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September  2019, 1(3): 307-327. doi: 10.3934/fods.2019014

Power weighted shortest paths for clustering Euclidean data

Daniel Mckenzie 1,, and  Steven Damelin 2,

1. 

Department of Mathematics, University of California, Los Angeles, Los Angeles CA 90095, USA
2. 

Department of Mathematics, University of Michigan, Ann Arbor MI 48109, USA

* Corresponding author: mckenzie@math.ucla.edu

Published  September 2019

Fund Project: The first author gratefully acknowledges the support of the Department of Mathematics, The University of Georgia, where the first author was a graduate student while this work was completed. The second author thanks the Department of Mathematics, The University of Michigan for their support. Both authors thank the anonymous reviewer for many useful suggestions.

We study the use of power weighted shortest path metrics for clustering high dimensional Euclidean data, under the assumption that the data is drawn from a collection of disjoint low dimensional manifolds. We argue, theoretically and experimentally, that this leads to higher clustering accuracy. We also present a fast algorithm for computing these distances.

Keywords: Clustering, shortest path distance, manifold hypothesis, unsupervised learning.
Mathematics Subject Classification: Primary: 62H30, 58F15; Secondary: 05C85.
Citation: Daniel Mckenzie, Steven Damelin. Power weighted shortest paths for clustering Euclidean data. Foundations of Data Science, 2019, 1 (3) : 307-327. doi: 10.3934/fods.2019014
References:
[1]

M. Alamgir and U. Von Luxburg, Shortest path distance in random k-nearest neighbor graphs, arXiv preprint, arXiv: 1206.6381. Google Scholar

[2]

A. Aldroubi, K. Hamm, A. Koku and A. Sekmen, Cur decompositions, similarity matrices, and subspace clustering, Front. Appl. Math. Stat., 4 (2019), p65. doi: 10.3389/fams.2018.00065.  Google Scholar

[3]

E. Arias-Castro, Clustering based on pairwise distances when the data is of mixed dimensions, IEEE Transactions on Information Theory, 57 (2011), 1692-1706.  doi: 10.1109/TIT.2011.2104630.  Google Scholar

[4]

R. Basri and D. Jacobs, Lambertian reflectance and linear subspaces, IEEE Transactions on Pattern Analysis & Machine Intelligence, 2002,218–233. doi: 10.1109/ICCV.2001.937651.  Google Scholar

[5]

J. L. Bentley, Multidimensional binary search trees used for associative searching, Communications of the ACM, 18 (1975), 509-517.  doi: 10.1145/361002.361007.  Google Scholar

[6]

A. Beygelzimer, S. Kakade and J. Langford, Cover trees for nearest neighbor, in Proceedings of the 23rd International Conference on Machine Learning, ACM, 2006, 97–104. doi: 10.1145/1143844.1143857.  Google Scholar

[7]

A. Bijral, N. Ratliff and N. Srebro, Semi-supervised learning with density based distances, in Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence, AUAI Press, 2011, 43–50. Google Scholar

[8]

H. Chang and D.-Y. Yeung, Robust path-based spectral clustering, Pattern Recognition, 41 (2008), 191-203.  doi: 10.1016/j.patcog.2007.04.010.  Google Scholar

[9]

T. Chu, G. Miller and D. Sheehy, Exploration of a graph-based density sensitive metric, arXiv preprint, arXiv: 1709.07797. Google Scholar

[10]

R. Coifman and S. Lafon, Diffusion maps, Applied and Computational Harmonic Analysis, 21 (2006), 5-30.  doi: 10.1016/j.acha.2006.04.006.  Google Scholar

[11] T. Cormen, C. Leiserson, R. Rivest and C. Stein, Introduction to Algorithms, MIT press, 2009.   Google Scholar
[12]

J. Costeira and T. Kanade, A multibody factorization method for independently moving objects, International Journal of Computer Vision, 29 (1998), 159-179.   Google Scholar

[13]

K. Diaz-Chito, A. Hernández-Sabaté and A. López, A reduced feature set for driver head pose estimation, Applied Soft Computing, 45 (2016), 98-107.  doi: 10.1016/j.asoc.2016.04.027.  Google Scholar

