On a Neural Implementation of Brenier’s Polar Factorization
In 1991, Brenier proved a theorem that generalizes the polar decomposition for square matrices — factored as PSD ×times× unitary — to any vector field F:Rd→RdF:mathbb{R}^drightarrow mathbb{R}^dF:Rd→Rd. The theorem, known as the polar factorization theorem, states that any field FFF can be recovered as the composition of the gradient of a convex function uuu with a measure-preserving map MMM, namely F=∇u∘MF=nabla u circ MF=∇u∘M. We propose a practical implementation of this far-reaching theoretical result, and explore possible uses within machine learning. The theorem is closely related…
The ever-increasing parameter counts of deep learning models necessitate effective compression techniques for deployment on resource-constrained devices. This paper explores the application of information geometry, the study of density-induced metrics on parameter spaces, to analyze existing methods within the space of model compression, primarily focusing on operator factorization. Adopting this…
This paper was accepted at the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA) 2025 Non-negative Matrix Factorization (NMF) is a powerful technique for analyzing regularly-sampled data, i.e., data that can be stored in a matrix. For audio, this has led to numerous applications using time-frequency…