Mean field models for large data–clustering problems

Michael Herty; Lorenzo Pareschi; Giuseppe Visconti

doi:10.3934/nhm.2020027

Article Contents

2020, Volume 15, Issue 3: 463-487. Doi: 10.3934/nhm.2020027

This issue Previous Article Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration Next Article Bounded confidence dynamics and graph control: Enforcing consensus

Mean field models for large data–clustering problems

1.
RWTH Aachen University, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52062 Aachen, Germany
2.
University of Ferrara, Mathematics and Computer Science Department, Via Machiavelli 35, 44121 Ferrara, Italy

^* Corresponding author: Lorenzo Pareschi
^* Corresponding author: Lorenzo Pareschi

Received: July 2019

Revised: March 2020

Early access: September 2020

Published: September 2020

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Abstract

We consider mean-field models for data–clustering problems starting from a generalization of the bounded confidence model for opinion dynamics. The microscopic model includes information on the position as well as on additional features of the particles in order to develop specific clustering effects. The corresponding mean–field limit is derived and properties of the model are investigated analytically. In particular, the mean–field formulation allows the use of a random subsets algorithm for efficient computations of the clusters. Applications to shape detection and image segmentation on standard test images are presented and discussed.

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Mathematics Subject Classification: Primary: 82C40, 94A08; Secondary: 68U10.

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Figure 1. Trend to the steady–state of the one–dimensional Hegselmann–Krause model (1) with $ n = 100 $ agents equally spaced at initial time and non–symmetric interactions (top row) and of the mean–field model (12) computed with Algorithm 1 (bottom row) up to final time $ T = 20 $. Left panels show the case for $ \epsilon_1 = 0.5 $, the right panels show the case for $ \epsilon_1 = 0.15 $

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Figure 2. Evolution in time of the first moment (left) and of the second moment (right) for the two values of the bounded confidence level $ \epsilon_1 = 0.5 $ (dashed lines) and $ \epsilon_1 = 0.15 $ (solid lines)

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Figure 3. Left: trend to the steady–state of the mean–field model (12) computed with Algorithm 1 with $ N = 2\times10^4 $, $ M = 2 $, $ \epsilon_1 = 0.15 $ and up to final time $ T = 20 $. Right: energy decay of the mean–field model (12) for several values of interacting particles $ M $

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Figure 4. Particle solution (left plots) with $ N = 10000 $ and kinetic density (right plots). Results are provided at time $ t = 4 $ (top row) and final time $ T = 50 $ (bottom row). The bounded confidence level is $ \epsilon_1 = 0.15 $

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Figure 5. Evolution in time of the two–dimensional first moments (left) and second moments (right) for the bounded confidence level $ \epsilon_1 = 0.15 $

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Figure 6. Top row: particles and kinetic density at initial time (left plot) and at equilibrium (right plot). Bottom row: at left, analysis of the distances between clusters in $ x $ (blue line with circle markers) and $ c $ direction (red line with triangle markers); at right, plot of the marginals. Confidence levels are $ \epsilon_1 = 0.15 $ and $ \epsilon_2 = 0.025 $

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Figure 7. Particle and kinetic density at equilibrium with confidence levels $ \epsilon_1 = 0.15 $, $ \epsilon_2 = 0.1 $ (left) and $ \epsilon_1 = 1 $, $ \epsilon_2 = 0.025 $ (right)

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Figure 8. Shape detection of the letter "A" initialized with $ 10\% $ of a uniform additive noise. Top left panel shows the initial condition. We show clusters obtained with bounded confidence values $ \epsilon_1 = 0.06 $ (top right), $ \epsilon_1 = 0.08 $ (bottom left) and $ \epsilon_1 = 0.1 $ (bottom right)

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Figure 9. Shape detection of the letter "A" initialized with $ 5.5\% $ of a Gaussian additive noise. Top left panel shows the initial condition. We show clusters obtained with bounded confidence values $ \epsilon_1 = 0.05 $ (top right), $ \epsilon_1 = 0.0675 $ (bottom left) and $ \epsilon_1 = 0.08 $ (bottom right)

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Figure 10. Left panel: initial image of $ 4096 $ pixels with four regions with different gray intensity. Middle panel: red dots show the positions of the clusters at equilibrium. Right panel: segmentation of the initial image in two regions

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Figure 11. Image segmentation of $ 174\times73 $ gray scale image taken by the data–set [6]

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Figure 12. Image segmentation of $ 93\times93 $ gray scale image taken by the data–set [24]

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Figure 13. Image segmentation of $ 128\times94 $ gray scale image taken by the data–set [32]

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Figure 14. Image segmentation of $ 67\times67 $ gray scale image taken by the data–set [32]

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Figure 15. Image segmentation of $ 132\times106 $ gray scale image taken by the data–set [32]

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Table 1. Number of clusters and errors depending on the bounded confidence value and the percentage of the noise when uniformly distributed

$ \alpha = 5\% $			$ \alpha = 7.5\% $			$ \alpha = 10\% $
$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $
0.03	1.25e-02	30	0.03	3.47e-02	51	0.06	2.64e-02	11
0.04	4.10e-03	16	0.05	1.21e-02	14	0.07	1.48e-02	8
0.05	4.00e-03	12	0.07	7.70e-03	8	0.08	1.12e-02	8
0.06	4.60e-03	9	0.09	7.90e-03	8	0.09	1.63e-02	5
0.07	5.40e-03	8	0.11	1.66e-02	3	0.10	1.63e-02	5

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Table 2. Number of clusters and errors depending on the bounded confidence value and the percentage of the noise when normally distributed

$ \alpha = 5\% $			$ \alpha = 5.5\% $			$ \alpha = 6\% $
$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $
0.05	4.44e-02	23	0.05	4.73e-02	24	0.05	6.37e-02	30
0.06	1.36e-02	11	0.06	2.62e-02	13	0.06	4.16e-02	16
0.065	6.40e-03	9	0.065	1.63e-02	11	0.07	2.12e-02	9
0.07	6.70e-03	7	0.0675	7.40e-03	10	0.075	9.70e-03	7
0.08	8.50e-03	7	0.07	8.00e-03	9	0.08	9.20e-03	7
0.09	1.00e-02	6	0.08	9.80e-03	7	0.085	1.10e-02	5

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