[14]

D. Dua and C. Graff, UCImachine learning repository, 2017, http://archive.ics.uci.edu/ml. Google Scholar

[15]

C. Fefferman, S. Mitter and H. Narayanan, Testing the manifold hypothesis, Journal of the American Mathematical Society, 29 (2016), 983-1049.  doi: 10.1090/jams/852.  Google Scholar

[16]

B. Fischer and J. Buhmann, Path-based clustering for grouping of smooth curves and texture segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 (2003), 513-518.  doi: 10.1109/TPAMI.2003.1190577.  Google Scholar

[17]

S. Har-Peled, Computing the k nearest-neighbors for all vertices via dijkstra, arXiv preprint, arXiv: 1607.07818. Google Scholar

[18]

J. Ho, M.-H. Yang, J. Lim, K.-C. Lee and D. Kriegman, Clustering appearances of objects under varying illumination conditions, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003, 11–18. doi: 10.1109/CVPR.2003.1211332.  Google Scholar

[19]

C. D. Howard and C. Newman, Geodesics and spanning trees for euclidean first-passage percolation, Annals of Probability, 29 (2001), 577-623.  doi: 10.1214/aop/1008956685.  Google Scholar

[20]

S. Hwang, S. Damelin and A. Hero Ⅲ, Shortest path through random points, The Annals of Applied Probability, 26 (2016), 2791-2823.  doi: 10.1214/15-AAP1162.  Google Scholar

[21]

M. Jacobs, E. Merkurjev and S. Esedoḡlu, Auction dynamics: A volume constrained mbo scheme, Journal of Computational Physics, 354 (2018), 288-310.  doi: 10.1016/j.jcp.2017.10.036.  Google Scholar

[22]

A. Little, M. Maggioni and J. Murphy, Path-based spectral clustering: Guarantees, robustness to outliers, and fast algorithms, arXiv preprint, arXiv: 1712.06206. Google Scholar

[23]

A. Moscovich, A. Jaffe and B. Nadler, Minimax-optimal semi-supervised regression on unknown manifolds, in Artificial Intelligence and Statistics, 2017,933–942. Google Scholar

[24]

S. Nene, S. Nayar, H. Murase et al., Columbia object image library (coil-20). Google Scholar

[25]

A. Ng, M. Jordan and Y. Weiss, On spectral clustering: Analysis and an algorithm, in Advances in Neural Information Processing Systems, 2002,849–856. Google Scholar

[26]

A. Orlitsky and Sajama, Estimating and computing density based distance metrics, in Proceedings of the 22nd International Conference on Machine Learning, ACM, 2005,760–767. Google Scholar

[27]

J. Tenenbaum, V. De Silva and J. Langford, A global geometric framework for nonlinear dimensionality reduction, Science, 290 (2000), 2319-2323.  doi: 10.1126/science.290.5500.2319.  Google Scholar

[28]

P. Vincent and Y. Bengio, Density-sensitive Metrics and Kernels, Snowbird Learning Workshop, 2003. Google Scholar

[29]

K. Yin and X.-C. Tai, An effective region force for some variational models for learning and clustering, Journal of Scientific Computing, 74 (2018), 175-196.  doi: 10.1007/s10915-017-0429-4.  Google Scholar

[30]

L. Zelnik-Manor and P. Perona, Self-tuning spectral clustering, in Advances in Neural Information Processing Systems, 2005, 1601–1608. Google Scholar

show all references

References:
[1]

M. Alamgir and U. Von Luxburg, Shortest path distance in random k-nearest neighbor graphs, arXiv preprint, arXiv: 1206.6381. Google Scholar

[2]

A. Aldroubi, K. Hamm, A. Koku and A. Sekmen, Cur decompositions, similarity matrices, and subspace clustering, Front. Appl. Math. Stat., 4 (2019), p65. doi: 10.3389/fams.2018.00065.  Google Scholar

[3]

E. Arias-Castro, Clustering based on pairwise distances when the data is of mixed dimensions, IEEE Transactions on Information Theory, 57 (2011), 1692-1706.  doi: 10.1109/TIT.2011.2104630.  Google Scholar

