Mad Max: Affine Spline Insights into Deep Learning
This provides a theoretical framework for interpreting deep learning models, with potential benefits for researchers and practitioners in improving model robustness and performance, though it builds on prior work.
The paper tackles the problem of understanding deep networks by connecting them to spline functions and operators, showing that many DNs can be expressed as compositions of max-affine spline operators, which reveals signal-dependent templates for classification. It proposes a penalty term to enforce template orthogonality, leading to significantly improved classification performance and reduced overfitting without architectural changes.
We build a rigorous bridge between deep networks (DNs) and approximation theory via spline functions and operators. Our key result is that a large class of DNs can be written as a composition of max-affine spline operators (MASOs), which provide a powerful portal through which to view and analyze their inner workings. For instance, conditioned on the input signal, the output of a MASO DN can be written as a simple affine transformation of the input. This implies that a DN constructs a set of signal-dependent, class-specific templates against which the signal is compared via a simple inner product; we explore the links to the classical theory of optimal classification via matched filters and the effects of data memorization. Going further, we propose a simple penalty term that can be added to the cost function of any DN learning algorithm to force the templates to be orthogonal with each other; this leads to significantly improved classification performance and reduced overfitting with no change to the DN architecture. The spline partition of the input signal space that is implicitly induced by a MASO directly links DNs to the theory of vector quantization (VQ) and $K$-means clustering, which opens up new geometric avenue to study how DNs organize signals in a hierarchical fashion. To validate the utility of the VQ interpretation, we develop and validate a new distance metric for signals and images that quantifies the difference between their VQ encodings. (This paper is a significantly expanded version of A Spline Theory of Deep Learning from ICML 2018.)