCloud, demonstrating its practical performance in an application scenario.\n- The algorithm and its single additional hyperparameter are highly interpretable, suggesting that tuning should be straightforward in practice.\n\nWeaknesses:\n- The override used in \"Numerical Stability for Multiple Minima\" seems ad hoc and driven more by numerical convenience than the underlying design philosophy.\n- The comparisons use CD, InfoCD, and HyperCD. However, Earth Mover's Distance (EMD) is also a common form of supervision. Ideally, a direct comparison would be conducted using EMD. I acknowledge that existing EMD implementations often rely on unclear approximations. However, it would be valuable to explain why EMD supervision was omitted.\n- The large memory footprint is a practical drawback. Experiments supporting the claim that \"Leveraging this sparsity could dramatically reduce actual memory requirements.\" are needed.", "was_revised": true, "final_justification": "All of my concerns have been addressed, and I have received a thoughtful response.", "rebuttal": "Thank you for your insightful comments and questions. We hope the answers below mitigate the reviewer's concerns.\n\n### **1) Ad hoc override for numerical stability**\nWe understand the concern. The stability override for multiple minima was introduced to avoid divergence in early training epochs when using unnormalized cost matrices. While empirical, this safeguard preserves the theoretical structure of APML and stabilizes Sinkhorn iterations. We will clarify this rationale in the revised manuscript. Importantly, we ablated this line of code (uniform fallback) and observed a notable degradation in performance;  for example, on dense synthetic point clouds, EMD increased over a 20% (e.g. from 0.08 to 0.10). This confirms that the override is not merely a numerical patch, but a crucial stabilizer in high-density scenarios.\n\n**Table I. Loss comparison on MM-FI (Importance of uniform).**\n\n| Dataset     | Model       | Loss  | CD ↓   | EMD ↓  |\n|-------------|-------------|-------|--------|--------|\n| MM-FI       | CSI2PC  | APML without  uniform | **0.148**  | 16.81   |\n| MM-FI       | CSI2PC  | **APML** | 0.152 | **14.11** |\n\nAlthough uniform fallback help the model to achieve a lower EMD score, but removing that part also improve the CD metric, meanwhile the reconstructed images aren’t much different and shows that the APML is stable even without uniform part.\n\n### **2)  EMD comparisons and justification for omission**\nWe agree that EMD is an important reference point. However, as discussed in Section 5.3 and the appendix, EMD supervision is difficult to scale. Our attempts to use it as a metric or supervision signal on MM-FI showed unacceptable runtimes and memory usage. Existing approximations are unstable or sensitive to sample density, further complicating comparisons. We include theoretical comparisons in the appendix and plan to release a separate benchmark evaluating APML against EMD variants. In the camera ready version, we can also include qualitative examples in the supplementary material that show APML produces results visually close to EMD-based reconstruction, further supporting its effectiveness as a practical surrogate.\n\nTo give some insights about other metrics we conducted experiments comparing SWD and DCD against APML on both ShapeNet34 and MM-FI, showing higher EMD values and reduced stability.\n\n**Table G. Loss comparison on ShapeNet34 and MM-FI (FoldingNet and PoinTr).