Study spectral distances on compact RCD spaces.
problem Understanding spectral convergence in RCD spaces.
method Established relationships between different spectral convergences and constructed a spectral approximation map.
result Found canonical spectral approximation map for RCD spaces.
Study geometric quantization on K3 surfaces, showing spectral convergence.
problem Quantization of K3 surfaces from spectral perspective.
method Special Lagrangian fibrations and hyper-Kähler structures.
result Spectral convergence of ∂ˉ-Laplacians on prequantum line bundles. Study approximates sub-Riemannian structures with Riemannian metrics and analyzes spectral convergence.
problem Approximating sub-Riemannian structures for analysis.
method Constructing Riemannian metrics tailored to sub-Riemannian structures and studying spectral convergence.
result Riemannian volumes converge to Popp's volume and spectral convergence of Laplace operators is studied.
Spheres' spectral structure converges to Gaussian space's as dimensions grow.
problem Understanding spectral convergence between high-dimensional spheres and Gaussian spaces.
method Proving spectral convergence using projections and eigenvalues.
result Spectral structure on high-dimensional spheres converges to Gaussian space's as dimensions increase.
Unified framework for analyzing graph neural operators converging to graph limits.
problem Analyzing convergence of graph neural operators to graph limits.
method Develops a unified spectral framework for graph neural operators under various graphon assumptions.
result Unified framework enables direct comparison of convergence rates and tradeoffs.
This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the…
Laplacian Eigenvectors of the graph constructed from a data set are used in many spectral manifold learning algorithms such as diffusion maps and spectral clustering. Given a graph constructed from a random sample of a d-dimensional compact submanifold M in RD, we establish the spectral 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.
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.
In this note we give necessary and sufficient conditions for the validity of the local spectral convergence, in balls, on the RCD∗-setting.
In this paper, we show the spectral convergence result of ∂-Laplacians when (X,ω) is a compact toric symplectic manifold equipped with the natural prequantum line bundle L. We consider a family {Js}s of ω-compatible complex structures tending to the large complex structure limit, and ob…
Uniform counting formulas for orthogeodesics in Kleinian groups converge.
problem Counting orthogeodesics in Kleinian groups converging to a limit.
method Spectral gap of the limit manifold and geodesic flow mixing property.
result Asymptotically uniform counting formulas for orthogeodesics.
It is shown that under mild conditions, Benjamini-Schramm convergence of lattices in locally compact groups is equivalent to spectral convergence. Next both notions are extended to the relative case and are then expressed in terms of relative L2-theory.
The abstract discusses a spectral sequence for Lie algebroids.
problem The abstract tackles the spectral sequence of Lie algebroids.
method The abstract presents a spectral sequence for Lie algebroids, generalizing classical constructions.
result The spectral sequence converges to Lie algebroid cohomology for wide Lie subalgebroids and to formal Lie algebroid cohomology for Lie subalgebroids over proper submanifolds.
We develop a new approach to geometric quantization using the theory of convergence of metric measure spaces. Given a family of Kähler polarizations converging to a non-singular real polarization on a prequantized symplectic manifold, we show the spectral convergence result of ∂ˉ-Laplacians, as well as th…
Muon optimizer simplifies matrix optimization with spectral orthogonalization.
problem Matrix optimization challenges, especially with large condition numbers.
method Simplified Muon optimizer using spectral orthogonalization of gradients.
result Simplified Muon converges linearly with independent scalar sequences, outperforming gradient descent and Adam.
We develop a method of calculation for the symplectic Floer homology of composite knots. The symplectic Floer homology of knots defined in \cite{li} naturally admits an integer graded lifting, and it formulates a filtration and induced spectral sequence. Such a spectral sequence converges to the symplectic homology of …
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.
Improved spectral convergence bounds for diffusion maps on tori.
problem Weak theoretical error bounds for diffusion maps.
method Spatial Hardy space estimates, PDE spectral stability, Sinkhorn weights.
result Matched pointwise error bounds for spectral data and operator convergence.
New algorithms improve spectral clustering for finite mixture models.
problem Issues with EM algorithm in spectral clustering.
method Spectral decomposition and non-parametric bootstrap sampling.
result Improved convergence and avoidance of poor solutions.
Graph Laplacians adapt to different manifold dimensions, while Dirichlet energies converge to a tensorized Dirichlet energy.
problem Understanding machine learning methods for data with varying intrinsic dimensions.
method Γ-convergence of graph Dirichlet energies and spectral convergence of graph Laplacians on intersecting manifolds of varying dimensions.
result Normalized Dirichlet energy converges to a tensorized Dirichlet energy that adapts to all dimensions simultaneously.
Accelerates optimal transport computation by 10x with spectral insights.
problem Exponential slow-down of convergence in Entropic Optimal Transport as regularization weakens.
method Spectral insights and spectral warm-start strategy to mitigate convergence issues.
result Faster convergence compared to the reference method Sinkhorn algorithm.
We extend the Fourier cosine method to discrete probability distributions, achieving faster convergence rates.
problem Extending Fourier cosine method to discrete probability distributions.
method Spectral filters and convergence rates analysis.
result Spectral filters achieve one order faster convergence rates than previously recognized.
Proof of wall-crossing formula using spectral networks.
problem Proving the Kontsevich-Soibelman wall-crossing formula.
method Path-lifting rules for spectral networks, convergence justification.
result Definition and justification of path lifting rules for spectral networks.
Paper analyzes convergence of Laplacian eigenmaps on submanifolds with singularities.
problem Analyzing convergence of Laplacian eigenmaps on submanifolds with singularities.
method Using ε-neighborhood graphs constructed from random points on the submanifold, the paper provides a spectral approximation result for the Laplacian.
result The convergence rate for the eigenvalue of the Laplacian is \( O\left(\left(\log n/n
ight)^{1/(m+2)}
ight) \), where \( m \) and \( n \) are the dimension of the manifold and the sample size, respectively.
