New DRGP models improve prediction accuracy for sequential data.
problem Modeling sequential data for applications like autonomous driving.
method Introduces Deep recurrent Gaussian process (DRGP) models based on Sparse Spectrum Gaussian process (SSGP) and variational Sparse Spectrum Gaussian process (VSSGP).
result Improves prediction accuracy compared to current state of the art methods.
The paper confirms a Weyl law for the p-Laplacian on closed Riemannian manifolds.
problem The variational spectrum of the p-Laplacian on closed Riemannian manifolds.
method Based on ideas of Gromov and Liokumovich, Marques, Neves.
result A Weyl law holds for the variational spectrum of the p-Laplacian.
Standard sparse pseudo-input approximations to the Gaussian process (GP) cannot handle complex functions well. Sparse spectrum alternatives attempt to answer this but are known to over-fit. We suggest the use of variational inference for the sparse spectrum approximation to avoid both issues. We model the covariance fu…
Modeling sequential data has become more and more important in practice. Some applications are autonomous driving, virtual sensors and weather forecasting. To model such systems so called recurrent models are used. In this article we introduce two new Deep Recurrent Gaussian Process (DRGP) models based on the Sparse Sp…
While much research effort has been dedicated to scaling up sparse Gaussian process (GP) models based on inducing variables for big data, little attention is afforded to the other less explored class of low-rank GP approximations that exploit the sparse spectral representation of a GP kernel. This paper presents such a…
In this article we prove a generalization of Weyl's criterion for the spectrum of a self-adjoint nonnegative operator on a Hilbert space. We will apply this new criterion in combination with Cheeger-Fukaya-Gromov and Cheeger-Colding theory to study the k-form essential spectrum over a complete manifold with vanishing…
Multi-parameter cognition in a cognitive radio network (CRN) provides a more thorough understanding of the radio environments, and could potentially lead to far more intelligent and efficient spectrum usage for a secondary user. In this paper, we investigate the multi-parameter cognition problem for a CRN where the pri…
Introduces tunable basis functions for Gaussian processes.
problem Reduces computational complexity in Gaussian process approximations.
method Introduces tunable, local, and bounded basis functions for kernel approximation.
result Demonstrates superior performance compared to state-of-the-art methods, especially with poorly chosen kernel functions.
Improved model for non-smooth signals with complex spectra.
problem Current models struggle with non-smooth signals and complex spectral structures.
method CGPCM and RGPCM models with causality and Bayesian nonparametric interpretations, improved variational inference.
result Proposed models show better performance on synthetic and real-world data.
We analyze the frequency spectrum of quantum neural networks using algebraic methods and prove maximality results.
problem Understanding the frequency spectrum and maximality properties of quantum neural networks.
method Using Minkowski sums and algebraic descriptions, we prove maximality results for QNN architectures.
result We establish spectral invariance under area-preserving transformations, showing the frequency spectrum depends only on the area A=RL. Study eigenvalues of Dirac operator on surfaces, proving existence and deriving inequalities.
problem Finding optimal bounds for Dirac eigenvalues on spin surfaces.
method Minimization problem within a fixed conformal class, focusing on surfaces.
result Derive isoperimetric inequalities for the Dirac operator on the sphere, complete conformal spectrum characterization.
Researchers compute the spectrum of Hodge-Laplacian on 1-forms for SU(2) and SO(3).
problem Computing the spectrum of the Hodge-Laplacian on 1-forms for homogeneous 3-spheres.
method Explicit computation of eigenvalues for Berger 3-spheres and general homogeneous metrics on SU(2) and SO(3).
result The spectrum on 1-forms determines the metric up to isometry.
Formula derived for Laplace-Beltrami spectrum on homogeneous spaces.
problem Calculating the spectrum of the Laplace-Beltrami operator on homogeneous spaces.
method Formula derivation based on eigenvalues of a generalized Casimir operator and spherical representations.
result First detailed computation and investigation of the spectrum for a family of metrics on the Aloff-Wallach manifold.
VAE improves MCMC efficiency by generating diverse prior proposals.
problem Inefficient MCMC methods in Bayesian inverse problems, especially subsurface flow modeling.
method Uses Variational Autoencoder (VAE) to generate broader-spectrum prior proposals.
result VAE achieves comparable accuracy to Karhunen-Loève Expansion (KLE) and outperforms it when correlation length is unknown.
