VQ-Kernels: Unraveling Deep Learning of High-Dimensional Data Geometry

Deep Networks (DNs) are state-of-the-art predictors, able to navigate billion dimensional spaces to produce compressed embeddings of datasets. While most of the focus has been on improving the performance of these embeddings, we ask instead a different question: {\em how can Deep Networks teach us about the data geometry.} Through the spline theory of DNs, we derive a novel kernel that characterizes DNs as vector quantizers implementing affine functions over a partition of the domain, where the regions are coupled in a manner not immediately obvious from the partition geometry. We employ this kernel in the interpretation of DNs, illustrating their internalization of the training data geometry.