arXiv subjects

Ronald Katende

Publications and source records attributed to Ronald Katende.

29 records · Page 2Linked to original sources

arXiv2025

Structured Variational $D$-Decomposition for Accurate and Stable Low-Rank Approximation

We introduce the $D$-decomposition, a non-orthogonal matrix factorization of the form $A \approx P D Q$, where $P \in \mathbb{R}^{n \times k}$, $D \in \mathbb{R}^{k \times k}$, and $Q \in \mathbb{R}^{k \times n}$. The decomposition is defined variationally by minimizing a regularized Frobenius loss, allowing control over rank, sparsity, and conditioning. Unlike algebraic factorizations such as LU or SVD, it is computed by alternating minimization. We establish existence and perturbation stability of the solution and show that each update has complexity $\mathcal{O}(n^2k)$. Benchmarks against truncated SVD, CUR, and nonnegative matrix factorization show improved reconstruction accuracy on MovieLens, MNIST, Olivetti Faces, and gene expression matrices, particularly under sparsity and noise.

math.NA↗

arXiv2025

Unified theoretical guarantees for stability, consistency, and convergence in neural PDE solvers from non-IID data to physics-informed networks

We establish a unified theoretical framework addressing the stability, consistency, and convergence of neural networks under realistic training conditions, specifically, in the presence of non-IID data, geometric constraints, and embedded physical laws. For standard supervised learning with dependent data, we derive uniform stability bounds for gradient-based methods using mixing coefficients and dynamic learning rates. In federated learning with heterogeneous data and non-Euclidean parameter spaces, we quantify model inconsistency via curvature-aware aggregation and information-theoretic divergence. For Physics-Informed Neural Networks (PINNs), we rigorously prove perturbation stability, residual consistency, Sobolev convergence, energy stability for conservation laws, and convergence under adaptive multi-domain refinements. Each result is grounded in variational analysis, compactness arguments, and universal approximation theorems in Sobolev spaces. Our theoretical guarantees are validated across parabolic, elliptic, and hyperbolic PDEs, confirming that residual minimization aligns with physical solution accuracy. This work offers a mathematically principled basis for designing robust, generalizable, and physically coherent neural architectures across diverse learning environments.

cs.LG↗

arXiv2025

Topological Eigenvalue Theorems for Tensor Analysis in Multi-Modal Data Fusion

This paper presents a novel framework for tensor eigenvalue analysis in the context of multi-modal data fusion, leveraging topological invariants such as Betti numbers. Traditional approaches to tensor eigenvalue analysis often extend matrix theory, whereas this work introduces a topological perspective to enhance the understanding of tensor structures. By establishing new theorems that link eigenvalues to topological features, the proposed framework provides deeper insights into the latent structure of data, improving both interpretability and robustness. Applications in data fusion demonstrate the theoretical and practical significance of this approach, with potential for broad impact in machine learning and data science.

stat.ML↗

arXiv2024

Adaptive Error-Bounded Hierarchical Matrices for Efficient Neural Network Compression

This paper introduces a dynamic, error-bounded hierarchical matrix (H-matrix) compression method tailored for Physics-Informed Neural Networks (PINNs). The proposed approach reduces the computational complexity and memory demands of large-scale physics-based models while preserving the essential properties of the Neural Tangent Kernel (NTK). By adaptively refining hierarchical matrix approximations based on local error estimates, our method ensures efficient training and robust model performance. Empirical results demonstrate that this technique outperforms traditional compression methods, such as Singular Value Decomposition (SVD), pruning, and quantization, by maintaining high accuracy and improving generalization capabilities. Additionally, the dynamic H-matrix method enhances inference speed, making it suitable for real-time applications. This approach offers a scalable and efficient solution for deploying PINNs in complex scientific and engineering domains, bridging the gap between computational feasibility and real-world applicability.

cs.LG↗

arXiv2024

A Novel Framework for Analyzing Structural Transformation in Data-Constrained Economies Using Bayesian Modeling and Machine Learning

Structural transformation, the shift from agrarian economies to more diversified industrial and service-based systems, is a key driver of economic development. However, in low- and middle-income countries (LMICs), data scarcity and unreliability hinder accurate assessments of this process. This paper presents a novel statistical framework designed to address these challenges by integrating Bayesian hierarchical modeling, machine learning-based data imputation, and factor analysis. The framework is specifically tailored for conditions of data sparsity and is capable of providing robust insights into sectoral productivity and employment shifts across diverse economies. By utilizing Bayesian models, uncertainties in data are effectively managed, while machine learning techniques impute missing data points, ensuring the integrity of the analysis. Factor analysis reduces the dimensionality of complex datasets, distilling them into core economic structures. The proposed framework has been validated through extensive simulations, demonstrating its ability to predict structural changes even when up to 60\% of data is missing. This approach offers policymakers and researchers a valuable tool for making informed decisions in environments where data quality is limited, contributing to the broader understanding of economic development in LMICs.

