A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach

Christophe Fotso; Duplex Elvis Houpa Danga; Daniel Tieudjo

doi:doi:10.11648/j.engmath.20250901.11

Research Article |

| Peer-Reviewed

A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach

Christophe Fotso^*

, Duplex Elvis Houpa Danga

, Daniel Tieudjo

Published in Engineering Mathematics (Volume 9, Issue 1)

Received: 27 March 2025 Accepted: 8 April 2025 Published: 6 May 2025

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Integrating Mean-sets theory employing generalized graph or group-theoretic tools and techniques into adaptive graph filtering can lead to more effective resilient filtering processes, particularly in challenging environments with clutter or uncertainty. In this paper, we show that under some crucial smoothing assumptions, the generalized Mean-sets theory developped for negatively curved convex combination Polish metric spaces following the formal Means-sets probability theory’s approach from Natalia Mosina, provides a new useful system for some secure adaptive graph filtering processes. We use convex combination operations (in the sense of Terán and Molchanov) on both individual input graph signals and filters. Individual adaptive graph filters being independently adapted by space (dataset)-valued random variables, while the convexification operator on the underlying dataset acts as a flexible theoretical instrument for preserving some good features of the standard scheme, like privacy of their informative trends, and looks more robust to changes. We exhibit a graph matrix model from a system of convex combination of two adaptive finite impulse response (FIR) graph filters processing a sampled-weighted mean-set (expectation) of some transversal graph signals with finite length N ≥ 2.

Published in	Engineering Mathematics (Volume 9, Issue 1)
DOI	10.11648/j.engmath.20250901.11
Page(s)	1-15
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Graph Signal, Convex Combination, Mean-set, Convexification Operator, Secure Adaptive Filtering

References

[1]	A. Ahidar-Coutrix, T. Le Gouic and Q. Paris. (2019). On the rate of convergence of empirical barycentres in metric spaces: curvature, convexity and extendible geodesics. HAL Id: hal-01810530. Available at https://hal.archives- ouvertes.fr/hal-01810530v3
[2]	J. Arenas-García, A. R. Figueiras-Vidal and A. H. Sayed. (2005). Tracking properties of a convex combination of two adaptive filters, IEEE/SP 13th Workshop on Statistical Signal Processing, 109–114.
[3]	J. Arenas-García, A. R. Figueiras-Vidal and A. H. Sayed. (2006). Mean-square performance of a convex combination of two adaptive filters. IEEE transactions on signal processing, 54(3), 1078–1090.
[4]	A. Bator and W. Ziéba. (2009). On some definition of expectation of random element in metric space. Annales Universitatis Mariae Curie-Sklodowska, 63(1), 39.
[5]	C.-S. Chang, W. Liao, Y.-S. Chen and L.-H. Liou. (2016). A mathematical theory for clustering in metric spaces. IEEE Transactions on Network Science and Engineering, 3(1), 2–16.
[6]	Douglas S. C. (1999). Introduction to Adaptive Filters in Digital Signal Processing Handbook by Ed. Vijay K. Madisetti and Douglas B. Williams, CRC Press LLC.
[7]	C. Fotso, E. D. Houpa Danga and D. Tieudjo. (2024). An Approach of Mean-sets Theory for Negatively Curved Convex Combination Polish (NCCCP) Metric spaces, Advances in Mathematics: Scientific Journal, 13(02), 145-180.
[8]	M. Fréchet. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l’institut Henri Poincaré, 10(4), 215–310.
[9]	C. E. Ginestet, A. Simmons and E. D. Kolaczyk. (2012). Weighted Fréchet means as convex combinations in metric spaces: properties and generalized median inequalities. Statistics and Probability Letters, 82(10), 1859–1863.
[10]	W. Herer. (1992). Mathematical expectation and strong law of large numbers for random variables with values in a metric space of negative curvature. Proba. Math. Statist, 13(1), 59–70.
[11]	E. Isufi, F. Gama, D. I. Shuman and S. Segarra. (2024). Graph Filters for Signal Processing and Machine Learning on Graphs. IEEE Transactions on Signal Processing.
[12]	D. Jin, J. Chen, and J. Chen. (2018). Convex combination of diffusion strategies over distributed networks. Proc. Asia-Pacific Signal Inf. Process. Association, 224-228.
[13]	M. Junaid. (2022). Signal Processing on Graphs: a state of art. HAL Id: hal-03813429.
[14]	J. Lember. (2003). On minimizing sequences for k- centres. Journal of Approximation Theory, 120(1), 20– 35.
[15]	S. Memon, T. L. Song and T. H. Kim. (2016). Smoothing data association for target trajectory estimation in cluttered environments. Eurasip Journal on Advances in Signal Processing, 2016(21), 1–13.
[16]	N. Mosina. (2009). Probability on graphs and groups: theory and applications, Ph D. Thesis, Columbia University. Available at https://www.math.columbia.edu/ thaddeus/theses/2009/mosina.pdf
[17]	N. Mosina and A. Ushakov. (2010). Mean-set attack: cryptanalysis of Sibert et al. authentication protocol. Journal of Mathematical Cryptology, 4(2), 149–174.
[18]	N. Mosina and A. Ushakov. (2011). Strong law of large numbers on graphs and groups. Groups Complexity Cryptology, 3(1), 67–103.
[19]	A. Ortega, P. Frossard, J. Kovačević, J. M. F. Moura and P. Vandergheynst. (2018). Graph Signal Processing: Overview, Challenges, and Applications. Proceedings of the IEEE — Vol. 106(5), 808-828.
[20]	D. Pollard. (1981). Strong consistency of k-means clustering. The Annals of Statistics, 135–140.
[21]	D. Pollard. (1982). Quantization and the method of k- means. IEEE Transactions on Information theory, 28(2), 199–205.
[22]	Y. S. Soh. (2019). Fitting convex sets to data: Algorithms and Applications. Ph.D Thesis, California Institute of Technology, 193 pages.
[23]	K.-T. Sturm. (2003). Probability measures on metric spaces of nonpositive curvature, Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces. Lecture Notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs: April 16-July 13, 2002, Emile Borel Centre of the Henri Poincaré Institute, 338-357.
[24]	P. Terán and I. Molchanov. (2006). The law of large numbers in a metric space with a convex combination operation. Journal of theoretical probability, 19(4), 875– 898.
[25]	M.Thorpe, F.Theil, A.M.JohansenandN.Cade. (2015). Convergence of the k-means minimization problem using Γ-convergence. SIAM Journal on Applied Mathematics, 75(6), 2444–2474.
[26]	N. Tran Thuan. (2015). An approach for metric space with a convex combination operation and applications. arXiv e-prints, arXiv–1501.
[27]	J. R. Troncoso-Pastoriza and F. Pérez-González. (2011). Secure adaptive filtering. IEEE Transactions on Information Forensics and Security, 6(2), 469-485.
[28]	J. Wodecki, A. Michalak and P. Stefaniak. (2018). Review of smoothing methods for enhancement of noisy data from heavy-duty LHD mining machines. E3S Web of Conferences, 29(00011).
[29]	W. Yamaka, R. Phadkantha and P. Maneejuk. (2021). A Convex Combination Approach for Artificial Neural Network of Interval Data. Appl. Sci. 11, 3997.

