Spectral density matrix estimation of multivariate time series is a classical problem in time series and signal processing. In modern neuroscience, spectral density based metrics are commonly used for analyzing functional connectivity among brain regions. In this paper, we develop a non-asymptotic theory for regularize…
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48 results for “spectral density”
Dynamics of the Number of Trades of Financial Securitiescond-mat.stat-mech
We perform a parallel analysis of the spectral density of (i) the logarithm of price and (ii) the daily number of trades of a set of stocks traded in the New York Stock Exchange. The stocks are selected to be representative of a wide range of stock capitalization. The observed spectral densities show a different power-…
Optimally adaptive Bayesian spectral density estimation for stationary and nonstationary processesstat.ME
Optimizes spectral density estimation for stationary and nonstationary processes.
problem Estimating spectral density of time series with complex structure.
method Optimally adaptive Bayesian spectral density estimation using smoothing spline covariance structure.
result Optimal eigendecomposition provides superior performance compared to alternative covariance functions.
Deep learning speeds spectral density estimation for large 2D/3D grids.
problem Computational challenges in estimating spectral densities for large grids.
method Deep learning neural network for spectral density estimation.
result Deep learning estimator is a universal approximator and faster than existing methods.
Sketching reduces data size for accurate spectral estimation.
problem Estimating spectral density from large simulation datasets.
method Sketching for dimensionality reduction and data compression.
result Sketching provides 90% accurate spectral density estimate with 10% data.
A new method for nonstationary Gaussian processes using Fourier features.
problem Efficient simulation of nonstationary Gaussian processes with high-dimensional distributions.
method Discretizes the spectral representation of nonstationary processes, avoiding probability measure assumptions.
result An efficient low-rank approximation of nonstationary spectral densities, consistent and positive semi-definite.
This work uses neural density estimation to analyze laser-induced breakdown spectroscopy data, enabling accurate predictions and uncertainty quantification.
problem Inference of probability densities in high-dimensional spectral data is often intractable.
method Normalizing flows on structured spectral latent spaces for density estimation and uncertainty quantification.
result The approach enables generation of realistic spectral samples and accurate prediction of state vectors with well-calibrated uncertainties.
Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and tot…
For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin-Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density fun…
Interactive privacy mechanisms improve spectral density estimation under local differential privacy.
problem Estimating spectral density of Gaussian time series with local differential privacy constraints.
method Two-stage process: Laplace mechanism followed by privatized sample analysis.
result Interactive mechanisms achieve faster rates for spectral density estimation.
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.
New matrix ensembles better match deep neural network spectral densities.
problem Theoretical spectral density models for deep networks do not match empirical observations.
method Introduced new matrix ensemble classes to better fit observed spectral densities.
result Theoretical models for deep networks are significantly flawed.
When it comes to clustering nonconvex shapes, two paradigms are used to find the most suitable clustering: minimum cut and maximum density. The most popular algorithms incorporating these paradigms are Spectral Clustering and DBSCAN. Both paradigms have their pros and cons. While minimum cut clusterings are sensitive t…
Lie PCA improves density estimation on symmetric manifolds.
problem Density estimation for symmetric manifolds.
method Spectral method to approximate Lie algebra of symmetry group.
result Improved sample complexity and density estimation on various data sets.
New method extrapolates spectral densities from smaller models to larger ones.
problem Limited practical computations for large machine learning models.
method Algebraic spectral curve theory for free decompression.
result Framework enables extrapolation of spectral densities with multiple or multi-modal bulks.
Method reduces categorical data to lower dimensions using density matrices.
problem Dimensionality reduction for categorical data.
method Density-matrix construction from class-conditional frequencies; spectral embedding.
result Low-dimensional spectral embeddings with controlled rank.
We consider the renormalized Bochner Laplacian acting on tensor powers of a positive line bundle on a compact symplectic manifold. We derive an explicit local formula for the spectral density function in terms of coefficients of the Riemannian metric and symplectic form.
