compute chamfer distance

This paper addresses the inefficiency of approximating the Chamfer distance between high-dimensional point clouds. We propose the first $(1+varepsilon)$-approximation algorithm with time complexity $Oig(nd(loglog n + log(1/varepsilon))/varepsilon^2ig)$. Our method integrates efficient approximate nearest neighbor search, hierarchical random sampling, and geometric pruning, underpinned by a rigorous error-control analysis framework. When $varepsilon$ is constant, the logarithmic factor improves from $O(log n)$ to $O(loglog n)$, significantly narrowing the gap between the current upper bound and the theoretical lower bound $Omega(dn)$. This represents the first substantial reduction in that gap. Compared to the state-of-the-art result from NeurIPS 2023, our algorithm achieves substantially faster runtime, offering both stronger theoretical guarantees and improved practical performance for large-scale point cloud similarity measurement.

This paper addresses the fully dynamic (1+ε)-approximate maintenance of the Chamfer distance between two evolving point sets A and B in ℝᵈ under arbitrary insertions and deletions. We propose the first fully dynamic algorithm for Chamfer distance approximation under ℓ₁/ℓ₂ norms, reducing the problem to lightweight approximate nearest neighbor (ANN) queries. Our method integrates locality-sensitive hashing with a dynamic grid index, achieving sublinear update time: Õ(ε⁻ᵈ) for (1+ε)-approximation and Õ(d n^{ε²} ε⁻⁴) for O(1/ε)-approximation. Empirical evaluation demonstrates substantial speedup over naive recomputation. This work breaks the traditional static computation paradigm, providing the first theoretically guaranteed and practically efficient solution for dynamic Chamfer distance maintenance—enabling real-time analysis of evolving point clouds in applications such as 3D vision and robotics.

This work presents the first systematic study of minimizing the Chamfer distance between point sets under translation. The authors propose an exact quadratic-time algorithm for the one-dimensional case and efficient approximation algorithms for higher dimensions, achieving approximation ratios of (2+ε) and (1+ε) with nearly quadratic time complexity in certain regimes. Their approach integrates geometric approximation, grid discretization, and refined complexity analysis, and includes a fast decision procedure under a separation assumption. This study not only fills a theoretical gap in understanding Chamfer distance optimization under translation but also provides a practical algorithmic framework for high-dimensional point set registration.

This paper addresses the efficiency degradation caused by diversity constraints in approximate nearest neighbor (ANN) search. We propose the first end-to-end graph neural indexing framework explicitly designed for diversity-aware retrieval. Unlike conventional two-stage paradigms (coarse retrieval followed by post-hoc diversification), our method integrates diversity optimization directly into graph construction and local traversal, guiding dynamic pruning and path selection in NSG- and HNSW-style graphs via a greedy diversity criterion. We theoretically prove that, under low intrinsic dimensionality, the query time complexity is (O(k log Delta))—matching the optimal bound for unconstrained ANN. Experiments demonstrate that our approach achieves high diversity (as measured by MaxMin and Intra-List Distance metrics) while maintaining latency close to single-point ANN search, significantly outperforming state-of-the-art two-stage baselines.

Existing similarity search methods are largely confined to metric spaces, rendering them inadequate for non-metric, topologically heterogeneous, sparse, or skewed data distributions. This paper introduces the first distributed approximate similarity search framework supporting arbitrary distance functions. Our approach addresses these challenges through three core contributions: (1) a multi-level clustering index structure that enhances robustness against outliers and data imbalance; (2) a clustering-driven embedding mechanism for non-metric distances, coupled with a compatibility layer ensuring consistent neighborhood semantics; and (3) a distributed nearest-neighbor retrieval architecture integrating scalable hashing with dynamic load balancing. Evaluated on diverse topological benchmark datasets, our framework achieves an average 2.3× higher query throughput and a 37% reduction in recall error compared to state-of-the-art ANN methods, significantly improving both efficiency and accuracy for large-scale similarity search in non-metric spaces.

Existing point cloud generation evaluation metrics—such as Chamfer Distance—are highly sensitive to geometric imperfections and lack robustness, failing to accurately quantify both local shape consistency and global fidelity. To address these limitations, this work proposes: (1) two novel evaluation metrics—Density-Aware Chamfer Distance (DCD) and Surface Normal Consistency (SNC)—designed to better discriminate sampling non-uniformity and normal vector distortion; and (2) Diffusion Point Transformer, a diffusion-based generative architecture leveraging serialized patch-wise attention, augmented with sample alignment preprocessing to enhance local structural modeling. Evaluated on ShapeNet, our method achieves state-of-the-art generation quality, significantly outperforming leading baselines across multiple metrics. The implementation is publicly available.

This study addresses the inefficiency of nearest-neighbor search for points, lines, and triangles in three-dimensional space, where balancing storage requirements with query performance remains challenging. To overcome this limitation, the proposed method leverages combinatorial algorithms based on arrangements of three- and four-dimensional surfaces, combined with vertical decomposition techniques, to construct highly efficient data structures. This work breaks through existing theoretical bounds by achieving query times of $O^*(n^{1/2})$ or $O^*(1)$, thereby establishing an optimal trade-off between storage space and query complexity. The results significantly surpass the best previously known solutions, offering efficient query schemes under linear storage constraints as well as scalable structures that support constant-time queries.

This work proposes a parameter-free local topographic descriptor for the comparison and rigid alignment of three-dimensional structured point patterns. The method decomposes each point pattern into multiple arms and introduces a normalized finite difference operator along each arm to capture the local variation of height components relative to the underlying planar geometry, thereby integrating fine-grained geometric details with global structural information. By combining Wasserstein distance with Procrustes analysis, the approach enables efficient distributional comparison and precise alignment of point clouds. The proposed descriptor preserves salient local topographic features while significantly enhancing the robustness and accuracy of point pattern matching.

This work proposes the first data structure for approximate nearest neighbor (ANN) queries in two-dimensional polygonal domains with obstacles that supports dynamic insertions and deletions. The structure efficiently answers $(1+\varepsilon)$-approximate nearest neighbor queries along with the corresponding shortest-path distances. By integrating geometric decomposition, approximate distance fields, a hierarchical layout, and a logarithmic-scale spatial index, the method achieves a query time of $O\left(\frac{1}{\varepsilon^2} \log n + \frac{1}{\varepsilon} \log n \log m + \frac{1}{\varepsilon} \log^2 m\right)$ and amortized update time within the same bound, while using $O\left(\frac{n+m}{\varepsilon} \log n + \frac{m}{\varepsilon} \log m\right)$ space, where $n$ is the complexity of the polygonal domain and $m$ is the number of sites. This design balances spatial efficiency with strong dynamic performance.

本文通过近似最近邻搜索方法，在一般度量空间中以亚二次时间构造具有特定失真的度量生成器，解决了高效生成器构建问题。