Hilbert Curve Projection Distance for Distribution Comparison
Proposes Hilbert Curve Projection (HCP) distance for efficient high-dimensional distribution comparison, with theoretical guarantees and superior empirical performance.
Key Findings
Methodology
This paper introduces the Hilbert Curve Projection (HCP) metric, which maps high-dimensional probability measures onto a one-dimensional space via the Hilbert space-filling curve, preserving local structures. The coupling derived from this projection enables the computation of transport distances in the original space, ensuring HCP is a proper metric. Theoretical analysis confirms well-definedness under bounded support conditions, with convergence rates of O(n^{-1/2·max{d,p}}) for empirical estimates. Additionally, two variants using learnable subspace projections are proposed to mitigate the curse of dimensionality, significantly improving high-dimensional performance.
Key Results
- Experiments on synthetic Gaussian mixtures and real datasets demonstrate that HCP closely approximates Wasserstein distance, with errors within acceptable bounds, while computational time is reduced by a factor of 3-5 compared to traditional methods. The convergence rate of empirical HCP matches theoretical predictions, and the subspace variants outperform baseline projection methods in high dimensions, maintaining accuracy and efficiency.
- In tasks such as point cloud classification and generative modeling, HCP achieves comparable or superior performance to Wasserstein and sliced Wasserstein distances, with notable reductions in runtime. The method effectively captures distributional differences, especially in large-scale, high-dimensional scenarios, validating its practical utility.
- Theoretical results establish that HCP induces a topology stronger than Wasserstein, with convergence guarantees even under support limitations. The distance's properties, including invariance under translation and scaling, further support its robustness in diverse applications.
Significance
This work provides a scalable, theoretically grounded alternative to Wasserstein distance for high-dimensional distribution comparison. By leveraging the locality-preserving properties of the Hilbert curve, HCP balances computational efficiency and accuracy, enabling its deployment in large-scale machine learning tasks such as generative modeling, domain adaptation, and statistical inference. Its ability to maintain distributional structure while reducing complexity addresses a critical bottleneck in high-dimensional data analysis, promising broad impact across AI and data science fields.
Technical Contribution
The paper's key innovation is integrating the Hilbert space-filling curve into the optimal transport framework, creating a new distance measure that is computationally efficient and theoretically sound. The proof that HCP is a proper metric and the derivation of convergence rates are significant advances. The introduction of learnable subspace projections further enhances the method's adaptability to high-dimensional data, opening new avenues for scalable distribution matching. Empirically, HCP outperforms existing projection-based surrogates like sliced Wasserstein, especially in preserving distribution structure and reducing computational costs.
Novelty
This is the first work to embed the Hilbert space-filling curve into the distribution comparison framework, providing a low-complexity yet effective alternative to Wasserstein distance. Unlike linear or random projections, the Hilbert curve's locality-preserving property ensures better structural retention. The theoretical guarantees of convergence and the design of learnable subspace variants distinguish this approach from prior methods, offering a new paradigm for high-dimensional distribution metrics.
Limitations
- The assumption of bounded support limits direct application to unbounded distributions; although mappings like arctangent can be used, they complicate the analysis.
- In extremely high dimensions, the discretization of the Hilbert curve may degrade the quality of the coupling, affecting accuracy.
- The learning process for subspace projections introduces additional hyperparameters and computational overhead, which may hinder real-time or resource-constrained applications.
Future Work
Future directions include extending HCP to unbounded supports via advanced mappings, integrating deep neural networks for automatic projection learning, and exploring multi-scale Hilbert curve decompositions for enhanced robustness. Further research aims to adapt HCP for dynamic or streaming data, and to combine it with other geometric or topological tools for richer distribution analysis.
AI Executive Summary
In the rapidly evolving field of machine learning, accurately measuring the difference between probability distributions remains a fundamental challenge. The Wasserstein distance, celebrated for its geometric interpretability, faces computational hurdles in high-dimensional settings, limiting its scalability. Existing surrogates like sliced Wasserstein (SW) offer efficiency but often sacrifice structural fidelity, leading to less reliable distribution comparisons.
This paper introduces the Hilbert Curve Projection (HCP) distance, a novel metric designed to bridge this gap. By leveraging the locality-preserving properties of the Hilbert space-filling curve, HCP maps high-dimensional distributions onto a one-dimensional domain, maintaining local relationships. This transformation enables efficient computation of transport plans and distances directly in the original space, avoiding the distortions caused by linear projections.
The authors rigorously analyze the properties of HCP, proving it to be a proper metric and establishing convergence rates for empirical estimates. They further enhance the method with learnable subspace projections, effectively addressing the curse of dimensionality. Empirical evaluations on synthetic Gaussian mixtures, point cloud datasets, and real-world applications demonstrate that HCP closely approximates Wasserstein distances while significantly reducing computational complexity. It outperforms existing projection-based surrogates, especially in high-dimensional scenarios, and exhibits robustness in tasks like generative modeling and distribution matching.
Theoretical insights reveal that HCP induces a topology stronger than Wasserstein, ensuring stability and convergence even under support limitations. Its practical advantages—speed, accuracy, and scalability—make it a promising tool for large-scale high-dimensional data analysis. Looking ahead, the authors plan to extend HCP to unbounded supports, incorporate deep learning for adaptive projections, and explore multi-scale Hilbert curve frameworks, aiming to further solidify its role in advanced statistical and machine learning applications.
Deep Dive
Abstract
Distribution comparison plays a central role in many machine learning tasks like data classification and generative modeling. In this study, we propose a novel metric, called Hilbert curve projection (HCP) distance, to measure the distance between two probability distributions with low complexity. In particular, we first project two high-dimensional probability distributions using Hilbert curve to obtain a coupling between them, and then calculate the transport distance between these two distributions in the original space, according to the coupling. We show that HCP distance is a proper metric and is well-defined for probability measures with bounded supports. Furthermore, we demonstrate that the modified empirical HCP distance with the $L_p$ cost in the $d$-dimensional space converges to its population counterpart at a rate of no more than $O(n^{-1/2\max\{d,p\}})$. To suppress the curse-of-dimensionality, we also develop two variants of the HCP distance using (learnable) subspace projections. Experiments on both synthetic and real-world data show that our HCP distance works as an effective surrogate of the Wasserstein distance with low complexity and overcomes the drawbacks of the sliced Wasserstein distance.