In this work a robust clustering algorithm for stationary time series is proposed. The algorithm is based on the use of estimated spectral densities, which are considered as functional data, as the basic characteristic of stationary time series for clustering purposes. A robust algorithm for functional data is then app…
Polynomial density theorem for specific subgroup orbits in quotient spaces.
problem Effective density of orbits in arithmetic quotients of and .
method Use of Margulis function, incidence geometry tools, and spectral gap of ambient space.
result Proved effective density theorems with polynomial error rate.
Paper connects probability density cuts to graph theory eigenfunctions.
problem Developing sparse cuts for probability densities.
method Defines sparse cuts and principal eigenfunctions for probability densities, proving Cheeger and Buser inequalities.
result No such inequalities hold for prior definitions, proving new inequalities for probability densities.
noise from the sequence of nonoverlapping rectangular pulsescond-mat.stat-mech
Study reveals noise in signals made from nonoverlapping rectangular pulses.
problem Analyzing noise in signals composed of nonoverlapping pulses.
method Derived a general formula for power spectral density, analyzed rectangular pulse case.
result Observed pure noise until very low frequencies with long pulse durations.
Auto-regressive conditionally heteroskedastic (ARCH) family models are still used, by practitioners in business and economic policy making, as a conditional volatility forecasting models. Furthermore ARCH models still are attracting an interest of the researchers. In this contribution we consider the well known GARCH(1…
The paper computes metrics and Einstein tensors on even-dimensional manifolds.
problem Computing metrics and Einstein tensors on even-dimensional Riemannian manifolds.
method Development of spectral Einstein functionals and equivariant Bismut Laplacian.
result Explicit computation of the equivariant noncommutative residue density.
We study the connections between spectral clustering and the problems of maximum margin clustering, and estimation of the components of level sets of a density function. Specifically, we obtain bounds on the eigenvectors of graph Laplacian matrices in terms of the between cluster separation, and within cluster connecti…
JNR Monopolesmath.DG
We review the theory of JNR, mass 1/2 hyperbolic monopoles in particular their spectral curves and rational maps. These are used to establish conditions for a spectral curve to be the spectral curve of a JNR monopole and to show that that rational map of a JNR monopole monopole arises by scattering using results of Ati…
Graph spectral techniques for measuring graph similarity, or for learning the cluster number, require kernel smoothing. The choice of kernel function and bandwidth are typically chosen in an ad-hoc manner and heavily affect the resulting output. We prove that kernel smoothing biases the moments of the spectral density.…
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.
We propose non-stationary spectral kernels for Gaussian process regression. We propose to model the spectral density of a non-stationary kernel function as a mixture of input-dependent Gaussian process frequency density surfaces. We solve the generalised Fourier transform with such a model, and present a family of non-…
Motivated by the local formulae for asymptotic expansion of heat kernels in spectral geometry, we propose a definition of Ricci curvature in noncommutative settings. The Ricci operator of an oriented closed Riemannian manifold can be realized as a spectral functional, namely the functional defined by the zeta function …
Following Hartigan, a cluster is defined as a connected component of the t-level set of the underlying density, i.e., the set of points for which the density is greater than t. A clustering algorithm which combines a density estimate with spectral clustering techniques is proposed. Our algorithm is composed of two step…
Enhances GPLVM for multi-view data with scalable latent representation learning.
problem Limited kernel expressiveness and computational inefficiency in multi-view GPLVM.
method Introduces a new duality between spectral density and kernel function, uses NG-SM kernel, and applies random Fourier feature approximation for scalability.
result Consistently outperforms state-of-the-art models in learning meaningful latent representations across diverse datasets.
DPSM clusters nodes in data and graph spaces via density propagation and subcluster merging.
problem Automatic clustering of nodes in data and graph spaces.
method Density-based node clustering with propagation process and spectral clustering on subclusters.
result DPSM effectively clusters nodes in both data and graph spaces.
FSPA bypasses eigenvalue estimation for quantum PCA, achieving optimal complexity and robustness.
problem Quantum PCA eigenvalue estimation is computationally expensive and prone to errors.
method Filtered Spectral Projection Algorithm (FSPA) that projects onto the dominant spectral subspace directly.
result FSPA achieves optimal complexity and robustness, outperforming classical methods.
