Title: Fully Dynamic Algorithms for Chamfer Distance
Venue: NeurIPS 2025
Accepted: yes
Keywords: Chamfer distance

dynamic algorithms
Abstract: We study the problem of computing Chamfer distance in the fully dynamic setting, where two sets of points $A, B \subset \mathbb{R}^{d}$, each of size up to $n$, dynamically evolve through point insertions or deletions and the goal is to efficiently maintain an approximation to $dist_{\mathrm{CH}}(A,B) = \sum_{a \in A} \min_{b \in B} dist(a,b)$, where $dist$ is a distance measure. Chamfer distance is a widely used dissimilarity metric for point clouds, with many practical applications that require repeated evaluation on dynamically changing datasets, e.g., when used as a loss function in machine learning. In this paper, we present the first dynamic algorithm for maintaining an approximation of the Chamfer distance under the $\ell_p$ norm for $p \in$ {$1,2$}.
Our algorithm reduces to approximate nearest neighbor (ANN) search with little overhead. Plugging in standard ANN bounds, we obtain $(1+\epsilon)$-approximation in $\tilde{O}(\epsilon^{-d})$ update time and $O(1/\epsilon)$-approximation in $\tilde{O}(d n^{\epsilon^2} \epsilon^{-4})$ update time.
We evaluate our method on real-world datasets and demonstrate that it performs competitively against natural baselines.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis manuscript proposes a near-linear time approximated algorithm for estimating Chamfer Distance (CD) for two dynamic point sets $A$ and $B$. \n\nThe closest literature [BIJ+23] solves the static problem through approximating the closest distance from $a\\in A$ to $B$ with LSH and performing importance sampling for a bounded-variance estimation of CD. This paper follows this method and tackles the ‘linear-time update barrier’ when transferring to the dynamic setting, i.e., maintaining the $\\alpha$-approximate assignment from A to B costs $\\Omega(N)$ time even when only reporting the recourse (L70). \n\nThe authors bypass the challenge by proposing a dynamic sampler that no longer maintains the specific closest distances, but can yield samples $a\\in A$ obeying the desired distribution probability proportional to importance (i.e., closest distance from $a$ to $B$).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1.\tThe paper is well written and well organized. The idea is original and results bears great significance in terms of fast algorithms for point cloud processing. All conclusions are accompanied with sufficient proof and backed up with experimental results.\n\n2.\tThe core idea of turning the explicit point assignment into an implicit assignment is very intuitive once fully explained. This trick may be of great use in other problems as well.\n\n3.\tThe experimental results well support the claims of bounded-error approximation, achieving outstanding robustness to outliers compared to the baseline methods.\n\nWeaknesses:\n\nN/A. I am not familiar with this field of research and may not be able to justify any doubts.\n\nQUESTIONS:\n1.\tHow does the algorithm perform on lower-dimensional datasets, such as point cloud datasets? I think this is important since you mentioned point cloud completion and up-scaling in L26 as potential applications.\n\n2.\tHow can we quantify the distributional uniformity of a certain dataset apart from the visualization in Fig. 4? Furthermore, can we assess the model’s performance with different degrees of uniformity (likely through a simulated dataset)?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of fully dynamic Chamfer Distance approximation. The Chamfer Distance is a “distance” (quotes since not actually a metric) between two sets of points A,B in R^d. It is a well known relaxation of the Earth Mover Distance, which can be computed easily in quadratic time via a simple all-pairs brute force distance computation. This metric is also well studied in its own right in the computer vision and ML literature. The similarity version of Chamfer (where distances are replaced with dot product, which is essentially equivalent for normalized vectors), is also very important in information retrieval, especially for the training of multi-vector embedding models, where it is the measure by which two sets of embeddings are compared. Thus, algorithmic improvements for handling the Chamfer distance are quite important.\n\nThe current paper studies the maintenance of an approximation of the Chamfer distance in the fully dynamic setting, where points can be added and removed from both A and B. The main result is an algorithm with O(1/eps) approximation and n^{eps^2} update time (for any eps in (0,1)). In fact, their update time is the same as the query time of an approximate nearest neighbor (ANN) search data structure in (R^d, ell_2); if that runtime is tau, their algorithm has update time tau*poly(log n). This is a significant improvement over a naive tilde{O}(n) update time required to update all the mappings from points in A to nearest neighbors in B (for each point, one could store all their neighbors in B in sorted order, and update each sorted order after each insertion or deletion).