Binary Classification as a Phase Separation Process

We propose a new binary classification model in Machine Learning, called Phase Separation Binary Classifier (PSBC). It consists of a discretization of a nonlinear reaction-diffusion equation (the Allen-Cahn equation), coupled with an ODE. Unlike many feedforward networks that are said to mimic brain or cortical cells functioning, the PSBC is inspired by fluid behavior, namely, on how binary fluids phase separate. Thus, (hyper)parameters have physical meaning, whose effects are carefully studied in several different scenarios: for instance, diffusion introduces interaction among features, whereas reaction plays an active role in classification. PSBC's coefficients are trainable weights, chosen according to a minimization problem using Gradient Descent; optimization relies on a classical Backpropagation Algorithm using weight sharing. Moreover, the model can be seen under the framework of feedforward networks, and is endowed with a nonlinear activation function that is linear in trainable weights but polynomial in other variables. In view of the model's connection with ODEs and parabolic PDEs, forward propagation amounts to an initial value problem. Thus, stability conditions are established through meshgrid constraints, discrete maximum principles, and, overall, exploiting the concept of Invariant regions, as developed in the work of Chueh, Conway, Smoller, and particularly in the application of their theory to finite-difference methods in the work of Hoff. The PSBC also has interesting model compression properties which are thoroughly discussed. We apply the model to the subset of numbers "0" and "1" of the classical MNIST database, where we are able to discern individuals from both classes with more than 94\% accuracy, sometimes using less than $80$ variables, a feature that is out of reach of Artificial Neural Networks without weight sharing or feature engineering.