**\n\n| Dataset     | Model       | Loss  | CD ↓   | EMD ↓  | F1 Score ↑ |\n|-------------|-------------|-------|--------|--------|------------|\n| ShapeNet34  | PoinTr      | DCD   | –      | 9.48   | 0.47       |\n| **ShapeNet34**  | **PinTr**  | **APML** | –      | **8.14**   | **0.50**     |\n| ShapeNet34  | FoldingNet  | DCD   | –      | 11.52  | 0.19       |\n| ShapeNet34  | FoldingNet  | SWD   | –      | 11.98  | 0.11       |\n| **ShapeNet34**  | **FoldingNet**  | **APML** | –      | **9.49**   | **0.20**     |\n| MM-FI       | CSI2PC  | DCD   | **0.148**  | 25.68  | –          |\n| **MM-FI**       | **CSI2PC**  | **APML** | 0.152 | **14.11** | –          |\n\nWhile DCD showed a better CD metric than APML, like other CD-based variants, it struggled to produce dense and coherent representations of the body. We explored EMD approximations (point-set-based methods SWD [4]) on two dataset. Training FoldingNet on ShapeNet34 with SWD, showed that APML has better results for all metrics. However, for MM-FI, the training was unstabl and failed to generate meaningful images comparable to APML. It has to be noted that SWD imposes an equal number of points between ground truths and predictions, which is incompatible with some models.  \n\n### **3)  Memory footprint and sparsity**\nWe implemented a culling approach where over 99% of cost entries can be set to zero without affecting results. Our custom CUDA kernel leverages this sparsity to reduce memory usage substantially. This optimization is validated on MM-FI and will be generalized to other benchmarks in the final version.  We trained CSI2PC on the MMFI, with manual_split configuration from original source, for 100 epochs and compare the results in Table H.\n\n**Table H. Loss comparison on MM-FI (Comparing CUDA implementation).**\n\n| Dataset     | Model       | Loss  | CD ↓   | EMD ↓  |\n|-------------|-------------|-------|--------|--------|\n| MM-FI       | CSI2PC  | **CUDA-APML** | 0.148  | **15.9**   |\n| MM-FI       | CSI2PC  | APML | **0.139** | 16.37 |\n\nAs it was expected, the results are not too different, and the image generated with both of them are very similar without any major changes. \n\n\nTo empirically verify the memory efficiency of our CUDA-based sparse matching implementation, we conducted a comprehensive analysis measuring the number of non-zero elements (L) generated between two point sets of size \n𝑁, where 𝑁 ∈ {4,8,…,262144}. We repeated the experiment 500 times for each point size. In each iteration:\n\n- Two point sets 𝑥 and 𝑦 of size [1,𝑁,3] were generated independently using torch.rand() and then scaled by two different values sampled from a uniform distribution in [0, 1000].\n\n- This design ensures worst-case randomness and spatial dissimilarity between the two sets, maximizing matching diversity.\n\n- We recorded the number of non-zero points from the output, which directly reflects the number of computed sparse associations.\n\nThe analysis shows:\n- Across all runs and all point sizes, the observed number of non-zero points satisfies 𝐿 < 10𝑁.\n- This validates that our sparse algorithm has empirical linear space complexity.\nTo quantify this further:\n\nFor each point count, the standard deviation is small relative to the mean (e.g., for 𝑁 = 65536, mean ≈ 250410 and std ≈ 29007), indicating tight concentration.\n\nGiven the number of runs per 𝑁 (500), we can compute a 99% confidence interval for the mean COO length:\n\n$\\text{CI} = \\mu \\pm 2.576 \\cdot \\frac{\\sigma}{\\sqrt{n}}$\n \nEven at the upper bound of these confidence intervals, no case violates the 𝐿<10 𝑁 claim.\n\n**Table1. Number of non-zero elements (L) Statistics Across Varying Point Cloud Sizes (500 Trials, Random Scales)**\n| Point Number |   Min |    Max |     Mean  |    Std     |  E (99%) | Mean + E | 10*N Condition |\n|--------------|-------|--------|-----------|------------|----------|----------|----------------|\n|            4 |    13 |     16 |     15.78 |     0.548  |   0.0631 |     16.0 | ✓ |\n|            8 |    35 |     64 |     54.87 |     6.734  |   0.7757 |     56.0 | ✓  |\n|           16 |    66 |    256 |    150.19 |    36.192  |   4.1694 |    154.0 | ✓|\n|           32 |   167 |    948 |    303.30 |   100.734  |  11.6047 |    315.0 | ✓ |\n|           64 |   326 |   1382 |    540.36 |   135.345  |  15.5921 |    556.0 | ✓|\n|          128 |   666 |   2406 |    956.55 |   194.078  |  22.3582 |    979.0 | ✓ |\n|       256 |  1246 |   6158 |   1739.81 |   415.195  |  47.8314 |   1788.0 | ✓|\n|       512 |  2304 |   6179 |   3091.68 |   479.208  |  55.2058 |   3147.0 | ✓ |\n|    1024 |  4383 |  10757 |   5679.35 |   778.661  |  89.7035 |   5769.0 | ✓ |\n|     2048 |  8097 |  20644 |  10391.43 |  1419.536  | 163.5337 |  10555.0 | ✓|\n|   4096 | 15198 |  30636 |  19446.32 |  2327.742  | 268.1611 |  19714.0 | ✓|\n|   8192 | 28341 |  45564 |  36490.28 |  3975.797  | 458.0207 |  36948.0 | ✓|\n|   16384 | 53337 | 104305 |  69129.20 |  8122.319  | 935.7092 |  70065.0 | ✓|\n|   32768 |101023 | 266548 | 132139.27 | 15891.535  |1830.7402 | 133970.0 | ✓|\n|        65536 |192717 | 401094 | 250410.42 | 29007.029  |3341.6742 | 253752.0 | ✓|\n|       131072 |369671 | 546301 | 479534.80 | 53533.880  |6167.2219 | 485702.0 | ✓|\n|       262144 |393429 | 526578 | 482152.21 | 41436.765  |4773.6074 | 486926.0 | ✓|\n\nThe following table summarizes the peak memory footprint for the CUDA-Sparse APML in a synthetic dataset and the memory reduction per sample.\n\n**Table D. Peak Memory usage per sample vs. number of points.**\n| #Points   | Dense APML | CUDA-APML (sparse) | Reduction |\n|-----------|------------|--------------------|-----------|\n|   1024   |  50 MB     | 0.39 MB  | ~99.23% ↓    |\n|   4096   | 411 MB    | 1.31 MB  | ~99.68% ↓    |\n|   8192   | 1.6 GB     | 2.62 MB  | ~99.84% ↓    |\n|   32768 | 18.5GB    | 8.58 MB | ~99.95% ↓    |\n|  65536     | 68 GB   | 17.MB    | 99.97%   ↓      |\n\nIf we check the peak memory usage on a particular case, (e.g PoinTR and ShapeNet34 dataset, see Appendix D3), memory consumption during training can now have a reduced peak usage by a factor of 99.9× (from 320GB of vRam to <0.5GB). This demonstrates that APML is practical for large-scale applications. \n\n### **4)  Clarification of hyperparameter p_min and Sinkhorn design**\nAs the reviewer pointed out, the single interpretable parameter refers to p_min. To show the small influence of the parameter, we conducted now ablation studies on synthetic datasets and MM-FI showing consistent performance across a range of values. Specifically, we use $p_{\\min}$∈{0.01, 0.1, 0.5, 0.8}. \n\n**Table B. p_min sweep on MMFI.**\n| p_min Value | CD ↓   | EMD ↓  |\n|-------------|--------|--------|\n| 0.01       | 0.1431 | 0.1686 |\n| 0.1         | 0.1398 | 0.1661 |\n| 0.5         | 0.1371 | 0.1671 |\n| 0.8         | 0.1392 | 0.1637 |\n\nAdditional experiments on a synthetic dataset (see reviewer dY8D), show that across a extense range of p_min values and different sparsity configurations, show variations smaller than 1% in both CD and EMD metrics.\n\nRegarding the independent contributions of Sinkhorn and temperature, we agree this should be tested more explicitly. We will include an ablation separating these components in our extended analysis. However, we now believe this decoupling is not conceptually meaningful: Sinkhorn acts as a regularized transport distance and the temperature directly controls its sharpness. They are not independent components but form a single probabilistic matching formulation. We will clarify this dependency and rationale in the camera-ready version."}, {"review_id": "vQY3YhPCoo", "reviewer": "Reviewer_ahb3", "summary": "The authors propose a loss function for 3D point cloud shape completion and generation. This loss function is based on the Sinkhorn Distance, where it uses a temperature-scaled and data-driven parameter T to replace the control parameter $\\lambda$ for the entropy value in the Sinkhorn Distance.\n\nThis loss function achieves near-quadratic runtime, comparable to Chamfer-based losses, and approximates the matching quality of losses based on the Earth Mover's Distance.\n\nThe paper performs quantitative and qualitative evaluations on point cloud completion (ShapeNet and PCN) and generation (MM-Fi) benchmarks.", "questions": "1. Add an ablation experiment for parameter T that the authors propose.\n\n2. Add experiments based on EMD loss to directly compare the performance between the loss the authors propose and EMD loss.\n\n3. Add experiments that show results when choosing different $p_{min}$ values and explain why to choose them and how they influence the 3D point cloud tasks.\n\n4. How to choose the value of the data-driven parameter T that the authors propose? There are no experiments about it.