# American Institute of Mathematical Sciences

March  2021, 3(1): 49-66. doi: 10.3934/fods.2021005

## A topological approach to spectral clustering

 Centro de Investigación en Matemáticas, Calle Jalisco S/N, Colonia Valenciana, Guanajuato C.P 36023, GTO, Mexico

* Corresponding author: Antonio Rieser

Received  July 2020 Revised  January 2021 Published  March 2021 Early access  March 2021

Fund Project: This research was supported in part by the TOPOSYS project FP7-ICT-318493-STREP, Cátedras CONACYT 1076, and Grant #N62909-19-1-2134 from the US Office of Naval Research Global and the Southern Office of Aerospace Research and Development of the US Air Force Office of Scientific Research

We propose two related unsupervised clustering algorithms which, for input, take data assumed to be sampled from a uniform distribution supported on a metric space $X$, and output a clustering of the data based on the selection of a topological model for the connected components of $X$. Both algorithms work by selecting a graph on the samples from a natural one-parameter family of graphs, using a geometric criterion in the first case and an information theoretic criterion in the second. The estimated connected components of $X$ are identified with the kernel of the associated graph Laplacian, which allows the algorithm to work without requiring the number of expected clusters or other auxiliary data as input.

Citation: Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021, 3 (1) : 49-66. doi: 10.3934/fods.2021005
##### References:
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##### References:
 [1] M. Belkin and P. Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation, Neural Comput., 15 (2003), 1373-1396.  doi: 10.1162/089976603321780317. [2] G. Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 255-308.  doi: 10.1090/S0273-0979-09-01249-X. [3] F. Chazal, L. J. Guibas, S. Y. Oudot and P. Skraba, Persistence-based clustering in Riemannian manifolds, J. ACM, 60 (2013), 38pp. doi: 10.1145/2535927. [4] F. R. K. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/cbms/092. [5] R. R. Coifman and S. Lafon, Diffusion maps, Appl. Comput. Harmon. Anal., 21 (2006), 5-30.  doi: 10.1016/j.acha.2006.04.006. [6] D. L. Donoho and C. Grimes, Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data, Proc. Natl. Acad. Sci. USA, 100 (2003), 5591-5596.  doi: 10.1073/pnas.1031596100. [7] R. B. Ghrist, Barcodes: The persistent topology of data, Bull. Amer. Math. Soc. (N.S.), 45 (2008), 61-75.  doi: 10.1090/S0273-0979-07-01191-3. [8] T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning. Data Mining, Inference, and Prediction, Springer Series in Statistics, Springer, New York, 2009 doi: 10.1007/978-0-387-84858-7. [9] J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-21298-7. [10] S. T. Roweis and L. K. Saul, Nonlinear dimensionality reduction by locally linear embedding, Science, 290 (2000), 2323-2326.  doi: 10.1126/science.290.5500.2323. [11] R. Sánchez-Garcia, M. Fennelly, S. Norris, N. Wright, G. Niblo, J. Brodzki and J. Bialek, Hierarchical spectral clustering of power grids, IEEE Transactions on Power Systems, 29 (2014), 2229-2237.  doi: 10.1109/TPWRS.2014.2306756. [12] W.-J. Shen, H.-S. Wong, Q.-W. Xiao, X. Guo and S. Smale, Introduction to the peptide binding problem of computational immunology: New results, Found. Comput. Math., 14 (2014), 951-984.  doi: 10.1007/s10208-013-9173-9. [13] J. B. Tenenbaum, V. de Silva and J. C. Langford, A global geometric framework for nonlinear dimensionality reduction, Science, 290 (2000), 2319-2323.  doi: 10.1126/science.290.5500.2319. [14] A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete Comput. Geom., 33 (2005), 249-274.  doi: 10.1007/s00454-004-1146-y.
