Abstract compares two norms in holomorphic quadratic differentials.
problem Comparing two norms in holomorphic quadratic differentials.
method Comparison between Avila-Gouëzel-Yoccoz norm and Teichmüller norm.
result Comparison of two norms in holomorphic quadratic differentials.
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…
We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number bigger than one generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many …
We study a regularizer which is defined as a parameterized infimum of quadratics, and which we call the box-norm. We show that the k-support norm, a regularizer proposed by [Argyriou et al, 2012] for sparse vector prediction problems, belongs to this family, and the box-norm can be generated as a perturbation of the fo…
New L0 norm added to TDA for market analysis.
problem Improving TDA tools for market prediction.
method Defined and applied L0 norm in TDA for four markets.
result Enhanced TDA tools for market analysis.
The k-support norm is a regularizer which has been successfully applied to sparse vector prediction problems. We show that it belongs to a general class of norms which can be formulated as a parameterized infimum over quadratics. We further extend the k-support norm to matrices, and we observe that it is a special …
CNN layers with large norms are still robust to adversarial attacks.
problem Understanding the relationship between layer norms and adversarial robustness in CNNs.
method Theoretical analysis of ℓ1 and ℓ∞ norms, norm decay method, adversarial training frameworks. result Adversarially robust CNNs can have comparable or larger layer norms than non-adversarially robust ones.
Study on minimal hypersurfaces in a special normed space.
problem Characterizing minimal hypersurfaces in a specific normed space.
method Investigate translation and separable minimal hypersurfaces.
result New insights into the properties of minimal hypersurfaces.
The paper defines minimal norm tensors for curvature and divergence tensors, explaining Weyl and Cotten tensors.
problem Understanding curvature tensors and their minimal norm.
method Analyzing minimal norm tensors for third and fourth covariant tensors, including Riemannian curvature and divergence.
result Weyl tensor and Cotten tensor are identified as minimal norm tensors of Riemannian curvature and divergence tensors, respectively.
Paper uses social norms to teach robots human behavior.
problem Teaching robots to behave like humans in society.
method Captured social norms guide reinforcement learning towards normative behavior.
result Robots can learn normative behavior through automatic norm reward system.
A result of Bangert states that the stable norm associated to any Riemannian metric on the 2-torus T2 is strictly convex. We demonstrate that the space of stable norms associated to metrics on T2 forms a proper dense subset of the space of strictly convex norms on R2. In particular, given a strictly convex …
The spectral k-support norm enjoys good estimation properties in low rank matrix learning problems, empirically outperforming the trace norm. Its unit ball is the convex hull of rank k matrices with unit Frobenius norm. In this paper we generalize the norm to the spectral (k,p)-support norm, whose additional para…
Paper finds conditions for different norms to produce same billiard paths.
problem Conditions for different norms to define the same billiard reflection law.
method Extending previous works by Milena Radnović and Serge Tabachnikov, the paper establishes conditions for two different non-symmetric norms to define the same billiard reflection law.
result Conditions for two different norms to define the same billiard reflection law.
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…
Optimization problems with rank constraints appear in many diverse fields such as control, machine learning and image analysis. Since the rank constraint is non-convex, these problems are often approximately solved via convex relaxations. Nuclear norm regularization is the prevailing convexifying technique for dealing …
The study connects norms and filtrations on section rings of projective manifolds.
problem Understanding norms and filtrations on section rings of polarized projective manifolds.
method Analyzes submultiplicative norms and their equivalence to sup-norms, discusses applications to spectral theory and holomorphic extension.
result Injective and projective tensor norms on symmetric algebras are asymptotically equivalent.
Functorial semi-norms on singular homology give refined "size" information on singular homology classes. A fundamental example is the l^1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the l^1-semi-norm.
The paper proves inequalities linking geometric norms and Thurston norms in hyperbolic 3-manifolds.
problem Inequalities linking geometric norms and Thurston norms in hyperbolic 3-manifolds.
method Analyzes geometric L2-norms, Thurston norms, and Lipschitz maps to prove inequalities. result Proves an inequality between geometric L2-norm and Thurston norm, qualitatively sharp. Article provides polytopes as dual unit balls of Thurston norms on 3-manifolds.
problem Understanding the dual unit ball shape of Thurston norms.
method Introduced a family of polytopes in Z^2g that can be dual unit balls of Thurston norms on 3-manifolds.
result Polytopes with mod 2 congruent vertices can be realized as dual unit balls of Thurston norms.
