arXiv research

Extends importance sampling to nonlinear models using adjoint operators.

New analysis proves sketching operators' RIP guarantees for mixture models without importance sampling.

GJMS operators connect geometry, analysis, and physics.

CR Paneitz operator on non-embeddable tori has infinitely many negative eigenvalues

In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of Jacobi forms of higher degree. We present some explicit invariant differential oper…

We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein-Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.

Operating envelope is an important concept in industrial operations. Accurate identification for operating envelope can be extremely beneficial to stakeholders as it provides a set of operational parameters that optimizes some key performance indicators (KPI) such as product quality, operational safety, equipment effic…

We give a method of decomposing bundle-valued polynomials compatible with the action of the Lie group $Spin(n)$, where important tools are $Spin(n)$-equivariant operators and their spectral decompositions. In particular, the top irreducible component is realized as an intersection of kernels of these operators.

Attention operators have been widely applied in various fields, including computer vision, natural language processing, and network embedding learning. Attention operators on graph data enables learnable weights when aggregating information from neighboring nodes. However, graph attention operators (GAOs) consume exces…

The paper calculates the noncommutative residue for a rescaled Dirac operator on 6D manifolds.

Survey on strong convergence in random matrices and its applications.

We give an overview of the generalized Calderón-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. $L^p-L^q$ of…

Study on CR Paneitz operator on non-embeddable CR manifolds.

The challenge of assigning importance to individual neurons in a network is of interest when interpreting deep learning models. In recent work, Dhamdhere et al. proposed Total Conductance, a "natural refinement of Integrated Gradients" for attributing importance to internal neurons. Unfortunately, the authors found tha…

Kernel method approximates dynamical operators from data.

We consider the crossed product $G_{ga}$ by $\R_+^*$ of the adiabatic groupoid associated with any Lie groupoid $G$. We construct an explicit Morita equivalence between the exact sequence of order 0 pseudodifferential operators on $G$ and (a restriction of) the natural exact sequence associated with $G_{ga}$. As an imp…

New operations defined on moduli spaces for bundles with orientations.

The paper establishes a majorization result for symmetric matrices.

This paper studies gl-regular Nijenhuis operators and their properties.

New methods for clustering graphs using spectral analysis.

Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on such spaces are, for instance, required to embed conditional probability distributi…

Develops tools for studying intersections of elliptic operators, focusing on $J$-holomorphic maps.

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. I…

Develops inference combinators for probabilistic programs using neural networks.

We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the $k$th eigenvalue by the lower eigenvalues,…

We provide a thorough construction of a system of compatible determinant line bundles over spaces of Fredholm operators, fully verify that this system satisfies a number of important properties, and include explicit formulas for all relevant isomorphisms between these line bundles. We also completely describe all possi…

We review origins and main properties of the most important bracket operations appearing canonically in differential geometry and mathematical physics in the classical, as well as the supergeometric setting. The review is supplemented by a few new concepts and examples.

In this paper we introduce a new family of operator-valued distributions on Euclidian space acting by convolution on differential forms. It provides a natural generalization of the important Riesz distributions acting on functions, where the corresponding operators are $(-Δ)^{-α/2}$, and we develop basic analogous prop…

This article investigates local properties of the further generalized Weierstrass relations for a spin manifold $S$ immersed in a higher dimensional spin manifold $M$ from viewpoint of study of submanifold quantum mechanics. We show that kernel of a certain Dirac operator defined over $S$, which we call submanifold Dir…

Proves Green function rigidity for specific operators and obtains new ADM mass formula.

The Atiyah-Singer index theorem is a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their…

Computer algebra methods are applied to investigation of spectral asymptotics of elliptic differential operators on curved manifolds with torsion and in the presence of a gauge field. In this paper we present complete expressions for the second coefficient (E_2) in the heat kernel expansion for nonminimal operator on m…

User response prediction makes a crucial contribution to the rapid development of online advertising system and recommendation system. The importance of learning feature interactions has been emphasized by many works. Many deep models are proposed to automatically learn high-order feature interactions. Since most featu…

In this paper we introduce the Dirac and spin-Dirac operators associated to a connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac operators associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time) and calculate the square of these operators, which play an important role i…

We study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume.

Using a K-theory point of view, Bott related the Atiyah-Singer index theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This article explains how to prove the local index theorem for compact homogenous spaces using Lie algebra methods. The method follows in outline the proof of …

This paper extends transfer operator theory to McKean-Vlasov equations.

The article constructs Feynman propagators for normally hyperbolic operators on curved spacetimes.

Researchers calculate the second coefficient in the expansion of a Toeplitz operator.

Study geometric inequalities and boundary estimates for Einstein-type manifolds with boundary.

The paper studies eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.

This study analyzes NAS benchmarks and finds that only a subset of operations is crucial for generating high-performing architectures.

We develop the first basic Operational Risk perspective on key risk management issues associated with the development of new forms of electronic currency in the real economy. In particular, we focus on understanding the development of new risks types and the evolution of current risk types as new components of financia…

Automates neural network design for diverse tasks.

Transfer operators such as the Perron--Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We …

Simulates DeLend Platform behavior to optimize operational parameters.

Necessary and sufficient conditions for the exponentiation of finite-dimensional real Lie algebras of linear operators on complete Hausdorff locally convex spaces are obtained, focused on the equicontinuous case - in particular, necessary conditions for exponentiation to compact Lie groups are established. Applications…

Construct opers with apparent singularities from λ-connections on Riemann surfaces.