Journals Information
Mathematics and Statistics Vol. 13(6), pp. 439 - 460
DOI: 10.13189/ms.2025.130602
Reprint (PDF) (2653Kb)
Applications of Inner Product and Hilbert Spaces in Machine Learning with Data Analysis
Md. Abdul Mannan 1,*, Md. Amanat Ullah 1, Md. Amzad Hossain 2, Siful Islam 3, Md. Shafikul Islam 4, Mohammad Makchudul Alam 5, Md Atiqur Rahman 6, Sahib Jada Eyakub Khan 7, Muzibur Rahman Mozumder 8
1 Department of Mathematics, Uttara University, Bangladesh
2 Department of Education, Uttara University, Bangladesh
3 Management Information System, Lamar University, USA
4 Department of Computer Science and Engineering, Uttara University, Bangladesh
5 Cyber Threat Intelligence & Security Operations, BGD e-GOV CIRT, Bangladesh Computer Council, Bangladesh
6 Department of Physics, Michigan University, USA
7 Department of Physics and Astronomy, Ball State University, USA
8 Department of Mathematics, Aligarh Muslim University, Aligarh (U.P.), India
ABSTRACT
This study presents a comprehensive exploration of inner product spaces and their completion into Hilbert spaces, examining their foundational roles in both pure mathematics and modern machine learning. Inner product spaces introduce geometric notions such as orthogonality, angle, and norm, while Hilbert spaces, being complete IPS, extend these ideas to infinite dimensional settings. This paper develops key theoretical concepts including the Cauchy鈥揝chwarz inequality, Bessel's inequality, Parseval's identity, the polarization identity, and orthogonal projections. The discussion further explores the functional enrichment that Hilbert spaces provide over normed and Banach spaces, particularly in contexts requiring convergence and projection-based optimization. The practical relevance of Hilbert spaces is demonstrated through their role in machine learning algorithms such as Support Vector Machines (SVM), Principal Component Analysis (PCA), and kernel methods using Reproducing Kernel Hilbert Spaces (RKHS). Numerical simulations and MATLAB visualizations are employed to aid understanding and demonstrate the application of inner product theory in data driven tasks such as classification and dimensionality reduction. The results show how the geometry of Hilbert spaces naturally supports core operations in machine learning, making them indispensable in theoretical development and algorithmic design.
KEYWORDS
Hilbert Space, Inner Product Space, RKHS, Kernel Methods, SVM, PCA, Data Classification
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Md. Abdul Mannan , Md. Amanat Ullah , Md. Amzad Hossain , Siful Islam , Md. Shafikul Islam , Mohammad Makchudul Alam , Md Atiqur Rahman , Sahib Jada Eyakub Khan , Muzibur Rahman Mozumder , "Applications of Inner Product and Hilbert Spaces in Machine Learning with Data Analysis," Mathematics and Statistics, Vol. 13, No. 6, pp. 439 - 460, 2025. DOI: 10.13189/ms.2025.130602.
(b). APA Format:
Md. Abdul Mannan , Md. Amanat Ullah , Md. Amzad Hossain , Siful Islam , Md. Shafikul Islam , Mohammad Makchudul Alam , Md Atiqur Rahman , Sahib Jada Eyakub Khan , Muzibur Rahman Mozumder (2025). Applications of Inner Product and Hilbert Spaces in Machine Learning with Data Analysis. Mathematics and Statistics, 13(6), 439 - 460. DOI: 10.13189/ms.2025.130602.