arXiv research

Adversarial online nonparametric regression achieves optimal rates with locally adaptive learning.

General lower bounds on neural network approximation in L^p norm.

Study approximates unknown function levels with queries.

We consider the problem of online nonparametric regression with arbitrary deterministic sequences. Using ideas from the chaining technique, we design an algorithm that achieves a Dudley-type regret bound similar to the one obtained in a non-constructive fashion by Rakhlin and Sridharan (2014). Our regret bound is expre…

The purpose of these notes is to explain parts of Gromov's survey of Carnot-Carathedory spaces, in the light of subsequent results of M. Rumin. Among the rich material provided by Gromov, most of which pertains to analysis on metric spaces, we choose to concentrate on the H{ö}lder equivalence problem for Carnot manifol…

A variant of Gromov's H{ö}lder-equivalence problem, motivated by a pinching problem in Riemannian geometry, is discussed. A partial result is given. The main tool is a general coarea inequality satisfied by packing energies of maps.

In the context of stochastic continuum-armed bandits, we present an algorithm that adapts to the unknown smoothness of the objective function. We exhibit and compute a polynomial cost of adaptation to the H{ö}lder regularity for regret minimization. To do this, we first reconsider the recent lower bound of Locatelli an…

We study finite energy classes of quasiplurisubharmonic (qpsh) functions in the setting of toric compact K{ä}hler manifolds. We characterize toric qpsh functions and give necessary and sufficient conditions for them to have finite (weighted) energy, both in terms of the associated convex function in R n , and through t…

Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\mathcal{T}(S)$ its Teichm{ü}ller space. For any isotopy class of closed curves $γ$, we compute the first three derivatives of the length function $\ell\_γ:\mathcal{T}(S)\rightarrow\mathbf{R}\_+$ in the shearing coordinates associated to a maxim…

In this paper we provide an alternative framework to tackle the first-best Principal-Agent problem under CARA utilities. This framework leads to both a proof of existence and uniqueness of the solution to the Risk-Sharing problem under very general assumptions on the underlying contract space. Our analysis relies on an…

Validates economic scenarios using statistical tests on stochastic processes.

We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. The flow exists for a uniform amount of time, during which the curvature stays bounded below by a controllable negative number. The curvature conditions we consider include 2-non-negative and weakly $\t…

Study shows zero-shot super-resolution in neural operators is impossible in many cases.

Efficient algorithms for contextual bandits with smooth regret in continuous action spaces.

This paper selects features in deep neural networks with theoretical guarantees.

New algorithms for interactive learning match minimax bounds efficiently.

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.

The paper explores why a specific type of predictor works well in noisy data.

Graph manifolds' Thurston norms are sums of linear functionals, and every such norm can be realized.

Proves Ricci flow extensibility with integral norms.