Green-ELM: Efficient Analytic Learning via High-Dimensional Random Projections

We present Green-ELM, a non-iterative neural architecture that replaces gradient-based optimization of the output layer with a closed-form analytic solution over a fixed, high-dimensional random feature representation. By projecting input manifolds into a high-dimensional, random feature space ($d \gg 784$), our results show that complex class boundaries can be effectively untangled without the computational overhead of backpropagation. Utilizing the Moore-Penrose pseudoinverse, LU and Cholesky decomposition to solve for the output layer in a single analytic step, Green-ELM achieves a classification accuracy of 98.10% on MNIST ($d=4000$) and 86.63% on Fashion-MNIST. Furthermore, we experiment with a pre-trained ``frozen-backbone'' based on ResNet-18 to extract high-quality features and show that these one-shot solvers are effective beyond simple datasets. Notably, our baseline CPU configuration on MNIST ($d=2000$) achieves 97.15% accuracy in 1.5s, representing a 11.6$\times$ reduction in reported training time over an SGD baseline while maintaining comparable performance. We observe a near-logarithmic scaling behavior between dimensionality and accuracy, where the accuracy increases approximately logarithmically with hidden dimensionality over the tested range, suggesting that feature-space expansion contributes substantially to performance in these experiments. . This one-shot linear matrix solver approach offers a viable alternative for real-time Edge AI, where the traditional training phase is bypassed in favor of non-iterative manifold representation and readout. Finally, we propose an Empirical Scaling Hypothesis, a framework that models accuracy bounds as a function of high dimensionality and intrinsic dataset complexity.