arXiv research

Scheme minimizes $p$-elastic energy of curves over time.

Study on extremizers for Sobolev inequality on curved manifolds.

The purpose of this paper is to give a self-contained proof that a complete manifold with more than one end never supports an $L^{q,p}$-Sobolev inequality ($2 \leq p$, $q\leq p^{*}$), provided the negative part of its Ricci tensor is small (in a suitable spectral sense). In the route, we discuss potential theoretic pro…

The paper proves $L^p$-Sobolev inequalities for minimal submanifolds.

This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine $L^{p}-$Sobolev inequalities. The logarithmic version of affine $L^{p}-$Sobolev inequalities is verified. Moreover, An alternative proof of the affine Mo…

In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the $W^{1,2}$ Sobolev inequality along the Ricci flow …

We concerns here with the continuity on the geometry of the second Riemannian L^p-Sobolev best constant B_0(p,g) associated to the AB program. Precisely, for 1 <= p <= 2, we prove that B_0(p,g) depends continuously on g in the C^2-topology. Moreover, this topology is sharp for p = 2. From this discussion, we deduce som…

We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty)$ in a synthetic sense, the so called $CD(K,N)$ spaces. We first establish a Polya-Szego type inequality stating that the $W^{…

The study establishes inequalities for functions on manifolds using Green function estimates.

The paper proves the concavity of entropy power for diffusion equations and applies it to new inequalities.

Paper proves inequalities for forms on sub-Riemannian manifolds.

We introduce the concept of Calderón-Zygmund inequalities on Riemannian manifolds. For $1<p<\infty$, these are inequalities of the form $\left\Vert \mathrm{Hess}\left( u\right) \right\Vert _{L^p}\leq C_{1}\left\Vert u\right\Vert _{L^p}+C_{2}\left\Vert Δu\right\Vert _{L^p},$ valid a priori for all smooth functions $…

The paper proves inequalities for twisted differential forms on manifolds.

On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of the integral kernel of the resolvent $R_q(k):=(Δ_q+k^2)^{-1}$ of the Hodge Laplacian $Δ_q$ on $q$-forms as the spectral parameter $k$ approaches zero, assuming that 0 is not a resonance. The first application we give is an $L^p$ Sobolev estimate…

Study establishes Pólya-Szegő inequalities on submanifolds with small total mean curvature.

Let $C^{[M]}$ be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence $M=(M_k)$ is log-convex and has moderate growth. We prove that the groups ${\operatorname{Diff}}\mathcal{B}^{[M]}(\mathbb{R}^n)$, ${\operatorname{Diff}}W^{[M],p}(\mathbb{R}^n)$, ${\operatorname{Diff}}{\mathcal{S}}{}_…

Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.

Sharp bounds for approximating Sobolev functions by ridge functions and networks.

We prove the exponential law $\mathcal A(E \times F, G) \cong \mathcal A(E,\mathcal A(F,G))$ (bornological isomorphism) for the following classes $\mathcal A$ of test functions: $\mathcal B$ (globally bounded derivatives), $W^{\infty,p}$ (globally $p$-integrable derivatives), $\mathcal S$ (Schwartz space), $\mathcal D$…

Abstract compares 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.

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.

Study on minimal hypersurfaces in a special normed space.

The paper defines minimal norm tensors for curvature and divergence tensors, explaining Weyl and Cotten tensors.

A result of Bangert states that the stable norm associated to any Riemannian metric on the $2$-torus $T^2$ is strictly convex. We demonstrate that the space of stable norms associated to metrics on $T^2$ forms a proper dense subset of the space of strictly convex norms on $\R^2$. 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.

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 …

We provide recovery guarantees for compressible signals that have been corrupted with noise and extend the framework introduced in \cite{bafna2018thwarting} to defend neural networks against $\ell_0$-norm, $\ell_2$-norm, and $\ell_{\infty}$-norm attacks. Our results are general as they can be applied to most unitary tr…

The study connects norms and filtrations on section rings of projective manifolds.

Future robots should follow human social norms in order to be useful and accepted in human society. In this paper, we leverage already existing social knowledge in human societies by capturing it in our framework through the notion of social norms. We show how norms can be used to guide a reinforcement learning agent t…

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.

Article provides polytopes as dual unit balls of Thurston norms on 3-manifolds.

Paper introduces new risk norms based on ES with flexible distortion functions.

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.

A new PCA method using T$\ell_1$-norm outperforms existing methods.

The study explores special surfaces in a normed space.

Study improves image classifier robustness to random p-norm corruptions.

Study calculates stable norm of slit tori using Farey sequence.

This work shows how penalising bias terms in norm regularisation leads to sparse solutions.