arXiv research

StoSOO optimistically maximizes noisy, locally smooth functions.

In this paper we aim for a generalisation of the Steenrod Approximation Theorem from, concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalisation is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a c…

Solves local minima problems on smooth manifolds.

We show how to construct the nonstandard hull of certain infinite-dimensional Lie algebras in order to generalize a theorem of Pestov on the enlargeability of Banach-Lie algebras. In the process, we consider a nonstandard smoothness condition on functions between locally convex spaces to ensure that the induced functio…

New subharmonicity concept proves conjecture on Riemannian manifolds.

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

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

Proves local bi-integrability of bi-Hamiltonian systems on real smooth manifolds.

The paper extends a variance gamma model to quadratic functions, reducing arbitrage and computational costs.

Novel method for shape optimization of non-smooth PDEs.

Smooth surfaces can always be locally described by Hessians.

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains $C$, with non-smooth boundary, in possibly non-compact manifolds. Assuming $C$ is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restrictio…

Decomposes smooth manifolds into algebraic submanifolds.

We identify the 2-groupoid of deformations of a gerbe on a smooth manifold with the Deligne 2-groupoid of a corresponding twist of the DGLA of local Hochschild cochains on infinite jets of smooth functions.

The Thurston spine's properties are studied in relation to Morse-Smale complexes.

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…

The paper studies special contact metric manifolds and their properties.

Locally convex classes on manifolds linked to Ricci curvature bounds.

Extends curve theory to non-smooth data with finite curvature and torsion.

We consider the non-parametric regression problem under Huber's $ε$-contamination model, in which an $ε$ fraction of observations are subject to arbitrary adversarial noise. We first show that a simple local binning median step can effectively remove the adversary noise and this median estimator is minimax optimal up t…

Surrogate-based analysis of interactions via local effect smooths

The paper tackles learning smooth distance functions using query-based methods.

We study the problem of optimizing a function under a \emph{budgeted number of evaluations}. We only assume that the function is \emph{locally} smooth around one of its global optima. The difficulty of optimization is measured in terms of 1) the amount of \emph{noise} $b$ of the function evaluation and 2) the local smo…

Paper optimizes multi-fidelity function with fast learning rates.

We design a stochastic algorithm to train any smooth neural network to $\varepsilon$-approximate local minima, using $O(\varepsilon^{-3.25})$ backpropagations. The best result was essentially $O(\varepsilon^{-4})$ by SGD. More broadly, it finds $\varepsilon$-approximate local minima of any smooth nonconvex function in …

We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to Hamiltonian vector fields to the setting of non-smooth geometry. For mildly sin…

New GP model estimates piecewise continuous functions.

The paper studies local heat kernel properties on smooth manifolds.

Convolutional Neural Networks (CNN) and the locally connected layer are limited in capturing the importance and relations of different local receptive fields, which are often crucial for tasks such as face verification, visual question answering, and word sequence prediction. To tackle the issue, we propose a novel loc…

Book on infinite-dimensional Lie groups, covering basics and various classes.

We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

New algorithm POO optimizes noisy, unknown-smooth functions.

New algorithm improves gradient-based ERM for smooth convex losses.

Using the reformulation in divergence form of the Euler-Lagrange equation for the Willmore functional as it was developed in "Analysis of the Willmore Functional" by T. Riviere (Invent. Math. 174), we study the limit of a local Palais-Smale sequence of weak Willmore immersions with locally square-integrable second fund…

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${…

New SGD covering technique yields dimension-independent generalization bounds.

This paper introduces SRPR for robust phase retrieval with smoothed loss functions.

Smooth functions preserve Zygmund class on curves.

Recently, Petrik et al. demonstrated that L1Regularized Approximate Linear Programming (RALP) could produce value functions and policies which compared favorably to established linear value function approximation techniques like LSPI. RALP's success primarily stems from the ability to solve the feature selection and va…

We consider the problem of finding local minimizers in non-convex and non-smooth optimization. Under the assumption of strict saddle points, positive results have been derived for first-order methods. We present the first known results for the non-smooth case, which requires different analysis and a different algorithm…

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

Paper addresses ERM in LDP, reducing sample complexity for smooth and convex losses.

We consider the problem of global optimization of an unknown non-convex smooth function with zeroth-order feedback. In this setup, an algorithm is allowed to adaptively query the underlying function at different locations and receives noisy evaluations of function values at the queried points (i.e. the algorithm has ac…

FedProx algorithm improved for non-smooth and heterogeneous data.

PF-LaCG removes the need for knowing smoothness and strong convexity parameters for locally accelerated CG.

This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic functi…

In this paper we find strictly locally convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an applicatio…

Smooth manifolds from locally homogeneous spaces.