arXiv research

Deep neural networks are susceptible to adversarial manipulations in the input domain. The extent of vulnerability has been explored intensively in cases of $\ell_p$-bounded and $\ell_p$-minimal adversarial perturbations. However, the vulnerability of DNNs to adversarial perturbations with specific statistical properti…

We study perturbations of a partially hyperbolic toral automorphism L which is diagonalizable over C and has a dense center foliation. For a small perturbation of L with a smooth center foliation we establish existence of a smooth leaf conjugacy to L. We also show that if a small perturbation of an ergodic irreducible …

The paper smooths out equations on 4-manifolds to show smooth moduli spaces.

We prove a Freed-Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa--Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the $PU(2)$-monopole equations on smooth closed four-manifolds. We introduce a set of perturbation terms…

Study perturbations of submodules in Drury-Arveson space, finding smooth vector bundles with Hermitian connections.

In this work we prove the fact that, for a short time, it is possible to construct a smooth parametrized family of isometric embeddings of an arbitrary smooth parametrized family of Riemannian metrics on a smooth closed manifold into an Euclidean space. In order to prove this statement we work out stability estimates w…

This paper establishes transversality for perturbed Vafa-Witten moduli spaces on 4-manifolds.

Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.

New framework improves adversarial robustness certification for various perturbations.

Extends randomized smoothing to certify robustness against various threat models and adversarial perturbations.

Introduces a new geometric framework for non-perturbative BV-theory.

Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.

The paper studies heat kernels on modified manifolds and bounds their properties.

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 ${…

We introduce a new stochastic smoothing perspective to study adversarial contextual bandit problems. We propose a general algorithm template that represents random perturbation based algorithms and identify several perturbation distributions that lead to strong regret bounds. Using the idea of smoothness, we provide an…

The problem of attribution is concerned with identifying the parts of an input that are responsible for a model's output. An important family of attribution methods is based on measuring the effect of perturbations applied to the input. In this paper, we discuss some of the shortcomings of existing approaches to pertur…

New method enhances neural network robustness against adversarial attacks.

Generic smooth minimal hypersurfaces exist in 8D manifolds.

Local minimizers are convex and close to Wulff shapes.

Modified perturbation method removes non-smoothness in solving Black-Scholes equations.

Despite achieving impressive performance, state-of-the-art classifiers remain highly vulnerable to small, imperceptible, adversarial perturbations. This vulnerability has proven empirically to be very intricate to address. In this paper, we study the phenomenon of adversarial perturbations under the assumption that the…

The paper proves smoothness of event horizons in Kerr spacetime perturbations.

Work on making classifiers robust against adversarial attacks for top-k predictions.

Study a modified Laplacian equation in spacetime.

We show how to turn any classifier that classifies well under Gaussian noise into a new classifier that is certifiably robust to adversarial perturbations under the $\ell_2$ norm. This "randomized smoothing" technique has been proposed recently in the literature, but existing guarantees are loose. We prove a tight robu…

New invariant from non-acyclic flat connections.

Improved perturbation reduces matrix condition number to O(n) with minimal storage.

New method improves solving combinatorial optimization problems with smoothed policies.

Singularities of area minimizing hypersurfaces can be smoothed in dimensions 9 and 10.

New metrics produce discrete zero sets for nondegenerate harmonic forms.

The traceless $SU(2)$ character variety $R(S^2,\{a_i,b_i\}_{i=1}^n)$ of a $2n$-punctured 2-sphere is the symplectic reduction of a Hamiltonian $n$-torus action on the $SU(2)$ character variety of a closed surface of genus $n$. It is stratified with a finite singular stratum and a top smooth symplectic stratum of dimens…

SIGNNAP learns stable and identifiable node representations in GNNs against graph perturbations.

The paper proves stability of certain singularities in integrable systems.

The paper studies invariant complex manifolds in holomorphic slow-fast systems.

We present batch virtual adversarial training (BVAT), a novel regularization method for graph convolutional networks (GCNs). BVAT addresses the shortcoming of GCNs that do not consider the smoothness of the model's output distribution against local perturbations around the input. We propose two algorithms, sample-based…

The Yamabe flow can blow up in infinite time with small perturbations.

Smooths out complex shapes into simpler forms.

Smooth knots with odd Conway polynomial terms have inscribed trefoils.

Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a seri…

Hierarchical randomized smoothing improves model robustness for complex data.

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…

Improved online Lasso reduces regret in sparse linear contextual bandits.

The large $k$ asymptotics (perturbation series) for integrals of the form $\int_{\cal F}μe^{i k S}$, where $μ$ is a smooth top form and $S$ is a smooth function on a manifold ${\cal F}$, both of which are invariant under the action of a symmetry group ${\cal G}$, may be computed using the stationary phase approximation…

We define a novel family of algorithms for the adversarial multi-armed bandit problem, and provide a simple analysis technique based on convex smoothing. We prove two main results. First, we show that regularization via the \emph{Tsallis entropy}, which includes EXP3 as a special case, achieves the $Θ(\sqrt{TN})$ minim…

In the framework of fibred cusp operators on a manifold $X$ associated to a boundary fibration $Φ: \pa X\to Y$, the homotopy groups of the space of invertible smoothing perturbations of the identity are computed in terms of the K-theory of $T^{*}Y$. It is shown that there is a periodicity, namely the odd and the even h…

Improved learning algorithms with privacy using smoothed analysis.

$α$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $α$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $α>1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in …

Paper defends deep learning classifiers against channel-aware adversarial attacks.