Some Neural Networks Inherently Preserve Subspace Clustering Structure

It has long been conjectured and empirically observed that neural networks tend to preserve clustering structure. This paper formalizes this conjecture. Specifically, we establish precise conditions for cluster structure preservation and derive bounds to quantify its extent. Through this analysis we are able to show that certain neural networks are learning parameters that preserve the clustering structure of the original data in their embeddings, without the need to impose mechanisms to promote this behavior. Extensive numerical analysis and experiments validate our results. Our findings offer deeper insight into neural network behavior, explaining why certain data types (such as images, audio, and text) benefit more from deep learning. Beyond theory, our findings guide better initialization, feature encoding, and regularization strategies.