New method to study group invariants using divergence spectra.
problem Understanding group invariants through divergence.
method Introducing divergence spectrum to compare classical notions and study relatively hyperbolic groups.
result Existence of groups with exponential divergence but different divergence spectra.
Study shows how a strip's twisting increases at infinity, affecting its spectrum.
problem Understanding the spectrum of a strip with diverging twisting.
method Analyzing the Dirichlet Laplacian in a two-dimensional strip with segments rotating at increasing velocity.
result Essential spectrum forms a three-dimensional tube at infinity, with discrete eigenvalues possible if the tube's cross-section is a disk.
Graph Laplacian spectrum serves as a robust feature representation.
problem Difficulties in analyzing and comparing graphs due to their structure.
method Proposes using the graph Laplacian spectrum (GLS) as a feature representation.
result Graph Laplacian spectrum (GLS) preserves structural information and is consistent under deformation and invariance under isomorphism.
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.
A new robust PCA estimator combining M-estimators and minimum divergence estimators.
problem Adverse effect of outlying observations in PCA for high-dimensional data.
method Minimum density power divergence estimator combined with a computationally efficient algorithm.
result High breakdown guarantee regardless of data dimension with theoretical support and practical applications.
RNNs struggle with chaotic dynamics due to exploding gradients, but we found a way to optimize training.
problem Challenging training of RNNs with chaotic dynamics due to exploding gradients.
method Relating loss gradients to Lyapunov spectrum to optimize training on chaotic data.
result RNNs with chaotic dynamics always have diverging gradients, while stable ones have bounded gradients.
We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eige…
Stability of Einstein metrics under Ricci iteration studied.
problem Stability of Einstein metrics under Ricci iteration.
method Sufficient condition based on Lichnerowicz Laplacian spectrum.
result Stability of several Einstein manifolds including symmetric spaces.
New method estimates covariance matrices without restrictive assumptions.
problem Estimating high-dimensional covariance matrices under restrictive assumptions.
method Distributionally robust covariance estimation problems with mild conditions.
result Robust estimators are efficient, consistent, and perform well.
Prove first-band large-diameter asymptotics for Dirichlet spectrum on horoconvex domains in real hyperbolic space.
problem Prove first-band large-diameter asymptotics for Dirichlet spectrum on horoconvex domains in real hyperbolic space.
method Prove first-band large-diameter asymptotics for Dirichlet spectrum on horoconvex domains in real hyperbolic space.
result Prove first-band large-diameter asymptotics for Dirichlet spectrum on horoconvex domains in real hyperbolic space.
This paper presents a general iterative bias correction procedure for regression smoothers. This bias reduction schema is shown to correspond operationally to the L2 Boosting algorithm and provides a new statistical interpretation for L2 Boosting. We analyze the behavior of the Boosting algorithm applied to commo…
New approach categorizes objective functions for embodied agents.
problem Understanding how objectives relate to each other and discovering new objectives.
method Introducing Action Perception Divergence (APD) to categorize objective functions.
result Introduces a spectrum of objectives from narrow to general, explaining various unsupervised objectives.
Paper analyzes Annealed Langevin Dynamics for multimodal sampling stability.
problem Ensuring stability of Annealed Langevin Dynamics across dimensions.
method Uniform-in-dimension analysis of ALD for Gaussian-mixture targets.
result ALD achieves prescribed accuracy in KL divergence with spectral conditions.
The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …
The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …
We define a new spectrum for compact length spaces and Riemannian manifolds called the "covering spectrum" which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers δ>0 which identify the distinct δ covers of the space. We investigat…
We propose a new method of learning a sparse nonnegative-definite target matrix. Our primary example of the target matrix is the inverse of a population covariance or correlation matrix. The algorithm first estimates each column of the target matrix by the scaled Lasso and then adjusts the matrix estimator to be symmet…
This paper shows a unique spectrum for hyperbolic surfaces.
problem The rigidity of marked length spectrum for closed hyperbolic surfaces is not true for unmarked spectra.
method Introducing the length-angle spectrum and proving its uniqueness.
result The length-angle spectrum determines the surface uniquely.