Proofs for spectral and geometric properties of hyperbolic surfaces.
problem Spectral and geometric properties of hyperbolic surfaces.
method Streamlined proofs for isospectral and quasi-Fuchsian groups.
result Generically, isospectral hyperbolic surfaces are isometric.
Researchers prove spectral rigidity of Liouville tori under specific conditions.
problem Spectral rigidity of Liouville tori under generic conformal classes.
method Noncancellation of wave trace and analysis of second order variational formula for energy.
result Laplace isospectral deformations of Liouville metrics on torus are trivial.
Deep autoencoder network solves spectrum sharing problems efficiently.
problem Resource allocation in wireless communications and D2D networks.
method Generative neural network (autoencoder) for solving linear sum assignment problems.
result Hybrid autoencoder architecture outperforms other methods in accuracy and speed.
Quantum models can approximate any function if data encoding allows for a rich enough frequency spectrum.
problem Theoretical properties of quantum machine learning models, particularly their expressive power.
method Investigated how data encoding affects the expressive power of parametrized quantum circuits.
result Quantum models can access increasingly rich frequency spectra by repeating data encoding gates, potentially making them universal function approximators.
The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution.
problem Studying spectral properties of the horizontal Laplacian
method Interpreting the horizontal Laplacian as a twisted Laplacian acting on a flat vector bundle
result The horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold
Prove first-band large-diameter asymptotics for Dirichlet spectrum on horoconvex domains in real hyperbolic space.
problem Prove first-band large-diameter asymptotics for Dirichlet spectrum on horoconvex domains in real hyperbolic space.
method Prove first-band large-diameter asymptotics for Dirichlet spectrum on horoconvex domains in real hyperbolic space.
result Prove first-band large-diameter asymptotics for Dirichlet spectrum on horoconvex domains in real hyperbolic space.
Paper uses non-Euclidean analysis to classify brain structure variations.
problem Classifying joint variations in multi-object brain structures.
method Combines non-Euclidean statistics and non-parametric integrative analysis.
result Effective, robust, and interpretable joint structure found.
Heavy-tailed regularization improves deep neural network performance.
problem Improving generalization of deep neural networks.
method Introducing Heavy-Tailed Regularization, using differentiable penalty terms and Bayesian statistics.
result Heavy-tailed regularization outperforms conventional regularization techniques.
Study on spectral stability of an embedded annulus under curve shortening and Ricci flows.
problem Spectral stability of Dirichlet eigenvalues on an evolving annulus.
method Variational formulas, Rellich-type identities, and harmonic capacity methods.
result Established quantitative bounds comparing the spectrum of the evolving annulus with a flat cylinder.
The miltifractal properties and scaling behaviour of the exchange rate variations of the Iranian rial against the US dollar from a daily perspective is numerically investigated. For this purpose the multifractal detrended fluctuation analysis (MF-DFA) is used. Through multifractal analysis, the scaling exponents, gener…
We consider an analytic family of Riemannian metrics on a compact smooth manifold M. We assume the Dirichlet boundary condition for the η-Laplacian and obtain Hadamard type variation formulas for analytic curves of eigenfunctions and eigenvalues. As an application, we show that for a subset of all Cr Riemannian …
Study on null-torsion holomorphic curves in 6-sphere, focusing on their second variation.
problem Characterize the second variation of area for null-torsion holomorphic curves in the round 6-sphere.
method Analyzing the spectrum of the Jacobi operator for compact null-torsion holomorphic curves.
result For g≤6, the multiplicity of the lowest eigenvalue λ1=−2 is exactly 4d. We define a new spectrum for compact length spaces and Riemannian manifolds called the "covering spectrum" which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers δ>0 which identify the distinct δ covers of the space. We investigat…
The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …
The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …
There is a large body of work, built on tools developed in mathematics and physics, demonstrating that financial market prices exhibit self-similarity at different scales. In this paper, we explore the use of analytical topology to characterize financial price series. While wavelet and Fourier transforms decompose a si…
New biharmonic Steklov problem on forms yields eigenvalue estimates.
problem Eigenvalue estimates for differential forms with curvature quantities.
method Introduced a new biharmonic Steklov problem and proved existence of a discrete spectrum.
result Established Kuttler-Sigillito inequalities connecting eigenvalues of differential forms.
