[Paper Review] A Near-Linear Time Algorithm for the Chamfer Distance

Ainesh Bakshi, Piotr Indyk+3 more|arXiv (Cornell University)|Jul 6, 2023

Data Management and Algorithms4 citations

TL;DR

This paper presents the first $(1+\varepsilon)$-approximate algorithm for Chamfer distance estimation with near-linear time complexity $\mathcal{O}(nd\log n/\varepsilon^2)$, using importance sampling and Locality-Sensitive Hashing (LSH). It achieves high accuracy and speed on large-scale, high-dimensional datasets, while proving that reporting an exact mapping is likely sub-quadratically hard under standard complexity assumptions.

ABSTRACT

For any two point sets $A,B \subset \mathbb{R}^d$ of size up to $n$, the Chamfer distance from $A$ to $B$ is defined as $ ext{CH}(A,B)=\sum_{a \in A} \min_{b \in B} d_X(a,b)$, where $d_X$ is the underlying distance measure (e.g., the Euclidean or Manhattan distance). The Chamfer distance is a popular measure of dissimilarity between point clouds, used in many machine learning, computer vision, and graphics applications, and admits a straightforward $O(d n^2)$-time brute force algorithm. Further, the Chamfer distance is often used as a proxy for the more computationally demanding Earth-Mover (Optimal Transport) Distance. However, the \emph{quadratic} dependence on $n$ in the running time makes the naive approach intractable for large datasets. We overcome this bottleneck and present the first $(1+ε)$-approximate algorithm for estimating the Chamfer distance with a near-linear running time. Specifically, our algorithm runs in time $O(nd \log (n)/\varepsilon^2)$ and is implementable. Our experiments demonstrate that it is both accurate and fast on large high-dimensional datasets. We believe that our algorithm will open new avenues for analyzing large high-dimensional point clouds. We also give evidence that if the goal is to \emph{report} a $(1+\varepsilon)$-approximate mapping from $A$ to $B$ (as opposed to just its value), then any sub-quadratic time algorithm is unlikely to exist.

Motivation & Objective

• To overcome the quadratic time bottleneck of the standard Chamfer distance computation, which limits its use on large point clouds.
• To design a practical, efficient algorithm that approximates the Chamfer distance within a $1+\varepsilon$ factor in near-linear time.
• To demonstrate that estimating the Chamfer distance is significantly easier than reporting a corresponding mapping, under complexity-theoretic assumptions.
• To evaluate the algorithm's performance on real and synthetic datasets, including large-scale benchmarks like Big-ANN-Benchmarks.

Proposed method

• The algorithm uses importance sampling to estimate the minimum distance from each point in set $A$ to any point in set $B$, with sampling probabilities proportional to the inverse of the expected minimum distance.
• It leverages Locality-Sensitive Hashing (LSH) for efficient approximate nearest neighbor queries under $\ell_1$ and $\ell_2$ norms.
• The method combines randomized sampling with concentration bounds to ensure a $(1+\varepsilon)$-approximation with high probability ($99\%$).
• The algorithm is implemented using a hierarchical LSH data structure, with performance depending on the LSH parameters and the underlying metric.
• Theoretical analysis shows that the expected number of samples per point is $\mathcal{O}(1/\varepsilon^2)$, leading to total time $\mathcal{O}(nd\log n/\varepsilon^2)$.
• A conditional lower bound is derived using the Hitting Set Conjecture, showing that any algorithm reporting a $(1+\varepsilon)$-approximate mapping must run in $\Omega(n^{2-\delta})$ time.

(a) ShapeNet

Experimental results

Research questions

• RQ1 Can the Chamfer distance be approximated within a $1+\varepsilon$ factor in near-linear time for large, high-dimensional point clouds?
• RQ2 Is it possible to design a practical and efficient algorithm for Chamfer distance estimation that outperforms brute-force and uniform sampling?
• RQ3 Why is estimating the Chamfer distance value significantly easier than reporting a corresponding approximate mapping?
• RQ4 What is the theoretical limit on the running time for reporting a $(1+\varepsilon)$-approximate mapping from $A$ to $B$?

Key findings

• The proposed algorithm runs in $\mathcal{O}(nd\log n/\varepsilon^2)$ time and estimates the Chamfer distance within a $1\pm\varepsilon$ factor with $99\%$ probability for $\ell_1$ and $\ell_2$ norms.
• On billion-scale datasets from Big-ANN-Benchmarks, the algorithm runs up to 50x faster than optimized brute-force methods.
• The algorithm is more than 5x faster than brute force (even with KD-trees) and 5–10x more sample-efficient than uniform sampling on high-variance datasets.
• For the ShapeNet dataset, the algorithm achieves over $98\%$ recall@30 in nearest neighbor search, reducing the search space by over 1000x compared to brute force.
• Under the Hitting Set Conjecture, any algorithm that reports a $(1+\varepsilon)$-approximate mapping must run in $\Omega(n^{2-\delta})$ time, even for $d=\Theta(\log^2 n)$ and $\varepsilon=\Theta(1)/d$.
• The algorithm adapts to distance variance across datasets, outperforming uniform sampling when distances from points in $A$ to $B$ vary significantly.

(b) Federalist Papers