Study on cosine function in smooth normed spaces, proving symmetry and characterizing planes.
problem Understanding cosine function properties in smooth normed spaces.
method Proved symmetry and derived cosine function in terms of norm's Gateaux derivative.
result Cosine function is symmetric if and only if space is Euclidean.
Kernel interpolation is inconsistent for norms with smoothness above a constant.
problem Inconsistency of kernel interpolation in reproducing kernel Hilbert spaces.
method Lower bounds for generalization error in Sobolev norms.
result Kernel interpolation is always inconsistent for norms with smoothness above a constant.
Study on Santaló point for convex bodies in normed spaces.
problem Exploring Santaló point for convex bodies in normed spaces.
method Existence and uniqueness proof for C1 norms, dual Santaló point for smooth curved unit balls. result Existence and uniqueness of Santaló point for convex bodies in normed spaces.
Novel framework improves randomized smoothing for various norms.
problem Developing robust defenses against adversarial attacks.
method Proposed a novel framework for devising and analyzing randomized smoothing schemes.
result Significantly improved certified accuracy in ℓ1 on standard datasets.
Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone K, a norm ∥⋅∥ and a smooth convex function f, we want either 1) to minimize the norm over the intersection of the cone and a level set of f, or 2) to minimize over the…
New Brownian motion defined in Minkowski normed spaces.
problem Constructing Brownian motion in non-Euclidean spaces.
method Singular McKean--Vlasov stochastic differential equation.
result Pathwise uniqueness of solutions to the stochastic differential equation.
Study shows horofunction compactification's topology matches dual norm's unit ball.
problem Global topology of horofunction compactification of Finsler manifolds.
method Construct explicit homeomorphisms for various spaces.
result Horofunction compactification homeomorphic to dual norm's unit ball.
We consider immersions admitting uniform graph representations over the affine tangent space over a ball of fixed radius r>0. We show that for sufficiently small C^0-norm of the graph functions, each graph function is smooth with small C^1-norm.
The real homology of a compact Riemannian manifold M is naturally endowed with the stable norm. The stable norm on H1(M,R) arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finite-dimensional vector space H1(M,R) are st…
Smooth models refine topological K-theory for differential extensions.
problem Constructing a smooth version of differential K-theory.
method Using norm completions of stable Grassmannian and stable unitary group models.
result Isomorphic to unique differential extension of K-theory.
New methods improve online matrix optimization with reduced computational cost.
problem Online matrix optimization with operator norm constraints.
method Gradient-based prediction scheme with smoothed potentials for nuclear norm.
result Adaptive matrix optimizers match Shampoo's regret up to a constant factor.
New inequalities link probability density norms to Sobolev norms and Kantorovich distances.
problem Bounding probability density norms on smooth weighted Riemannian manifolds.
method Refining and generalizing interpolation inequalities under CD(0,∞) condition. result Established new inequalities linking Lp norms to Sobolev norms and Kantorovich distances. In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete Ck Finsler manifold M is determined by the normed algebra Cbk(M) of all real-valued, bounded and Ck smooth functions with bounded derivative defined on M. As a consequence, we obtain: (i) the Finsler structu…
Geodesics in Kähler metrics connect metrics with constant scalar curvature.
problem Deriving geodesics for relatively Kähler metrics on fibrations.
method Deriving geodesic equation, proving uniqueness, convexity of log-norm functional.
result Fibrations with optimal symplectic connections are polystable.
New method improves signal estimation by convexifying ℓ0-norm constraints.
problem Signal estimation with sparsity and smoothness priors.
method Iterative convex conic quadratic relaxations exploiting ℓ0-norm and smoothness terms. result Significantly better estimators than ℓ1-norm approaches and interpretable parameters. The paper extends the convolution operator to non-smooth valuations using geometric inequalities.
problem Extending the convolution operator to non-smooth valuations.
method Using geometric inequalities derived from optimal transport methods.
result Constructing a continuous extension of the convolution operator on smooth valuations to non-smooth valuations.
The study analyzes convergence rates for sparse pivotal estimators in high-dimensional regression.
problem Sparse pivotal estimation in high-dimensional regression problems.
method Theoretical analysis and comparison of non-smooth + non-smooth optimization problems, including smoothing techniques.
result Minimax sup-norm convergence rates for square-root Lasso-type estimators are derived.