Cite This Article

Plain Text BibTeX RIS

APA Style

Fotso, C., Danga, D. E. H., Tieudjo, D. (2025). A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach. Engineering Mathematics, 9(1), 1-15. https://doi.org/10.11648/j.engmath.20250901.11

Copy | Download

ACS Style

Fotso, C.; Danga, D. E. H.; Tieudjo, D. A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach. Eng. Math. 2025, 9(1), 1-15. doi: 10.11648/j.engmath.20250901.11

Copy | Download

AMA Style

Fotso C, Danga DEH, Tieudjo D. A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach. Eng Math. 2025;9(1):1-15. doi: 10.11648/j.engmath.20250901.11

Copy | Download

@article{10.11648/j.engmath.20250901.11,
  author = {Christophe Fotso and Duplex Elvis Houpa Danga and Daniel Tieudjo},
  title = {A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach
},
  journal = {Engineering Mathematics},
  volume = {9},
  number = {1},
  pages = {1-15},
  doi = {10.11648/j.engmath.20250901.11},
  url = {https://doi.org/10.11648/j.engmath.20250901.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20250901.11},
  abstract = {Integrating Mean-sets theory employing generalized graph or group-theoretic tools and techniques into adaptive graph filtering can lead to more effective resilient filtering processes, particularly in challenging environments with clutter or uncertainty. In this paper, we show that under some crucial smoothing assumptions, the generalized Mean-sets theory developped for negatively curved convex combination Polish metric spaces following the formal Means-sets probability theory’s approach from Natalia Mosina, provides a new useful system for some secure adaptive graph filtering processes. We use convex combination operations (in the sense of Terán and Molchanov) on both individual input graph signals and filters. Individual adaptive graph filters being independently adapted by space (dataset)-valued random variables, while the convexification operator on the underlying dataset acts as a flexible theoretical instrument for preserving some good features of the standard scheme, like privacy of their informative trends, and looks more robust to changes. We exhibit a graph matrix model from a system of convex combination of two adaptive finite impulse response (FIR) graph filters processing a sampled-weighted mean-set (expectation) of some transversal graph signals with finite length N ≥ 2.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach

