arXiv research

New Transformers maintain Lipschitz continuity for robustness.

The paper proves Lipschitz continuity of cut times in spacetimes.

GroupSort neural networks can approximate Lipschitz continuous functions.

Constructs a map with prescribed local Lipschitz constants on a subset of a manifold.

The paper studies Lipschitz bounds for integral kernels under differentiability assumptions.

Investigates Lipschitz continuity in neural networks across various settings.

Proves existence of curved surfaces in hyperbolic space.

We show that the isoperimetric profile $h_{g(t)}(ξ)$ of a compact Riemannian manifold $(M,g)$ is jointly continuous when metrics $g(t)$ vary continuously. We also show that, when $M$ is a compact surface and $g(t)$ evolves under normalized Ricci flow, $h^2_{g(t)}(ξ)$ is uniform Lipschitz continuous and hence $h_{g(t)}(…

This research explores principles of Lipschitz continuity in neural networks for robustness and generalization.

We examine the impact of learning Lipschitz continuous models in the context of model-based reinforcement learning. We provide a novel bound on multi-step prediction error of Lipschitz models where we quantify the error using the Wasserstein metric. We go on to prove an error bound for the value-function estimate arisi…

In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map $α:{\mathcal H}\rightarrow\mathbb{R}^m$ is injective, with $(α(x))_k=|<x,f_k>|^2$, where $…

We model how Lipschitz continuity changes during neural network training.

Lipschitz continuity recently becomes popular in generative adversarial networks (GANs). It was observed that the Lipschitz regularized discriminator leads to improved training stability and sample quality. The mainstream implementations of Lipschitz continuity include gradient penalty and spectral normalization. In th…

Proves conditions for Fourier transforms in rank 1 symmetric spaces.

In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder continuity …

New method improves optimization algorithms without Lipschitz smoothness.

We study continuous maps between differential manifolds from a microlocal point of view. In particular, we characterize the Lipschitz continuity of these maps in terms of the microsupport of the constant sheaf on their graph. Furthermore, we give lower and upper bounds on the microsupport of the graph of a continuous m…

Maximal causal curves for Lipschitz metrics are either lightlike or timelike.

In binary classification and regression problems, it is well understood that Lipschitz continuity and smoothness of the loss function play key roles in governing generalization error bounds for empirical risk minimization algorithms. In this paper, we show how these two properties affect generalization error bounds in …

Study on residual Monge-Ampère mass of complex functions with directional Lipschitz continuity.

Lipschitz normalization boosts deep attention models, especially for graph neural networks.

Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(Ω, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

Study shows prior Lipschitz continuity can improve adversarial robustness of Bayesian Neural Networks.

Improved robustness of 1D CNNs for heart arrhythmia classification.

The paper proves topological stability between RCD spaces and Riemannian manifolds.

We provide an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The construction is based on results of A. Brudnyi and Yu. Brudnyi concerning linear e…

Proposes DMOC for more nuanced neural network robustness.

Efficient algorithm for global optimization of multivariate Lipschitz functions.

LALR adapts learning rate for faster convergence in regression and neural nets.

We prove the existence and uniqueness of solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free martingales. The main tool in our reasoning is Picard's iterative procedure and a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous martingale…

We consider Aubry-Mather theory for a subclass of class A spacetimes, i.e. compact vicious spacetimes with globally hyperbolic Abelian cover. In this subclass, called class A_1, we obtain improved results on timelike maximizers and Lipschitz continuity of the time separation of the Abelian cover on the i.g. optimal sub…

We prove a Frobenius-type theorem for singular distributions generated by a family of locally Lipschitz continuous vector fields satisfying almost everywhere a quantitative finite type condition.

Lipschitz RNNs improve stability and performance in various tasks.

Efficient binary sampling method for global optimization of univariate functions with low regret.

A generalization of the Flow-box Theorem is given. The assumption of continuous differentiability of the vector field is relaxed to a local Lipschitz condition. The theorem holds in any Banach space.

The paper quantifies the regularity of attention operations.

New method balances multivariate model fitting for mixed likelihoods.

In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation φ_y+ [φ^{2}/2]_t=w, where w is a bounded function depending o…

New methods solve MI problems with locally Lipschitz operators, improving solution efficiency.

ECP optimizes expensive functions without knowing Lipschitz constant.

Transfer learning for bandits with latent Lipschitz continuity.

In this paper, we investigate the underlying factor that leads to failure and success in the training of GANs. We study the property of the optimal discriminative function and show that in many GANs, the gradient from the optimal discriminative function is not reliable, which turns out to be the fundamental cause of fa…

New framework tightens certified robustness gaps in machine learning models.

Study examines stability of image-reconstruction algorithms using variational regularization.

We investigate the effect of explicitly enforcing the Lipschitz continuity of neural networks with respect to their inputs. To this end, we provide a simple technique for computing an upper bound to the Lipschitz constant---for multiple $p$-norms---of a feed forward neural network composed of commonly used layer types.…

New algorithms sample from log concave distributions without gradient Lipschitz continuity.