Study cryptocurrency price dynamics using adaptive EMD and spectral analysis.
problem Analyze the time-varying volatility of cryptocurrency prices.
method Adaptive complementary ensemble empirical mode decomposition (ACE-EMD) and Hilbert spectral analysis.
result Reveal the properties of various timescales in cryptocurrency price dynamics.
Defines and computes a generalized spectral action for Lorentz warped products.
problem Computing spectral actions for Lorentz warped products.
method Defines and computes the bimetric spectral Einstein-Hilbert action for Lorentz warped products.
result Derives a Kastler-Kalau-Walze type theorem for Lorentz warped products.
Extends Einstein-Hilbert action to higher-order spectral triples.
problem No specific problem stated; focuses on extending action.
method Introduced two second-order spectral triples and computed their Einstein-Hilbert actions.
result Demonstrated applicability of the theoretical framework.
New methods avoid spectral pollution in transfer operators for accurate analysis.
problem Spectral pollution in finite-dimensional approximations of transfer operators.
method Algorithms for computing spectral properties of transfer operators without spectral pollution.
result Accurate spectral estimation across various applications, including protein folding models.
Optimizes learning Hilbert-Schmidt operators between Sobolev spaces.
problem Statistical limits of learning mappings between infinite-dimensional function spaces.
method Minimax optimal regularization and multilevel training.
result Multilevel kernel operator learning achieves optimal learning rate.
New spectral triples defined for SU(1,1) using harmonic analysis.
problem Defining new spectral triples for SU(1,1).
method Using harmonic analysis of SU(1,1) to construct pseudo-Riemannian and indefinite spectral triples.
result Triple (A,H,D) forms both pseudo-Riemannian and indefinite spectral triples. New Hilbert bundles with ends defined from indexed bases.
problem Defining new structures in Hilbert bundles.
method Indexed bases and unitary operators of finite propagation.
result Characteristic classes of Hilbert bundles with ends.
Harmonic gauge simplifies geometric analysis of Riemannian metrics.
problem Analyzing the Hilbert-Einstein functional and its stability.
method Developed a harmonic gauge to eliminate divergence terms and induce elliptic structure.
result Positivity of curvature operator implies spectral stability of the functional.
Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.
Novel framework for learning infinitesimal generator of stochastic processes.
problem Challenges in learning infinitesimal generator due to unbounded nature and state space dimensionality.
method Introduces a novel framework based on energy functional, integrates physical priors, and uses reduced-rank estimator in RKHS.
result Learning bounds independent of state space dimension and non-spurious spectral estimation.
New algorithms compute Koopman operators on RKHSs efficiently and accurately.
problem Data-driven spectral analysis of Koopman operators on RKHSs.
method General, provably convergent algorithms for RKHSs.
result Optimal algorithms with error control and spectral measures.
We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity…
Enhances Koopman operator estimation with intrinsic observables in RKHS.
problem Accurate estimation of Koopman operator and its spectrum.
method Jet Extended Dynamic Mode Decomposition (JetEDMD) leveraging RKHS jets.
result Proves JetEDMD's superiority with error bounds and convergence rate.
Spectral algorithms improve under covariate shift with novel weighted techniques.
problem Improving spectral algorithms' performance under covariate shift.
method Analysis of spectral algorithms in non-parametric regression over RKHS, proposing a weighted spectral algorithm with clipped weights.
result Normalized weighted spectral algorithm achieves optimal capacity-independent convergence rates, and clipped weights can approach optimal capacity-dependent rates.
We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with C2 boundaries. We show that for an n-dimensional geometry, the spectral gap is bounded above by (n−1)2/4, which we prove to be the infimum of the essential spectrum. We also construct examples of c…
Two proofs of Melrose-Piazza theorem on spectral sections.
problem Analytic index of families of Fredholm operators.
method Two independent proofs of the theorem, generalizing and clarifying the analytic index definition.
result Generalization and clarification of the Melrose-Piazza theorem on spectral sections.
Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety of engineering and scientific fields. Dynamic mode decomposition (DMD), which is a numerical algorithm for the spectral analysis of Koopman operators, has been attracting attention as a way of obtaining global modal descriptions of NLDSs with…
Estimates kernel eigenvalues for compositional dot-product kernels.
problem Improving estimates for kernel eigenvalues.
method Eigenvalue decay estimates of integral operators associated with dot-product kernels.
result Improved estimates for kernel volumes in reproducing kernel Hilbert spaces.
Random feature approximation speeds up spectral methods and improves learning rates.
problem Improving the efficiency and generalization of spectral methods in large-scale algorithms.
method Combining random feature approximation with spectral regularization methods.
result Optimal learning rates for estimators over various regularity classes, including those not in the RKHS.
Paper analyzes spectral algorithms under covariate shift, providing convergence rates.
problem Addressing distributional mismatch in regression models.
method Incorporates importance weights into spectral algorithms in RKHS.
result Establishes minimax-optimal convergence rates for misspecified cases.
Extends random feature analysis to spectral methods and improves learning rates.
problem Improving generalization properties of spectral methods in large-scale learning.
method Extends random feature analysis to a broad class of spectral regularization techniques, including gradient descent and Nesterov method.
result Obtains optimal learning rates for regularity classes, including those not in the RKHS.
In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral/regularized algorithms, including ridge regression, principal component regression, and gradient methods. We pro…
HHT feature generation enhances financial time series forecasting.
problem Forecasting nonstationary financial time series.
method CEEMD and HHT for decomposition, machine learning integration.
result HHT-enhanced models outperform traditional models in forecasting.
Develops an ℓ_p theory for PCA and spectral clustering.
problem Lack of precise characterizations of PCA scores for low-dimensional embedding.
method An ℓ_p perturbation theory for PCA in Hilbert spaces, analyzing eigenvectors and Gram matrix.
result Optimal recovery results for Gaussian mixture and stochastic block models.
The report analyzes infinite-dimensional output space regression.
problem Learning theory in vector-valued RKHS regression.
method Integral operator technique with spectral theory for non-compact operators.
result Results with minimal assumptions using Chebyshev's inequality.
In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the N=(2,2) 2d Landau-Ginzburg theory in models describing link embeddings in R3 to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert sp…
We consider a continuous curve of self-adjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain Dm and fixed {\it intermediate} domain DW. Our main example is a family of symmetric generalized operators of Dirac type on a compact manifold with boundary with varying well-posed…
We study learning properties of accelerated gradient descent methods for linear least-squares in Hilbert spaces. We analyze the implicit regularization properties of Nesterov acceleration and a variant of heavy-ball in terms of corresponding learning error bounds. Our results show that acceleration can provides faster …
In this paper we introduce the curvature of densely defined universal connections on Hilbert C∗-modules relative to a spectral triple (or unbounded Kasparov module), obtaining a well-defined curvature operator. Fixing the spectral triple, we find that modulo junk forms, the curvature only depends on the represente…
The paper studies convergence of kernel autocovariance operators for stationary processes.
problem Estimating autocovariance operators of stationary processes on Polish spaces.
method Investigates convergence of empirical estimates of autocovariance operators under various conditions.
result Provides consistency results for kernel PCA and spectral analysis methods.
We address the problem of estimating the mixing time of a Markov chain from a single trajectory of observations. Unlike most previous works which employed Hilbert space methods to estimate spectral gaps, we opt for an approach based on contraction with respect to total variation. Specifically, we estimate the contracti…
Let K be a compact Lie group, endowed with a bi-invariant Riemannian metric. The complexification G of K inherits a Kaehler structure having twice the kinetic energy of the metric as its potential, and left and right translation turn the Hilbert space of square-integrable holomorphic functions on G relative to a suitab…
In this paper we study the spectral asymmetry of (possibly nonselfadjoint) elliptic PsiDO's in terms of the difference of zeta functions coming from different cuttings. Refining previous formulas of Wodzicki in the case of odd class elliptic PsiDO's, our main results have several consequence concerning the local indepe…
Researchers approximate conditional expectation operators using kernel methods.
problem Statistical approximation of conditional expectation operators under minimal assumptions.
method Modifying the domain of the operator, approximating it by Hilbert-Schmidt operators in a reproducing kernel Hilbert space.
result The nonparametric estimate of the operator converges to a specific limiting object.