\n\n5. The loss that the authors propose is based on the Sinkhorn loss and modifies its control parameter $\\lambda$. This paper should add experiments showing what will happen if there are different $\\lambda$ values when they are used in 3D point cloud tasks to verify the necessity of the change.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. This paper proposes an Adaptive Probabilistic Matching Loss, which mitigates common shortcomings of losses based on Chamfer Distance, such as point clumping and density bias.\n2. The qualitative evaluation of APML is nearly identical to losses based on Earth Mover's Distance, while APML runtime is lower than EMD. APML combines the advantages of CD and EMD while avoiding their disadvantages in 3D point cloud tasks.\n\nWeaknesses:\n\n1. This paper lacks a comparison with EMD-based losses, which are among the most closely related losses to the proposed one, and serve as the primary motivation.\n2. This paper lacks an ablation experiment on the parameter T, which the authors propose to replace the control parameter $\\lambda$ of the entropy value in the Sinkhorn Distance.\n3. This paper lacks a detailed explanation about how to choose the value of parameter T and how the parameter T influences 3D point cloud tasks.", "was_revised": true, "final_justification": "With the additioanl experiemnts on EMD and parameter analysis, I am overall satisfied on this work, remaining positive.", "rebuttal": "Thank you for your helpful and constructive review. We think that the following experiments and clarifications can greatly improve the paper.\n\n### **1. Comparison with EMD-based losses:**\nWe fully agree that EMD is an important baseline. However, its high computational cost prevents its use at training scale. As a metric, it significantly slows down evaluation even on small batches. We attempted several EMD approximations (e.g., [3], [4]) but observed instability, especially in high-density settings like MM-FI. Qualitatively, our results are on par with EMD-style reconstructions. We include these visual comparisons in the supplementary material. Preliminary attempts at using EMD for training on even moderate-sized toy datasets confirmed the infeasibility due to long runtimes and memory saturation. This practical limitation supports the use of APML as a scalable, effective proxy.\nWe will include additional quantitative and visual comparisons in the camera-ready version.\n\nTo give some insights about other metrics we conducted experiments comparing SWD and DCD against APML on both ShapeNet34 and MM-FI, showing higher EMD values and reduced stability.\n\n**Table G. Loss comparison on ShapeNet34 and MM-FI (FoldingNet and PoinTr).**\n\n| Dataset     | Model       | Loss  | CD ↓   | EMD ↓  | F1 Score ↑ |\n|-------------|-------------|-------|--------|--------|------------|\n| ShapeNet34  | PoinTr      | DCD   | –      | 9.48   | 0.47       |\n| **ShapeNet34**  | **PinTr**  | **APML** | –      | **8.14**   | **0.50**     |\n| ShapeNet34  | FoldingNet  | DCD   | –      | 11.52  | 0.19       |\n| ShapeNet34  | FoldingNet  | SWD   | –      | 11.98  | 0.11       |\n| **ShapeNet34**  | **FoldingNet**  | **APML** | –      | **9.49**   | **0.20**     |\n| MM-FI       | CSI2PC  | DCD   | **0.148**  | 25.68  | –          |\n| **MM-FI**       | **CSI2PC**  | **APML** | 0.152 | **14.11** | –          |\n\nWhile DCD showed a better CD metric than APML, like other CD-based variants, it struggled to produce dense and coherent representations of the body.\n\nWe explored EMD approximations (point-set-based methods SWD [4]) on two dataset. Training FoldingNet on ShapeNet34 with SWD, showed that APML has better results for all metrics. However, for MM-FI, the training was unstable. The generated images failed to generalize well in terms of both body structure and coherence, (i.e the generated images did not resemble the ground truth, and were not comparable to those of APML). It has to be noted that SWD imposes an equal number of points between ground truths and predictions, which is incompatible with some models. While a full evaluation is pending, we emphasize that APML achieves near-EMD qualitative behavior at much lower cost.