In this paper we improve the spectral convergence rates for graph-based approximations of Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the same as the pointwise consist…
Local limit theorem for random walks on hyperbolic groups with parabolic subgroups.
problem Analyzing the behavior of random walks on relatively hyperbolic groups.
method Study of convergent random walks with finite derivative of Green function at spectral radius.
result Proves a local limit theorem for the probability of returning to the origin.
Paper tackles functional linear regression using spectral algorithms with discrete observations.
problem Functional linear regression problem with discretely observed data.
method Combines distributed spectral algorithms with Sobolev kernels for regularization.
result Derives matching upper and lower bounds for convergence in Sobolev norm.
A spectral sequence converging to Khovanov homology is constructed which is applied to calculate the rational Khovanov homology of (3,q)-torus links.
We analyze convergence of Fermat distances and their application in clustering.
problem Understanding convergence properties of Fermat distances on Riemannian manifolds.
method Geometric and statistical arguments in percolation theory, leveraging novel arguments for non-uniform densities and curved domains.
result Discrete, sample-based Fermat distances converge to their continuum analogues with a precise rate dependent on intrinsic dimensionality.
The purpose of this paper is to compare two spectral sequences converging to the cohomology of a configuration space. The collapsing of these spectral sequences is established, in some cases, using Massey products.
A large class of machine learning techniques requires the solution of optimization problems involving spectral functions of parametric matrices, e.g. log-determinant and nuclear norm. Unfortunately, computing the gradient of a spectral function is generally of cubic complexity, as such gradient descent methods are rath…
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…
The paper analyzes spectral properties of connection Laplacian on tori, proving convergence to real torus.
problem Spectral analysis of connection Laplacian on tori.
method Employing parallel orthonormal basis in pullback bundle, examining eigenvalues of connection Laplacian on real and discrete tori.
result Eigenvalues of connection Laplacian on discrete tori converge to those on real torus, with unique twist in torsion matrix.
Extends spectral number variance convergence to random matrix ensembles for twisted Laplacians.
problem Spectral number variance convergence for twisted Laplacians and Dirac operators.
method Extends Rudnick's approach to Gaussian ensembles for twisted Laplacians and Dirac operators.
result Convergence to Gaussian ensembles for twisted Laplacians and Dirac operators.
The paper studies Hodge Laplacians from point clouds, proving spectral convergence and harmonic form consistency.
problem Analyzing Riemannian submanifolds from point cloud data.
method Constructing deformed Hodge Laplacians and empirical operators from point clouds, proving convergence properties.
result Empirical spectral cluster contains the k-th Betti number and converges to harmonic k-forms. New Hermite approximations accelerate convergence with adaptive coordinate transformations.
problem Accelerating convergence of spectral approximations for Hermite expansions.
method Using normalizing flows for adaptive coordinate transformations and deriving error estimates.
result Error estimates for Hermite expansions under adaptive coordinate transformations.
In this dissertation we propose alternative analysis of distributed stochastic gradient descent (SGD) algorithms that rely on spectral properties of the data covariance. As a consequence we can relate questions pertaining to speedups and convergence rates for distributed SGD to the data distribution instead of the regu…
For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which in…
Spectral clustering is a technique that clusters elements using the top few eigenvectors of their (possibly normalized) similarity matrix. The quality of spectral clustering is closely tied to the convergence properties of these principal eigenvectors. This rate of convergence has been shown to be identical for both th…
The paper proves spectral convergence rates for graph Laplacian to manifold Laplace-Beltrami operator.
problem Spectral convergence of graph Laplacian to manifold Laplace-Beltrami operator.
method Analysis of Dirichlet form convergence and construction of approximate eigenfunctions via manifold heat kernel.
result Proves spectral convergence rates for Gaussian kernelized graph Laplacian.
SC-Net learns interpretable filters for inverse problems, achieving optimal convergence and super-resolution.
problem Solving ill-posed inverse problems with effective regularization and interpretability.
method SC-Net operates in the spectral domain, learning a pointwise adaptive filter function based on signal-to-noise ratio.
result SC-Net achieves optimal convergence rate and zero-shot super-resolution, matching theoretical bounds.
Mini-batch SGD with momentum is a fundamental algorithm for learning large predictive models. In this paper we develop a new analytic framework to analyze noise-averaged properties of mini-batch SGD for linear models at constant learning rates, momenta and sizes of batches. Our key idea is to consider the dynamics of t…
Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eige…
Proposes CRG_IMSC for better clustering of multi-view data.
problem Lack of effective connectivity in clustering results.
method Directly obtains clustering result with nonnegative constraint; constructs connectivity matrix based on spectral clustering result; uses multiplicative update algorithm.
result Improves clustering performance on benchmark datasets.
We prove the existence of a spectral sequence for Lagrangian Floer homology which converges to the Floer homology of the image of a Lagrangian submanifold under multiple fibred Dehn twists. The E1 term of the sequence is given by the hypercube of "resolutions" of the Dehn twists involved. The proof relies on the exa…
Study shows deterministic equivalent for neural network kernel convergence.
problem Understanding convergence of neural network kernels.
method Analyzes empirical spectral distribution of Conjugate Kernel, proving convergence to a deterministic limit.
result Obtains a deterministic equivalent for the Stieltjes transform and resolvent of the Conjugate Kernel.
Improves learning of spectral mixture kernels with approximate Bayesian inference.
problem Difficult optimization of large number of SM kernel parameters.
method Approximate Bayesian inference using variational distribution of spectral points and random Fourier features.
result Accelerates convergence and leads to better optimal parameters.