We show how spectral filters can improve the convergence of numerical schemes which use discrete Hilbert transforms based on a sinc function expansion, and thus ultimately on the fast Fourier transform. This is relevant, for example, for the computation of fluctuation identities, which give the distribution of the maxi…
We study the bottom of the spectrum in Hilbert geometries, we show that it is zero if and only if the geometry is amenable, in other words if and only if it admits a Fölner sequence. We also show that the bottom of the spectrum admits an upper bound, which depends only on the dimension and which is the bottom of the sp…
We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restriction…
Spectral algorithms on manifolds using diffusion kernels improve convergence rates.
problem The limitations of existing spectral algorithms in RKHSs for data on manifolds.
method Integrating manifold structure into spectral algorithms using heat kernel diffusion spaces.
result Spectral algorithms converge to the target function and its derivatives in a strong sense, with rates dependent on manifold intrinsic dimension.
New scalable geometric framework for SPD matrices.
problem Costly spectral computations in SPD matrix analysis.
method Efficient computation of extreme generalized eigenvalues through Hilbert and Thompson geometries of the semidefinite cone.
result Existence and uniqueness of a novel iterative mean of SPD matrices.
Paper proves uniqueness of solutions to a geometric inequality problem.
problem Uniqueness of solutions to the isotropic Lp Minkowski problem. method Analysis of the Hilbert-Brunn-Minkowski operator LK to derive stability estimates. result Uniqueness of S2-isotropic solutions to the isotropic Lp Minkowski problem in Rn for specific ranges of p. We study generalization properties of distributed algorithms in the setting of nonparametric regression over a reproducing kernel Hilbert space (RKHS). We first investigate distributed stochastic gradient methods (SGM), with mini-batches and multi-passes over the data. We show that optimal generalization error bounds c…
Paper introduces RKHM for more explicit variable structures analysis.
problem Explicitly analyzing structures among variables.
method Orthonormal systems in Hilbert C∗-modules, RKHM. result Theoretical and practical procedures for RKHM orthonormalization.
Study uses SGD to learn operators in Hilbert spaces with convergence analysis.
problem Learning operators in general Hilbert spaces with SGD.
method Proposes weak and strong regularity conditions for convergence analysis.
result SGD converges to best linear approximation of nonlinear operators.
Stochastic gradient descent achieves polynomial convergence rates for noiseless linear models.
problem Convergence analysis of stochastic gradient descent in noiseless linear models.
method Fixed step-size stochastic gradient descent on least-square risk.
result Polynomial convergence rates depend on the regularities of the optimum and feature vectors.
Unified analysis of kernel-based and locally adaptive bandit optimization methods.
problem Performance of bandit optimization algorithms in RKHS functions.
method Investigates the relationship between kernel regularity and algorithmic performance, characterizing spectral properties of various kernels.
result Unified framework for analyzing kernel-based and locally adaptive bandit algorithms, deriving explicit regret bounds.
Method finds compatible features for subsets of data.
problem Selecting relevant features for subsets of data.
method Reframe feature selection as finding sections of quiver representations, using quiver Laplacians.
result Eigenvectors of quiver Laplacian yield compatible features.
To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose ``Riemannian'' aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson-Sullivan measure. We prove …
Study of conformal logarithmic Laplacian on sphere, connecting Yamabe problems and Sobolev spaces.
problem Yamabe-type problems and Sobolev spaces on the sphere.
method Detailed spectral analysis, conformal invariance, and Hilbert space introduction.
result Established precise connection between sphere and \(\mathbb{R}^N\) logarithmic Laplacian.