Paper improves estimates for solutions to a specific equation.
problem Establishing uniform C2,θ estimates for stable solutions. method Combines infinite dimensional reduction and small regularity theorems.
result Uniform C2,θ estimates for stable solutions in dimensions ≤10. Describes 3-manifolds by families of singularly fibered surfaces.
problem Understanding the structure of 3-manifolds through fibered surfaces.
method Explicitly describes a fibration of the complement of a homogeneous braid.
result Each fiber intersects every cross-section of S3. Study proves solutions concentrate on a capillary surface in a manifold.
problem Existence of solutions for a nonlinear Neumann boundary condition equation.
method Inspired by Pacard and Ritoré, constructs solutions concentrating to a capillary surface.
result Solutions concentrate asymptotically to a given volume nondegenerate capillary hypersurface.
Given (M, g0) we consider the problem -ε^2Delta_{g0+h}u + u = (u+)^{p-1} with (ε, h) \in (0, ε0) \times Bρ. Here Bρ is a ball centered at 0 with radius ρ in the Banach space of all Ck symmetric covariant 2-tensors on M. Using the Poincaré polynomial of M, we give an estimate on the number of nonconstant solutions with …
Defines connections for singularly foliated bundles.
problem Understanding connections in bundles with singular foliations.
method Introduces connections compatible with singular foliations, defines holonomy groupoids.
result Holonomy groupoids of trivial bundles are quotients of Androulidakis-Skandalis groupoids.
In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation α2Δu+u(1−u2)=0,in Ω⊂RN where N=3, Ω is a smooth bounded domain and $\A>0$ is a small parameter. We provide asymptotic behavior which shows that, as α→0, the level sets of the soluti…
We consider the following singularly perturbed Neumann problem \begin{eqnarray*} \ve^2 Δu -u +u^p = 0 \, \quad u>0 \quad {\mbox {in}} \quad Ω, \quad {\partial u \over \partial ν}=0 \quad {\mbox {on}} \quad \partial Ω, \end{eqnarray*} where p>2 and Ω is a smooth and bounded domain in R2. We construct a new class…
During this last decades, several attempts to construct slow invariant manifold of the Lorenz-Krishnamurthy five-mode model of slow-fast interactions in the atmosphere have been made by various authors. Unfortunately, as in the case of many two-time scales singularly perturbed dynamical systems the various asymptotic p…
Let $(\MM ,{\tilde g})$ be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation $$ ε^2Δ_{ {\tilde g}} {u}\,+\, (1 - {u}^2)u \,=\,0\quad \mbox{in } \MM, $$ where ε is a small parameter. Let $\KK\subset \MM$ be an (N−1)-dimensional smooth minimal submanifold …
We prove that finite Morse index solutions to the Allen-Cahn equation in R2 have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf curvature decay estimate} on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second order regularit…
Generalizes pseudo-product structures with abnormal extremals.
problem Finiteness of symmetry algebras for non-degenerate pseudo-product structures.
method Modified universal prolongation of graded nilpotent Lie algebras and generalized finiteness criterion.
result Distributions with singularly transitive properties have finite-dimensional symmetries.
Generic 3D vector fields have singularly hyperbolic transitive sets.
problem Understanding the dynamics of generic three-dimensional vector fields.
method Analyzing C1 generic vector fields on closed 3-manifolds. result Generic vector fields have singularly hyperbolic transitive sets.
Given two univalent harmonic mappings f1 and f2 on D, which lift to minimal surfaces via the Weierstrass-Enneper representation theorem, we give necessary and sufficient conditions for f3=(1−s)f1+sf2 to lift to a minimal surface for s∈[0,1]. We then construct such mappings from Enneper's surfa…
The paper studies learning dynamics in two-layer neural networks.
problem Learning dynamics and time scales in two-layer neural networks.
method Gradient flow dynamics of a wide two-layer neural network in high-dimension, with data following a single-index model.
result The learning dynamics exhibit separation of timescales and intermittency.