[4]

R. Basri and D. Jacobs, Lambertian reflectance and linear subspaces, IEEE Transactions on Pattern Analysis & Machine Intelligence, 2002,218–233. doi: 10.1109/ICCV.2001.937651.  Google Scholar

[5]

J. L. Bentley, Multidimensional binary search trees used for associative searching, Communications of the ACM, 18 (1975), 509-517.  doi: 10.1145/361002.361007.  Google Scholar

[6]

A. Beygelzimer, S. Kakade and J. Langford, Cover trees for nearest neighbor, in Proceedings of the 23rd International Conference on Machine Learning, ACM, 2006, 97–104. doi: 10.1145/1143844.1143857.  Google Scholar

[7]

A. Bijral, N. Ratliff and N. Srebro, Semi-supervised learning with density based distances, in Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence, AUAI Press, 2011, 43–50. Google Scholar

[8]

H. Chang and D.-Y. Yeung, Robust path-based spectral clustering, Pattern Recognition, 41 (2008), 191-203.  doi: 10.1016/j.patcog.2007.04.010.  Google Scholar

[9]

T. Chu, G. Miller and D. Sheehy, Exploration of a graph-based density sensitive metric, arXiv preprint, arXiv: 1709.07797. Google Scholar

[10]

R. Coifman and S. Lafon, Diffusion maps, Applied and Computational Harmonic Analysis, 21 (2006), 5-30.  doi: 10.1016/j.acha.2006.04.006.  Google Scholar

[11] T. Cormen, C. Leiserson, R. Rivest and C. Stein, Introduction to Algorithms, MIT press, 2009.   Google Scholar
[12]

J. Costeira and T. Kanade, A multibody factorization method for independently moving objects, International Journal of Computer Vision, 29 (1998), 159-179.   Google Scholar

[13]

K. Diaz-Chito, A. Hernández-Sabaté and A. López, A reduced feature set for driver head pose estimation, Applied Soft Computing, 45 (2016), 98-107.  doi: 10.1016/j.asoc.2016.04.027.  Google Scholar

[14]

D. Dua and C. Graff, UCImachine learning repository, 2017, http://archive.ics.uci.edu/ml. Google Scholar

[15]

C. Fefferman, S. Mitter and H. Narayanan, Testing the manifold hypothesis, Journal of the American Mathematical Society, 29 (2016), 983-1049.  doi: 10.1090/jams/852.  Google Scholar

[16]

B. Fischer and J. Buhmann, Path-based clustering for grouping of smooth curves and texture segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 (2003), 513-518.  doi: 10.1109/TPAMI.2003.1190577.  Google Scholar

[17]

S. Har-Peled, Computing the k nearest-neighbors for all vertices via dijkstra, arXiv preprint, arXiv: 1607.07818. Google Scholar

[18]

J. Ho, M.-H. Yang, J. Lim, K.-C. Lee and D. Kriegman, Clustering appearances of objects under varying illumination conditions, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003, 11–18. doi: 10.1109/CVPR.2003.1211332.  Google Scholar

[19]

C. D. Howard and C. Newman, Geodesics and spanning trees for euclidean first-passage percolation, Annals of Probability, 29 (2001), 577-623.  doi: 10.1214/aop/1008956685.  Google Scholar

[20]

S. Hwang, S. Damelin and A. Hero Ⅲ, Shortest path through random points, The Annals of Applied Probability, 26 (2016), 2791-2823.  doi: 10.1214/15-AAP1162.  Google Scholar

[21]

M. Jacobs, E. Merkurjev and S. Esedoḡlu, Auction dynamics: A volume constrained mbo scheme, Journal of Computational Physics, 354 (2018), 288-310.  doi: 10.1016/j.jcp.2017.10.036.  Google Scholar

[22]

A. Little, M. Maggioni and J. Murphy, Path-based spectral clustering: Guarantees, robustness to outliers, and fast algorithms, arXiv preprint, arXiv: 1712.06206. Google Scholar

[23]

A. Moscovich, A. Jaffe and B. Nadler, Minimax-optimal semi-supervised regression on unknown manifolds, in Artificial Intelligence and Statistics, 2017,933–942. Google Scholar

