Related papers: The Curvature Rate {\lambda}: A Scalar Measure of …

Understanding the curvature evolution of the loss landscape is fundamental to analyzing the training dynamics of neural networks. The most commonly studied measure, Hessian sharpness ($\lambda_{\max}^H$) -- the largest eigenvalue of the…

The highly non-linear nature of deep neural networks causes them to be susceptible to adversarial examples and have unstable gradients which hinders interpretability. However, existing methods to solve these issues, such as adversarial…

Hyperbolic neural networks (HNNs) have demonstrated notable efficacy in representing real-world data with hierarchical structures via exploiting the geometric properties of hyperbolic spaces characterized by negative curvatures. Curvature…

Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations…

Curvature information -- particularly, the largest eigenvalue of the loss Hessian, known as the sharpness -- often forms the basis for learning rate tuners. However, recent work has shown that the curvature information undergoes complex…

Network data, commonly used throughout the physical, social, and biological sciences, consist of nodes (individuals) and the edges (interactions) between them. One way to represent network data's complex, high-dimensional structure is to…

Modern neural networks can achieve high accuracy while remaining poorly calibrated, producing confidence estimates that do not match empirical correctness. Yet calibration is often treated as a post-hoc attribute. We take a different…

We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver [Larios-C\'ardenas and Gibou, J. Comput. Phys. (May 2022),…

Deep neural networks have been demonstrated to achieve phenomenal success in many domains, and yet their inner mechanisms are not well understood. In this paper, we investigate the curvature of image manifolds, i.e., the manifold deviation…

Toward achieving robust and defensive neural networks, the robustness against the weight parameters perturbations, i.e., sharpness, attracts attention in recent years (Sun et al., 2020). However, sharpness is known to remain a critical…

We propose a deep learning strategy to estimate the mean curvature of two-dimensional implicit interfaces in the level-set method. Our approach is based on fitting feed-forward neural networks to synthetic data sets constructed from…

We consider the problem of estimating curvature where the data can be viewed as a noisy sample from an underlying manifold. For manifolds of dimension greater than one there are multiple definitions of local curvature, each suggesting a…

Quantization, a commonly used technique to reduce the memory footprint of a neural network for edge computing, entails reducing the precision of the floating-point representation used for the parameters of the network. The impact of such…

Statistical neurodynamics studies macroscopic behaviors of randomly connected neural networks. We consider a deep layered feedforward network where input signals are processed layer by layer. The manifold of input signals is embedded in a…

Sharpness of minima is a promising quantity that can correlate with generalization in deep networks and, when optimized during training, can improve generalization. However, standard sharpness is not invariant under reparametrizations of…

We introduce an intrinsic estimator for the scalar curvature of a data set presented as a finite metric space. Our estimator depends only on the metric structure of the data and not on an embedding in $\mathbb{R}^n$. We show that the…

Multi-layer feedforward networks have been used to approximate a wide range of nonlinear functions. An important and fundamental problem is to understand the learnability of a network model through its statistical risk, or the expected…

The scaling of model and data sizes has reshaped the AI landscape, establishing finetuning pretrained models as the standard paradigm for solving downstream tasks. However, dominant finetuning methods typically rely on weight adaptation,…

We propose Curved Inference - a geometric Interpretability framework that tracks how the residual stream trajectory of a large language model bends in response to shifts in semantic concern. Across 20 matched prompts spanning emotional,…

Overparameterized shallow neural networks admit substantial parameter redundancy: distinct parameter vectors may represent the same predictor due to hidden-unit permutations, rescalings, and related symmetries. As a result, geometric…