\n\n### **2. Ablation on T / (data-driven) parameter and effect of p_min hyperparameter:**\n\nT (or p_min) controls the entropy regularization in the Sinkhorn optimization. Our analytic formulation ensures that it scales appropriately with point cloud resolution and distance magnitude. As such, it avoids the need for manual tuning and makes APML stable across tasks and datasets. We emphasize that the entropy parameter is not heuristically set, but derived from point-wise distance statistics. This principled approach ensures consistency and robustness across input conditions.\nThis means that the proposed temperature parameter T is computed analytically using dataset-level statistics (hence its data-driven nature) and cannot be chosen. However, the hyperparameter p_min has effect on the values that T can take. To test its effect, we conducted a toy experiment where p_min is varied broadly. Results show consistent performance across this range. Specifically we tested p_min values on MM-FI and found negligible performance differences. These observations confirm the robustness of APML to its sole hyperparameter. \n\n**Table B. p_min sweep on MMFI.**\n| p_min Value | CD ↓   | EMD ↓  |\n|-------------|--------|--------|\n| 0.01       | 0.1431 | 0.1686 |\n| 0.1         | 0.1398 | 0.1661 |\n| 0.5         | 0.1371 | 0.1671 |\n| 0.8         | 0.1392 | 0.1637 |\n\nAdditional experiments on a toy dataset (see reviewer dY8D), show that across a extense range of p_min values and different sparsity configurations, show variations smaller than 1% in both CD and EMD metrics.\n\nWe also plotted the resulting temperature T as a function of p_min (for a fixed dataset size of 8192 points). The relationship is smooth and weakly non-linear for most of the relevant range (p_min ∈ [0.1, 0.9]), reinforcing the stability of the learned matching under different settings. These results empirically validate the robustness and explain why the choice of p_min is not sensitive. We will show this plot in the camera ready version of the supplementary material. \n\n\n### **3. Clarification of Sinkhorn design and hyperparameter p_min:**\nThank you for this question. To clarify, we do not use the entropic-regularized variant of Sinkhorn that involves a fixed λ (or ϵ) parameter in log-space. Instead, our method applies the standard Sinkhorn algorithm that alternates row and column normalization in the primal (cost) space, commonly referred to as Sinkhorn-Knopp normalization. In this setting, the softness of the resulting assignment matrix is controlled by scaling the input cost matrix before normalization.\n\nIn APML, we modulate this scaling via a data-driven temperature parameter T (as explained before), computed analytically from local distance gaps. This approach eliminates the need for a global hyperparameter and allows the method to adapt to each point's levels of ambiguity: large cost gaps produce sharp assignments, while small gaps yield softer distributions.\n\nThis confirms that the data-driven temperature mechanism is not a cosmetic change but a necessary design component for stable and generalizable matchings. As noted above, the single interpretable parameter in our method refers to p_min, which governs the adaptive scaling behavior."}]

Author rebuttal:

Meta-review: The authors did a great job in addressing most of the reviewers's concerns, with added experiments/analysis, clarification etc. 

The manuscript should be revised based on the comments from reviewers, the added experiments/analysis and clarification, for example, 
1) the cost analysis, 
2) more experiments, 
3) ablation study, etc.


[SAC Update] 
This submission was positively assessed by reviewers and Area Chair. 
After calibrating this submission in the larger pool of accepted papers, and given its contributions, it is more appropriate to present it as a poster. Please update the meta-review accordingly.
Decision: Accept (poster)