Study characterizes cryospheric spectral feature space using joint PC+t-SNE approach.
problem Characterize cryospheric spectral feature space for remote sensing applications.
method Compare and contrast two approaches for identifying feature space basis vectors via dimensionality reduction (PCA and t-SNE).
result Joint characterization reveals distinct continua and clusters of ice reflectance properties.
The paper characterizes Besse and Zoll Reeb flows on specific manifolds.
problem Characterizing Besse and Zoll Reeb flows on different manifolds.
method Using spectral invariants and Ekeland-Hofer capacities.
result Characterizations of Besse and Zoll Reeb flows for specific manifolds.
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.
Spectral methods improve parameter estimation in structured GLMs.
problem Parameter estimation in high-dimensional generalized linear models with structured data.
method Spectral methods using the principal eigenvector of a data-dependent matrix, with preprocessing for optimal performance.
result Precise asymptotic performance characterization and optimal preprocessing identified.
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.
A method of computation of its terms is presented together with some stabilization results. As an application a characterization of symplectic harmonic manifolds is given and a relationship with the C-spectral sequence is indicated.
Developed a spectral theory for sinh-Gordon equation solutions.
problem Solving spectral data for simply periodic solutions of the sinh-Gordon equation.
method Defined spectral data, solved inverse problem, constructed Jacobi variety and Abel map.
result Spectral theory for sinh-Gordon equation solutions is developed and solved.
Study spectral estimators for multi-index models to recover low-dimensional signal subspaces.
problem Recovering low-dimensional signal subspaces in multi-index models.
method Spectral estimators for multi-index models.
result Precise asymptotic characterization of spectral methods' performance, revealing a phase transition for weak recovery.
Developed spectral theory for sinh-Gordon solutions, solving inverse problem.
problem Solving spectral theory for simply periodic solutions of the sinh-Gordon equation.
method Asymptotic estimates and Jacobi variety construction.
result Spectral data defined and inverse problem solved for sinh-Gordon solutions.
This research optimizes Andrews plots for better visual clarity in high-dimensional data.
problem Visualizing high-dimensional datasets with clarity and aesthetics.
method Developed a method to add spectral smoothing to Andrews plots to reduce visual clutter.
result Optimal spatial-spectral smoothing leads to more aesthetically pleasing and clutter-free visualizations.
Let (M,g) be a compact Einstein manifold with smooth boundary. We consider the spectrum of the p form valued Laplacian with respect to a suitable boundary condition. We show that certain geometric properties of the boundary may be spectrally characterized in terms of this data where we fix the Einstein constant.
The paper extends a spectral evolution model for link prediction in evolving networks.
problem Link prediction in evolving networks.
method Approximated eigenvalue trajectories using Rayleigh quotient and extrapolation.
result Learning algorithms based on approximated trajectories outperform traditional methods.
We obtain geometric characterizations of isospectral minimal Riemannian Legendre foliations on compact Sasakian manifolds of constant φ-sectional curvature.
Proposes a probabilistic framework for stationary topological signals on simplicial complexes.
problem Complex data structures require new models and tools.
method Generalizes stationarity to topological signals on simplicial complexes.
result Defines topological power spectral density (PSD) for stationary signals.
Paper proposes AMP with spectral initialization for robust signal estimation.
problem Signal estimation from generalized linear model measurements with correlated initialization.
method Approximate message passing (AMP) with spectral initialization.
result Characterization of AMP with spectral initialization in high-dimensional limit.
In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension n+1>3. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold (X,g) is less than $\ndemi -1$ if and only …
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.
New algorithms optimize spectral risk measures, improving interpolation between average and worst-case performance.
problem Optimizing spectral risk measures for learning systems.
method Developed stochastic algorithms to optimize spectral risk measures by characterizing their subdifferential and addressing challenges like biasedness of subgradient estimates and non-smoothness.
result Our approach outperforms out-of-the-box stochastic subgradient and dual averaging methods in optimizing spectral risk measures.
