ISSN (Online): 2321-3418
server-injected
Mathematics and Statistics
Open Access

Mathematical foundation of High-Dimensional Data Analysis: Leveraging Topology and Geometry for Enhanced Model Interpretability in AI

DOI: 10.18535/ijsrm/v12i11.m01· Pages: 546-557· Vol. 12, No. 11, (2024)· Published: November 16, 2024
PDF
Views: 773 PDF downloads: 268

Abstract

One of the most important challenges for modern AI and machine learning is the analysis of high-dimensional data. Traditional methods face serious complications in such cases due to high complexity of datasets: the curse of dimensionality, overfitting, and lack of transparency of model behavior. In this paper, we adopt a novel approach to analyze high-dimensional data; topological and geometric techniques will be exploited, taking advantage of better model interpretability and deeper insights into the structure. Precisely, we discuss Topological Data Analysis, mainly Persistent Homology  (Edelsbrunner et al., 2002), which allows the extraction of topological features-like loops and connected components that enable the extracting knowledge about the global structure of data. We also see how some concepts of differential geometry and Riemannian geometry (Do Carmo, 1976) can be used to cast light on manifold data structure lying at the heart of any attempt at modeling intrinsic patterns in high-dimensional spaces.

We will review how these mathematical pillars, combined with state-of-the-art techniques for dimensionality reduction like t-SNE, UMAP, Principal Component Analysis, are able to provide interpretable and low-dimensional representations of high-dimensional data that can be used to understand models and make decisions. Case studies are also included, which explain the practical working of these methods in AI systems and show how much complex models can be made transparent using these, especially in domains that are very critical, such as healthcare  (Caruana et al., 2015), finance (Chen et al., 2018), and autonomous systems ( Wang et al., 2019).

We also discuss some of the difficulties in using these methods for practical applications: computational complexity; the need for large-scale data processing (Bengio et al., 2007); and integration of topological and geometric intuition with the rest of the machine learning pipeline (Zhu et al., 2020). We conclude with possible future directions of research toward fine-tuning these methods and exploring their broader applicability to AI in its quest for more robust, interpretable, and reliable AI models. Given this work, we focus on how linking topology, geometry, and AI bears great promise for solving one of today's critical challenges: model interpretability in high-dimensional data analysis.

