For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from to a compact Riemannian manifold without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…
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48 results for “piecewise Lipschitz”
GraN-GAN normalizes gradients for better GAN performance.
problem Improving image generation in GANs with piecewise linear discriminators.
method Piecewise Gradient Normalization (GraN) for input-dependent normalization.
result Significant performance gains in image generation across various datasets.
Theorem proves integrability for piecewise-smooth distributions.
problem Integrability of piecewise-smooth distributions.
method Generalizations of Frobenius integrability theorem.
result Sufficient criteria for complete integrability with bi-Lipschitz coordinates.
New cones in 4D space found with minimal mass.
problem Finding mass-minimizing piecewise linear cones in 4D space.
method Mass minimization via Lipschitz maps, classification of candidates.
result No additional mass-minimizing cones found outside five known cases.
New algorithms minimize regret in changing environments for piecewise Lipschitz functions.
problem Optimizing in unpredictable, time-varying conditions for piecewise Lipschitz functions.
method Introduced shifting regret as a metric, and provided bounds for .
result Improved bounds for -dispersed functions, with empirical validation in online clustering.
Corners can be identified by a drum's sound spectrum.
problem Determining the presence of corners in a drum's shape from its sound.
method Proving spectral invariance of corners in domains with Lipschitz, piecewise smooth boundaries.
result Corners are uniquely determined by a drum's spectrum among domains with fixed genus.
We study the Lipschitz simplicial volume, which is a metric version of the simplicial volume. We introduce the piecewise straightening procedure for singular chains, which allows us to generalize the proportionality principle and the product inequality to the case of complete Riemannian manifolds of finite volume with …
The study examines metrics on Riemannian spaces with bounded properties and finds conditions for Lipschitz and uniform bounds.
problem Investigating bounded rough Riemannian metrics and their properties.
method Analyzing the structure of bounded rough Riemannian metrics and finding conditions for Lipschitz and uniform bounds.
result Weak conditions are identified for Lipschitz and uniform bounds on the metrics.
We develop a method to learn neural network activations with controlled Lipschitz constant.
problem Increase neural network capacity while controlling Lipschitz constant.
method Variational framework to learn activation functions with piecewise-linear constraints.
result Proves existence of solutions with continuous and piecewise-linear activations.
We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipsc…
We develop some basic Lipschitz homotopy technique and apply it to manifolds with finite asymptotic dimension. In particular we show that the Higson compactification of a uniformly contractible manifold is mod acyclic in the finite dimensional case. Then we give an alternative proof of the Higher Signature Novikov …
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.
New method uses DC functions for piecewise linear regression.
problem Regression with piecewise linear constraints.
method Estimates piecewise linear convex functions using a difference of convex functions.
result Method achieves close to minimax statistical risk and comparable performance to existing methods.
The paper designs neural networks with assurance for controlling nonlinear systems.
problem Designing neural networks with assurance for nonlinear system control.
method Bounding the number of affine functions needed for a CPWA function, connecting it to a TLL NN architecture.
result The TLL NN architecture is parameterized by the number of affine functions in the CPWA function it realizes.
In this article we investigate a first order reparametrization-invariant Sobolev metric on the space of immersed curves. Motivated by applications in shape analysis where discretizations of this infinite-dimensional space are needed, we extend this metric to the space of Lipschitz curves, establish the wellposedness of…
New findings show non-open chronological futures in low regularity spacetimes.
problem Breakdown of Lorentzian causality theory in low regularity spacetimes.
method Refined notion of causal bubble and analysis of locally Lipschitz curves.
result Chronological futures may be non-open and differ from those defined via piecewise -curves.
Two-layer neural networks must be robust, even with arbitrary weights.
problem Proving the robustness of two-layer neural networks with arbitrary weights.
method Developed a new function-space covering method to prove the robustness law, replacing parameter-space covering.
result Proved the conjectured law for two-layer networks with arbitrary real weights, biases, and affine skip connections.