In four and higher dimensions, we show that any stationary admissible Yang-Mills field can be gauge transformed to a smooth field if the L2 norm of the curvature is sufficiently small. There are three main ingredients. The first is Price's monotonicity formula, which allows us to assert that the curvature is small n…
Characterizes isometries between non-reversible Finsler manifolds.
problem Understanding isometries in non-reversible Finsler manifolds.
method Generalization of Myers-Nakai Theorem for Riemannian manifolds, modification of function spaces to accommodate asymmetric structure.
result Functional characterization of isometries between non-reversible Finsler manifolds.
We consider a complete noncompact smooth metric measure space (Mn,g,e−fdv) and the associated drifting Laplacian. We find sufficient conditions on the geometry of the space so that every nonnegative f-subharmonic function with bounded weighted L1 norm is constant.
Study on smooth moduli space of Riemann surfaces with uniformization theorem.
problem Understanding the smooth moduli space of Riemann surfaces and their uniformization.
method Developed techniques of rational norm of homological marking and decomposition of probability measures.
result Closed Riemann surfaces are uniformizable by Schottky groups of Hausdorff dimension less than one.
New curvature measures characterize non-convex Wulff shapes in normed spaces.
problem Characterizing non-convex sets with curvature measures.
method Extending curvature measures to non-convex and non-smooth sets in normed spaces.
result Finite unions of disjoint Wulff shapes are the only sets with proportional curvature measures.
Certified robustness for ImageNet models with randomized smoothing.
problem Creating robust models against adversarial attacks.
method Randomized smoothing with Gaussian noise.
result Certified top-1 accuracy of 49% on ImageNet under small ℓ2 perturbations. SGD achieves a O(ε−4) bound for minimizing gradient norm of smooth functions.
problem Finding stationary points with SGD for gradient norm minimization.
method Stochastic Gradient Descent (SGD) for smooth, possibly nonconvex functions.
result The O(ε−4) bound for gradient norm minimization cannot be improved upon. Study shows smooth holomorphic structures can be approximated from weak connections.
problem Approximating smooth holomorphic structures from weak connections.
method Proves connections with specific properties can be approximated in Sobolev norms.
result Strong approximations of smooth holomorphic structures from weak connections.
Improved algorithms solve ℓp-norm regression problems efficiently.
problem Efficiently solving ℓp-norm regression problems for p∈(1,2)∪(2,∞). method Iterative refinement scheme using smoothed ℓp-norms to improve solutions. result Solves ℓp-norm regression to 1/extpoly(n) accuracy in ildeOp(m31) iterations. Smooth approximation of integral cycles in manifolds.
problem Approximating integral cycles in Riemannian manifolds.
method Approximation of integral cycles by smooth submanifolds with controlled area and singularities.
result Integral cycles can be approximated by smooth submanifolds with controlled area and singularities.
The study solves the isoperimetric problem for Heisenberg group norms.
problem Solving the isoperimetric problem for anisotropic norms in the Heisenberg group.
method Representation formula for perimeter, foliation property, differential equation characterization, approximation procedure.
result Characterization of isoperimetric sets as sub-Finsler analogues of Pansu's bubbles.
Unified scalable equivalent formulations for Schatten quasi-norms improve efficiency.
problem Efficiently solving Schatten quasi-norm minimization problems for large-scale matrices.
method Proved equivalence between Schatten-p quasi-norm and product/sum of Schatten-p1 and p2 norms of factor matrices.
result Transformed SQNM problems into simpler, more efficient algorithms for p>1/2.
Study shows momentum-based optimizers like Muon and MomentumGD bias towards KKT points in smooth homogeneous models.
problem Understanding the implicit bias of momentum-based optimizers on smooth homogeneous models.
method Analysis of Muon, MomentumGD, Signum, and Adam optimizers under decaying learning rate schedules.
result Momentum-based optimizers approximate steepest descent trajectories and bias towards KKT points of margin maximization problems.
The paper proves density of smooth functions in Sobolev spaces on certain manifolds.
problem Density of smooth functions in Sobolev spaces on manifolds with unbounded geometry.
method Distance-like function with bounded gradient and mild growth of Hessian, proving density results.
result Smooth compactly supported functions are dense in W2,p on the considered manifolds. The curvature-dimension condition fails in sub-Finsler geometry, extending previous results in sub-Riemannian geometry.
problem The failure of the curvature-dimension condition in sub-Finsler geometry.
method Non-trivial adaptation of Juillet's work, introduction of new tools and ideas.
result The CD(K,N) condition does not hold in sub-Finsler geometry for various norms and measures. Proposes a filtering method for cluster analysis using ℓ0-norm regularization.
problem Improving cluster analysis by filtering data.
method Minimizes a least squares function with a weighted ℓ0-norm penalty, approximated by smooth non-convex functions. result The proposed method can enhance existing clustering techniques.
