arXiv research

Paper proves non-triviality of Johnson kernel torsion subgroup.

Gradient descent benefits from tangent kernel advantages under specific conditions.

We investigate the prescribed Q-curvature flow for GJMS operators with non-trivial kernel on compact manifolds of even dimension. When the total Q-curvature is negative, we identify a conformally invariant condition on the nodal domains of functions in the kernel of the GJMS operator, allowing us to prove the global ex…

We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.

Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.

New kernels from ELU and GELU networks reveal non-trivial fixed points.

Quantum kernel methods can lead to trivial models due to exponential concentration of kernel values.

The paper proves conditions for the triviality of $L^2$-harmonic forms on Riemannian manifolds.

Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct …

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We show explicitly that the obtaine…

We show that a non-trivial, non-central normal subgroup of the braid groups contains a braid whose closure is a hyperbolic knot with arbitrary large genus. This shows that non-faithfulness of a quantum representation implies that the corresponding quantum invariant fails to detect the unknot. The proof utilizes the Deh…

We prove that the first complex homology of the Johnson subgroup of the Torelli group $T_g$ is a non-trivial unipotent $T_g$-module for all $g\ge 4$ and give an explicit presentation of it as a $\Sym H_1(T_g,\C)$-module when $g\ge 6$. We do this by proving that, for a finitely generated group $G$ satisfying an assumpti…

Detecting the emergence of abrupt property changes in time series is a challenging problem. Kernel two-sample test has been studied for this task which makes fewer assumptions on the distributions than traditional parametric approaches. However, selecting kernels is non-trivial in practice. Although kernel selection fo…

The X-ray transform on the periodic slab $[0,1]\times\mathbb T^n$, $n\geq0$, has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless $n=0$. We characterize t…

Bayesian nonparametric models, such as Gaussian processes, provide a compelling framework for automatic statistical modelling: these models have a high degree of flexibility, and automatically calibrated complexity. However, automating human expertise remains elusive; for example, Gaussian processes with standard kerne…

It is an open problem whether Kirk's $σ$ invariant is the complete obstruction to a link map $S^2\cup S^2\to S^4$ being link homotopically trivial. With the objective of constructing counterexamples, Li proposed a link homotopy invariant $ω$ that is defined on the kernel of $σ$ and also obstructs link nullhomotopy. We …

New bounded cohomology classes found for exact forms on curved manifolds.

A well-recognized limitation of kernel learning is the requirement to handle a kernel matrix, whose size is quadratic in the number of training examples. Many methods have been proposed to reduce this computational cost, mostly by using a subset of the kernel matrix entries, or some form of low-rank matrix approximatio…

In this paper we investigate the small time heat kernel asymptotics on the cut locus on a class of surfaces of revolution, which are the simplest 2-dimensional Riemannian manifolds different from the sphere with non trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate poi…

We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2,C) connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even component of the space of flat PSL(2,C) connections. The non-trivial part consists …

This work challenges the Neural Tangent Kernel's role in overparameterized neural networks, especially with large width and depth.

This paper presents a new exact sequence for orbifold braid groups and mapping class groups.

We study the Rarita-Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel. For positive quaternion Kähler manifolds and symmetric spaces with spin structure we giv…

A novel algorithm for unbiased graph kernel estimation with subquadratic time complexity.

New bounds for KRR condition number reveal overfitting phenomena.

We consider the problem of Bayesian optimization (BO) in one dimension, under a Gaussian process prior and Gaussian sampling noise. We provide a theoretical analysis showing that, under fairly mild technical assumptions on the kernel, the best possible cumulative regret up to time $T$ behaves as $Ω(\sqrt{T})$ and $O(\s…

We study the effect of Nielsen moves and their geometric counterparts, handle slides, on good boundary links. A collection of links, universal for 4-dimensional surgery, is shown to admit Seifert surfaces with trivial Lagrangian. They are good boundary links, with Seifert matrices of a more general form than in known c…

Paper proves existence of Dirac-harmonic maps with trivial index.

New method uses kernel methods to approximate ground states of quantum Hamiltonians efficiently.

Kernel adaptive filters, a class of adaptive nonlinear time-series models, are known by their ability to learn expressive autoregressive patterns from sequential data. However, for trivial monotonic signals, they struggle to perform accurate predictions and at the same time keep computational complexity within desired …

We prove that the sequence of projective representations of the mapping class group obtained from the projective flat connection in the SU(n)-Verlinde bundles over Teichmuller space is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genu…

Zero-inflated datasets, which have an excess of zero outputs, are commonly encountered in problems such as climate or rare event modelling. Conventional machine learning approaches tend to overestimate the non-zeros leading to poor performance. We propose a novel model family of zero-inflated Gaussian processes (ZiGP) …

Learning linear combinations of multiple kernels is an appealing strategy when the right choice of features is unknown. Previous approaches to multiple kernel learning (MKL) promote sparse kernel combinations to support interpretability and scalability. Unfortunately, this 1-norm MKL is rarely observed to outperform tr…

Calculates Dehn twist actions on conformal blocks for modular categories.

This paper generalizes regularized regression problems in a hyper-reproducing kernel Hilbert space (hyper-RKHS), illustrates its utility for kernel learning and out-of-sample extensions, and proves asymptotic convergence results for the introduced regression models in an approximation theory view. Algorithmically, we c…

Proofs for flows of linear vector fields and their applications.

New RL method handles large state-action spaces with complex models.

The paper explores how kernel eigenalignments affect generalization in KRR.

We introduce and study new invariants associated with Laplace type elliptic partial differential operators on manifolds. These invariants are constructed by using the off-diagonal heat kernel; they are not pure spectral invariants, that is, they depend not only on the eigenvalues but also on the corresponding eigenfunc…

New method tackles high-dimensional contextual bandits with flexible kernel models.

The paper analyzes heat kernel asymptotics for real powers of Laplacians on manifolds.

Study kernels of mapping class group representations on surface configuration spaces.

In this paper, we introduce a geometric structure called top, which is a trivialized bundle of plane pencils over a Riemannian 3-manifold, defined as the set of kernels of a circle of 1-forms (e.g. of contact and integrable forms) with particular properties with respect to the metric. We classify the manifolds which ad…

Uniform heat kernel and diffusion bridge asymptotics for sub-Riemannian geometry.

Proves properties of Torelli Lie algebra for surfaces.

PAC-Bayesian bounds improve understanding of K-NN classifier performance.

New algorithm optimizes noisy function evaluations with delayed feedback.

Improves regression efficiency by separating material and immaterial parts of responses.