New Advances in the Mathematical Foundations of Modern Statistical Machine Learning
A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "D1: Probability and Statistics".
Deadline for manuscript submissions: 27 January 2027 | Viewed by 131
Special Issue Editor
Interests: statistical learning theory; Bayesian and computational statistics; algebraic and ring-theoretic learning; category-theoretic foundations of statistical learning; representation learning and kernel alignment; ensemble and generative learning; generalizability and stability of learning machines; information geometry and topological data analysis; statistical foundations of artificial intelligence and large language models (LLMs); robustness, interpretability, and uncertainty quantification in data science
Special Issue Information
Dear Colleagues,
The current research landscape in machine learning is undergoing a profound "structural turn." As we move beyond empirical success, the need to formalize the mathematical principles governing high-dimensional models—particularly the theoretical foundations of deep neural networks—has never been more critical. At the heart of this evolution lies the centrality of the composition of functions and the role of morphisms in preserving structural integrity across learning tasks. Understanding the isomorphisms between disparate model architectures and statistical frameworks provides the key to unlocking model behavior, phase transitions, and reliability.
This Special Issue aims to provide a rigorous forum for researchers investigating the deep mathematical underpinnings of learning machines. We are particularly interested in works that synthesize classical statistical learning with modern abstractions, such as sheaf-theoretic approaches, topological data analysis (TDA), and the categorical foundations of deep learning. By exploring the "Ninefold Path" to statistical machine learning, this collection seeks to map the journey from foundational data geometry to the sophisticated frontiers of generative and structural intelligence.
The topical scope of this Special Issue includes, but is not limited to, the following:
- Algebraic and structural foundations: Ring-theoretic and category-theoretic learning, the role of morphisms and isomorphisms in representation learning, and the functorial nature of deep architectures.
- Singular learning theory (SLT): Algebraic geometry in statistics, resolution of singularities in model selection, and the study of non-identifiable parameter spaces.
- Geometric and topological insights: Sheaf-theoretic data fusion, topological data analysis (TDA), and the manifold structure of learning machines.
- Deep learning theory: Mathematical foundations of neural networks, the geometry of function composition, and the convergence of over-parameterized models.
- The ninefold path to SML: Research addressing the nine essential dimensions of statistical learning—from data representation and optimization to generalizability and ethical foundations.
- Statistical mechanics of AI: Foundations of large language models (LLMs), kernel alignment, ensemble learning, and the generalizability of complex learning machines.
- Computational statistics: Advanced Bayesian inference, variational methods, and the interplay between computational complexity and statistical efficiency.
We invite original research, conceptual frameworks, and comprehensive reviews that bridge the gap between abstract mathematical theory and the evolving paradigms of data science.
Prof. Dr. Ernest Fokoué
Guest Editor
Manuscript Submission Information
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Keywords
- statistical learning theory
- singular learning theory
- category-theoretic machine learning
- morphisms and isomorphisms in learning
- sheaf theory and topology
- deep learning theory
- information geometry
- the ninefold path to SML
- representation learning
- algebraic statistics
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