We prove that the Yamabe invariant of any simply connected smooth manifold of dimension n greater than four is non-negative. Equivalently that the infimum of the L^{n/2} norm of the scalar curvature, over the space of all Riemannian metrics on the manifold, is zero.
Kähler information manifolds for signal filters in weighted Hardy spaces are explored.
problem Developing a geometric framework for signal processing filters in weighted Hardy spaces.
method Introducing weighted Hardy spaces and smooth transformations of transfer functions, demonstrating the Kähler manifold structure.
result The Riemannian geometry of weighted Hardy norms for transfer functions forms a Kähler manifold.
We introduce a new family of matrix norms, the "local max" norms, generalizing existing methods such as the max norm, the trace norm (nuclear norm), and the weighted or smoothed weighted trace norms, which have been extensively used in the literature as regularizers for matrix reconstruction problems. We show that this…
The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…
This paper assesses Gaussian and Exponential mechanisms for certifying adversarial robustness.
problem Certifying adversarial robustness using randomized smoothing mechanisms.
method Proposes a generic framework to assess the appropriateness of randomized smoothing mechanisms.
result Gaussian mechanism is an appropriate option for certifying both ℓ2-norm and ℓ∞-norm robustness. This paper optimizes exclusive sparsity norm minimization with random groupings.
problem Sparse feature selection with even distribution across groups.
method Developed efficient algorithms for exclusive sparsity norm minimization with smooth and non-smooth losses, and proposed random grouping scheme for unknown group information.
result Achieved optimal convergence rate for exclusive sparsity norm minimization.
Control data constructed for smooth weak deformation retraction of stratified spaces.
problem Construct control data for smooth weak deformation retraction of stratified spaces.
method Show smooth local triviality with conical fibers, construct control data, use fiber-wise scalar multiplications.
result Obtain neighbourhood smooth weak deformation retraction of stratified spaces.
Optimal rates for vector-valued regression on various norms.
problem Optimal rates for vector-valued ridge regression on continuous norms.
method Combining standard capacity assumptions with tensor product constructions of vector-valued interpolation spaces.
result Optimal rates for vector-valued ridge regression, independent of output space dimension.
Paper proves an equivalence theorem for a class of norms.
problem Investigating equivalence of norms in Finsler geometry.
method Detailed analysis of Cartan tensors and proof of equivalence theorem.
result General (α,β)-metrics on manifolds with vanishing Landsberg curvatures are Berwald manifolds.
Smooth approximation of integral cycles mod 2 in Riemannian manifolds.
problem Approximating mod 2 integral cycles by smooth submanifolds.
method Approximation of mod 2 integral cycles by smooth submanifolds with controlled singularities.
result Every mod 2 integral cycle can be approximated by a smooth submanifold with a controlled singular set.
High-dimensional smoothing techniques struggle with robustness guarantees against various attacks.
problem Challenges in extending randomized smoothing to other attack models in high-dimensional space.
method Analysis of isotropic and generalized Gaussian smoothing distributions, proving bounds on certified robustness radii.
result Certifiable robustness radii decrease as $O(1/d^{ rac{1}{2} - rac{1}{p}})$ with dimension d for p>2. The paper generalizes Riemann curvature for manifolds with discontinuous metrics.
problem Generalizing Riemann curvature for manifolds with discontinuous metrics.
method Proposes a generalized Riemann curvature tensor combining angle defects and jumps in second fundamental forms.
result The generalized curvature tensor approximates classical curvature for smooth approximations of metrics.
New approach to solving minimal surface system Dirichlet problem on smooth domains.
problem Solving Dirichlet problem for minimal surface system on smooth domains.
method Using mean curvature flow (MCF) with boundary conditions and non-negative Ricci curvature assumption.
result Existence of long-time mean curvature flow and existence result for exterior Dirichlet problem.
Proves smoothness of certain Lagrangian submanifolds in complex space.
problem Smoothness of Hamiltonian stationary Lagrangian submanifolds in complex space.
method Morrey-type theorem and analysis of nonlinear fourth order equation.
result Proves smoothness of weakly harmonic Lagrangian phase submanifolds.
Noiseless KRR achieves optimal rates and exhibits saturation effects.
problem Understanding optimal rates and saturation phenomena in noiseless kernel ridge regression.
method Comprehensive study of noiseless KRR, establishing minimax optimal rates and uncovering phenomena of extra-smoothness and saturation.
result Noiseless KRR achieves minimax optimal rates and exhibits saturation effects.