Spectral clustering identifies clusters of multivariate extremes.
problem Analyzing the dependence structure of multivariate extremes.
method Spectral clustering based on a random k-nearest neighbor graph. result Spectral clustering can consistently identify clusters of multivariate extremes under certain conditions.
To a closed Riemannian manifold, we associate a set of (special values of) a family of Dirichlet series, indexed by functions on the manifold. We study the meaning of equality of two such families of spectral Dirichlet series under pullback along a map. This allows us to give a spectral characterization of when a smoot…
Paper proposes a forecasting model combining autoregressive models with spectral attention.
problem Time series forecasting across various domains.
method Combines deep autoregressive models with Spectral Attention (SA) module.
result SAAM consistently demonstrates improved forecasting accuracy compared to state-of-the-art approaches.
The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.
problem Understanding spectral networks in symplectic topology and their role in Lagrangian fillings.
method Analytic results on adiabatic degeneration of Floer trajectories and explicit computation of continuation strips.
result Established equivalence between Family Floer functor and non-abelianization functor for Lagrangian fillings with spectral networks.
The study of spectral-tightness in Riemannian manifolds and its topological implications.
problem Understanding the spectral properties of Riemannian manifolds and their coverings.
method Analyzing the fundamental group and the Euclidean local de Rham factor to characterize spectral-tightness.
result Spectral-tightness is a topological property of the fundamental group, and it can be characterized by the dimension of the Euclidean local de Rham factor.
A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3--sphere S3=R3∪{∞}. The class of Willmore tori obtained this way is given a spectral theoretic characterization as the class of Willmore tori with reducible spectral curve. A spectral curve of this type…
Study reveals learning curves and benign overfitting in spectral algorithms for large dimensions.
problem Understanding learning curves and benign overfitting in spectral algorithms for large-dimensional data.
method Analysis of learning curves and benign overfitting in spectral algorithms for inner-product kernels on the sphere and general domains.
result Characterization of three distinct regimes: over-regularized, under-regularized, and interpolation regimes, revealing benign overfitting across both under-regularized and interpolation regimes.
Characterizes lamination spaces of graphs on a pair of pants.
problem Understanding lamination spaces of graphs on a pair of pants.
method Identifying lamination spaces as lattice polytopes and using graph exploration technique.
result Characterizes the polytopes that arise as lamination spaces of graphs on a pair of pants.
If a curve in R^3 is closed, then the curvature and the torsion are periodic functions satisfying some additional constraints. We show that these constraints can be naturally formulated in terms of the spectral problem for a 2x2 matrix differential operator. This operator arose in the theory of the self-focusing Nonlin…
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 …
The paper proves geodesic balls maximize the first Steklov eigenvalue in non-compact symmetric spaces.
problem Characterizing geodesic balls in non-compact rank one symmetric spaces.
method Utilizing a weighted isoperimetric inequality on harmonic manifolds, the paper proves the maximization of the first Steklov eigenvalue.
result Geodesic balls uniquely maximize the first Steklov eigenvalue among domains of fixed volume in non-compact rank one symmetric spaces.
The study determines if eigenvalues of a Laplacian can reveal the curvature of Kähler manifolds.
problem Can eigenvalues of the Laplacian determine the holomorphic sectional curvature of Kähler manifolds?
method Analyzes cohomologically Einstein and Fano Einstein conditions, showing constancy of curvature for most pairs (p,n).
result Characterizes standard complex projective spaces using a single spectral set under cohomological Einstein conditions.
The paper finds extremum values for mixed Laplacian eigenvalues on triangles and trapezoids.
problem Finding extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
method Characterizations obtained under suitable geometric constraints.
result Characterizations of extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
Categorifies symmetric functions and computes invariants of tangles.
problem Computing and characterizing invariants of tangles.
method Using homotopy categories and symmetric monoidal categories.
result Existence of symmetric group actions and spectral sequences.
