New spectral triples for higher-rank graphs linked to wavelet decompositions.
problem Creating spectral triples for higher-rank graph C∗-algebras. method Generalizing spectral triples from Cuntz-Krieger algebras to higher-rank graph C∗-algebras and connecting them to wavelet decompositions. result Wavelet decompositions describe eigenspaces of Dirac operators in these spectral triples.
Proposes neural dynamic mode decomposition for end-to-end modeling of nonlinear dynamics.
problem Understanding and modeling nonlinear dynamical systems.
method Trains neural networks to minimize forecast error based on spectral decomposition in the lifted space.
result Demonstrates effectiveness in eigenvalue estimation and forecast performance.
Efficient spectral clustering using autoencoders and landmarks.
problem High computational complexity of spectral clustering.
method Build adjacency matrix using landmarks, define efficient Laplacian matrix, perform eigen decomposition using autoencoder.
result Overall complexity of O(np), where n is data points and p is landmarks. SPEDER extracts state-action abstraction from dynamics for reinforcement learning.
problem Curse of dimensionality and limited applicability of spectral methods.
method Spectral Decomposition Representation (SPEDER) that extracts state-action abstraction from dynamics without policy dependence.
result Theoretical analysis establishes sample efficiency in online and offline settings.
Many modern datasets can be represented as graphs and hence spectral decompositions such as graph principal component analysis (PCA) can be useful. Distinct from previous graph decomposition approaches based on subspace projection of a single topological feature, e.g., the Fiedler vector of centered graph adjacency mat…
Analysis of DPPs and k-DPPs via spectral decomposition reveals identifiable parameters and non-identifiability gaps.
problem Identifying parameters of DPPs and k-DPPs through spectral decomposition.
method Spectral decomposition of the covariance matrix, analysis of invariances, and counting arguments.
result Identifiability of parameters changes fundamentally for k-DPPs, with specific invariances and non-identifiability gaps.
Galerkin method outperforms graph-based methods in spectral decompositions.
problem Improving spectral decomposition methods in machine learning.
method Restricting study to a small set of test functions using the Galerkin method.
result Statistical and computational superiority of Galerkin method over graph-based approaches.
We introduce a new parameterization method for deep learning layers using spectral tensor train decomposition.
problem Efficiency and stability in deep learning models with weight matrix compression.
method Spectral Tensor Train Parameterization (STTP) of weight matrices.
result Improved compression and training stability in neural networks.
Spectral learning extends matrix methods to tensors for better latent variable modeling.
problem Limitations of matrix-based spectral methods in capturing non-Gaussian data.
method Extend spectral decomposition to tensor-based methods for higher-order moments.
result Tensor decomposition can identify latent effects missed by matrix methods.
New method deforms function algebras on manifolds using spectral decomposition.
problem Deforming function algebras on compact Riemannian manifolds.
method Introducing a bilinear product on the finite spectral core of smooth functions using unimodular phases.
result The product extends to a Sobolev algebra and admits iteration under certain conditions.
Paper proposes a fully data-driven method for Koopman spectral analysis.
problem Manual preparation of nonlinear observables is often required for Koopman spectral analysis.
method Learning Koopman invariant subspaces from observed data using linear least-squares regression.
result Performance evaluated using nonlinear dynamical systems and applications.
New algorithm for latent variable models using spectral decomposition.
problem Unsupervised learning of latent variable models from unlabeled data.
method Spectral decomposition for robust unsupervised learning.
result Efficient technique to learn parameters of text mining models.
FreDN separates trends and periodicities in non-stationary time series forecasts.
problem Spectral entanglement and computational burden in frequency-domain methods for non-stationary time series.
method FreDN introduces a learnable Frequency Disentangler module to separate trend and periodic components directly in the frequency domain, and uses a ReIm Block to reduce complexity.
result FreDN outperforms state-of-the-art methods by up to 10% on long-term forecasting benchmarks.
Paper analyzes spectral clustering for large graphs using random signals.
problem Complex eigen decomposition for large graphs.
method Graph filtering of random signals for approximate spectral embedding.
result Consistency of spectral clustering in stochastic block model.
Study rigidity of spectral gap on Finsler manifolds with specific curvature bounds.
problem Rigidity of spectral gap on Finsler manifolds with Ricci curvature bound.
method Analysis of spectral gap, splitting phenomena, and needle decomposition.
result Rigidity results for spectral gap, logarithmic Sobolev, and Bakry-Ledoux inequalities.
