Uniform Lipschitz continuity of isoperimetric profiles in evolving surfaces.
problem Uniform Lipschitz continuity of isoperimetric profiles in evolving surfaces.
method Normalized Ricci flow on compact surfaces.
result Uniform Lipschitz continuity of isoperimetric profiles under normalized Ricci flow.
Characterizes Lipschitz continuity using microlocal sheaf theory.
problem Understanding Lipschitz continuity on differential manifolds.
method Microlocal sheaf theory applied to the graph of continuous maps.
result Characterization of Lipschitz continuity in microlocal terms.
New Transformers maintain Lipschitz continuity for robustness.
problem Ensuring robustness in Transformers for safety-sensitive applications.
method Introducing gradient-descent-type in-context Transformers with explicit Euler steps of negative gradient flows.
result Universal approximation theorem for Lipschitz continuous Transformers.
Proves existence of curved surfaces in hyperbolic space.
problem Finding surfaces with specific curvature and boundary conditions.
method Proves existence using Weingarten curvature and asymptotic boundary conditions.
result Proves existence of locally Lipschitz continuous hypersurfaces.
The paper proves Lipschitz continuity of cut times in spacetimes.
problem Lipschitz continuity of cut times in globally hyperbolic spacetimes.
method Adapted Itoh-Tanaka method to Lorentzian setting.
result Lipschitz continuity of cut times with quantitative estimates.
GroupSort neural networks can approximate Lipschitz continuous functions.
problem Understanding and improving the expressive power of neural networks with Lipschitz constraints.
method Introduced and studied GroupSort neural networks with constraints on weights, proving their ability to approximate Lipschitz continuous functions.
result GroupSort networks can represent any Lipschitz continuous piecewise linear functions and are well-suited for approximating general Lipschitz continuous functions.
This research explores principles of Lipschitz continuity in neural networks for robustness and generalization.
problem Ensuring robustness and generalization in neural networks, especially to small input perturbations and out-of-distribution data.
method Two complementary perspectives: internal (training dynamics) and external (frequency signal propagation).
result Advances in understanding the principles of Lipschitz continuity in neural networks.
We model how Lipschitz continuity changes during neural network training.
problem Understanding how Lipschitz continuity evolves during training.
method We use a system of stochastic differential equations to capture the dynamics of Lipschitz continuity under SGD.
result We identify three factors driving the evolution of Lipschitz continuity: gradient flow projection, gradient noise, and Hessian projection.
New method improves GAN training stability and quality.
problem Improving training stability and sample quality in GANs.
method Proposes a new method for Lipschitz continuity in GANs that is efficient and unbiased.
result Demonstrates the effectiveness of the new method in various GAN training scenarios.
The paper explores how Lipschitz-continuity improves GAN training stability and quality.
problem Failure and instability in GAN training due to unreliable gradient from optimal discriminative function.
method Investigates the property of optimal discriminative function and proves Lipschitz-continuity is a solution.
result Lipschitz-continuity condition ensures convergence and leads to more stable and higher quality generated samples.
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 shows prior Lipschitz continuity can improve adversarial robustness of Bayesian Neural Networks.
problem Improving adversarial robustness of Bayesian Neural Networks.
method Analysis of i.i.d., zero-mean Gaussian priors and posteriors approximated via mean-field variational inference.
result Adversarial robustness is sensitive to the prior variance.
Investigates Lipschitz continuity in neural networks across various settings.
problem Understanding the Lipschitz behavior of neural networks.
method Empirical investigation of Lipschitz bounds in different neural network architectures and datasets.
result Remarkable fidelity of the lower Lipschitz bound and a Double Descent trend in both upper and lower bounds.
We study Lipschitz models in reinforcement learning to bound prediction and value-function errors.
problem Bounding errors in reinforcement learning models with Lipschitz continuity constraints.
method We provide bounds on multi-step prediction error and value-function estimate using the Wasserstein metric for Lipschitz models.
result Lipschitz models lead to bounded errors in prediction and value-function estimates.
We enforce Lipschitz continuity in neural networks to improve performance.
problem Improving neural network performance with limited data.
method Enforcing Lipschitz continuity as a regularisation technique.
result Models with bounded Lipschitz constants outperform other regularisers.