Paper introduces new risk norms based on ES with flexible distortion functions.
problem Risk quantification and anomaly detection in financial data.
method Developed generalized Expected-Shortfall (ES) norms using distortion risk measures and duality theory.
result Unified analytical framework for risk quantification and practical applications.
We propose a set of convex low rank inducing norms for a coupled matrices and tensors (hereafter coupled tensors), which shares information between matrices and tensors through common modes. More specifically, we propose a mixture of the overlapped trace norm and the latent norms with the matrix trace norm, and then, w…
General (α,β) norms are an important class of Minkowski norms which contains the original (α,β) norms. In this note, by studying the behavior of the Darboux curves of the indicatrix, we give a characterization of 3-dimensional general (α,β) norms. By studying the isoperimetric properties of the indicatrix, as …
Uniform convergence of interpolators proven for Gaussian data.
problem Interpolation learning in high-dimensional linear regression with Gaussian data.
method Generic uniform convergence guarantee in terms of Gaussian width.
result Consistency of interpolators for minimum-norm and near-minimal-norm cases.
A new PCA method using Tℓ1-norm outperforms existing methods.
problem Outliers and noise sensitivity in classical PCA.
method PCA based on Tℓ1-norm maximization. result The method outperforms PCA-ℓp, ℓpSPCA, and PCA in numerical experiments. The study explores special surfaces in a normed space.
problem Constant Gaussian and mean curvature surfaces in normed spaces.
method Analyzes rotational surfaces with specific curvature properties.
result Generalizes catenoid, pseudo-sphere, and Delaunay surfaces.
Guarantees recovery of compressible signals from adversarial noise.
problem Recovering compressible signals from noise and adversarial attacks.
method Extends adversarial defense framework to ℓ0, ℓ2, and ℓ∞ norms. result Recovery guarantees for various signal recovery methods under different noise types.
Study improves image classifier robustness to random p-norm corruptions.
problem Improving robustness of image classifiers to real-world imperceptible corruptions.
method Training and testing with random p-norm corruptions, evaluating robustness against different p-norms.
result Training with a combination of p-norm corruptions significantly improves robustness.
Study calculates stable norm of slit tori using Farey sequence.
problem Computing the stable norm of slit tori.
method Explicit computations using the Farey sequence and gluing slit tori.
result Estimates the asymptotic counting of simple homology classes.
This work shows how penalising bias terms in norm regularisation leads to sparse solutions.
problem Understanding the relation between parameter norm regularization and the sparsity of neural network solutions.
method Analyzes one hidden ReLU layer networks with unidimensional data, showing the norm required for function representation and the importance of the bias term's norm.
result Penalising the bias terms in regularisation leads to sparse solutions, enforcing the uniqueness and sparsity of the minimal norm interpolator.
The paper explores why a specific type of predictor works well in noisy data.
problem Understanding why a specific type of predictor (minimum-norm interpolator) works well in noisy data.
method The paper uses uniform convergence and zero-error predictors in a norm ball to explain the success of the minimum-norm interpolator.
result The minimum-norm interpolator is consistent, and this can be explained by uniform convergence of zero-error predictors in a norm ball.
Infinite width ReLU networks can approximate functions with bounded Euclidean norm.
problem Functions that can be approximated by ReLU networks with bounded Euclidean norm.
method Analyzing the minimal network norm required to approximate a given function.
result The minimal network norm for representing a function \( f \) is \( \max(\int |f''(x)| dx, |f'(-\infty) + f'(+\infty)|) \).
The paper proves that the support norm of tight contact structures adds up.
problem Understanding the support norm of contact structures.
method Analyzing open books supporting contact structures.
result Additivity of the support norm for tight contact structures.
Graph manifolds' Thurston norms are sums of linear functionals, and every such norm can be realized.
problem Understanding Thurston norms of graph manifolds and their realizability.
method Analyzing the structure of Thurston norms as sums of linear functionals and showing realizability.
result Every Thurston norm of a graph manifold can be expressed as a sum of absolute values of linear functionals with rational coefficients.