AU  - Christophe Fotso
AU  - Duplex Elvis Houpa Danga
AU  - Daniel Tieudjo
Y1  - 2025/05/06
PY  - 2025
N1  - https://doi.org/10.11648/j.engmath.20250901.11
DO  - 10.11648/j.engmath.20250901.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 1
EP  - 15
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20250901.11
AB  - Integrating Mean-sets theory employing generalized graph or group-theoretic tools and techniques into adaptive graph filtering can lead to more effective resilient filtering processes, particularly in challenging environments with clutter or uncertainty. In this paper, we show that under some crucial smoothing assumptions, the generalized Mean-sets theory developped for negatively curved convex combination Polish metric spaces following the formal Means-sets probability theory’s approach from Natalia Mosina, provides a new useful system for some secure adaptive graph filtering processes. We use convex combination operations (in the sense of Terán and Molchanov) on both individual input graph signals and filters. Individual adaptive graph filters being independently adapted by space (dataset)-valued random variables, while the convexification operator on the underlying dataset acts as a flexible theoretical instrument for preserving some good features of the standard scheme, like privacy of their informative trends, and looks more robust to changes. We exhibit a graph matrix model from a system of convex combination of two adaptive finite impulse response (FIR) graph filters processing a sampled-weighted mean-set (expectation) of some transversal graph signals with finite length N ≥ 2.

VL  - 9
IS  - 1
ER  -

Copy | Download

Author Information

Christophe Fotso

Department of Mathematics and Computer Science, The University of Ngaoundere, Ngaoundere, Cameroon

Contact Email

http://orcid.org/0000-0002-0383-4751
Duplex Elvis Houpa Danga

Department of Mathematics and Computer Science, The University of Ngaoundere, Ngaoundere, Cameroon

Contact Email

http://orcid.org/0000-0003-4779-5800
Daniel Tieudjo

Department of Mathematics and Computer Science, National School of Agro-Industrial Science, Ngaoundere, Cameroon

Contact Email

http://orcid.org/0009-0008-2647-7428

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Fotso, C., Danga, D. E. H., Tieudjo, D. (2025). A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach. Engineering Mathematics, 9(1), 1-15. https://doi.org/10.11648/j.engmath.20250901.11

Copy | Download

ACS Style

Fotso, C.; Danga, D. E. H.; Tieudjo, D. A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach. Eng. Math. 2025, 9(1), 1-15. doi: 10.11648/j.engmath.20250901.11

Copy | Download

AMA Style

Fotso C, Danga DEH, Tieudjo D. A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach. Eng Math. 2025;9(1):1-15. doi: 10.11648/j.engmath.20250901.11

Copy | Download

@article{10.11648/j.engmath.20250901.11,
  author = {Christophe Fotso and Duplex Elvis Houpa Danga and Daniel Tieudjo},
  title = {A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach
},
  journal = {Engineering Mathematics},
  volume = {9},
  number = {1},
  pages = {1-15},
  doi = {10.11648/j.engmath.20250901.11},
  url = {https://doi.org/10.11648/j.engmath.20250901.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20250901.11},
  abstract = {Integrating Mean-sets theory employing generalized graph or group-theoretic tools and techniques into adaptive graph filtering can lead to more effective resilient filtering processes, particularly in challenging environments with clutter or uncertainty. In this paper, we show that under some crucial smoothing assumptions, the generalized Mean-sets theory developped for negatively curved convex combination Polish metric spaces following the formal Means-sets probability theory’s approach from Natalia Mosina, provides a new useful system for some secure adaptive graph filtering processes. We use convex combination operations (in the sense of Terán and Molchanov) on both individual input graph signals and filters. Individual adaptive graph filters being independently adapted by space (dataset)-valued random variables, while the convexification operator on the underlying dataset acts as a flexible theoretical instrument for preserving some good features of the standard scheme, like privacy of their informative trends, and looks more robust to changes. We exhibit a graph matrix model from a system of convex combination of two adaptive finite impulse response (FIR) graph filters processing a sampled-weighted mean-set (expectation) of some transversal graph signals with finite length N ≥ 2.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A Matrix Model for Adaptive Graph Filtering Using a Generalized Mean-sets Theory’s Approach

AU  - Christophe Fotso
AU  - Duplex Elvis Houpa Danga
AU  - Daniel Tieudjo
Y1  - 2025/05/06
PY  - 2025
N1  - https://doi.org/10.11648/j.engmath.20250901.11
DO  - 10.11648/j.engmath.20250901.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 1
EP  - 15
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20250901.11
AB  - Integrating Mean-sets theory employing generalized graph or group-theoretic tools and techniques into adaptive graph filtering can lead to more effective resilient filtering processes, particularly in challenging environments with clutter or uncertainty. In this paper, we show that under some crucial smoothing assumptions, the generalized Mean-sets theory developped for negatively curved convex combination Polish metric spaces following the formal Means-sets probability theory’s approach from Natalia Mosina, provides a new useful system for some secure adaptive graph filtering processes. We use convex combination operations (in the sense of Terán and Molchanov) on both individual input graph signals and filters. Individual adaptive graph filters being independently adapted by space (dataset)-valued random variables, while the convexification operator on the underlying dataset acts as a flexible theoretical instrument for preserving some good features of the standard scheme, like privacy of their informative trends, and looks more robust to changes. We exhibit a graph matrix model from a system of convex combination of two adaptive finite impulse response (FIR) graph filters processing a sampled-weighted mean-set (expectation) of some transversal graph signals with finite length N ≥ 2.

VL  - 9
IS  - 1
ER  -

Copy | Download