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…
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 : H → R m α:{\mathcal H}\rightarrow\mathbb{R}^m α : H → R m is injective, with ( α ( x ) ) k = ∣ < x , f k > ∣ 2 (α(x))_k=|<x,f_k>|^2 ( α ( x ) ) k = ∣ < x , f k > ∣ 2 , where $…
New method for RL tasks transfer using Lipschitz continuity.
problem Knowledge transfer in RL tasks over time.
method Established Lipschitz continuity between MDPs and applied it to RL.
result Improved convergence rate and no negative transfer with high probability.
Study on residual Monge-Ampère mass of complex functions with directional Lipschitz continuity.
problem Estimating the residual Monge-Ampère mass of plurisubharmonic functions.
method General decomposition formula under Sasakian structure, L 1 L^{1} L 1 -apriori estimate, upper-bound estimate on residual mass. result Upper-bound estimate on residual mass for uniformly directional Lipschitz continuity confirmed.
New theorem for deep neural networks improves classification margins.
problem Improving classification margins in deep neural networks.
method Local class-purity theorem and margin p-values for training and testing samples.
result Enhanced understanding and computation of classification margins.
Proposes a weaker version of Strong Cosmic Censorship with curvature bounds.
problem The original Strong Cosmic Censorship conjecture.
method Weakens the conjecture to allow manifolds with bounded curvature and Lipschitz continuity of metrics.
result Proves the conjecture with bounded curvature for sufficiently large p (p>4 with uniform bounds, p>2 without uniform bounds).
Lipschitz normalization boosts deep attention models, especially for graph neural networks.
problem Gradient explosion in deep graph attention networks leads to poor performance.
method Enforcing Lipschitz continuity by normalizing attention scores.
result Deep GAT models with LipschitzNorm achieve state-of-the-art results for tasks with long-range dependencies.
The paper studies Lipschitz bounds for integral kernels under differentiability assumptions.
problem Understanding the Lipschitz continuity of feature maps associated with integral kernels.
method Analyzes differentiability assumptions to derive explicit formulas for Lipschitz constants and conditions for non-Lipschitz continuity.
result Explicit formulas and conditions for Lipschitz continuity of feature maps associated with various kernels.
For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from ( Ω , g ) (Ω, g) ( Ω , g ) to a compact Riemannian manifold ( N , h ) ⊂ R k (N,h)\subset\mathbb R^k ( N , h ) ⊂ R k without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…
New methods solve MI problems with locally Lipschitz operators, improving solution efficiency.
problem Solving monotone inclusions with locally Lipschitz continuous operators.
method Primal-dual extrapolation methods using backtracking line search.
result Improved operation complexity for solving MI problems.
Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.
problem Proving Lipschitz continuity of sub-elliptic harmonic maps between singular spaces.
method Analyzing sub-elliptic harmonic maps from the Heisenberg group into CAT(0) spaces.
result Local Lipschitz continuity established for sub-elliptic harmonic maps.
The paper quantifies the regularity of attention operations.
problem Quantifying the regularity of attention operations.
method Proposes a new mathematical framework using measure theory and integral operators.
result Proves attention operation is Lipschitz continuous on compact domains and provides an estimate of its Lipschitz constant.
New framework tightens certified robustness gaps in machine learning models.
problem Persistent gap between theoretical certified robustness and empirical accuracy.
method Leverages Lipschitz continuity and novel confidence intervals.
result Improves robust accuracy, compressing the gap between theory and practice.
A scattering transform defines a signal representation which is invariant to translations and Lipschitz continuous relatively to deformations. It is implemented with a non-linear convolution network that iterates over wavelet and modulus operators. Lipschitz continuity locally linearizes deformations. Complex classes o…
New algorithm verifies deep neural networks with guaranteed bounds.
problem Verifying deep neural networks' correctness is hard.
method Adaptive nested optimisation for reachability analysis.
result Efficiently verifies a broader class of DNNs than previous methods.
New method improves optimization algorithms without Lipschitz smoothness.
problem Improving optimization algorithms in the absence of Lipschitz smoothness.
method Dual kernel conditioning (DKC) to provide dual Lipschitz continuity.
result First complexity bounds and iterate convergence for random reshuffling mirror descent.
Proves Lipschitz continuity for solutions of certain nonlinear elliptic equations.
problem Interior Lipschitz regularity for continuous viscosity solutions of nonlinear degenerate elliptic equations.
method Establishes interior Lipschitz regularity using a weak form of the strong comparison principle.
result Weak form of the strong comparison principle (principle of propagation of touching points) for specific operators.
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 …
Optimization algorithms' regret bounds guide choice based on data and loss function.
problem Choosing the best optimization algorithm for a given dataset and loss function.
method Assumed convex and Lipschitz continuous loss function; compared traditional vs adaptive algorithms.
result Traditional algorithms' regret bounds are different from adaptive ones, providing a guide.
