Emergence of Latent Binary Encoding in Deep Neural Network Classifiers

We observe the emergence of binary encoding within the latent space of deep-neural-network classifiers. Such binary encoding is induced by introducing a linear penultimate layer, which is equipped during training with a loss function that grows as $\exp(\vec{x}^2)$, where $\vec{x}$ are the coordinates in the latent space. The phenomenon we describe represents a specific instance of a well-documented occurrence known as \textit{neural collapse}, which arises in the terminal phase of training and entails the collapse of latent class means to the vertices of a simplex equiangular tight frame (ETF). We show that binary encoding accelerates convergence toward the simplex ETF and enhances classification accuracy.

Extrapolation to complete basis-set limit in density-functional theory by quantile random-forest models

The numerical precision of density-functional-theory (DFT) calculations depends on a variety of computational parameters, one of the most critical being the basis-set size. The ultimate precision is reached with an infinitely large basis set, i.e., in the limit of a complete basis set (CBS). Our aim in this work is to find a machine-learning model that extrapolates finite basis-size calculations to the CBS limit. We start with a data set of 63 binary solids investigated with two all-electron DFT codes, exciting and FHI-aims, which employ very different types of basis sets. A quantile-random-forest model is used to estimate the total-energy correction with respect to a fully converged calculation as a function of the basis-set size. The random-forest model achieves a symmetric mean absolute percentage error of lower than 25% for both codes and outperforms previous approaches in the literature. Our approach also provides prediction intervals, which quantify the uncertainty of the models' predictions.