# APML: Adaptive Probabilistic Matching Loss for Robust 3D Point Cloud Reconstruction

arXiv: https://arxiv.org/abs/2509.08104 · HTML: https://mlsanity.com/paper/2509.08104/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 39 references that are in the map.
- First posted on arXiv 9 Sep 2025.

## Builds on (most cited first, 21 of 21)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [ScanNet: Richly-annotated 3D Reconstructions of Indoor Scenes](https://mlsanity.com/paper/1702.04405/) (2017)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [SuperGlue: Learning Feature Matching with Graph Neural Networks](https://mlsanity.com/paper/1911.11763/) (2019)
- [A Point Set Generation Network for 3D Object Reconstruction from a Single Image](https://mlsanity.com/paper/1612.00603/) (2016)
- [Deep Learning for 3D Point Clouds: A Survey](https://mlsanity.com/paper/1912.12033/) (2019)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [PoinTr: Diverse Point Cloud Completion with Geometry-Aware Transformers](https://mlsanity.com/paper/2108.08839/) (2021)
- [Shape Completion using 3D-Encoder-Predictor CNNs and Shape Synthesis](https://mlsanity.com/paper/1612.00101/) (2016)
- [SnowflakeNet: Point Cloud Completion by Snowflake Point Deconvolution with Skip-Transformer](https://mlsanity.com/paper/2108.04444/) (2021)
- [Density-aware Chamfer Distance as a Comprehensive Metric for Point Cloud Completion](https://mlsanity.com/paper/2111.12702/) (2021)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [SeedFormer: Patch Seeds based Point Cloud Completion with Upsample Transformer](https://mlsanity.com/paper/2207.10315/) (2022)
- [MM-Fi: Multi-Modal Non-Intrusive 4D Human Dataset for Versatile Wireless Sensing](https://mlsanity.com/paper/2305.10345/) (2023)
- [Point-set Distances for Learning Representations of 3D Point Clouds](https://mlsanity.com/paper/2102.04014/) (2021)
- [Accurate Point Cloud Registration with Robust Optimal Transport](https://mlsanity.com/paper/2111.00648/) (2021)
- [Accelerating Sinkhorn Algorithm with Sparse Newton Iterations](https://mlsanity.com/paper/2401.12253/) (2024)
- [Importance Sparsification for Sinkhorn Algorithm](https://mlsanity.com/paper/2306.06581/) (2023)
- [Learnable Chamfer Distance for Point Cloud Reconstruction](https://mlsanity.com/paper/2312.16582/) (2023)
- [Fine-Grained Complexity of Earth Mover's Distance under Translation](https://mlsanity.com/paper/2403.04356/) (2024)

## Built on by (most cited first, 3 of 3)

- [Learning Permutation-invariant Macroscopic Dynamics](https://mlsanity.com/paper/2605.30812/) (2026)
- [From Theory to Throughput: CUDA-Optimized APML for Large-Batch 3D Learning](https://mlsanity.com/paper/2512.19743/) (2025)
- [Point-MF: Stabilizing One-Step Mean Flows for Single-Image Point Cloud Reconstruction](https://mlsanity.com/paper/2604.24586/) (2026)

## Abstract (arXiv)

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, and InfoCD 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 spatiotemporal Transformer (CSI2PC) that generates 3D 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.