Learnable Chamfer Distance (LCD)

• The paper introduces LCD, a reconstruction loss that extends Chamfer Distance by integrating learnable, per-point attention weights to emphasize discrepancies in 3D point clouds.
• LCD employs a dual-branch PointNet-style architecture and an adversarial min–max training scheme to dynamically highlight under-reconstructed regions, accelerating convergence.
• Empirical results show LCD reduces reconstruction errors and improves unsupervised classification accuracy on benchmarks like ShapeNet-Part and ModelNet40.

Learnable Chamfer Distance (LCD) is a reconstruction loss that extends the standard Chamfer Distance by integrating learnable, per-point attention weights predicted by neural networks. Designed for point cloud reconstruction tasks, LCD dynamically emphasizes discrepancies between the input and reconstructed point clouds through an adversarial training scheme, addressing the limitations of rigid, static matching criteria employed by classical approaches. LCD efficiently combines the inductive bias of the original Chamfer Distance with the adaptivity and expressiveness of a small, PointNet-style attention network, facilitating both faster convergence and improved representation learning (Huang et al., 2023).

1. Mathematical Definition and Formulation

The standard Chamfer Distance (CD) between two point sets $X, Y \subset \mathbb{R}^3$ is defined as

$$\mathrm{CD}(X,Y) = \frac{1}{|X|}\sum_{x\in X}\min_{y\in Y}\|x-y\|_2^2 + \frac{1}{|Y|}\sum_{y\in Y}\min_{x\in X}\|y-x\|_2^2$$

LCD, in contrast, replaces the uniform averaging in CD with learnable, non-negative per-point weights. For input point cloud $S_i$ (ground-truth) and reconstructed $S_o$, LCD introduces weights $W_i \in \mathbb{R}^{|S_i|}$, $W_o \in \mathbb{R}^{|S_o|}$ predicted by neural networks: $$L_R(S_i, S_o) = \frac{1}{2}\left( \frac{1}{|S_i|} \sum_{x \in S_i} W_i(x) \min_{y \in S_o} \|x - y\|_2 + \frac{1}{|S_o|} \sum_{y \in S_o} W_o(y) \min_{x \in S_i} \|y - x\|_2 \right)$$ Weights are derived through a mechanism involving global and local feature extraction followed by non-linear normalization. Specifically, they are formed by concatenating point features and global descriptors and processed with an MLP $g$, normalized via a Gaussian-shaped kernel: $$W_i^{(j)} = \frac{\sigma + \exp(-F_i^{(j)2})} {|S_i|\cdot\sigma + \sum_k \exp(-F_i^{(k)2})}$$ and similarly for $W_o$, where $F_i$ represents the attention scores, and $\sigma > 0$ is a small stabilization constant.

2. LCD Architecture: Weight Prediction Networks

The core of LCD’s adaptivity lies in its per-point weight modules built from the PointNet family of architectures:

• Global Feature Extraction ("SiaCon"): A shared PointNet-style encoder $f_1$, consisting of multilayer perceptrons (MLPs) with ReLU activations and symmetric max-pooling, produces descriptors $h_i, h_o$. These are concatenated into $F_{io} = [h_i \| h_o]$.
• Local Feature Extraction and Attention ("SiaAtt" + $g$): Using another PointNet-style encoder $f_2$, per-point features $p_i, p_o$ are derived for $S_i, S_o$.
• Attention-Score MLP ($g$): Each point is represented by the concatenation of its coordinates, its local feature, and the global pair descriptor. This vector is processed through two or three FC+ReLU layers to yield a scalar value per point. The normalization ensures the scores sum to one per cloud, forming a probability distribution over points.

This architectural setup supports the network's capacity to allocate attention where synthetic-reconstruction discrepancies are most pronounced.

