arXiv research

Novel framework improves randomized smoothing for various norms.

The study analyzes convergence rates for sparse pivotal estimators in high-dimensional regression.

Kernel interpolation is inconsistent for norms with smoothness above a constant.

In four and higher dimensions, we show that any stationary admissible Yang-Mills field can be gauge transformed to a smooth field if the $L^2$ 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…

Study on Santaló point for convex bodies in normed spaces.

Certified robustness for ImageNet models with randomized smoothing.

SGD achieves a $O(ε^{-4})$ bound for minimizing gradient norm of smooth functions.

Smooth approximation of integral cycles in manifolds.

The study solves the isoperimetric problem for Heisenberg group norms.

Study shows momentum-based optimizers like Muon and MomentumGD bias towards KKT points in smooth homogeneous models.

Signal estimation problems with smoothness and sparsity priors can be naturally modeled as quadratic optimization with $\ell_0$-"norm" constraints. Since such problems are non-convex and hard-to-solve, the standard approach is, instead, to tackle their convex surrogates based on $\ell_1$-norm relaxations. In this paper…

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 real homology of a compact Riemannian manifold $M$ is naturally endowed with the stable norm. The stable norm on $H_1(M,\mathbb{R})$ arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finite-dimensional vector space $H_1(M,\mathbb{R})$ are st…

New methods improve online matrix optimization with reduced computational cost.

This paper assesses Gaussian and Exponential mechanisms for certifying adversarial robustness.

Paper proves an equivalence theorem for a class of norms.

Smooth approximation of integral cycles mod 2 in Riemannian manifolds.

We study a hybrid conditional gradient - smoothing algorithm (HCGS) for solving composite convex optimization problems which contain several terms over a bounded set. Examples of these include regularization problems with several norms as penalties and a norm constraint. HCGS extends conditional gradient methods to cas…

Matrix completion works well for smooth non-linear structures, even without low-rank assumptions.

The Schatten-$p$ norm ($0<p<1$) has been widely used to replace the nuclear norm for better approximating the rank function. However, existing methods are either 1) not scalable for large scale problems due to relying on singular value decomposition (SVD) in every iteration, or 2) specific to some $p$ values, e.g., $1/…

Paper introduces a new $G^\star$ regret measure for online convex optimization with smooth losses.

New method turns optimization algorithms into uniformly stable learning algorithms for non-Euclidean norms.

The Schatten quasi-norm can be used to bridge the gap between the nuclear norm and rank function, and is the tighter approximation to matrix rank. However, most existing Schatten quasi-norm minimization (SQNM) algorithms, as well as for nuclear norm minimization, are too slow or even impractical for large-scale problem…

This work presents a general framework for solving the low rank and/or sparse matrix minimization problems, which may involve multiple non-smooth terms. The Iteratively Reweighted Least Squares (IRLS) method is a fast solver, which smooths the objective function and minimizes it by alternately updating the variables an…

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.

The problem of joint feature selection across a group of related tasks has applications in many areas including biomedical informatics and computer vision. We consider the l2,1-norm regularized regression model for joint feature selection from multiple tasks, which can be derived in the probabilistic framework by assum…

Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone $K$, a norm $\|\cdot\|$ 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…

We refine and generalize several interpolation inequalities bounding the $L^p$ norm of a probability density with respect to the reference measure $μ$ by its Sobolev norm and the Kantorovich distance to $μ$ on a smooth weighted Riemannian manifold satisfying $CD(0, \infty)$ condition.

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.

Geodesics in Kähler metrics connect metrics with constant scalar curvature.

Improved algorithms solve $\ell_p$-norm regression problems efficiently.

Many practical applications such as gene expression analysis, multi-task learning, image recognition, signal processing, and medical data analysis pursue a sparse solution for the feature selection purpose and particularly favor the nonzeros \emph{evenly} distributed in different groups. The exclusive sparsity norm has…

ERM with square loss achieves sublinear error for learnable function classes with smoothed data.

New framework assesses regularization norms in ill-posed problems, revealing L2 instability and proposing adaptive fractional RKHS solutions.

New method stabilizes machine learning for physics-informed inverse problems.

Two definitions quantify $C^{2,α}$ regularity of Riemannian surfaces.

We do further investigation in a certain cosine function defined for smooth Minkowski spaces. We prove that such function is symmetric if and only if the referred space is Euclidean, and also that it can be given in terms of the Gateaux derivative of the norm. As an application we use it to study the ratio between the …

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 …

Generalizes smoothness conditions for optimization methods.

The paper provides approximation guarantees for neural networks trained with gradient flow.

Equivalence of norms on manifolds with curvature bounds established.

Estimates the dual Thurston norm for foliations on negative curvature 3-manifolds.

New optimization method combines gradient clipping and non-Euclidean smoothness.

New optimizers control network width scaling, improving stability and transfer across different model sizes.

Smooth models refine topological K-theory for differential extensions.

The curvature-dimension condition fails in sub-Finsler geometry, extending previous results in sub-Riemannian geometry.

Deforms Minkowski norms using linear forms and functions.

The paper proves conjectures about Minkowski norms with specific symmetry groups.