We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obt…
Improved singular value approximation for convolutional layers.
problem Improving accuracy of singular value approximation for linear convolutional layers.
method Developed a new spectral density matrix method for singular value approximation with improved accuracy and reduced computational complexity.
result Obtained moderate improvement in singular value distribution compared to circular approximation.
The study shows how geometric Weyl bulk-density exponent rigidifies spectral encodings in O-regularly varying classes.
problem Understanding spectral encodings under Weyl growth conditions.
method Analyzing geometric Weyl bulk-density exponent and proving spectral rigidity.
result The geometric Weyl bulk-density exponent rigidifies spectral encodings in the O-regularly varying class, leading to unique admissible exponents and scaling laws.
Paper characterizes optimal graph clustering limits under a new model.
problem Graph clustering under varying edge density signals.
method Introduced Popularity-Adjusted Block Model (PABM) to address SBM and DCBM limitations.
result Cluster recovery possible even when edge density signals vanish, highlighting local connectivity differences.
We consider an elliptic self-adjoint first order differential operator acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of our operator is assumed to be trace-free. We study the spectral function which is the sum of squar…
The paper analyzes the variance of different shuffling methods in stochastic gradient descent.
problem Understanding the variance of different shuffling methods in stochastic gradient descent.
method Power spectral density analysis to study the noise sequences of stochastic gradients.
result The stationary variances of iterates decrease in the order of SGD, SGD-RR, and SGD-SO.
Study spectral functionals on manifolds with torsion.
problem Extending classical results to geometries with torsion.
method Using Wodzicki residue, investigate spectral functionals of Dirac and Laplace-type operators.
result Local densities recover fundamental geometric tensors.
Interpolates mean shift and spectral clustering on graphs.
problem Data clustering algorithms.
method Fokker-Planck equations on data graphs.
result New theoretical insights on diffusion maps and mean shift dynamics.
Study uses spectral risk for learning with heavy-tailed data.
problem Learning with heavy-tailed loss distributions.
method Spectral risk with Lipschitz-continuous density, derivative-free learning.
result Excess risk guarantees and improved performance over traditional methods.
Study spectral density of neural networks using resolvent method.
problem Investigate spectral density of neural networks with random feature matrices.
method Use resolvent method from random matrix theory, cumulant expansion.
result Impossible to preserve singular value distribution with additive bias.
Learning HMMs with Nonparametric Emissions via Spectral Decompositions of Continuous Matricesstat.ML
Recently, there has been a surge of interest in using spectral methods for estimating latent variable models. However, it is usually assumed that the distribution of the observations conditioned on the latent variables is either discrete or belongs to a parametric family. In this paper, we study the estimation of an $m…
Bayesian method estimates Kronecker graphical models from autoregressive processes.
problem Estimating Kronecker graphical models from autoregressive Gaussian processes.
method Bayesian approach to estimate Kronecker graphical models.
result Effectiveness demonstrated through numerical experiments and real-world data application.
Spectral Independence Criterion helps infer cause-effect relationships in time series.
problem Distinguishing cause from effect in time series data.
method Spectral Independence Criterion (SIC) based on PSD and frequency response.
result SIC provides a robust method for causal inference in time series data.
Regularized Variational and Spectral Log-Density-Ratio Estimation in the Gaussian Location Modelcs.LG
Estimates Gaussian location model with ridge regularization, comparing variational and spectral methods.
problem Estimating parameters in Gaussian location model with regularization.
method Ridge-regularized log-density-ratio estimation, variational and spectral approaches.
result Regularized variational estimator has lower risk with many observations, spectral estimator with fewer observations.
Free Probability for predicting the performance of feed-forward fully connected neural networksstat.ML
New method predicts neural network performance using free probability theory.
problem Stability and performance prediction of feed-forward neural networks.
method Free Probability Theory and homotopy method for Jacobian spectral density computation.
result FPT metrics correlate highly with final test accuracies of neural networks.