An accurate classification of $500$ points using the Average Local Volume Ratio method, illustrated by color. The classification produced by the Average Relative Entropy method was identical in this experiment
Average Relative Entropy vs. Scale, for the experiment in Figure 6.1. Note that local maxima appear where the smaller circles join to a larger cluster
Average Local Volume Ratio vs. Scale for the experiment in Figure 6.1. Note that local minima appear where the smaller circles join with the larger circle
Example classification of 500 points using the Average Local Volume Ratio method, illustrated by color, for an experiment where the algorithm reported 4 clusters. One can see that the vast majority of points are correctly classified, although a small group of points in the circle on the left are assigned to their own cluster
The classification of 500 points using the Average Local Volume Ratio method, illustrated by color, for an experiment where the algorithm reported 5 clusters. One can see that the majority of points are correctly classified, even though the horizontal circle is split into three groups
Example classification of 1000 points using the Average Local Volume Ratio method, illustrated by color, for an experiment where the algorithm reported 4 clusters. One can see that the vast majority of points are correctly classified, although one outlier is assigned to its own cluster
Example classification of 1000 points using the Average Local Volume Ratio method, illustrated by color, for an experiment where the algorithm reported 5 clusters. One can see that the vast majority of points are correctly classified, and there are two outliers visible in different shades of light blue, one partially obscured by the circle on the left
Example classification of 1000 points using the Average Local Volume Ratio method, illustrated by color, for an experiment where the algorithm reported 6 clusters. As in Figures A.3 and A.4, the vast majority of points were classified correctly, and the additional clusters are from (partially obstructed) outliers
The classification of 500 points using the Average Relative Entropy method, illustrated by color, for an experiment where the algorithm reported 4 clusters. One can see that the vast majority of points are correctly classified, although there are three outliers at least partially visible: the darker green point on the left circle, a light violet point in the center circle, and a dark violet point which is mostly obscured by the center cicle
The classification of 500 points using the Average Relative Entropy method, illustrated by color, for an experiment where the algorithm reported 5 clusters. The majority of points are correctly classified, although there are two extra groups assigned to the slightly separated groups of points in the circle on the left
The classification of 500 points using the Average Relative Entropy method, illustrated by color, for an experiment where the algorithm reported 6 clusters. As in Figure A.6 and A.6, the majority of points were classified correctly, although the circle on the right is split into top and bottom parts, there is an outlier (in blue) on the horizontal circle, as well a small group of separated points (in indigo) on the horzontal circle
Example classification of 1000 points using the Average Relative Entropy method, illustrated by color, for an experiment where the algorithm reported 4 clusters. One can see that the vast majority of points are correctly classified, although there is an outlier on the right circle given its own cluster
Example classification of 1000 points using the Average Relative Entropy Method, illustrated by color, for an experiment where the algorithm reported 5 clusters. One can see that the vast majority of points are correctly classified, although the center circle is split into 2 subgroups, and there is an outlier which is given its own cluster
