arXiv research

Improved algorithm speeds up generation of universal adversarial perturbations.

We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion that depends on the geometry of the manifold and the ambient space. We use these…

New counterexample shows curved surfaces can deform geodesics without diffeomorphism.

The purpose of this paper is to study finite dimensional equivariant moduli problems from the viewpoint of stratification theory. We show that there exists a stratified obstruction system for a finite dimensional equivariant moduli problem. In addition, we define a coindex for a G-vector bundle which is determined by t…

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

Formula calculates invariant for 3-manifolds with torus boundaries.

Study analyzes perturbations in singular subspaces under random noise.

We prove a theorem which asserts that the Lie algebra of all holomorphic vector fields on a compact Kähler manifold with a perturbed extremal metric has the structure similar to the case of an unperturbed extremal Kähler metric proved by Calabi.

In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set $\fc$ of $m$ vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then there is a pertu…

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

Fast algorithm for rescaling vectors with clipping, improving training efficiency.

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

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

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

Study local perturbations of vector bundles with polynomial curvature solutions.

Learnable token perturbations boost extrapolation in LLMs.

We propose an extension of the recently-proposed volume conjecture for closed hyperbolic 3-manifolds, to all orders in perturbative expansion. We first derive formulas for the perturbative expansion of the partition function of complex Chern-Simons theory around a hyperbolic flat connection, which produces infinitely-m…

New invariant counts graph configurations in 3D manifolds.

Deterministic bounds for tensor singular values and vectors, differing from matrix cases.

The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.

The paper proves new theorems about specific types of operator perturbations.

The paper studies vector fields on manifolds and their embeddings into tangent bundles.

We developed a perturbation model for affine gravity theories.

We investigate the orientability of a class of vector bundles over flag manifolds of real semi-simple Lie groups, which include the tangent bundle and also stable bundles of certain gradient flows. Closed formulas, in terms of roots, are provided.

We consider perturbed quadharmonic operators, $Δ^4 + V$, acting on sections of a Hermitian vector bundle over a complete Riemannian manifold, with the potential $V$ satisfying a bound from below by a non-positive function depending on the distance from a point. Under a bounded geometry assumption on the Hermitian vecto…

Study vector fields with complex singularities, proving bounds and formulas.

Complex functional maps link tangent bundles, preserving orientation and angles.

We quantify the sensitivity of the Eisenberg-Noe clearing vector to estimation errors in the bilateral liabilities of a financial system in a stylized setting. The interbank liabilities matrix is a crucial input to the computation of the clearing vector. However, in practice central bankers and regulators must often es…

Study proves 3-manifolds with parallel vector fields have odd Betti numbers.

A perturbative SU(3) Casson invariant $Λ_{SU(3)}(X)$ for integral homology 3-spheres is defined. Besides being fully perturbative, it has nice properties: (1) $4 . Λ_{SU(3)}(X)$ is an integer. (2) It is preseved under orientation change. (3) A connected sum formula holds. Explicit calculations of the invariant for $1/k…

For a closed oriented 3-manifold $Y$ we define $n(Y)$ to be the minimal non-negative number such that in each homotopy class of non-singular vector fields of $Y$ there is a Morse-Smale vector field with less or equal to $n(Y)$ periodic orbits. We combine the construction process of Morse-Smale flows given in [2] with h…

New deep learning method preserves orientation in shape matching.

Geometrically proves twisted Poincaré duality for orientable Poisson manifolds.

Beyond existing multi-view clustering, this paper studies a more realistic clustering scenario, referred to as incomplete multi-view clustering, where a number of data instances are missing in certain views. To tackle this problem, we explore spectral perturbation theory. In this work, we show a strong link between per…

We show that the Spivak normal fibration of an orientable 4-dimensional Poincaré complex has a vector bundle reduction.

Given a discrete subgroup $Γ$ of finite co-volume of $\mathrm{PGL}(2,\mathbb{R})$, we define and study parabolic vector bundles on the quotient $Σ$ of the (extended) hyperbolic plane by $Γ$. If $Γ$ contains an orientation-reversing isometry, then the above is equivalent to studying real and quaternionic parabolic vecto…

The paper calculates expected distances on partially oriented flag manifolds.

The paper modifies Vafa-Witten equations on 4-manifolds for better solution estimates.

Proposes an efficient method for ordered counterfactual explanations.

We consider remodeling the planar search patterns, in the presence of the river-type perturbation represented by the weak vector field, basing on the time-optimal paths as Finslerian solutions to the Zermelo navigation problem via Randers metric.

The paper calculates spectral torsion for rescaled Dirac operators on manifolds.

For any compact oriented manifold $M$, we show that that the top degree multi-vector fields transverse to the zero section of $\wedge^{\text{top}}TM$ are classified, up to orientation preserving diffeomorphism, in terms of the topology of the arrangement of its zero locus and a finite number of numerical invariants. Th…

We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.

Study linear perturbations in Schwarzschild black hole spacetime.

Given a state-of-the-art deep neural network text classifier, we show the existence of a universal and very small perturbation vector (in the embedding space) that causes natural text to be misclassified with high probability. Unlike images on which a single fixed-size adversarial perturbation can be found, text is of …

A new method for reconstructing flows from perturbed distributions.

Given a state-of-the-art deep neural network classifier, we show the existence of a universal (image-agnostic) and very small perturbation vector that causes natural images to be misclassified with high probability. We propose a systematic algorithm for computing universal perturbations, and show that state-of-the-art …

We elaborate on an idea of M. Abouzaid of equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an $A_\infty$-algebra. This is a variation on K. Fukaya's definition of Morse-$A_\infty$-categories for closed oriented manifolds involving families of Morse funct…