Uniform small energy regularity for fractional geometric problems proved.
problem Proving regularity for fractional geometric problems.
method Analyzing parabolic boundary reaction Ginzburg-Landau problems and fractional harmonic maps to spheres.
result Uniform small energy regularity results for s∈(0,1), answering a posed question. Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.
problem Bounding curvature of regularized metrics with constraints on Ricci tensor and injectivity radius.
method Mollification of riemannian metrics, uniform W2,p-harmonic radius bounds, Ricci tensor bounds, injectivity radius bounds. result Uniform estimate on the change of sectional curvature for regularized metrics.
The paper proposes a uniformity regularization scheme to improve deep neural network transferability.
problem Improving deep neural network transferability and adaptation to new tasks.
method Introduces a uniformity regularization scheme to encourage high uniformity in embedding space.
result Uniformity regularization consistently offers benefits over baseline methods and achieves state-of-the-art performance in Deep Metric Learning and Meta-Learning.
We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem (P,C) is equivalent to the uniform regularity of the natural family (Px,Cx) of associated boundary value …
New approach finds minima of geodesic lengths for non-uniform fillings.
problem Finding minima of geodesic length functions for non-uniform fillings.
method Elementary optimization for 4-regular topological fillings, analysis of fat graphs and optimization techniques.
result Minima of geodesic length functions are found to be at triangle surfaces in both analyzed classes of non-uniform fillings.
The study explores maps of 2- and 3-uniform tilings on the torus.
problem Understanding the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
method Analyzing the quotient maps of 2- and 3-uniform tilings on the torus.
result Bounds on the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
We design and mathematically analyze sampling-based algorithms for regularized loss minimization problems that are implementable in popular computational models for large data, in which the access to the data is restricted in some way. Our main result is that if the regularizer's effect does not become negligible as th…
We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of intrinsic elliptic Lagrangians which are sub-critical or critical. They are derived using uniform ε−regularity estimates whic…
Entropy for uniform hypergraphs defined via tensor theory.
problem Entropy calculation for uniform hypergraphs.
method Probability distribution of generalized singular values from Laplacian tensors, Shannon entropy formula.
result Tensor entropy is a measure of regularity for uniform hypergraphs.
We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with…
Adapting \cite{strz3}, we define generalized p-harmonic maps into Riemannian homogeneous targets, a notion of solutions not belonging to the energy space. Restricting our attention to the subcritical range p greater than the domain dimension n, we show a uniform C1,α-regularity result for a sequence of such …
In continuing the study of harmonic mapping from 2-dimensional Riemannian simplicial complexes in order to construct minimal surfaces with singularity, we obtain an a-priori regularity result concerning the real analyticity of the free boundary curve. The free boundary is the singular set along which three disk-type mi…
The method to derive uniform bounds with Gaussian and Rademacher complexities is extended to the case where the sample average is replaced by a nonlinear statistic. Tight bounds are obtained for U-statistics, smoothened L-statistics and error functionals of l2-regularized algorithms.
We prove that Riemannian metrics with a uniform weak norm can be smoothed to having arbitrarily high regularity. This generalizes all previous smoothing results. As a consequence we obtain a generalization of Gromov's almost flat manifold theorem. A uniform Betti number estimate is also obtained.
Proves regularity of geodesic equation on Hermitian manifolds.
problem Regularity of geodesic equation in mixed volume forms space.
method Ellipticity conditions, uniform Laplacian estimates, explicit subsolutions.
result Existence of unique C1,1 solution to Donaldson equation. The paper proves rigidity and uniformization theorems for infinite circle patterns and convex polyhedra in hyperbolic 3-space.
problem Characterize infinite circle patterns and convex polyhedra in hyperbolic 3-space.
method Extends techniques from previous work to prove rigidity and uniformization theorems for infinite circle patterns and convex polyhedra.
result Establishes existence and rigidity of infinite regular circle patterns and convex trivalent polyhedra.
Given a flag in each of the vertex-transitive tessellations of the Euclidean plane by regular polygons, we determine the flag stabilizer under the action of the automorphism group of a regular cover. In so doing we give a presentation of these tilings as quotients of regular (infinite) polyhedra.
Fiedler regularization uses spectral graph theory to improve neural network performance.
problem Improving neural network performance by penalizing weights based on connectivity.
method Uses the Fiedler value of the neural network's graph as a regularization tool, providing theoretical and computational methods.
result Demonstrates Fiedler regularization's effectiveness in improving neural network performance.
Unified geometric description of Kepler flow across all energies.
problem Understanding the Kepler flow across different energy levels.
method Revisiting Ligon--Schaaf regularization and identifying geometric origins of anomalies.
result Unified geometric description of Kepler flow for all energies.
We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.
We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein met…
The paper analyzes convergence of neural SDEs as sample size increases.
problem Understanding the limiting behavior of neural SDEs as sample size grows.
method Analyzes Hamilton-Jacobi-Bellman equation and uses stochastic maximum principle.
result Convergence of minima and optimal parameters of neural SDEs as sample size increases.
Uniform K-homology theory applied to elliptic operators on manifolds with boundary.
problem Developing a theory to study boundary conditions for elliptic operators on non-compact manifolds.
method Theory of relative uniform K-homology, developing a relative index map.
result Uniform K-homology classes of boundary conditions and their connection to the higher ρ-invariant.
Deep networks adapt to function regularity and data distribution.
problem Understanding deep learning's adaptability to function regularity and data distribution.
method Developed nonparametric approximation and estimation theories for a broad class of functions using deep ReLU networks.
result Deep neural networks are adaptive to different regularity of functions and nonuniform data distributions.