stat.AP↗

arXiv2024

Decentralized Neural Networks for Robust and Scalable Eigenvalue Computation

This paper introduces a novel method for eigenvalue computation using a distributed cooperative neural network framework. Unlike traditional techniques that face scalability challenges in large systems, our decentralized algorithm enables multiple autonomous agents to collaboratively estimate the smallest eigenvalue of large matrices. Each agent employs a localized neural network, refining its estimates through communication with neighboring agents. Our empirical results confirm the algorithm's convergence towards the true eigenvalue, with estimates clustered closely around the true value. Even in the presence of communication delays or network disruptions, the method demonstrates strong robustness and scalability. Theoretical analysis further validates the accuracy and stability of the proposed approach, while empirical tests highlight its efficiency and precision, surpassing traditional centralized algorithms in large-scale eigenvalue computations.

cs.LG↗

arXiv2024

Symmetry-Enriched Learning: A Category-Theoretic Framework for Robust Machine Learning Models

This manuscript presents a novel framework that integrates higher-order symmetries and category theory into machine learning. We introduce new mathematical constructs, including hyper-symmetry categories and functorial representations, to model complex transformations within learning algorithms. Our contributions include the design of symmetry-enriched learning models, the development of advanced optimization techniques leveraging categorical symmetries, and the theoretical analysis of their implications for model robustness, generalization, and convergence. Through rigorous proofs and practical applications, we demonstrate that incorporating higher-dimensional categorical structures enhances both the theoretical foundations and practical capabilities of modern machine learning algorithms, opening new directions for research and innovation.

cs.LG↗

arXiv2024

A Nonlinear Generalization of the Bauer-Fike Theorem and Novel Iterative Methods for Solving Nonlinear Eigenvalue Problems

Nonlinear eigenvalue problems (NEPs) present significant challenges due to their inherent complexity and the limitations of traditional linear eigenvalue theory. This paper addresses these challenges by introducing a nonlinear generalization of the Bauer-Fike theorem, which serves as a foundational result in classical eigenvalue theory. This generalization provides a robust theoretical framework for understanding the sensitivity of eigenvalues in NEPs, extending the applicability of the Bauer-Fike theorem beyond linear cases. Building on this theoretical foundation, we propose novel iterative methods designed to efficiently solve NEPs. These methods leverage the generalized theorem to improve convergence rates and accuracy, making them particularly effective for complex NEPs with dense spectra. The adaptive contour integral method, in particular, is highlighted for its ability to identify multiple eigenvalues within a specified region of the complex plane, even in cases where eigenvalues are closely clustered. The efficacy of the proposed methods is demonstrated through a series of numerical experiments, which illustrate their superior performance compared to existing approaches. These results underscore the practical applicability of our methods in various scientific and engineering contexts. In conclusion, this paper represents a significant advancement in the study of NEPs by providing a unified theoretical framework and effective computational tools, thereby bridging the gap between theory and practice in the field of nonlinear eigenvalue problems.

math.NA↗

arXiv2024

Cross-Country Comparative Analysis of Climate Resilience and Localized Mapping in Data-Sparse Regions

Climate resilience across sectors varies significantly in low-income countries (LICs), with agriculture being the most vulnerable to climate change. Existing studies typically focus on individual countries, offering limited insights into broader cross-country patterns of adaptation and vulnerability. This paper addresses these gaps by introducing a framework for cross-country comparative analysis of sectoral climate resilience using meta-analysis and cross-country panel data techniques. The study identifies shared vulnerabilities and adaptation strategies across LICs, enabling more effective policy design. Additionally, a novel localized climate-agriculture mapping technique is developed, integrating sparse agricultural data with high-resolution satellite imagery to generate fine-grained maps of agricultural productivity under climate stress. Spatial interpolation methods, such as kriging, are used to address data gaps, providing detailed insights into regional agricultural productivity and resilience. The findings offer policymakers tools to prioritize climate adaptation efforts and optimize resource allocation both regionally and nationally.

cs.NE↗

arXiv2024

Optimizing Neural Network Performance and Interpretability with Diophantine Equation Encoding

This paper explores the integration of Diophantine equations into neural network (NN) architectures to improve model interpretability, stability, and efficiency. By encoding and decoding neural network parameters as integer solutions to Diophantine equations, we introduce a novel approach that enhances both the precision and robustness of deep learning models. Our method integrates a custom loss function that enforces Diophantine constraints during training, leading to better generalization, reduced error bounds, and enhanced resilience against adversarial attacks. We demonstrate the efficacy of this approach through several tasks, including image classification and natural language processing, where improvements in accuracy, convergence, and robustness are observed. This study offers a new perspective on combining mathematical theory and machine learning to create more interpretable and efficient models.

cs.LG↗

arXiv2024

Strange Attractors in Fractional Differential Equations: A Topological Approach to Chaos and Stability

In this work, we explore the dynamics of fractional differential equations (FDEs) through a rigorous topological analysis of strange attractors. By investigating systems with Caputo derivatives of order $ α\in (0, 1) $, we identify conditions under which chaotic behavior emerges, characterized by positive topological entropy and the presence of homoclinic and heteroclinic structures. We introduce novel methods for computing the fractional Conley index and Lyapunov exponents, which allow us to distinguish between chaotic and non-chaotic attractors. Our results also provide new insights into the fractal and spectral properties of strange attractors in fractional systems, establishing a comprehensive framework for understanding chaos and stability in this context.

math.DS↗