Separability is not the best goal for machine learning

TL;DR

This paper argues that shifting the learning goal from linear separability to k-separability simplifies the transformation of complex data distributions in neural networks, potentially reducing the need for deep architectures.

Contribution

It introduces the concept of k-separability as an alternative learning goal, enabling simpler solutions for complex data and Boolean problems like parity.

Findings

k-separability simplifies learning complex data distributions

Linear projection combined with k-separability solves Boolean problems efficiently

Replacing deep layers with k-separability targets can streamline neural network training

Abstract

Neural networks use their hidden layers to transform input data into linearly separable data clusters, with a linear or a perceptron type output layer making the final projection on the line perpendicular to the discriminating hyperplane. For complex data with multimodal distributions this transformation is difficult to learn. Projection on $k\geq 2$ line segments is the simplest extension of linear separability, defining much easier goal for the learning process. Simple problems are 2-separable, but problems with inherent complex logic may be solved in a simple way by $k$-separable projections. The difficulty of learning non-linear data distributions is shifted to separation of line intervals, simplifying the transformation of data by hidden network layers. For classification of difficult Boolean problems, such as the parity problem, linear projection combined with \ksep is sufficient and provides a powerful new target for learning. More complex targets may also be defined, changing the goal of learning from linear discrimination to creation of data distributions that can easily be handled by specialized models selected to analyze output distributions. This approach can replace many layers of transformation required by deep learning models.