arXiv research

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

Study approximates unknown function levels with queries.

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…

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

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…

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

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…

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…

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.

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

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…

New algorithms for interactive learning match minimax bounds efficiently.

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

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…

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.

Study stabilizers of smooth functions on surfaces, focusing on Morse-Bott functions.

Characterizes smooth functions on manifolds with simple Reeb spaces.

Smooth approximations for continuous functions on orbit spaces.

Variant of previous work on smooth algebraic functions with compact and non-compact preimages.

Reidemeister's theorem proved using smooth functions and transversality.

We show that for any smooth Hausdorff manifolds M and N, which are not necessarily second countable, paracompact or connected, any isomorphism from the algebra of smooth (real or complex) functions on N to the algebra of smooth functions on M is given by composition with a unique diffeomorphism from M to N. An analogou…

In this paper, we discuss the problem of minimizing the sum of two convex functions: a smooth function plus a non-smooth function. Further, the smooth part can be expressed by the average of a large number of smooth component functions, and the non-smooth part is equipped with a simple proximal mapping. We propose a pr…

The paper links set cuspidality to function regularity and flatness.

New algorithm POO optimizes noisy, unknown-smooth functions.

Among all $C^\infty$-algebras we characterize those which are algebras of smooth functions on smooth separable Hausdorff manifolds.

We study a non-parametric multi-armed bandit problem with stochastic covariates, where a key complexity driver is the smoothness of payoff functions with respect to covariates. Previous studies have focused on deriving minimax-optimal algorithms in cases where it is a priori known how smooth the payoff functions are. I…

Smooth Busemann functions found in harmonic Finsler spaces.

The study finds a special type of smooth function on connected sums of manifolds.

Smooth manifolds have functions with exactly two critical values.

The paper explores the structure of Reeb spaces for smooth functions on manifolds.

Topologies on algebraic and equational theories are used to define germ determined, near-point determined, and point determined rings of smooth functions, without requiring them to be finitely generated. It is proved, that any commutative algebra morphism (without requiring continuity) between near-point determined rin…

The folk questions in Lorentzian Geometry, which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime $(M,g)$ admits a smooth time function $τ$ whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth…

Algebras of smooth functions help reconstruct bulk topological types.

Decomposes smooth manifolds into algebraic submanifolds.

Deep ReLU networks can approximate and learn smooth functions efficiently.

Smooth activations enable optimal error rates in neural networks for Sobolev function classes.

New method for faster convergence in non-convex optimization with unbounded smoothness.

We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of th…

Regularized MFPCA smooths multivariate functional data for clearer patterns.

For a smooth function on a smooth manifold of a suitable class, the space of all connected components of preimages is the graph and called the {\it Reeb graph}. Reeb graphs are fundamental tools in the algebraic and differential topological theory of Morse functions and more general functions which are not so wild. In …

The paper extends properties of smooth functions to closed sets and maps.

The paper explores different smooth map notions on convex sets and their relationships.

Deep, wide ConvResNets can approximate functions and their smoothness.

We generalize stochastic smoothing for gradient estimation of non-differentiable functions.

SFM generates smooth functional data without exposing real data.

In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is contin…

New method constructs smooth functions with specific Reeb graphs and preimages on 3D manifolds.