[24]

S. Nene, S. Nayar, H. Murase et al., Columbia object image library (coil-20). Google Scholar

[25]

A. Ng, M. Jordan and Y. Weiss, On spectral clustering: Analysis and an algorithm, in Advances in Neural Information Processing Systems, 2002,849–856. Google Scholar

[26]

A. Orlitsky and Sajama, Estimating and computing density based distance metrics, in Proceedings of the 22nd International Conference on Machine Learning, ACM, 2005,760–767. Google Scholar

[27]

J. Tenenbaum, V. De Silva and J. Langford, A global geometric framework for nonlinear dimensionality reduction, Science, 290 (2000), 2319-2323.  doi: 10.1126/science.290.5500.2319.  Google Scholar

[28]

P. Vincent and Y. Bengio, Density-sensitive Metrics and Kernels, Snowbird Learning Workshop, 2003. Google Scholar

[29]

K. Yin and X.-C. Tai, An effective region force for some variational models for learning and clustering, Journal of Scientific Computing, 74 (2018), 175-196.  doi: 10.1007/s10915-017-0429-4.  Google Scholar

[30]

L. Zelnik-Manor and P. Perona, Self-tuning spectral clustering, in Advances in Neural Information Processing Systems, 2005, 1601–1608. Google Scholar

Figure 1.  Three sample geodesics in the power weighted shortest path metric with $ p=2 $, for the data set "Three Lines" (see §6). Observe how the geodesics consist of many small hops, instead of several large hops. The total lengths of the red and green paths are significantly smaller than the length of the blue path
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Figure 2.  All three synthetic data sets, projected into $ \mathbb{R}^{2} $. From left to right: Three Lines, Three Moons and Three Circles
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Figure 3.  Varying $ p $ and recording the accuracy of spectral clustering on the Three Lines data set, for three different values of the ambient dimension
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Table 1.  Classification accuracy of spectral clustering. Note that $ A^{(1)} $ represents using the Euclidean metric
$ A^{(f, 1)} $ $ A^{(1)} $ $ A^{(2)} $ $ A^{(10)} $ $ A^{(\infty)} $
3 Lines $ 66.11\pm 0.94\% $ $ 66.35 \pm 3.73\% $ $ 66.87 \pm 3.37\% $ $ 95.38\pm 9.22\% $ $ \bf{95.38 \pm 9.1\%} $
3 Moons $ 85.90 \pm 1.13\% $ $ 94.40 \pm 1.48\% $ $ 94.40 \pm 1.48\% $ $ \bf{96.20 \pm 1.76\%} $ $ 94.35 \pm 3.34\% $
3 Circles $ 51.87 \pm 0.00\% $ $ 51.93 \pm 0.32\% $ $ 51.94 \pm 0.36\% $ $ 71.22 \pm 9.50\% $ $ \bf{73.61 \pm 10.47\%} $
$\mathtt{ DrivFace} $ $ 78.88\% $ $ 71.62\% $ $ 71.62\% $ $ 74.71\% $ $ \bf{85.38\%} $
