arXiv research

Helicoidal surfaces rotate and translate under mean curvature flow.

We describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal surfaces that rotate/translate with constant velocity under the flow. We look at…

Optimal transport aligns source and target distributions for linear regression in 2D.

New neural network processes 3D volumes with improved equivariance.

The study finds complete translating solitons for certain powers of Gaussian curvature in Riemannian products.

New model learns graph neural networks equivariant to various transformations.

Generating point clouds, e.g., molecular structures, in arbitrary rotations, translations, and enumerations remains a challenging task. Meanwhile, neural networks utilizing symmetry invariant layers have been shown to be able to optimize their training objective in a data-efficient way. In this spirit, we present an ar…

We extend randomized smoothing to cover parameterized transformations (e.g., rotations, translations) and certify robustness in the parameter space (e.g., rotation angle). This is particularly challenging as interpolation and rounding effects mean that image transformations do not compose, in turn preventing direct cer…

Deep Reinforcement Learning (DRL) has shown impressive performance on domains with visual inputs, in particular various games. However, the agent is usually trained on a fixed environment, e.g. a fixed number of levels. A growing mass of evidence suggests that these trained models fail to generalize to even slight vari…

We develop a family of techniques to align word embeddings which are derived from different source datasets or created using different mechanisms (e.g., GloVe or word2vec). Our methods are simple and have a closed form to optimally rotate, translate, and scale to minimize root mean squared errors or maximize the averag…

Invariances to translation, rotation and other spatial transformations are a hallmark of the laws of motion, and have widespread use in the natural sciences to reduce the dimensionality of systems of equations. In supervised learning, such as in image classification tasks, rotation, translation and scale invariances ar…

Even as deep neural networks (DNNs) have achieved remarkable success on vision-related tasks, their performance is brittle to transformations in the input. Of particular interest are semantic transformations that model changes that have a basis in the physical world, such as rotations, translations, changes in lighting…

The $H_\infty$ control design problem is considered for nonlinear systems with unknown internal system model. It is known that the nonlinear $H_\infty$ control problem can be transformed into solving the so-called Hamilton-Jacobi-Isaacs (HJI) equation, which is a nonlinear partial differential equation that is genera…

Study on Brownian motion on discrete curve spaces, proving stochastic completeness.

A rapidly growing area of work has studied the existence of adversarial examples, datapoints which have been perturbed to fool a classifier, but the vast majority of these works have focused primarily on threat models defined by $\ell_p$ norm-bounded perturbations. In this paper, we propose a new threat model for adver…

Researchers classify translators and rotators in hyperbolic 3-space for mean curvature flow.

New method reduces spatial graphs while preserving their topological features.

New method learns identity-preserving transformations on data manifolds without labels.

This paper addresses the model-free nonlinear optimal problem with generalized cost functional, and a data-based reinforcement learning technique is developed. It is known that the nonlinear optimal control problem relies on the solution of the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear partial differ…