Study the energy spectrum of metrics on surfaces and its relation to simple length spectrum.
problem Relate the energy spectrum to the simple length spectrum of metrics on surfaces.
method Analyze the energy spectrum of metrics on surfaces and their Teichmüller spaces, considering homotopy conditions.
result The energy spectrum determines the simple length spectrum under certain conditions.
The paper compares two spectrum definitions and finds stability in one modification.
problem Generalizing eigenvalues to arbitrary functionals with stability.
method Comparison of Gromov's homotopy significant spectrum and Krasnoskii spectrum, with a modified definition of the homotopy significant spectrum.
result The modified homotopy significant spectrum is stable, and Cheeger constant corresponds to Krasnoskii eigenvalue.
Proofs high-dimensional spectrum convergence of weighted sample covariance.
problem High-dimensional spectrum convergence of weighted sample covariance.
method Proposes a new, concise proof with stronger assumptions.
result Spectrum convergence proven for different weight distributions.
Study shows spectrum properties for specific Hadamard manifolds.
problem Spectrum properties of Hadamard manifolds.
method Absolute continuity and spectrum determination for two classes of Hadamard manifolds.
result Spectrum properties determined for specific Hadamard manifolds.
DeepCCG adapts classifiers to representation shifts in one step.
problem Adapting classifiers to shifts in continuous representation.
method Empirical Bayesian approach using class conditional Gaussian classifier and KL divergence for selection.
result DeepCCG reduces performance change due to representation shifts.
Iterative method 'Concent' corrects spectrum bias in covariance matrices.
problem Consistent bias in the spectrum of covariance matrices.
method 'Concent' iterative algorithm.
result Corrects spectrum bias for small and moderate dimensions.
Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.
problem Understanding transverse link invariants in the annular setting.
method Constructs a stable homotopy type for annular links and defines a map to the Khovanov skein spectrum.
result At extreme gradings, the map from the Khovanov spectrum to the Khovanov skein spectrum recovers the cohomotopy transverse invariant.
Constructs manifolds with specific spectral properties.
problem Spectral properties of Riemannian manifolds.
method Asymptotically hyperbolic manifolds with sharp curvature bounds.
result Embeds singular continuous spectrum into the essential spectrum of the Laplacian.
The spectrum of certain manifolds matches that of hyperbolic space if the bottom spectrum is maximal.
problem Investigating spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum.
method Analyzing the spectrum of the Laplacian on manifolds with specific Ricci curvature bounds.
result The spectrum of the manifold coincides with that of hyperbolic space if the bottom spectrum is maximal.
Lower bounds for Hodge-Laplacian spectrum on orbifolds.
problem Finding bounds for the essential spectrum of Hodge-Laplacian.
method Deriving lower bounds for the essential spectrum of the Hodge-Laplacian on geometrically finite orbifolds and their suborbifolds.
result Lower bounds for the essential spectrum of the Hodge-Laplacian.
Rigidity of spectral data for spherical manifolds with boundary.
problem Determining the length spectrum of spherically symmetric manifolds with boundary.
method Proving a trace formula and using it to show spectral rigidity.
result The Neumann spectrum uniquely determines the length spectrum for spherically symmetric manifolds with boundary.
A machine learning approach for efficient spectrum sharing in distributed DSA networks.
problem Effective spectrum sharing among secondary users (SUs) and primary users (PUs) in a distributed network.
method Deep reinforcement learning (DRL) combined with reservoir computing (RC) for distributed spectrum access decisions.
result The RC-based spectrum access strategy significantly reduces collision chances and outperforms other methods.
Trapezoids uniquely identified by their Dirichlet Laplace spectrum.
problem Identifying trapezoids based on their spectral properties.
method Analyzing the Dirichlet Laplace spectrum of non-obtuse trapezoids.
result Non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum.
In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a sp…
Survey on bottom of spectrum of Hodge Laplacian on complete noncompact Kähler manifolds
problem Bottom of the spectrum of Hodge Laplacian on complete noncompact Kähler manifolds
method Survey on Kähler hyperbolic manifolds and bounded symmetric domains
result Proposed several open problems
New divergences extend Bregman and skew Jensen, including f-divergences.
problem Developing new divergences to include f-divergences.
method Introducing g-Bregman and skew g-Jensen divergences, showing they include f-divergences.
result g-divergences generalize existing divergences and inequalities.