Study quantum diffusion on spectral triples and spinor bundles.
problem Characterize quantum diffusion on almost commutative spectral triples.
method Spin geometry, C *-Dirichlet forms, quantum stochastic flows.
result Existence of covariant quantum stochastic flows on spinor bundles.
Geometrically characterizes invariants of Higgs bundles.
problem Characterizing topological invariants of Higgs bundles.
method Using KO-theory and Langlands correspondence, define split orthogonal spectral data.
result Obtains a natural grading of the moduli space of SO(m,m+1)-Higgs bundles.
The angular power spectrum characterizes neural network complexity.
problem Characterizing the complexity of deep neural networks.
method Using the angular power spectrum of the limiting field to characterize network complexity.
result Classified neural networks as low-disorder, sparse, or high-disorder.
The spectral k-support norm enjoys good estimation properties in low rank matrix learning problems, empirically outperforming the trace norm. Its unit ball is the convex hull of rank k matrices with unit Frobenius norm. In this paper we generalize the norm to the spectral (k,p)-support norm, whose additional para…
Proposes a new definition of spacetimes in Noncommutative Geometry.
problem Defining spacetimes in Noncommutative Geometry.
method Extends Connes' spectral triple to Lorentzian setting.
result Characterizes the signature of the metric in terms of a time-orientation 1-form.
New method preserves spectral clustering performance under aggressive sparsification and quantization.
problem Maintaining spectral clustering performance with sparse and quantized data.
method Random matrix theory applied to eigenspectrum changes under sparsification and quantization.
result Spectral clustering performance is preserved even with aggressive sparsification and quantization.
Machine learning predicts extreme events from spectral data.
problem Predicting extreme events in nonlinear systems from limited data.
method Trained a neural network to correlate spectral and temporal properties of optical fibre modulation instability.
result Predicted temporal probability distribution from high-dynamic range spectral data.
Estimates spectral density of large implicit matrices efficiently.
problem Estimating eigenvalues of large implicit matrices efficiently.
method Combines randomized estimation techniques to construct unbiased estimators.
result Validated methods on large-scale problems in graph theory and random matrix theory.
Recursive training of generative models can lead to model collapse, and the recursion converges to a unique limiting distribution.
problem Model collapse in recursive training of generative models
method Recursive training on their own outputs
result Recursive training converges to a unique limiting distribution
New method accelerates smooth games using spectral shape analysis.
problem Accelerating optimization in smooth games with complex numerical challenges.
method Matrix iteration theory and spectral shape analysis to characterize and manipulate acceleration.
result Identified a continuum of optimization strategies from convex minimization to gradient descent.
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.
New spectral conditions ensure graph rigidity and global rigidity in the Euclidean plane.
problem Ensuring graph rigidity and global rigidity in the Euclidean plane.
method Improving algebraic connectivity bounds for graph rigidity and global rigidity.
result Every 6-connected graph is rigid and globally rigid if its algebraic connectivity exceeds specific thresholds.
The study characterizes wobbly rank-2 bundles on Riemann surfaces using spectral curves.
problem Characterizing wobbly rank-2 bundles on Riemann surfaces.
method Using spectral curves and direct images of line bundles, the study provides sufficient and necessary conditions for wobbly bundles.
result All rank-2 wobbly bundles can be characterized as twists of direct images of line bundles.
Paper improves understanding of random Fourier features for kernel ridge regression.
problem Understanding statistical properties of random Fourier features for kernel ridge regression.
method Spectral matrix approximation approach to analyze random Fourier features.
result Proves statistical guarantees for kernel ridge regression using random Fourier features.
Spectral clustering is a fast and popular algorithm for finding clusters in networks. Recently, Chaudhuri et al. (2012) and Amini et al.(2012) proposed inspired variations on the algorithm that artificially inflate the node degrees for improved statistical performance. The current paper extends the previous statistical…
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.