We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact n-dimensional submanifold …
Hierarchical pretraining with slow-fast ODEs
problem Causal self-attention vs. slow-fast ODEs
method Instantiating fast-slow ODE formalism as a concrete neural network
result Equilibrium manifold x=φ(y) is exactly the master-equation (ME) stationary distribution Consider a dihedral cover f:Y→X with X and Y four-manifolds and f branched along an oriented surface embedded in X with isolated cone singularities. We prove that only a slice knot can arise as the unique singularity on an irregular dihedral cover f:Y→S4 if Y is homotopy equivalent to $\mathbb{CP…
We study the optimal dividend problem for a firm's manager who has partial information on the profitability of the firm. The problem is formulated as one of singular stochastic control with partial information on the drift of the underlying process and with absorption. In the Markovian formulation, we have a 2-dimensio…
Solves inventory control with unknown demand trend using singular control.
problem Optimally managing inventory with an unknown demand trend.
method Formulates as a stochastic control problem under partial observation, solves equivalent separated problem using transition between formulations, and applies viscosity theory.
result Constructs an optimal control rule and shows bounded Lipschitz continuity of free boundaries.
Many modern machine learning models are trained to achieve zero or near-zero training error in order to obtain near-optimal (but non-zero) test error. This phenomenon of strong generalization performance for "overfitted" / interpolated classifiers appears to be ubiquitous in high-dimensional data, having been observed …
Normal forms and invariants for nondegenerate hypersurfaces in C^2.
problem Equivalence problem for nondegenerate real hypersurfaces in C^2.
method Equivariant moving frames and invariant differentiation.
result A single real differential invariant of order 7 generates the entire algebra of differential invariants for nondegenerate real hypersurfaces at singularly umbilic points.
Improved algorithm speeds up generation of universal adversarial perturbations.
problem Slow generation of universal adversarial perturbations.
method Optimized algorithm based on orientation of perturbation vectors.
result Significantly faster generation of universal perturbations with higher fooling rates.
This paper presents a new approach, called perturb-max, for high-dimensional statistical inference that is based on applying random perturbations followed by optimization. This framework injects randomness to maximum a-posteriori (MAP) predictors by randomly perturbing the potential function for the input. A classic re…
The paper tackles extrapolation of gene knockouts effects on RNA counts.
problem Modeling effects of gene knockouts on RNA counts for new perturbations.
method Formulated as a latent variable model with additive perturbation effects, proved identifiability, proposed PDAE for estimation.
result PDAE can accurately predict effects of unseen but identifiable perturbations.
Study shows transfer of adversarial robustness between different perturbation types is limited.
problem Understanding adversarial robustness across various perturbation types.
method Evaluated 32 attacks of 5 different types on models trained on a subset of ImageNet.
result Adversarial robustness transfer between perturbation types is limited and depends on the specific type of perturbation.
Novel geometry-informed irreversible perturbation accelerates Langevin dynamics convergence.
problem Accelerating convergence of Langevin dynamics for Bayesian computation.
method Geometry-informed irreversible perturbation of Riemannian manifold Langevin dynamics.
result Improves estimation performance over irreversible perturbations that ignore geometry.
Study linear perturbations in Schwarzschild black hole spacetime.
problem Linear perturbations of Schwarzschild black hole spacetime.
method Investigate linearised perturbation of constant mass aspect function foliation at null infinity.
result Linearised perturbations of Bondi energy and mass vanish, and all linear momentum can be achieved.
New research evaluates various perturbation methods for improving neural network robustness.
problem Understanding and improving robustness of Convolutional Neural Networks (CNNs) against adversarial attacks.
method Detailed evaluation of five main perturbation-based defenses, comparing random and deterministic approaches.
result Perturbation-based defenses are equivalent in efficacy, and attacks transfer between them.
EVILL uses randomised perturbations to improve exploration in bandit problems.
problem Improving exploration in structured stochastic bandit problems.
method Solves for the minimiser of a linearly perturbed regularised negative log-likelihood function.
result EVILL matches the performance of Thompson-sampling-style methods in theory and practice.
Adversarial training helps classifiers resist universal perturbations.
problem Vulnerability of classifiers to universal perturbations.
method Adversarial training with shared adversarial examples.
result Adversarial training reduces sensitivity to universal perturbations.
New bifurcation found in perturbations of non-generic closed self-shrinkers.
problem Understanding the behavior of perturbations in non-generic closed self-shrinkers.
method Analyzing the mean curvature flow singularity transitions.
result Different types of singularity transitions based on perturbation direction.
Adversarial perturbations fool deepfake detectors with high accuracy.
problem Improving deepfake detection accuracy against adversarial attacks.
method Used adversarial perturbations and two defenses: Lipschitz regularization and Deep Image Prior (DIP).
result Deepfake detectors achieved 27% accuracy on perturbed images, compared to 95% on unperturbed.