The paper extends decay estimates to graphs with positive spectrum.
problem Proving decay estimates for nonnegative functions on graphs.
method Sharp ℓ2 decay estimates for nonnegative generalized subharmonic functions. result Extends Li and Wang's result to graphs with positive Laplacian spectrum.
Upper bounds for essential spectrum of minimal submanifolds linked to volume growth.
problem Estimating the essential spectrum of minimal submanifolds.
method Using volume growth to bound the bottom of the essential spectrum.
result Improved essential spectrum estimate for minimal submanifolds.
Upper bounds for volume spectrum depend on volume, dimension, and a conformal invariant.
problem Bounding the volume spectrum of Riemannian manifolds.
method Proves upper bounds that depend on volume, dimension, and a conformal invariant.
result Upper bounds for the volume spectrum are established.
Study on magnetic Dirac operators and their spectrum.
problem Understanding the spectrum of magnetic Dirac operators.
method Analysis of magnetic Dirac operators over complete Riemannian manifolds.
result Find sufficient conditions for maximal or discrete spectrum.
Covering preserves bottom spectrum, implies amenable covering.
problem Spectral preservation in Riemannian coverings.
method Proving spectral properties of Schrödinger operators on coverings.
result Covering preserving bottom spectrum implies amenability.
Notes on continuity of discrete-spectrum Fredholm operators.
problem Continuity properties of discrete-spectrum families of Fredholm operators.
method Relates recent work on discrete-spectrum families to classical continuity properties.
result Establishes connections between new and classical concepts.
Paper shows how to break down a specific type of divergence into simpler parts.
problem Understanding and simplifying divergence functions.
method Decomposes the symmetric Bregman divergence into two types of Jensen divergences and a Bregman divergence, and extends this to include f-divergences.
result Sum decomposition of divergence into simpler parts is possible.
The paper proves wellposedness of flows on manifolds with bounded geometry.
problem Analyzing wellposedness of nonlinear flows on manifolds of bounded geometry.
method Establishing conditions for the operator to generate an analytic semigroup, proving existence of resolvent, and using geometric microlocal calculus.
result Wellposedness of nonlinear flows on manifolds of bounded geometry is proven.
Study shows ortho spectrum doesn't fully determine systolic length but limits the number of possible structures.
problem Determining the systolic length of hyperbolic surfaces with boundary.
method Analyzing the ortho spectrum of hyperbolic surfaces with totally geodesic boundary.
result There are only finitely many possibilities for the ortho spectrum and corresponding hyperbolic structures.
ManifoldFlow relaxes fixed-spectrum Stiefel layers to learn a positive spectrum.
problem Fixed-spectrum Stiefel layers impose rigid spectral constraints.
method Introduces ManifoldFlow, a relaxation that learns a positive spectrum while keeping the basis on the Stiefel manifold.
result Learnable SPD spectrum improves performance in various settings.
Study essential spectrum of differential operators on geometrically finite orbifolds.
problem Analyzing the essential spectrum of differential operators over specific geometric structures.
method Investigates first order and Laplace type elliptic differential operators on Riemannian vector bundles over geometrically finite orbifolds.
result Discovers properties of essential spectra for these operators.
Study on length spectrum of random hyperbolic 3-manifolds.
problem Understanding the length spectrum of random hyperbolic 3-manifolds.
method Modeling random hyperbolic 3-manifolds using truncated tetrahedra and analyzing their length spectrum as volume tends to infinity.
result The length spectrum converges in distribution to a Poisson point process with a computable intensity λ as volume increases.
Paper proves flat metrics from holomorphic quadratic differentials can be identified by length spectrum.
problem Identifying flat metrics from holomorphic quadratic differentials.
method Proved using length spectrum on closed oriented surfaces.
result Flat metrics from holomorphic quadratic differentials can be distinguished by their length spectrum.
Proves existence of solutions to Poisson equation on manifolds with positive spectrum.
problem Existence of solutions to Poisson equation on manifolds with positive essential spectrum.
method Sharp pointwise decay on source function, unbounded Ricci curvature, general spectrum and curvature bounds.
result Existence of solutions on manifolds with positive essential spectrum and unbounded Ricci curvature.