Efficient algorithm for global optimization of multivariate Lipschitz functions.
problem Global optimization of multivariate Lipschitz continuous functions.
method Proposes an efficient minimax optimal algorithm using a predetermined query creation rule.
result Achieves an average regret bound of O ( L n T − 1 n ) O(L\sqrt{n}T^{-\frac{1}{n}}) O ( L n T − n 1 ) , minimax optimal. This paper presents a novel nonmyopic adaptive Gaussian process planning (GPP) framework endowed with a general class of Lipschitz continuous reward functions that can unify some active learning/sensing and Bayesian optimization criteria and offer practitioners some flexibility to specify their desired choices for defi…
Study examines stability of image-reconstruction algorithms using variational regularization.
problem Stability and robustness of image-reconstruction algorithms in medical imaging.
method Review and novel stability results for ℓ p \ell_p ℓ p -regularized linear inverse problems, focusing on p ∈ ( 1 , ∞ ) p\in(1,\infty) p ∈ ( 1 , ∞ ) . result Guarantees Lipschitz continuity for small p p p and Hölder continuity for larger p p p in L p ( Ω ) L_p(Ω) L p ( Ω ) function spaces. New algorithms sample from log concave distributions without gradient Lipschitz continuity.
problem Sampling from log concave distributions without gradient Lipschitz continuity.
method Two algorithms based on monotone polygonal (tamed) Euler schemes.
result Non-asymptotic 2-Wasserstein distance bounds between the process and target measure.
The paper proves deep learning can be robust with certain loss functions.
problem The robustness of deep learning models under flawed data.
method Empirical-risk minimization with unbounded, Lipschitz-continuous loss functions.
result These loss functions provide efficient prediction under minimal data assumptions.
LALR adapts learning rate for faster convergence in regression and neural nets.
problem Finding optimal learning rates for faster convergence in regression and neural networks.
method Lipschitz continuity theory applied to Mean Absolute Error and Quantile loss functions.
result Adaptive learning rate policy enables up to 20x faster convergence.
Efficient binary sampling method for global optimization of univariate functions with low regret.
problem Global optimization of univariate loss functions.
method Binary sampling approach to circumvent hard-to-determine query points in traditional methods.
result At most L log ( 3 T ) L\log (3T) L log ( 3 T ) and 2.25 H 2.25H 2.25 H regret for L L L -Lipschitz continuous and H H H -Lipschitz smooth functions respectively. Paper proposes a new approach to improve WGAN training and achieves state-of-the-art results.
problem Difficulty in training GANs, especially Wasserstein GANs.
method Introduces a consistency term to enforce Lipschitz continuity in WGAN training.
result Achieves inception score of more than 5.0 with only 1,000 CIFAR-10 images and exceeds 90% accuracy on CIFAR-10 with 4,000 labeled images.
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.
Invertible DenseNets improve model efficiency and performance.
problem Improving model efficiency and performance in neural networks.
method Enforcing invertibility in DenseNets by satisfying the Lipschitz constraint and proposing a learnable concatenation.
result i-DenseNets outperform Residual Flows in negative log-likelihood on various datasets.
Study shows limitations of Lie bracket commutation for nonsmooth vector fields.
problem Limitations of Lie bracket commutation for nonsmooth vector fields.
method Analysis of nonsmooth vector fields, focusing on commutation of flows and Lie bracket conditions.
result Lie bracket commutation cannot be extended to general a.e. differentiable vector fields, but holds for certain Sobolev regular fields.
Transfer learning for bandits with latent Lipschitz continuity.
problem Learning to transfer structural information from prior tasks to new tasks.
method Proposes a framework to estimate Lipschitz constant from prior tasks and apply it to new tasks.
result Regret bound close to oracle algorithm with full knowledge of Lipschitz constant under mild assumptions.
AutoShuffleNet learns permutation matrices in CNNs for improved accuracy.
problem Manual design of channel shuffling in ShuffleNet.
method Learning permutation matrices via an exact Lipschitz continuous penalty in deep learning.
result Improved classification accuracies on CIFAR-10 and ImageNet datasets.
The curse of dimensionality affects neural network optimization, especially with smooth functions.
problem The curse of dimensionality in neural network optimization.
method Examined through the evolution of the parameter distribution under 2-Wasserstein gradient flow.
result The curse of dimensionality persists in neural network optimization, even with smooth functions.