Advances FTPL results for bandit problems with unbounded perturbations.
problem Improving analytical foundations of FTPL in bandit problems.
method Revisiting classical FTRL-FTPL duality for unbounded perturbations.
result Establishes Best-of-Both-Worlds (BOBW) results for FTPL under a broad family of asymmetric unbounded perturbations.
Universal perturbations misclassify text with high accuracy.
problem Vulnerability of text classifiers to small perturbations.
method Algorithm to compute universal adversarial perturbations.
result Deep neural networks are highly vulnerable to universal adversarial perturbations.
Adversarial training adds dynamic perturbations to neural networks for robustness.
problem Accuracy trade-off and lack of diversity in adversarial examples.
method Dynamic adversarial perturbations in the parameter space of neural networks, updating perturbation biases during training.
result Adversarial training with negligible cost and reduced accuracy trade-off.
Topological string theory derived from string geometry for non-perturbative effects.
problem Deriving non-perturbative effects in string theory.
method Formulating topological string geometry theory and deriving the partition function from fluctuations around a classical solution.
result Perturbative partition function of topological string theory derived.
Generative Intervention Models predict perturbation effects without knowing the underlying mechanisms.
problem Predicting perturbation effects when the mechanisms are unknown.
method Generative Intervention Models (GIM) that map perturbation features to distributions over atomic interventions in a causal model.
result GIMs achieve robust out-of-distribution predictions and infer underlying perturbation mechanisms.
Identifies all perturbative vacua in bosonic string theory.
problem Identifying all perturbative vacua in bosonic string theory.
method Completely identified perturbative vacua through string fluctuations.
result Derivation of path-integrals up to any order from fluctuations.
Develops new methods to create imperceptible image changes that fool classifiers.
problem Improving the robustness of image classifiers by creating subtle changes undetectable to humans.
method Two methods: Edge-Aware and Color-Aware, designed to reduce detectability of image perturbations.
result Demonstrated that the new methods effectively cause misclassification and are computationally efficient.
SmoothFool efficiently computes smooth adversarial perturbations for deep networks.
problem Vulnerability of deep neural networks to adversarial attacks with specific statistical properties.
method SmoothFool: a general and computationally efficient framework for computing smooth adversarial perturbations.
result Smoothness significantly enhances robustness against adversarial attacks and improves transferability.
The paper explores maximal perturbations to hide certain attributes in data while keeping the model's performance intact.
problem Protecting sensitive attributes from both model and human detection.
method Adversarial perturbations applied to raw data to conditionally damage model's classification of one attribute while preserving the rest.
result Maximal perturbations can hide certain attributes from both model and human detection, impacting model performance but not human perception.
Charge measurements for instantons and gravitational perturbations.
problem Evaluating charges in Hermitian non-Kähler Einstein 4-manifolds and their perturbations.
method Evaluation of charges via Killing spinors and perturbation analysis of gravitational instantons.
result Generic gravitational perturbations admit a closed 2-form measuring the charge change.
Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.
problem Understanding how Steklov eigenvalues respond to boundary changes.
method Analyzing smooth boundary perturbations of Steklov eigenvalues.
result Steklov eigenvalues are generically simple under such perturbations.
Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.
problem Linear perturbations of Spin(7) metrics.
method Applying the method of linear perturbations to Spin(7)-structures.
result Only rank one nilpotent matrices determine nontrivial perturbations.
Simple perturbation of Vafa-Witten equations leads to transversality.
problem Transversality of Vafa-Witten moduli space.
method Simple perturbation of Vafa-Witten equations, proving transversality for SU(2) or SO(3) structure groups. result For generic perturbation parameter, the full rank part of the moduli space satisfies transversality.
We developed a perturbation model for affine gravity theories.
problem Cosmological perturbations in theories without metric.
method Segregated perturbations into symmetric and antisymmetric components, decomposing into irreducible elements.
result Fully addressed gauge freedom in affine gravity theories.
Perturbative GAN reduces training complexity and improves image quality.
problem Training complexity and image quality in GANs.
method Replaces convolution layers with perturbation layers that add fixed noise masks.
result Higher inception score and faster convergence of generated images.
Unified analysis of perturbation-based strategies in stochastic and adversarial bandit problems.
problem Optimality of perturbation-based strategies in multi-armed bandit problems.
method Unified regret analysis for stochastic and adversarial settings, using perturbations of sub-Weibull and bounded support.
result Unified bounds for perturbations in both stochastic and adversarial settings, with optimal perturbations of Frechet-type.