$ \mathtt{ COIL-20}$ $ 63.24\% $ $ 75.28\% $ $ \bf{78.61\%} $ $ 77.45\% $ $ 60.92\% $
$ \mathtt{ OptDigits} $ $ 77.73\% $ $ 91.49\% $ $ \bf{91.54\%} $ $ 88.39\% $ $ 83.17\% $
$ \mathtt{ USPS} $ $ 48.65\% $ $ 65.05\% $ $ 65.02\% $ $ 76.20\% $ $ \bf{77.92\%} $
$\mathtt{ MNIST} $ - $ 76.11\% $ $ 75.63\% $ $ 84.54\% $ $ \bf{86.77\%} $
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$ A^{(f, 1)} $ $ A^{(1)} $ $ A^{(2)} $ $ A^{(10)} $ $ A^{(\infty)} $
3 Lines $ 66.11\pm 0.94\% $ $ 66.35 \pm 3.73\% $ $ 66.87 \pm 3.37\% $ $ 95.38\pm 9.22\% $ $ \bf{95.38 \pm 9.1\%} $
3 Moons $ 85.90 \pm 1.13\% $ $ 94.40 \pm 1.48\% $ $ 94.40 \pm 1.48\% $ $ \bf{96.20 \pm 1.76\%} $ $ 94.35 \pm 3.34\% $
3 Circles $ 51.87 \pm 0.00\% $ $ 51.93 \pm 0.32\% $ $ 51.94 \pm 0.36\% $ $ 71.22 \pm 9.50\% $ $ \bf{73.61 \pm 10.47\%} $
$\mathtt{ DrivFace} $ $ 78.88\% $ $ 71.62\% $ $ 71.62\% $ $ 74.71\% $ $ \bf{85.38\%} $
$ \mathtt{ COIL-20}$ $ 63.24\% $ $ 75.28\% $ $ \bf{78.61\%} $ $ 77.45\% $ $ 60.92\% $
$ \mathtt{ OptDigits} $ $ 77.73\% $ $ 91.49\% $ $ \bf{91.54\%} $ $ 88.39\% $ $ 83.17\% $
$ \mathtt{ USPS} $ $ 48.65\% $ $ 65.05\% $ $ 65.02\% $ $ 76.20\% $ $ \bf{77.92\%} $
$\mathtt{ MNIST} $ - $ 76.11\% $ $ 75.63\% $ $ 84.54\% $ $ \bf{86.77\%} $
Table 2.  Run time of spectral clustering, in seconds. Note that this includes the time rquired to construct the similarity matrix. $ A^{(1)} $ represents using the Euclidean metric
$ A^{(f, 1)} $ $ A^{(1)} $ $ A^{(2)} $ $ A^{(10)} $ $ A^{(\infty)} $
3 Lines $ 0.32 $ $ 0.16 $ $ 1.20 $ $ 1.22 $ $ 1.22 $
3 Moons $ 0.33 $ $ 0.17 $ $ 1.31 $ $ 1.30 $ $ 1.36 $
3 Circles $ 0.35 $ $ 0.16 $ $ 1.00 $ $ 1.06 $ $ 1.07 $
$\mathtt{ DrivFace} $ $ 0.37 $ $ 1.24 $ $ 1.55 $ $ 1.64 $ $ 1.64 $
$ \mathtt{ COIL-20}$ $ 0.57 $ $ 0.72 $ $ 1.57 $ $ 1.82 $ $ 1.78 $
$\mathtt{ OptDigits} $ $ 5.40 $ $ 1.41 $ $ 5.28 $ $ 5.58 $ $ 5.67 $
$\mathtt{ USPS} $ $ 27.40 $ $ 17.12 $ $ 26.75 $ $ 22.78 $ $ 23.79 $
$ \mathtt{ MNIST}$ - $ 2060.23 $ $ 2031.38 $ $ 1554.15 $ $ 1613.41 $
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$ A^{(f, 1)} $ $ A^{(1)} $ $ A^{(2)} $ $ A^{(10)} $ $ A^{(\infty)} $
3 Lines $ 0.32 $ $ 0.16 $ $ 1.20 $ $ 1.22 $ $ 1.22 $
3 Moons $ 0.33 $ $ 0.17 $ $ 1.31 $ $ 1.30 $ $ 1.36 $
3 Circles $ 0.35 $ $ 0.16 $ $ 1.00 $ $ 1.06 $ $ 1.07 $
$\mathtt{ DrivFace} $ $ 0.37 $ $ 1.24 $ $ 1.55 $ $ 1.64 $ $ 1.64 $
$ \mathtt{ COIL-20}$ $ 0.57 $ $ 0.72 $ $ 1.57 $ $ 1.82 $ $ 1.78 $
$\mathtt{ OptDigits} $ $ 5.40 $ $ 1.41 $ $ 5.28 $ $ 5.58 $ $ 5.67 $
$\mathtt{ USPS} $ $ 27.40 $ $ 17.12 $ $ 26.75 $ $ 22.78 $ $ 23.79 $
$ \mathtt{ MNIST}$ - $ 2060.23 $ $ 2031.38 $ $ 1554.15 $ $ 1613.41 $
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