\n\nNaively, the main result is very easy to obtain if B is fixed and only insertions or deletions are allowed to A: every time you insert a new point a to A, just run ANN and you are done. The hard part is when points are deleted or added to B, since this may change the nearest neighbors of O(n) points in A.\n\nRecently, Bakshi et. al. (NeurIPS 23’) showed that a (1+eps) approximation to the Chamfer distance can be obtained in roughly n*poly(1/eps) time using an importance sampling technique. First, rough approximate distances to B for each point a in A are obtained using a standard quadtree construction. Then, poly(1/eps)*log(n) points are sampling with probability proportional to these approximate distances. The true nearest neighbors for these points are then computed exactly via brute force. The current paper’s main contribution is a technique to implement this in a fully dynamic fashion. The main challenge is maintaining an algorithm that can dynamically sample from the distribution of approximate distances from A to B (when this distribution is changing). Once 1/eps points are sampled, one can just run ANN to get the approximate neighbors in B (giving the resulting ANN runtime guarantee). To solve the key challenge of maintaining a dynamic sampler, they show that they can implicitly sample from the distribution over A (without explicitly maintaining the weights) by keeping track of the count of points in A in each node in the quadtree that has no points in B. Since the highest node in the quadtree for which a point a in A collides with no other b in B determines its approximate distance (and therefore sampling probability), one can first sample a node from the quadtree (which has no B nodes in it) proportional to the number of nodes from A that are in it (and proportional to the 2^{level} of that node in the quadtree), and then sample a point uniformly from that node\n\nTechnically, this is a very clean idea, if perhaps somewhat straightforward. I’d venture to say that, if you asked someone who is an expert in ANN algorithms to solve this problem by making the algorithm of Bakshi et. al. dynamic, they probably would be able to do so without terrible difficulty. But hindsight is powerful (and the idea that making Bakshi et. al. dynamic is the right approach is non-trivial), thus the algorithm in this paper is quite interesting and important regardless. Moreover, it is quite practical, since the data structures are essentially no more complicated than the original quadtree data structure. It is also appealing because the final ANN algorithm can be implemented with any ANN algorithm in practice, such as HSNW / DiskANN, and does not need to be the theoretically optimal LSH based algorithms (that give the theoretical trade-offs). \n\nExperimentally, they outperform uniform sampling methods primarily on datasets where outliers are injected. This makes sense, since importance sampling is only different from uniform sampling when outliers exist. The fact that artificial injection of outliers was necessary is likely due to the choice of datasets; likely other datasets exist with outliers present, and it is clear that this technique is preferable for such datasets.\n\nOverall, this paper was very nicely written and enjoyable to read. The result is a strong theoretical advance for an important distance measure, and is likely to be useful in practice. Thus, I think this paper should be accepted.\n\nSTRENGTHS AND WEAKNESSES:\nIn Summary\n\nQUESTIONS:\nIt would be good to see experiments on datasets that are naturally dynamic, perhaps one that has build in temporal data. This would allow for a more interesting ANN algorithm to be used which searching for the nearest neighbors.\n\nHow often do the samples changes from update to update? If the distribution does not change significantly, then perhaps it is possible to reuse most of the samples from the last step. The samples between consecutive updates are no longer independent, but this should not be a big issue. In theory, in the worst case, the distribution can totally change, thus you get no gain. In practice though, you may have big wins since the distribution may not change much on each step.\n\nAlso, you can imagine other optimizations, such as storing the k-NN for each point in A, and using the \"living\" points in that set to answer ANN queries for that point (instead of re-querying the ANN data structure for that point from scratch). It would be interesting to see how these methods improve runtime (note that this would probably improve the runtime of uniform sampling even more).\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-05"}

comments: SUMMARY:
The paper studies the problem of computing Chamfer distance in the fully dynamic setting, and evaluates the proposed method on real-world datasets and demonstrates that it performs competitively against natural baselines. Actually, I am not an expert in this field, so my comments are based on low confidence. If I make basic mistakes, I hope it will not influence the meta-review at the final decision stage.