Proves Ricci flow extensibility with integral norms.
problem Ricci flow singularities under integral norms.
method Bounded integral norms of curvature and scalar curvature.
result Extends Ricci flow under certain conditions.
Classifies surfaces with special curvature properties.
problem Rotational surfaces with specific curvature conditions.
method Classifies surfaces with rotationally symmetric norms and linear curvature relations.
result Rotational surfaces with linearly related curvatures are classified.
A toolkit for path-norms enhances neural network generalization bounds.
problem Establishing generalization bounds for modern neural networks.
method Introducing a comprehensive toolkit for path-norms in ReLU networks with various operations.
result Established generalization bounds for modern neural networks that are the most widely applicable and recover/beat the sharpest known bounds.
Low-rank inducing unitarily invariant norms have been introduced to convexify problems with low-rank/sparsity constraint. They are the convex envelope of a unitary invariant norm and the indicator function of an upper bounding rank constraint. The most well-known member of this family is the so-called nuclear norm. To …
Study normal curves in sub-Finsler Lie groups with specific norms, focusing on branching and face stability.
problem Analyzing normal curves in sub-Finsler Lie groups with different norms.
method Using tools from convex analysis, the Pontryagin Maximum Principle is revisited to express the normal equation as a differential inclusion involving the subdifferential of the dual norm.
result Normal curves in polyhedral norms have controls that locally take values in a single face of a sphere with respect to the norm.
Study on minimal surfaces in a 3D space with 2m-norm.
problem Characterizing minimal surfaces in a specific geometric space.
method Examining translation, homothetical, and separable minimal surfaces.
result New insights into minimal surfaces in a 3D space with 2m-norm.
For sutured 3-manifolds M, there is a sutured Thurston norm due to Scharlemann. We show how depth one foliations of M and corresponding fibrations and the usual Thurston norm on the double of M are useful tools for computing this norm. In many examples, the faces of the unit ball of the sutured norm are related to cone…
Recently, l2,1 matrix norm has been widely applied to many areas such as computer vision, pattern recognition, biological study and etc. As an extension of l1 vector norm, the mixed l2,1 matrix norm is often used to find jointly sparse solutions. Moreover, an efficient iterative algorithm has been designed…
Efficiently regularizes deep learning models using Jacobian nuclear norm.
problem Regularizing deep learning models to prevent overfitting and improve generalization.
method Proposes a denoising-style approximation to penalize the Jacobian nuclear norm without computing the Jacobian matrix.
result Demonstrates that penalizing the average squared Frobenius norm of Jg and Jh is equivalent to penalizing the Jacobian nuclear norm for function compositions. In 1986, W. Thurston introduced a (possibly degenerate) norm on the first cohomology group of a 3-manifold. Inspired by this definition, Turaev introduced in 2002 a analogous norm on the first cohomology group of a finite 2-complex. We show that if N is the exterior of a link in a rational homology sphere, then the Thu…
We relate the Gromov norm on homology classes to the harmonic norm on the dual cohomology and obtain double sided bounds in terms of the volume and other geometric quantities of the underlying manifold. Along the way, we provide comparisons to other related norms and quantities as well.
In this paper, we study global existence and blow up properties to Lp norm preserving non-local heat flows. We first study two kinds of Lp norm preserving non-local flows and prove that these flows have the global solutions. Finally, we give a example to show that one kind of this heat flow may blow up in $L^{\in…
Improved LDA with capped l_{2,1}-norm reduces outlier sensitivity.
problem Outliers and noise sensitivity in classical LDA.
method Introducing capped l_{2,1}-norm and proposing CLDA.
result CLDA effectively removes outliers and suppresses noise.
Adversarial training linked to operator norm regularization, proving network sensitivity to attacks.
problem Robustifying neural networks against adversarial attacks.
method Theoretical link established between adversarial training and operator norm regularization.
result Adversarial training is equivalent to data-dependent operator norm regularization.
Characterizes functions representable by infinite-width ReLU networks with bounded weights.
problem Understanding function representation in overparameterized neural networks.
method Analyzes functions in Ws,1(R) spaces and their Radon transform. result All functions in Ws,1(R) can be represented with bounded norm.