Fast algorithms for tensor decomposition robust to errors, with applications to dictionary learning.
problem Tensor decomposition with robustness to errors and sparsity constraints.
method Spectral algorithms with tensor-mode rearrangements, achieving guarantees similar to sum-of-squares (SOS) semidefinite programming.
result Efficient algorithms with running time n5 for decomposing tensors and learning sparse dictionaries, matching or surpassing previous polynomial-time methods. The paper derives upper bounds on eigenvalues of Laplace-Beltrami operator on hyperbolic surfaces.
problem Finding upper bounds on eigenvalues of Laplace-Beltrami operator on hyperbolic surfaces.
method Using spectral decompositions and consistency conditions derived from quadruple overlap integrals in terms of triple overlap integrals.
result Derives upper bounds on eigenvalues, nearly saturated by the Bolza surface.
Authors find a Hodge-type decomposition for holomorphic Poisson cohomology on nilmanifolds.
problem Investigating conditions for spectral sequence degeneration in holomorphic Poisson cohomology.
method Analyzing spectral sequences associated with bi-complexes on nilmanifolds.
result A Hodge-type decomposition of holomorphic Poisson cohomology is established for a specific class of structures.
KCoreMotif clusters large networks efficiently by exploiting k-core decomposition and motifs.
problem Efficiently clustering large networks for trust evaluation.
method Exploits k-core decomposition and motifs to perform motif-based spectral clustering on k-core subgraphs.
result The proposed algorithm is accurate and efficient for large networks.
New spectral tensor network algorithms solve continuous tensor problems.
problem Continuous tensor decomposition and orbit recovery problems over infinite groups.
method Leverage tensor networks to design spectral algorithms.
result Solve continuous multi-reference alignment over infinite SO(2) group.
Lefschetz decompositions for Kähler manifold eigenforms identified.
problem Understanding the spectrum and eigenspaces of Laplacians on Kähler manifolds.
method Analyzing the eigenspaces of the Laplacian Δk on k-forms on a compact Kähler manifold. result The positive part of the spectrum of Δk lies in the spectrum of Δk+1. Geometrically decomposes Kähler functions on toric manifolds.
problem Decomposing Kähler functions on Kähler toric manifolds.
method Defining spectrum of Kähler functions and proving spectral decomposition theorem.
result Geometric spectral theory for Kähler functions established.
A robust method for decomposing spectral peaks robust to distortion and interference.
problem Decomposing spectral peaks in the presence of distortion and interference.
method Optimizing a nonparametric approach using pseudo-symmetric functions with nonincreasing behavior.
result Decomposed spectral peaks show pseudo-orthogonal behavior and power preserving equality.
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.
The paper decomposes spectral functions on marked tori strata.
problem Decomposing square-integrable functions on strata of differentials.
method Spectral decomposition and analysis of differential operators.
result The continuous spectrum of the foliated Laplacian is larger than Siegel-Veech transforms.
Paper identifies key function spaces for ReLU networks based on Fisher information.
problem Understanding the structure of Fisher information matrices in ReLU networks.
method Spectral decomposition of Fisher information matrices, focusing on the first three eigenspaces.
result The first three eigenspaces account for 97.7% of the trace of the Fisher information matrix, corresponding to spherical harmonic functions of order ≤2.
New theorem improves spectral gap for sampling from mixture distributions.
problem Sampling from multimodal distributions with simulated tempering.
method Introduced a decomposition theorem for the restricted spectral gap of simulated tempering.
result Lower bound on the restricted spectral gap for mixture distributions.
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.
Improved spectral methods of moments for robust latent variable model learning.
problem Limited robustness of spectral methods of moments to model misspecification.
method Hierarchical approach using approximate joint diagonalization instead of tensor decomposition.
result Our method outperforms previous tensor decomposition methods in speed and model quality.
New spectral method learns DNA methylation models efficiently.
problem Learning parameters of Binomial HMMs for DNA methylation data.
method Feature-map based approach exploiting Binomial HMM properties.
result The new algorithm provides theoretical guarantees and performs well on real data.
New method improves music transcription by treating frequency distributions holistically.
problem Small frequency shifts and variations in sound timbre harm traditional fit measures.
method Optimal transportation and new holistic frequency distribution measure.
result Simplified note templates lead to faster, state-of-the-art performance.
Bayesian parametric matrix models provide uncertainty quantification for spectral learning.
problem Uncertainty quantification in spectral learning for safety-critical applications.
method Bayesian parametric matrix models (B-PMMs) that extend PMMs to provide uncertainty estimates.
result B-PMMs achieve exceptional uncertainty calibration (ECE < 0.05) while maintaining favorable scaling.
