Algebraic σ-Based (Cekirge) Model for Deterministic and Energy-Efficient Unsupervised Machine Learning

Unsupervised learning is a fundamental branch of machine learning that operates without labeled outputs, aiming instead to uncover latent structures, intrinsic relationships, and patterns embedded in data. Unlike supervised approaches, which rely on explicit input-output mappings, unsupervised methods extract regularities directly from raw, often high-dimensional, datasets. Core methodological paradigms include clustering, dimensionality reduction, and anomaly detection. Clustering techniques partition data into groups according to similarity metrics; dimensionality reduction methods, such as Principal Component Analysis (PCA) and t-SNE, map high-dimensional inputs into lower-dimensional subspaces while preserving meaningful structure; and density estimation approaches model probability distributions to detect rare or anomalous events. A central concept is the latent space, in which data are encoded into compact representations that capture essential features. These representations may arise from empirical observations or serve as hypothetical abstractions. Weights and biases can be systematically organized using structured matrix formulations that parallel neural computation. Ultimately, unsupervised learning seeks to reveal intrinsic data regularities without external supervision, while its latent encodings provide a transferable foundation for downstream supervised tasks such as classification, regression, and prediction. Once a robust latent representation is obtained, these encoded datasets can serve as the foundation for downstream supervised learning tasks, enabling prediction, classification, or regression on previously unlabeled data. The Algebraic σ-Based (Cekirge) Model presented in this paper allows deterministic computation of neural network weights, including bias, for any number of inputs. Auxiliary σ perturbations ensure a nonsingular matrix, guaranteeing a unique solution. Compared to gradient descent, the Algebraic σ-Based (Cekirge) Model is orders of magnitude faster and consumes significantly less energy. Gradient descent is iterative, slower, and only approximates without careful tuning, resulting in higher energy usage. The method scales naturally with the number of inputs, requiring only a square system with perturbations. Biological neurons exhibit robust recognition, maintaining performance despite variations in orientation, illumination, or noise. Inspired by this, the Algebraic (Cekirge) Model, developed by Huseyin Murat Cekirge, deterministically computes neural weights in a closed-form, energy-efficient manner. This study benchmarks the model against conventional Gradient Descent (GD), a standard iterative method, highlighting efficiency, stability under perturbations, and accuracy. Results show that the Cekirge method produces weights nearly identical to GD while running over three orders of magnitude faster, demonstrating a robust and scalable alternative for neural network training.