Related papers: Signature Kernel and Schwinger-Dyson Kernel Equati…
Kernel methods are an extremely popular set of techniques used for many important machine learning and data analysis applications. In addition to having good practical performances, these methods are supported by a well-developed theory.…
In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is…
The paper introduces new sufficient conditions of strict positive definiteness for kernels on d-dimensional spheres which are not radially symmetric but possess specific coefficient structures. The results use the series expansion of the…
Signature tensors of paths are a versatile tool for data analysis and machine learning. Recently, they have been applied to persistent homology, by embedding barcodes into spaces of paths. Among the different path embeddings, the…
Randomized smoothing is a powerful tool for certifying robustness to adversarial perturbations, including poisoning attacks via randomized training and evasion attacks via randomized inference. Extending these guarantees to backdoor…
This work proposes four novel hybrid quadrature schemes for the efficient and accurate evaluation of weakly singular boundary integrals (1/r kernel) on arbitrary smooth surfaces. Such integrals appear in boundary element analysis for…
This paper presents a parametric family of compactly-supported positive semidefinite kernels aimed to model the covariance structure of second-order stationary isotropic random fields defined in the $d$-dimensional Euclidean space. Both the…
Kernel methods are an incredibly popular technique for extending linear models to non-linear problems via a mapping to an implicit, high-dimensional feature space. While kernel methods are computationally cheaper than an explicit feature…
The expected signature uniquely determines the law of a random rough path under a moment-growth condition, yet finite-sample bounds for estimating it from a single long dependent trajectory have been lacking. We study a stationary…
Boundary integral equations and Nystrom discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weakly-singular kernel arises, in which case specialized quadratures that…
We show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann-Radin crystallization theorem (R. C. Heitmann, C.…
We bring the theory of rough paths to the study of non-parametric statistics on streamed data. We discuss the problem of regression where the input variable is a stream of information, and the dependent response is also (potentially) a…
Any applied mathematical model contains parameters. The paper proposes to use kernel learning for the parametric analysis of the model. The approach consists in setting a distribution on the parameter space, obtaining a finite training…
The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel--Darboux form constructed from sequences of biorthogonal polynomials. For…
On a two dimensional Stein space with isolated, normal singularities, smooth finite type boundary, and locally algebraic Bergman kernel, we establish an estimate on the type of the boundary in terms of the local algebraic degree of the…
We present a derivation of the Schr\"odinger equation for a path integral of a point particle in a space with curvature and torsion which is considerably shorter and more elegant than what is commonly found in the literature.
We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schr\"odingerization, introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y.…
A pair of rooted tangents -- defining a quantum triangle -- with an associated quantum wave of spin 1/2 is proposed as the primitive to represent and compute symmetry. Measures of the spin characterize how "isosceles" or how "degenerate"…
The de Rham complex arises naturally when studying problems in electromagnetism and fluid mechanics. Stable numerical methods to solve these problems can be obtained by using a discrete de Rham complex that preserves the structure of the…
The path integral by which quantum field theories are defined is a particular solution of a set of functional differential equations arising from the Schwinger action principle. In fact these equations have a multitude of additional…