Training-Free Artificial Neural Networks

This paper presents a numerical scheme for the computation of Artificial Neural Networks' weights, without a laborious iterative procedure. The proposed algorithm adheres to the underlying theory, is highly fast, and results in remarkably low errors when applied for regression and classification of complex data-sets, such as the Griewank function of multiple variables $\mathbf{x} \in \mathbb{R}^{100}$ with random noise addition, and MNIST database for handwritten digits recognition, with $7\times10^4$ images. Interestingly, the same mathematical formulation found capable of approximating highly nonlinear functions in multiple dimensions, with low errors (e.g. $10^{-10}$) for the test set of the unknown functions, their higher-order partial derivatives, as well as numerically solving Partial Differential Equations. The method is based on the calculation of the weights of each neuron, in small neighborhoods of data, such that the corresponding local approximation matrix is invertible. Accordingly, the hyperparameters optimization is not necessary, as the neurons' number stems directly from the dimensions of the data, further improving the algorithmic speed. The overfitting is inherently eliminated, and the results are interpretable and reproducible. The complexity of the proposed algorithm is of class P with $\mathcal{O}(mn^3)$ computing time, which is linear for the observations and cubic for the features, in contrast with the NP-Complete class of standard algorithms for training ANNs. The performance of the method is high, for small as well as big datasets, and the test-set errors are similar or smaller than the train errors indicating the generalization efficiency. The supplementary computer code in Julia Language, may reproduce the validation examples, and run for other data-sets