Example classification of 1000 points using the Average Relative Entropy method, illustrated by color, for an experiment where the algorithm reported 6 clusters. One can see that the majority of points were classified correctly, although the left circle is split into two subgroups, and there are two outliers given their own cluster
Results for the Average Relative Entropy method, using $500$ and $1000$ sample points. This table shows the percentages, rounded to the nearest hundredth, of the $150$ trials using the standard deviation in the left-hand column in which the ARE algorithm returned a given number $\beta_0$ of clusters
 Average Relative Entropy Method, 500 sample points $SD\mid\beta_0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $\geq 10$ $0.01$ $0$ $0$ $64$ $18.67$ $10.67$ $5.33$ $0.67$ $0.67$ $0$ $0$ $0.02$ $0$ $0$ $51.33$ $27.33$ $12.67$ $4$ $2$ $2$ $0.67$ $0$ $0.03$ $0$ $0$ $54.67$ $26.67$ $10.67$ $4$ $3.33$ $0.67$ $0$ $0$ $0.04$ $0.67$ $2$ $44$ $23.33$ $13.33$ $5.33$ $3.33$ $2.67$ $2$ $3.33$ $0.05$ $6$ $26$ $18$ $20$ $8.67$ $4.67$ $4.67$ $4.67$ $2$ $5.33$ Average Relative Entropy Method, 1000 sample SD|β0 1 2 3 4 5 6 7 8 9 ≥ 10 0.01 0 0 62.67 26 7.33 0.67 2 1.33 0 0 0.02 0 0 59.33 22 8 9.33 0.67 0.67 0 0 0.03 0 0 23.33 28 22 10.67 4.67 1.33 0 10 0.04 0 0 4 9.33 12 2 5.33 4 5.33 58.01 0.05 0 0 1.33 5.33 2.67 4.67 2 5.33 2.67 76
 Average Relative Entropy Method, 500 sample points $SD\mid\beta_0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $\geq 10$ $0.01$ $0$ $0$ $64$ $18.67$ $10.67$ $5.33$ $0.67$ $0.67$ $0$ $0$ $0.02$ $0$ $0$ $51.33$ $27.33$ $12.67$ $4$ $2$ $2$ $0.67$ $0$ $0.03$ $0$ $0$ $54.67$ $26.67$ $10.67$ $4$ $3.33$ $0.67$ $0$ $0$ $0.04$ $0.67$ $2$ $44$ $23.33$ $13.33$ $5.33$ $3.33$ $2.67$ $2$ $3.33$ $0.05$ $6$ $26$ $18$ $20$ $8.67$ $4.67$ $4.67$ $4.67$ $2$ $5.33$ Average Relative Entropy Method, 1000 sample SD|β0 1 2 3 4 5 6 7 8 9 ≥ 10 0.01 0 0 62.67 26 7.33 0.67 2 1.33 0 0 0.02 0 0 59.33 22 8 9.33 0.67 0.67 0 0 0.03 0 0 23.33 28 22 10.67 4.67 1.33 0 10 0.04 0 0 4 9.33 12 2 5.33 4 5.33 58.01 0.05 0 0 1.33 5.33 2.67 4.67 2 5.33 2.67 76
Results for the Average Local Volume Ratio Method, using $500$ and $1000$ sample points.This table shows the percentages, rounded to the nearest hundredth, of the $150$ trials using the standard deviation in the left-hand column in which the ALVR algorithm returned a given number $\beta_0$ of clusters
 Average Local Volume Ratio Method, 500 Sample Points $SD\mid\beta_0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $\geq 10$ $0.01$ $0$ $0$ $96$ $3.33$ $0.67$ $0$ $0$ $0$ $0$ $0$ $0.02$ $18$ $0$ $76$ $6$ $0$ $0$ $0$ $0$ $0$ $0$ $0.03$ $65.33$ $0$ $32$ $2.67$ $0$ $0$ $0$ $0$ $0$ $0$ $0.04$ $95.33$ $1.33$ $2.67$ $0.67$ $0$ $0$ $0$ $0$ $0$ $0$ $0.05$ $91.33$ $7.33$ $1.33$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ Average Local Volume Ratio Method, 1000 Sample Points SD|β0 1 2 3 4 5 6 7 8 9 ≥ 10 0.01 0 0 98.67 1.33 0 0 0 0 0 0 0.02 0 0 89.33 8 1.33 1.33 0 0 0 0 0.03 0 0 45.33 28 16 6.67 2.67 1.33 0 0 0.04 1.33 0 13.33 21.33 26 11.33 12 6 3.33 0 0.05 75.33 11.33 2.67 4.67 1.33 2 0.67 1.33 0.67 5.38
 Average Local Volume Ratio Method, 500 Sample Points $SD\mid\beta_0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $\geq 10$ $0.01$ $0$ $0$ $96$ $3.33$ $0.67$ $0$ $0$ $0$ $0$ $0$ $0.02$ $18$ $0$ $76$ $6$ $0$ $0$ $0$ $0$ $0$ $0$ $0.03$ $65.33$ $0$ $32$ $2.67$ $0$ $0$ $0$ $0$ $0$ $0$ $0.04$ $95.33$ $1.33$ $2.67$ $0.67$ $0$ $0$ $0$ $0$ $0$ $0$ $0.05$ $91.33$ $7.33$ $1.33$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ Average Local Volume Ratio Method, 1000 Sample Points SD|β0 1 2 3 4 5 6 7 8 9 ≥ 10 0.01 0 0 98.67 1.33 0 0 0 0 0 0 0.02 0 0 89.33 8 1.33 1.33 0 0 0 0 0.03 0 0 45.33 28 16 6.67 2.67 1.33 0 0 0.04 1.33 0 13.33 21.33 26 11.33 12 6 3.33 0 0.05 75.33 11.33 2.67 4.67 1.33 2 0.67 1.33 0.67 5.38
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