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

Emi Zeger, Mert Pilanci

Keywords: deep neural networks, convex optimization, geometric algebra, Lasso model, sparsity

TL;DR: We derive equivalent convex formulations of deep neural networks and identify novel reflection planes using geometric algebra. We show that theoretically predicted Lasso features appear in Large Language Models.

Abstract: 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.

Primary Area: optimization

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Submission Number: 4741