Study the energy spectrum of metrics on surfaces and its relation to simple length spectrum.
problem Relate the energy spectrum to the simple length spectrum of metrics on surfaces.
method Analyze the energy spectrum of metrics on surfaces and their Teichmüller spaces, considering homotopy conditions.
result The energy spectrum determines the simple length spectrum under certain conditions.
Variational methods yield formulas for eigenvalues of elliptic operators, with applications to metric evolution.
problem Deriving formulas for eigenvalues of elliptic operators on compact manifolds.
method Variational methods applied to elliptic operators on compact Riemannian manifolds.
result Generic subsets of metrics yield simple spectra of elliptic operators.
The paper compares two spectrum definitions and finds stability in one modification.
problem Generalizing eigenvalues to arbitrary functionals with stability.
method Comparison of Gromov's homotopy significant spectrum and Krasnoskii spectrum, with a modified definition of the homotopy significant spectrum.
result The modified homotopy significant spectrum is stable, and Cheeger constant corresponds to Krasnoskii eigenvalue.
Study shows spectrum properties for specific Hadamard manifolds.
problem Spectrum properties of Hadamard manifolds.
method Absolute continuity and spectrum determination for two classes of Hadamard manifolds.
result Spectrum properties determined for specific Hadamard manifolds.
Proofs high-dimensional spectrum convergence of weighted sample covariance.
problem High-dimensional spectrum convergence of weighted sample covariance.
method Proposes a new, concise proof with stronger assumptions.
result Spectrum convergence proven for different weight distributions.
Iterative method 'Concent' corrects spectrum bias in covariance matrices.
problem Consistent bias in the spectrum of covariance matrices.
method 'Concent' iterative algorithm.
result Corrects spectrum bias for small and moderate dimensions.
Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.
problem Understanding transverse link invariants in the annular setting.
method Constructs a stable homotopy type for annular links and defines a map to the Khovanov skein spectrum.
result At extreme gradings, the map from the Khovanov spectrum to the Khovanov skein spectrum recovers the cohomotopy transverse invariant.
Unified method to compute Laplace spectra on homogeneous principal bundles.
problem Computing the Laplace-Beltrami spectrum on homogeneous principal bundles.
method Unified representation-theoretic approach using generalized canonical variations and spectral branching criterion.
result Explicit formulas for the full spectra of several geometric families.
The spectrum of certain manifolds matches that of hyperbolic space if the bottom spectrum is maximal.
problem Investigating spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum.
method Analyzing the spectrum of the Laplacian on manifolds with specific Ricci curvature bounds.
result The spectrum of the manifold coincides with that of hyperbolic space if the bottom spectrum is maximal.
Lower bounds for Hodge-Laplacian spectrum on orbifolds.
problem Finding bounds for the essential spectrum of Hodge-Laplacian.
method Deriving lower bounds for the essential spectrum of the Hodge-Laplacian on geometrically finite orbifolds and their suborbifolds.
result Lower bounds for the essential spectrum of the Hodge-Laplacian.
We construct Riemannian manifolds with singular continuous spectrum embedded in the absolutely continuous spectrum of the Laplacian. Our manifolds are asymptotically hyperbolic with sharp curvature bounds.
Trapezoids uniquely identified by their Dirichlet Laplace spectrum.
problem Identifying trapezoids based on their spectral properties.
method Analyzing the Dirichlet Laplace spectrum of non-obtuse trapezoids.
result Non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum.
We propose a flexible framework for spectral conversion (SC) that facilitates training with unaligned corpora. Many SC frameworks require parallel corpora, phonetic alignments, or explicit frame-wise correspondence for learning conversion functions or for synthesizing a target spectrum with the aid of alignments. Howev…
In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a sp…
Survey on bottom of spectrum of Hodge Laplacian on complete noncompact Kähler manifolds
problem Bottom of the spectrum of Hodge Laplacian on complete noncompact Kähler manifolds
method Survey on Kähler hyperbolic manifolds and bounded symmetric domains
result Proposed several open problems
The paper extends decay estimates to graphs with positive spectrum.
problem Proving decay estimates for nonnegative functions on graphs.
method Sharp ℓ2 decay estimates for nonnegative generalized subharmonic functions. result Extends Li and Wang's result to graphs with positive Laplacian spectrum.
Upper bounds for volume spectrum depend on volume, dimension, and a conformal invariant.
problem Bounding the volume spectrum of Riemannian manifolds.
method Proves upper bounds that depend on volume, dimension, and a conformal invariant.
result Upper bounds for the volume spectrum are established.