3. Training Paradigm: Adversarial Min–Max Optimization

LCD leverages a two-player min–max training scheme:

• The reconstruction network (e.g., an auto-encoder or FoldingNet) with parameters $\theta$ minimizes the weighted reconstruction loss $L_R$.
• The LCD module (parameters $\phi$) is updated to maximize an adversarial criterion:

$$L_{\mathrm{LCD}(\phi)} = -\log\left(L_R(S_i, S_o) + \sigma_r\right)$$

where $\sigma_r$ is a small positive constant for numerical stability.

Consequently, training alternates between:

• Ascending $\phi$ to emphasize residual errors.
• Descending $\theta$ to minimize these focused discrepancies.

This adversarial process encourages the LCD attention network to uncover "hard" or under-reconstructed regions until the reconstructor has minimized such errors everywhere.

The training loop is succinctly captured by the following pseudocode:

for t in range(T): # 1) reconstruct S_o = R(S_i) # 2) update LCD to spotlight residual defects φ += η_LCD * gradient_φ(-log(L_Rφ(S_i, S_o) + σ_r)) # 3) update reconstructor to remove weighted errors θ -= η_R * gradient_θ(L_Rφ(S_i, S_o))

4. Empirical Performance and Ablation Analysis

Extensive experiments, primarily on ShapeNet-Part and ModelNet40/10, establish LCD’s quantitative and qualitative benefits. Evaluation metrics include Multi-scale Chamfer Distance (MCD), Hausdorff Distance (HD), and SVM-based unsupervised classification accuracy. Results for PointNet-based AE on ShapeNet-Part are shown below:

Method	MCD ↓	HD ↓
CD (baseline)	0.32	1.87
EMD	0.25	2.23
DCD	0.28	1.75
PCLoss	0.23	1.66
LCD (ours)	0.22	1.51

Ablations reveal the incremental benefit of each component. Adding Siamese-Attention (SiaAtt) yields a significant drop in MCD, while SiaCon and adversarial log-loss further reduce HD:

Variant	MCD ↓	HD ↓
CD only	0.32	1.87
+ Siamese-Attention	0.22	1.98
++ Siamese-Concatenation	0.22	1.54
+++ Adversarial log-loss	0.22	1.51

LCD achieves lower errors and converges in 2–3× fewer iterations compared to PCLoss and CD. In terms of unsupervised classification via SVM, LCD codes achieve ∼88.4% accuracy on ModelNet40, outperforming PCLoss (86.4%) and CD (85.9%).

5. Computational Overhead, Efficiency, and Limitations

LCD’s additional computation consists of two small PointNet-style subnetworks and one per-point MLP, resulting in an average overhead of approximately 20 ms per iteration over bare CD but remains 10–15 ms faster than PCLoss on an NVIDIA 2080Ti + CPU. LCD runs at 43 ms/iter compared to CD/DCD (23 ms/iter), EMD (216 ms/iter), and PCLoss (57 ms/iter).

A principal limitation arises in low-data regimes where the LCD attention network may overfit, as well as a need for careful tuning of the min–max learning rates (optimal LCD learning rates found in [0.001, 0.005]). The adversarial setup may also require monitoring for training instability.

6. Potential Extensions and Broader Applicability

Several extensions to LCD are plausible:

• Replacing Gaussian weighting with alternative kernels, such as a learnable temperature softmax.
• Adapting the attention mechanism to other distances, including Earth Mover's Distance or deformable matching architectures.
• Conditioning the attention mechanism for partial-to-partial matching scenarios, such as completion tasks, by leveraging observed/unobserved point flags.

These directions suggest LCD’s design principles are transferable to a wider range of geometric matching and similarity assessment tasks in point cloud and 3D vision problems.

7. Interpretive Context and Significance

Learnable Chamfer Distance provides an interpretable and computationally efficient mechanism to localize and rectify reconstruction errors in deep point cloud models. By combining a lightweight, learnable attention architecture with adversarial loss concentration, LCD achieves both strong empirical performance and accelerated convergence without abandoning the structural priors embedded by classical Chamfer-matching (Huang et al., 2023). This suggests LCD offers a compelling synthesis of hand-engineered and learned similarity metrics that may generalize across multiple domains involving set-to-set comparison and geometric learning.