SRNF framework extends surface distance to Lipschitz surfaces.
problem Defining a distance metric for unparametrized surfaces.
method Square Root Normal Fields (SRNF) and Wasserstein Fisher Rao (WFR) metric.
result SRNF distance on Lipschitz surfaces is equivalent to WFR metric.
Unified framework for analyzing graph neural operators converging to graph limits.
problem Analyzing convergence of graph neural operators to graph limits.
method Develops a unified spectral framework for graph neural operators under various graphon assumptions.
result Unified framework enables direct comparison of convergence rates and tradeoffs.
The paper develops a new method for estimating non-parametric regression functions with spatio-temporal dependencies.
problem Estimating non-parametric regression functions with spatio-temporal dependencies.
method Locally Adaptive Regression Splines (LARS) with ADMM algorithm.
result The method shows superior performance compared to existing techniques.
New NN design for nonlinear systems control with guarantees.
problem Designing NN architectures for nonlinear system control with guarantees.
method Exploits system model to design NN architecture, uses TLL NN for approximation.
result Guaranteed NN architecture sufficient for implementing a controller.
Envelopes in Outer Spacemath.GT
Geometric study of Outer Space using envelopes and geodesics.
problem Understanding the geometry and structure of Outer Space .
method Study of envelopes in the asymmetric Lipschitz metric of .
result For almost all pairs of points in , their envelopes have dimension .
Improved robustness for deep neural networks with tighter bounds and attacks.
problem Loose upper bounds and prohibitive computation in existing adversarial robustness methods.
method Primal approach with exact Lipschitz certificates for ReLU networks and modern architectures, and novel Wasserstein Distributional Attacks.
result Tighter upper bounds and greater flexibility in attack points compared to existing methods.
Unified method for estimating properties of large domain distributions efficiently.
problem Estimating properties of distributions over large domains efficiently.
method Piecewise-polynomial approximation technique for constructing sample- and time-efficient estimators.
result Near-linear-time computable estimators with optimal and highly-concentrated approximation values.
The paper studies harmonic graphs in the Heisenberg group and their properties.
problem No analogous theorem exists for -minimal surfaces in the Heisenberg group.
method Introduced intrinsic Dirichlet energy and studied its critical points (contact harmonic graphs).
result Calibration condition and construction of energy-minimizing graphs with various singularities.
We study the differential geometric consequences of our previous result on the existence of fat triangulations, in conjunction with a result of Cheeger, Müller and Schrader, regarding the convergence of Lipschitz-Killing curvatures of piecewise-flat approximations of smooth Riemannian manifolds. A further application t…
PPGD solves nonconvex nonsmooth optimization problems without KL property.
problem Nonconvex and nonsmooth optimization problems in statistics and machine learning.
method Projective Proximal Gradient Descent (PPGD) for solving a class of nonconvex and nonsmooth problems.
result PPGD achieves a fast convergence rate of O(1/k^2) for k ≥ k_0.
In our previous paper [SIMAX 31 n.3 1491-1506(2010)], we studied the condition metric in the space of maximal rank matrices. Here, we show that this condition metric induces a Lipschitz-Riemann structure on that space. After investigating geodesics in such a nonsmooth structure, we show that the inverse of the smallest…
We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geome…
Paper tackles online learning with interval regret, achieving adaptive bounds.
problem Non-stationary online learning over time intervals.
method Two-layer online ensemble structure with gradient variation.
result Achieves strong theoretical guarantees with adaptive bounds.
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
problem Stability of piecewise flat Ricci flow.
method Linear stability analysis and numerical simulations.
result Adaptations avoided numerical instability and led to convergence to smooth solutions.
Neural networks can represent complex piecewise functions efficiently.
problem Representing continuous piecewise affine functions with neural networks.
method Two hidden layers with ReLU activation, neurons for pieces.
result CPA functions can be represented by a neural network with linear size.
Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.
problem Approximating local extrinsic curvature on discrete manifolds.
method Constructing discrete curvature forms on piecewise flat manifolds, using weighted sums of hinge angles.
result Converges to smooth curvature values as mesh refinement occurs, favorably comparing with other discrete approaches.
Study geometrically characterizes piecewise circular curves with decreasing curvature.
problem Characterizing piecewise circular curves with decreasing curvature.
method Introducing moduli spaces and relating them to Legendrian polygons.
result Proves the moduli space contains a connected component homeomorphic to the Fock-Goncharov space of positive flags.
Center and torsion of a quotient of the group of piecewise linear homeomorphisms of the real linemath.GT
The center of a quotient group of piecewise linear homeomorphisms is trivial.
problem Understanding the structure of a specific group of homeomorphisms.
method Analyzing a quotient of piecewise linear homeomorphisms of the real line.
result The center of the quotient group is trivial.
New algorithms cluster non-stationary time series data.
problem Clustering time series generated by piecewise stationary processes.
method Proposed a natural formulation and introduced a notion of consistency for clustering.
result Simple, efficient algorithms that work without additional assumptions.
The paper extends a variance gamma model to quadratic functions, reducing arbitrage and computational costs.
problem Creating an arbitrage-free interpolation for option pricing models.
method Generalizing the local variance gamma model to a piecewise quadratic local variance function.
result The quadratic model results in an arbitrage-free interpolation of class C3, reducing knots and computational cost.
New algorithm reveals piecewise affine structure of neural networks.
problem Lack of strong guarantees on deep neural networks' behavior in safety-critical applications.
method Developed a novel algorithm to compute the piecewise affine form of neural networks.
result Computed piecewise affine representations of neural networks with rectified linear unit activations.
This article provides an attempt to extend concepts from the theory of Riemannian manifolds to piecewise linear spaces. In particular we propose an analogue of the Ricci tensor, which we give the name of an Einstein vector field. On a given set of piecewise linear spaces we define and discuss (normalized) Ricci flows. …
We show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note …
PARC uses piecewise linear predictors for regression and classification.
problem Multivariate regression and classification problems.
method Alternates between ridge and softmax regression, and cluster assignment based on accuracy and separability.
result Converges to a local minimum in a finite number of steps.
Injectivity of geodesic ray transform for piecewise constants on compact manifolds.
problem Injectivity of geodesic ray transform for piecewise constant functions.
method Injectivity of geodesic ray transform on piecewise constant functions weighted by a continuous matrix weight.
result Injectivity of the geodesic X-ray transform on piecewise constant functions.
A piecewise flat Finsler metric on a triangulated surface is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex…
Neural network models improve survival analysis with reduced computation time.
problem Limited expressiveness of standard survival models.
method Piecewise neural network models of hazard and density functions.
result Models outperform state-of-the-art models with less computation time.
Estimates time-varying network connections using multi-stage smoothing.
problem Estimating edge probabilities of time-varying networks.
method Multi-stage smoothing: temporal local smoothing followed by node-domain smoothing.
result Captures both smooth temporal evolution and structural patterns in connectivity.
New GP model estimates piecewise continuous functions.
problem Piecewise continuous regression functions in scientific and engineering applications.
method Local Gaussian process model with partitioned local data and joint estimation of boundaries.
result Superior performance over conventional GP models in estimating piecewise regression functions.
Paper proposes algorithms to accurately identify breakpoints in piecewise regression.
problem Identifying accurate breakpoints in piecewise regression for better data fitting.
method Proposes novel greedy algorithms to minimize error and determine optimal breakpoints.
result The proposed algorithms outperform existing methods in accuracy and efficiency.
Simplicial, piecewise-flat discretizations of manifolds provide a clear path towards curvature analysis on discrete geometries and for solutions of PDE's on manifolds of complex topologies. In this manuscript we review and expand on discrete exterior calculus methods using hybrid domains. We then analyze the geometric …