References

  1. Liu, S., Wang, D., Maljovec, D., Anirudh, R., Thiagarajan, J. J., Jacobs, S. A., ... & Bremer, P. T. (2019). Scalable topological data analysis and visualization for evaluating data-driven models in scientific applications. IEEE transactions on visualization and computer graphics, 26(1), 291-300.Google Scholar ↗
  2. Rysavy, S. J., Bromley, D., & Daggett, V. (2014). DIVE: A graph-based visual-analytics framework for big data. IEEE computer graphics and applications, 34(2), 26-37.Google Scholar ↗
  3. Garth, C., Gueunet, C., Guillou, P., Hofmann, L., Levine, J. A., Lukasczyk, J., ... & Wetzels, F. (2021, October). Topological Analysis of Ensemble Scalar Data with TTK. In IEEE VIS Tutorials.Google Scholar ↗
  4. Bremer, P. T., Weber, G., Tierny, J., Pascucci, V., Day, M., & Bell, J. (2010). Interactive exploration and analysis of large-scale simulations using topology-based data segmentation. IEEE Transactions on Visualization and Computer Graphics, 17(9), 1307-1324.Google Scholar ↗
  5. Goodell, J. W., Kumar, S., Lim, W. M., & Pattnaik, D. (2021). Artificial intelligence and machine learning in finance: Identifying foundations, themes, and research clusters from bibliometric analysis. Journal of Behavioral and Experimental Finance, 32, 100577.Google Scholar ↗
  6. Cao, L. (2022). Ai in finance: challenges, techniques, and opportunities. ACM Computing Surveys (CSUR), 55(3), 1-38.Google Scholar ↗
  7. De Prado, M. L. (2018). Advances in financial machine learning. John Wiley & Sons.Google Scholar ↗
  8. Devarasetty, N. (2023). AI and Data Engineering: Harnessing the Power of Machine Learning in Data-Driven Enterprises. International Journal of Machine Learning Research in Cybersecurity and Artificial Intelligence, 14(1), 195-226.Google Scholar ↗
  9. Sabharwal, C. L. (2018). The rise of machine learning and robo-advisors in banking. IDRBT Journal of Banking Technology, 28.Google Scholar ↗
  10. Patil, D., Rane, N. L., Desai, P., & Rane, J. (2024). Machine learning and deep learning: Methods, techniques, applications, challenges, and future research opportunities. Trustworthy Artificial Intelligence in Industry and Society, 28-81.Google Scholar ↗
  11. Suthaharan, S. (2016). Machine learning models and algorithms for big data classification. Integr. Ser. Inf. Syst, 36, 1-12.Google Scholar ↗
  12. Gilpin, L. H., Bau, D., Yuan, B. Z., Bajwa, A., Specter, M., & Kagal, L. (2018, October). Explaining explanations: An overview of interpretability of machine learning. In 2018 IEEE 5th International Conference on data science and advanced analytics (DSAA) (pp. 80-89). IEEE.Google Scholar ↗
  13. Van der Maaten, L., & Hinton, G. (2008). Visualizing data using t-SNE. Journal of machine learning research, 9(11).Google Scholar ↗
  14. Wang, Y., Liu, M., Yang, J., & Gui, G. (2019). Data-driven deep learning for automatic modulation recognition in cognitive radios. IEEE Transactions on Vehicular Technology, 68(4), 4074-4077.Google Scholar ↗
  15. Wang, P., Li, Y., & Reddy, C. K. (2019). Machine learning for survival analysis: A survey. ACM Computing Surveys (CSUR), 51(6), 1-36.Google Scholar ↗
  16. Zhu, N., Zhang, D., Wang, W., Li, X., Yang, B., Song, J., ... & Tan, W. (2020). A novel coronavirus from patients with pneumonia in China, 2019. New England journal of medicine, 382(8), 727-733.Google Scholar ↗
  17. Guan, W. J., Ni, Z. Y., Hu, Y., Liang, W. H., Ou, C. Q., He, J. X., ... & Zhong, N. S. (2020). Clinical characteristics of coronavirus disease 2019 in China. New England journal of medicine, 382(18), 1708-1720.Google Scholar ↗
  18. Bellman, R. (1957). A Markovian decision process. Journal of mathematics and mechanics, 679-684.Google Scholar ↗
  19. Carriere, M., Cuturi, M., & Oudot, S. (2017, July). Sliced Wasserstein kernel for persistence diagrams. In International conference on machine learning (pp. 664-673). PMLR.Google Scholar ↗
  20. Carisson, B., Kindberg, E., & Buesa, J. (2009). The G428A nonsense mutation in FUT2 provides strong but not absolute protection against symptomatic GEL4 Norovirus infection. PLoS ONE, 4, e5593.Google Scholar ↗
  21. Hershey, S., Chaudhuri, S., Ellis, D. P., Gemmeke, J. F., Jansen, A., Moore, R. C., ... & Wilson, K. (2017, March). CNN architectures for large-scale audio classification. In 2017 ieee international conference on acoustics, speech and signal processing (icassp) (pp. 131-135). IEEE.Google Scholar ↗
  22. McInnes, L., Healy, J., & Melville, J. (2018). Umap: Uniform manifold approximation and projection for dimension reduction. arXiv preprint arXiv:1802.03426.Google Scholar ↗
  23. Caruana, A., Bandara, M., Musial, K., Catchpoole, D., & Kennedy, P. J. (2023). Machine learning for administrative health records: A systematic review of techniques and applications. Artificial Intelligence in Medicine, 102642.Google Scholar ↗
  24. Omata, M., Cheng, A. L., Kokudo, N., Kudo, M., Lee, J. M., Jia, J., ... & Sarin, S. K. (2017). Asia–Pacific clinical practice guidelines on the management of hepatocellular carcinoma: a 2017 update. Hepatology international, 11, 317-370.Google Scholar ↗
  25. Crawford, J., & Brownlie, I. (2019). Brownlie's principles of public international law. Oxford University Press, USA.Google Scholar ↗
  26. do Carmo Giordano, L., & Riedel, P. S. (2008). Multi-criteria spatial decision analysis for demarcation of greenway: A case study of the city of Rio Claro, Sao Paulo, Brazil. Landscape and urban planning, 84(3-4), 301-311.Google Scholar ↗
  27. Edelsbrunner, Letscher, & Zomorodian. (2002). Topological persistence and simplification. Discrete & computational geometry, 28, 511-533.Google Scholar ↗
  28. Akerib, D. S., Akerlof, C. W., Akimov, D. Y., Alsum, S. K., Araújo, H. M., Arnquist, I. J., ... & Saba, J. S. (2017). Identification of radiopure titanium for the LZ dark matter experiment and future rare event searches. Astroparticle Physics, 96, 1-10.Google Scholar ↗
  29. Jolliffe, I. T. (2002). Principal component analysis for special types of data (pp. 338-372). Springer New York.Google Scholar ↗
  30. McInnes, M. D., Moher, D., Thombs, B. D., McGrath, T. A., Bossuyt, P. M., Clifford, T., ... & Willis, B. H. (2018). Preferred reporting items for a systematic review and meta-analysis of diagnostic test accuracy studies: the PRISMA-DTA statement. Jama, 319(4), 388-396.Google Scholar ↗
  31. Mikolov, T., Sutskever, I., Chen, K., Corrado, G. S., & Dean, J. (2013). Distributed representations of words and phrases and their compositionality. Advances in neural information processing systems, 26.Google Scholar ↗
Author details
Jonathan Keningson
Independent Scholar
✉ Corresponding Author
👤 View Profile →