The study shows that certain graphs are regular at boundary points.
problem Boundary regularity of anisotropic minimal Lipschitz graphs.
method Proves regularity for graphs with bounded anisotropic mean curvature and atomic energy condition.
result Regularity at boundary points with density bounded above by 1/2 + σ.
Multiple kernel learning (MKL), structured sparsity, and multi-task learning have recently received considerable attention. In this paper, we show how different MKL algorithms can be understood as applications of either regularization on the kernel weights or block-norm-based regularization, which is more common in str…
Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly …
We study a generalized Abreu Equation in n-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform K-stability.
This work analyzes how to choose regularization norms for adversarial training in high dimensions.
problem Choosing the right regularization norm for adversarial training in high-dimensional settings.
method Derives asymptotic descriptions and uniform convergence bounds for robust, regularized empirical risk minimizers.
result Characterizes the relationship between perturbation size and optimal regularization choice.
We show that any d-Ahlfors regular subset of Rn supporting a weak (1,d)-Poincaré inequality with respect to surface measure is uniformly rectifiable.
The Sinkhorn-Knopp derivatives converge with linear rate.
problem Optimal transport problem with entropic regularization.
method Iterative proportional fitting procedure.
result Derivatives converge with linear rate.
Paper tackles estimating initial conditions of spatio-temporal processes from sparse data.
problem Estimating initial conditions of spatio-temporal advection-diffusion processes from sparse data.
method Regularized convex optimization problem with Alternating Direction Method of Multipliers.
result Efficient solutions for non-uniform and shifted uniform sampling schemes.
Variational dropout (VD) is a generalization of Gaussian dropout, which aims at inferring the posterior of network weights based on a log-uniform prior on them to learn these weights as well as dropout rate simultaneously. The log-uniform prior not only interprets the regularization capacity of Gaussian dropout in netw…
In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on Kähler manifolds. We first define a notion of weak solution of CSCK for an L∞ Kähler metric. The main result is to show that such a weak solution (with uniform L∞ bound…
The paper proposes a method for generating uniform interpolations on data manifolds.
problem Generating high-quality interpolations between data samples on complex manifolds.
method Autoencoder network with interpolation network, regularized by a Riemannian metric.
result The method generates interpolations that remain within the manifold's distribution.
In this paper, we study the structure of the pointed-Gromov-Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger-Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any Ricci shrinker limit space is co…
Uniform convexity in divisible domains leads to hyperbolic geometry.
problem Understanding the geometry of divisible convex sets in Finsler manifolds.
method Proving β-uniform convexity of a specific Finsler metric. result A strictly convex divisible domain induces a β-uniformly convex Finsler metric. This paper proves Hölder continuity for complex Monge-Ampère equations on Kähler varieties.
problem Establishing Hölder estimates on singular Kähler varieties.
method Geometric regularization based on partial C0 estimate. result Uniform Hölder continuity for complex Monge-Ampère equations on Kähler varieties.
Proves an ε-regularity theorem for Ricci flows, leading to new singularity estimates.
problem Analyzing singularity models in Fano Kähler-Ricci flows.
method Proves ε-regularity theorem and uses it to derive new estimates.
result Establishes new estimates for singularity models of Fano Kähler-Ricci flows.
Analytic networks with bounded coefficients can't outperform polynomial approximations.
problem Approximation limits of neural networks with analytic activation functions under coefficient constraints.
method Deterministic analysis using comparison argument and Bernstein-type estimates.
result Networks with analytic activation functions and controlled coefficients cannot outperform classical polynomial approximation rates on non-analytic targets.
Paper tackles low-rank matrix recovery with column ℓ2,0-norm regularization.
problem Low-rank matrix recovery problems with column sparsity constraints.
method Developed alternating majorization-minimization (AMM) methods with extrapolation and hybrid AMM.
result Global convergence analysis and superior performance in matrix completion problems.
Stability result for a popular algorithm in optimal transport.
problem Stability of the Iterative Proportional Fitting Procedure in time and metric.
method Uniform stability analysis in the 1-Wasserstein metric.
result Quantitative stability result for entropy-regularized Optimal Transport and Schrödinger bridges.
Flow adjusts curvature to avoid a fixed region, proving bounds and regularity.
problem Adjusting curvature flow to avoid a fixed region.
method Flow by powers of Gauss curvature, proving optimal curvature bounds and regularity.
result Proves optimal curvature bounds and long time existence for all dimensions and powers.
Study on Kähler-Ricci flow's Hölder regularity on compact manifolds.
problem Hölder regularity of Kähler-Ricci flow on compact Kähler manifolds.
method Adapting Hein-Tosatti's method for collapsing Calabi-Yau metrics, uniform spatial Hölder estimate obtained for all time.
result Uniform spatial Hölder estimate of Kähler-Ricci flow for all time.
Choquet regularization improves exploration in RL.
problem Improving exploration in reinforcement learning.
method Introducing Choquet regularizers to measure and manage exploration, reformulating RL problems and deriving explicit solutions.
result Explicit optimal distributions and Choquet regularizers for various exploratory samplers.
Uniform consistency proven for spatial distribution and depth estimators in any dimension.
problem Uniform consistency of spatial distribution and depth estimators in arbitrary dimensions.
method Proof of uniform L1-consistency using sample size n as the only dependency. result Consistency rate is independent of dimension d and sample size n. New methods improve convergence in non-convex non-smooth learning problems.
problem Sparse learning from high-dimensional data with non-convex, non-smooth regularizers.
method Stochastic proximal gradient methods with arbitrary sampling.
result Independent sampling improves performance over uniform sampling.
Spectral algorithm recovers community structure in sparse hypergraphs.
problem Community detection in sparse random hypergraphs with community structure and higher-order interactions.
method Spectral algorithm with three steps: hyperedge selection, spectral partition, and correction/merging.
result Weak consistency achieved for weak signal-to-noise ratio.