STRENGTHS AND WEAKNESSES:
Strengths:

1.	The speedup of Chamfer Distance is very useful.
2.	The experimental verification is relatively complete

Weaknesses:

1.	It is advised to present experiment results in some point cloud datasets such as ShapeNet because Chamfer Distance is widely used as loss function in point cloud processing.
2.	The current approximation is one-sided (from A to B). Would the same dynamic framework support symmetric Chamfer distance efficiently?
3.	While runtime speedups are demonstrated, it would be helpful to understand the algorithm’s space complexity, especially for high-dimensional datasets.

QUESTIONS:
See the weaknesses part.

LIMITATIONS:
The authors adequately addressed the limitations.
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents the first dynamic algorithm for maintaining approximations of the Chamfer distance between two evolving point sets A and B under L1 and L2 norms, achieving $(1+\\epsilon)$-approximation in $\\widetilde{O}(\\epsilon^{-d})$ update time for low dimensions, and $O({1}/{\\epsilon})$-approximation in $\\widetilde{O}\\left(dn^{\\epsilon^2}\\epsilon^{-4}\\right)$ update time for high dimensions. The authors address a fundamental gap in computational geometry by developing a data structure that can efficiently handle point insertions and deletions while maintaining distance approximations with provable guarantees.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- The paper addresses a genuine gap in dynamic computational geometry - no prior work existed for dynamic Chamfer distance. The problem is well motivated with several example use-cases and the need for a faster alternative to EMD.\n- Rigorous theoretical analysis - detailed proofs for approximation guarantees, time complexity, and correctness has been provided. Clever reduction to well-studied ANN data structures, allowing the algoritjm to inherit existing performance guarantees.\n- Sophisticated sampling scheme that samples cells proportionally to their size and point count, leading to provably good approximations. Well-designed integration of multiple dynamic samplers (TREE-SAMPLER and NODE-SAMPLER) with efficient maintenance procedures.\n\n**Weaknesses**\n- The paper uses the asymmetric definition of Chamfer distance, while the symmetric version is more commonly seen in practice. The authors should better emphasise this choice and discuss its implications for applications.\n- Experiments focus primarily on dynamic B with static A, not fully demonstrating the algorithm's dynamic capabilities on both sets.\n- There are some restrictive assumptions like bounded aspect ratio poly(n) and points in $[U]^d$, which may not hold in real applications with arbitrary point cloud data.\n- The algorithm suffers from exponential dependence $\\epsilon^{-d}$ in low dimensions, while high-dimensional guarantees only provide weak $O(1/\\epsilon)$-approximation, possibly limiting practical applicability for moderate dimensions.\n\nQUESTIONS:\n- What is the total memory overhead of the quad-tree structure and multiple dynamic samplers compared to storing the point sets explicitly?\n- Can you provide experiments where both sets A and B undergo frequent updates to demonstrate the complete dynamic capabilities?\n- Given that symmetric Chamfer distance is more common in practice, how efficiently could your approach compute                                      $dist_{CH}(A,B) + dist_{CH}(B,A)$  and would this require maintaining two separate data structures?\n- The paper motivates the work with applications like point cloud completion and medical imaging - can you provide experiments on these actual use-cases to demonstrate practical relevance beyond generic benchmark datasets?\n\nLIMITATIONS:\nWhile the authors discuss some theoretical limitations, they should better address practical concerns including memory overhead of maintaining multiple dynamic samplers and quad-tree structures. Additionally, the paper lacks analysis of algorithm sensitivity to the bounded aspect ratio assumption, or worst-case point distributions that might cause the importance sampling to degrade significantly. A brief Limitations section addressing these scalability and practical deployment challenges would strengthen the paper's contribution.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}