Brian Bell

Brian Bell, Michael Geyer, David Glickenstein et al.

We explore the equivalence between neural networks and kernel methods by deriving the first exact representation of any finite-size parametric classification model trained with gradient descent as a kernel machine. We compare our exact representation to the well-known Neural Tangent Kernel (NTK) and discuss approximation error relative to the NTK and other non-exact path kernel formulations. We experimentally demonstrate that the kernel can be computed for realistic networks up to machine precision. We use this exact kernel to show that our theoretical contribution can provide useful insights into the predictions made by neural networks, particularly the way in which they generalize.

Brian Bell

This work starts with the intention of using mathematics to understand the intriguing vulnerability observed by ~\citet{szegedy2013} within artificial neural networks. Along the way, we will develop some novel tools with applications far outside of just the adversarial domain. We will do this while developing a rigorous mathematical framework to examine this problem. Our goal is to build out theory which can support increasingly sophisticated conjecture about adversarial attacks with a particular focus on the so called ``Dimpled Manifold Hypothesis'' by ~\citet{shamir2021dimpled}. Chapter one will cover the history and architecture of neural network architectures. Chapter two is focused on the background of adversarial vulnerability. Starting from the seminal paper by ~\citet{szegedy2013} we will develop the theory of adversarial perturbation and attack. Chapter three will build a theory of persistence that is related to Ricci Curvature, which can be used to measure properties of decision boundaries. We will use this foundation to make a conjecture relating adversarial attacks. Chapters four and five represent a sudden and wonderful digression that examines an intriguing related body of theory for spatial analysis of neural networks as approximations of kernel machines and becomes a novel theory for representing neural networks with bilinear maps. These heavily mathematical chapters will set up a framework and begin exploring applications of what may become a very important theoretical foundation for analyzing neural network learning with spatial and geometric information. We will conclude by setting up our new methods to address the conjecture from chapter 3 in continuing research.

Brian Bell, Michael Geyer, David Glickenstein et al.

There are a number of hypotheses underlying the existence of adversarial examples for classification problems. These include the high-dimensionality of the data, high codimension in the ambient space of the data manifolds of interest, and that the structure of machine learning models may encourage classifiers to develop decision boundaries close to data points. This article proposes a new framework for studying adversarial examples that does not depend directly on the distance to the decision boundary. Similarly to the smoothed classifier literature, we define a (natural or adversarial) data point to be $(γ,σ)$-stable if the probability of the same classification is at least $γ$ for points sampled in a Gaussian neighborhood of the point with a given standard deviation $σ$. We focus on studying the differences between persistence metrics along interpolants of natural and adversarial points. We show that adversarial examples have significantly lower persistence than natural examples for large neural networks in the context of the MNIST and ImageNet datasets. We connect this lack of persistence with decision boundary geometry by measuring angles of interpolants with respect to decision boundaries. Finally, we connect this approach with robustness by developing a manifold alignment gradient metric and demonstrating the increase in robustness that can be achieved when training with the addition of this metric.

Jamie Mahowald, Brian Bell, Alex Ho et al.

Neural tangent kernel (NTK) methods are computationally limited by the need to evaluate large Jacobians across many data points. Existing approaches reduce this cost primarily through projecting and sketching the Jacobian. We show that NTK computation can also be reduced by compressing the data dimension itself using NTK-tuned dataset distillation. We demonstrate that the neural tangent space spanned by the input data can be induced by dataset distillation, yielding a 20-100$\times$ reduction in required Jacobian calculations. We further show that per-class NTK matrices have low effective rank that is preserved by this reduction. Building on these insights, we propose the distilled neural tangent kernel (DNTK), which combines NTK-tuned dataset distillation with state-of-the-art projection methods to reduce up NTK computational complexity by up to five orders of magnitude while preserving kernel structure and predictive performance.