In many areas of machine learning, it becomes necessary to find the eigenvector decompositions of large matrices. We discuss two methods for reducing the computational burden of spectral decompositions: the more venerable Nystom extension and a newly introduced algorithm based on random projections. Previous work has c…
This work provides a computationally efficient and statistically consistent moment-based estimator for mixtures of spherical Gaussians. Under the condition that component means are in general position, a simple spectral decomposition technique yields consistent parameter estimates from low-order observable moments, wit…
A new metric compares dynamical systems using operator eigenvalues.
problem Comparing and interpolating nonlinear dynamical systems from trajectory data.
method Representing systems as distributions of operator eigenvalues and projectors, defining a spectral-Grassmann Wasserstein metric.
result The proposed metric outperforms standard operator-based distances in machine learning applications.
Lobb observed in [arXiv:1103.1412] that each equivariant sl(N) Khovanov-Rozansky homology over C[a] admits a standard decomposition of a simple form. In the present paper, we derive a formula for the corresponding Lee-Gornik spectral sequence in terms of this decomposition. Based on this formula, we give a simple alter…
Develops a tensor decomposition method with side information.
problem Identifying the relationship between a high-dimensional tensor and side information.
method Supervised tensor decomposition incorporating multiple feature matrices.
result Captures effective dimension reduction of the data tensor in feature space.
Graph-based denoising framework for smooth manifolds.
problem Denoising of signals on smooth manifolds.
method Spectral Graph Wavelet transform applied to the graph Fourier frequency domain.
result Significantly outperforms state-of-the-art denoising methods.
The study examines Fisher information matrices and neural tangent kernels for simple ReLU networks with random weights.
problem Understanding the relationship between Fisher information matrices and neural tangent kernels for 2-layer ReLU networks.
method Analyzes Fisher information matrices and neural tangent kernels for 2-layer ReLU networks with random hidden weights, focusing on spectral decomposition and eigenfunctions.
result Obtained an approximation formula for functions represented by 2-layer neural networks.
End-to-end differentially private LDA using spectral algorithm with theoretical guarantees.
problem Learning LDA models with differential privacy.
method Spectral algorithm with noise injection for differential privacy, identifying subsets of edges (configurations) for privacy guarantees.
result End-to-end differentially private spectral algorithm for LDA with utility guarantees.
Study on rigidity of logarithmic Sobolev inequality on manifolds.
problem Rigidity of logarithmic Sobolev inequality on weighted Riemannian manifolds.
method Needle decomposition method.
result Splitting off of 1-dimensional Gaussian space when equality holds.
New framework explains neural network bias in solving differential equations.
problem Understanding and controlling the bias in PINNs for differential equations.
method Deriving an integro-differential equation from PINNs and GPR equivalence.
result PINN predictions are influenced by a kernel term reflecting architecture choices.
New approach speeds up DNA sequence alignment.
problem Efficiently estimating alignment scores for large sets of reads.
method Rank-one crowdsourcing models and multi-armed bandit algorithm.
result Adaptive algorithm identifies pairs with large alignment scores.
Gaussian processes classify graphs using vertex and edge features.
problem Graph classification in machine learning.
method Transform graph features into spectral Euclidean features, apply Hodge decomposition.
result Gaussian processes can classify graphs using vertex and edge features.
Improved clustering algorithm for large datasets.
problem Finding alternative partitions in large datasets.
method Iterative Spectral Method (ISM) for alternative clustering.
result Significantly improved scalability and computation time.
Derives adjoint formulas for matrix operations and applies them to specific cases.
problem Computing adjoints for matrix operations and specific matrix types.
method Derives adjoint formulas for matrix operations and applies them to specific cases.
result Closed-form expressions for adjoints in specific matrix types.
Sparse spectral decomposition identifies overlapping communities in networks.
problem Estimating overlapping community memberships in networks where nodes can belong to multiple communities.
method Sparse principal subspace estimation with iterative thresholding.
result The fixed point of the algorithm corresponds to correct node memberships under the stochastic block model.
Given an open book decomposition (Σ,τ) of a three manifold Y, Thurston and Winkelnkemper [TW] construct a specific contact form a on Y. Given a spin-c Dirac operator D on Y, the contact form naturally associates a one parameter family of Dirac operators $D_r = D - \frac{ir}{2}\cl(a)$ for r≥0. When $r>>…