Black Boxes and Looking Glasses: Multilevel Symmetries, Reflection Planes, and Convex Optimization in Deep Networks

We show that training deep neural networks (DNNs) with absolute value activation and arbitrary input dimension can be formulated as equivalent convex Lasso problems with novel features expressed using geometric algebra. This formulation reveals geometric structures encoding symmetry in neural networks. Using the equivalent Lasso form of DNNs, we formally prove a fundamental distinction between deep and shallow networks: deep networks inherently favor symmetric structures in their fitted functions, with greater depth enabling multilevel symmetries, i.e., symmetries within symmetries. Moreover, Lasso features represent distances to hyperplanes that are reflected across training points. These reflection hyperplanes are spanned by training data and are orthogonal to optimal weight vectors. Numerical experiments support theory and demonstrate theoretically predicted features when training networks using embeddings generated by Large Language Models. Recent advancements have demonstrated that deep neural networks are powerful models that can perform tasks including natural language processing, synthetic data and image generation, classification, and regression. However, research literature still lacks in intuitively understanding why deep networks are so powerful: what they "look for" in data, or in other words, how each layer extracts features. We are interested in the following question: Is there a fundamental difference in the nature of functions learned by deep networks, as opposed to shallow networks? We answer this question by transforming non-convex training problems into convex formulations and analyzing their structure.