Robust Point Matching with Distance Profiles
Abstract: We show the outlier robustness and noise stability of practical matching procedures based on distance profiles. Although the idea of matching points based on invariants like distance profiles has a long history in the literature, there has been little understanding of the theoretical properties of such procedures, especially in the presence of outliers and noise. We provide a theoretical analysis showing that under certain probabilistic settings, the proposed matching procedure is successful with high probability even in the presence of outliers and noise. We demonstrate the performance of the proposed method using a real data example and provide simulation studies to complement the theoretical findings. Lastly, we extend the concept of distance profiles to the abstract setting and connect the proposed matching procedure to the Gromov-Wasserstein distance and its lower bound, with a new sample complexity result derived based on the properties of distance profiles. This paper contributes to the literature by providing theoretical underpinnings of the matching procedures based on invariants like distance profiles, which have been widely used in practice but have rarely been analyzed theoretically.
- On convex relaxation of graph isomorphism. Proceedings of the National Academy of Sciences, 112(10):2942–2947.
- Gromov-Wasserstein alignment of word embedding spaces. arXiv preprint arXiv:1809.00013.
- Towards optimal transport with global invariances. In International Conference on Artificial Intelligence and Statistics, pages 1870–1879. PMLR.
- A User’s Guide to Optimal Transport, pages 1–155. Springer Berlin Heidelberg.
- Network analysis in the social sciences. Science, 323(5916):892–895.
- Brécheteau, C. (2019). A statistical test of isomorphism between metric-measure spaces using the distance-to-a-measure signature. Electronic Journal of Statistics, 13(1):795 – 849.
- Learning generative models across incomparable spaces. In International conference on machine learning.
- The Gromov–Wasserstein distance between networks and stable network invariants. Information and Inference: A Journal of the IMA, 8(4):757–787.
- Minimax rates in permutation estimation for feature matching. Journal of Machine Learning Research, 17(1):162–192.
- Word translation without parallel data. arXiv preprint arXiv:1710.04087.
- Efficient random graph matching via degree profiles. Probability Theory and Related Fields, 179:29–115.
- Dudley, R. M. (1969). The speed of mean Glivenko-Cantelli convergence. The Annals of Mathematical Statistics, 40(1):40–50.
- Dudley, R. M. (1979). Balls in ℝksuperscriptℝ𝑘\mathbb{R}^{k}blackboard_R start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT do not cut all subsets of k+2𝑘2k+2italic_k + 2 points. Advances in Mathematics, 31(3):306–308.
- On the rate of convergence in Wasserstein distance of the empirical measure. Probability Theory and Related Fields, 162(3-4):707–738.
- Optimal detection of the feature matching map in presence of noise and outliers. Electronic Journal of Statistics, 16(2):5720–5750.
- Unsupervised alignment of embeddings with Wasserstein procrustes. In International Conference on Artificial Intelligence and Statistics, pages 1880–1890. PMLR.
- Reversible Gromov-Monge sampler for simulation-based inference. arXiv preprint arXiv:2109.14090.
- Tight relaxation of quadratic matching. Computer Graphics Forum, 34(5):115–128.
- Assignment problems and the location of economic activities. Econometrica, pages 53–76.
- A spectral technique for correspondence problems using pairwise constraints. In International Conference on Computer Vision.
- A survey for the quadratic assignment problem. European Journal of Operational Research, 176(2):657–690.
- Mémoli, F. (2011). Gromov–Wasserstein distances and the metric approach to object matching. Foundations of Computational Mathematics, 11:417–487.
- Distance distributions and inverse problems for metric measure spaces. Studies in Applied Mathematics, 149(4):943–1001.
- Papadakis, P. (2014). The canonically posed 3d objects dataset. In Eurographics Workshop on 3D Object Retrieval, pages 33–36.
- Computational optimal transport: With applications to data science. Foundations and Trends® in Machine Learning, 11(5-6):355–607.
- Gromov-Wasserstein averaging of kernel and distance matrices. In International Conference on Machine Learning.
- Convergence properties of the softassign quadratic assignment algorithm. Neural Computation, 11(6):1455–1474.
- Entropic metric alignment for correspondence problems. ACM Transactions on Graphics (ToG), 35(4):1–13.
- The human connectome: a structural description of the human brain. PLoS Computational Biology, 1(4):e42.
- Sturm, K.-T. (2006). On the geometry of metric measure spaces. Acta Mathematica, 196(1):65 – 131.
- A survey on shape correspondence. Computer Graphics Forum, 30(6):1681–1707.
- Vershynin, R. (2018). High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge University Press.
- Villani, C. (2003). Topics in Optimal Transportation. American Mathematical Society.
- Distribution of distances based object matching: Asymptotic inference. Journal of the American Statistical Association, pages 1–14.
- Semidefinite programming relaxations for the quadratic assignment problem. Journal of Combinatorial Optimization, 2:71–109.
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