The paper identifies a new geometric and spectral phenomenon in the critical hyperbolic catenoid family.
problem The study investigates the critical hyperbolic catenoid family and its geometric and spectral properties.
method The approach involves analyzing the critical hyperbolic catenoid family, identifying parameter-criticality, and studying the Robin spectrum.
result The paper proves that at a parameter-critical value a♯, the Robin nullity of Σa♯ is at least 3, with an additional kernel element in mode k=0. New cutoff phenomenon found for geodesic paths on hyperbolic manifolds.
problem Understanding the cutoff phenomenon for geodesic paths on hyperbolic manifolds.
method Spectral strategy and detailed spectral analysis of the spherical mean operator.
result Geodesic paths on compact hyperbolic manifolds exhibit cutoff for spatially localized initial conditions.
Clustering is concerned with coherently grouping observations without any explicit concept of true groupings. Spectral graph clustering - clustering the vertices of a graph based on their spectral embedding - is commonly approached via K-means (or, more generally, Gaussian mixture model) clustering composed with either…
The paper explains how ReLU nets converge globally in high dimensions without strict assumptions.
problem Understanding global convergence of ReLU nets in very high dimensions.
method Fine-grained analysis of random activation matrices and detailed gradient norm and curvature analysis.
result Empirical loss function has favorable geometrical properties in the overparameterized setting.
Deep models store facts in geometric embeddings, not just associative memory.
problem Understanding how deep models store and utilize atomic facts.
method Identified geometric memory, contrasting with associative lookup.
result Geometric memory transforms hard reasoning into easy tasks.
The paper analyzes the latent geometry of generative diffusion models.
problem The manifold overfitting phenomenon in generative models.
method Statistical physics approach to analyze the spectrum of eigenvalues and singular values of the Jacobian of the score function.
result Three distinct qualitative phases during the generative process: trivial, manifold coverage, and consolidation phases.
Enhanced spectral clustering for geometric graphs improves clustering accuracy.
problem Ineffective standard spectral clustering for geometric graphs.
method Higher-order spectral clustering using higher-order eigenvectors.
result Established weak and strong consistency for Soft Geometric Block Model.
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 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.
Geometric framework explains and controls implicit bias in machine learning.
problem Understanding and controlling the selection of solutions in overparameterized models.
method Developed a theoretical and constructive framework based on geometric corrections induced by gradient noise and continuous symmetries of the loss.
result Computed the induced bias across various architectures and enabled inverse design to shape the bias.
Method detects neural network equivalence via matrix ensembles and spectral analysis.
problem Detecting equivalence among different deep learning architectures.
method Generating Mixed Matrix Ensembles (MMEs) and matching to conjugate circular ensembles.
result Empirical evidence shows vanishing differences in spectral densities with long tail decay rates.
Study on variance of Laplace eigenfunctions on manifolds.
problem Investigating the variance of Laplace eigenfunctions on compact manifolds.
method Combining Kac-Rice formula, Wiener-Itô chaos decompositions, and pointwise Weyl law analysis.
result Established a quantitative bound for the fluctuations of nodal volumes, improving existing results.
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.
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.
The paper proves spectral convergence for a specific type of geometric quantization.
problem Spectral convergence of ∂-Laplacians on toric symplectic manifolds. method Study of a family of compatible complex structures converging to the large complex structure limit.
result Spectral convergence of ∂-Laplacians acting on Lk. The aim of this paper is to introduce and study a geometric spectral sequence in Khovanov homology. The construction was motivated by a similar spectral sequence from Khovanov homology to Heegaard Floer homology.
Graph convolutional networks fail to use eigenvectors beyond the first, unlike spectral embedding.
problem Understanding when graph convolutional networks fail compared to spectral embedding.
method Presented a simple generative model to illustrate failure.
result Graph convolutional networks fail to use eigenvectors beyond the first in certain graphs.
When looking at Bott's original proof of his periodicity theorem for the stable homotopy groups of the orthogonal and unitary groups, one sees in the background a differential geometric periodicity phenomenon. We show that this geometric phenomenon extends to the standard inclusion of the orthogonal group into the unit…
Efficiently scales continuous kernels with sparse Fourier domain learning.
problem High computational and memory demands, spectral bias in continuous kernels.
method Sparse learning in the Fourier domain.
result Efficient scaling of continuous kernels, reduced computational and memory requirements, mitigated spectral bias.
Study connects spectral and algebraic torsion in geometric contexts.
problem Relating different torsion concepts in geometric settings.
method Example of product geometry, focusing on spin manifolds and two-point space.
result Established connection between spectral and algebraic torsion.
Root Laplacian Eigenmaps help in spectral embedding of graphs.
problem Efficient spectral embedding of graphs.
method Square root of graph-Laplacian operator.
result Improved spectral embedding techniques.
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.
Spectral clustering performance depends on eigenvector fluctuations, shown to be Gaussian.
problem Predicting the performance of spectral clustering.
method General spike random matrix model and rotational invariance of noise.
result Fluctuations of eigenvector entries are Gaussian in large-dimensional regime.
Study reveals uniform spectral gaps for random hyperbolic surfaces with few cusps.
problem Investigating spectral gaps for random hyperbolic surfaces with limited cusps.
method Analyzing Weil-Petersson random hyperbolic surfaces, showing no eigenvalues in specific intervals.
result Uniform lower bounds on spectral gaps for Weil-Petersson random hyperbolic surfaces, revealing a critical phenomenon of 'second order cancellation'.
High-dimensional curved diffusions show abrupt convergence at a critical time.
problem Understanding abrupt convergence in high-dimensional curved diffusions.
method Functional inequalities and spectral rigidity.
result Abrupt convergence (cutoff) occurs in high dimensions, linked to spectral rigidity.
We identify spectral conditions for reliable neural probe interpretation.
problem Unreliable performance of linear probes in interpreting neural representations.
method Formalized Spectral Identifiability Principle (SIP) based on eigengap and Fisher error.
result Reliability of neural probes depends on the eigengap relative to Fisher estimation error.
Proves new inequality linking spectral numbers of Lagrangians and their reductions.
problem Understanding spectral properties of Lagrangian submanifolds.
method Develops inverse reduction inequalities for spectral numbers.
result Proof of inequality between spectral numbers of Lagrangian and its reductions.
Persistent homology reveals a topological signature of grokking in neural networks.
problem Understanding how neural networks learn and generalize from modular arithmetic tasks.
method Persistent homology on point clouds derived from embedding matrices of models trained on modular arithmetic.
result A sharp increase in first homology persistence indicates grokking, with a dominant long-lived topological feature and structured secondary features.
Paper explores SNN for learning spectral geometric info from data.
problem Challenges in applying traditional eigensolvers to big data.
method Introduces Spectral Neural Networks (SNN) as an alternative.
result Investigates tradeoffs and optimization landscape of SNN.
A common difficulty in applications of machine learning is the lack of any general principle for guiding the choices of key parameters of the underlying neural network. Focusing on a class of recurrent neural networks - reservoir computing systems that have recently been exploited for model-free prediction of nonlinear…
Neural networks learn spectral representations for group composition.
problem Understanding structured emergence in neural network training.
method Lifting gradient flow to Fourier domain, proving convergence to irreducible representations.
result Neurons converge to single irreducible representations, cross-layer coefficients align.
New approach to geometric quantization for symplectic manifolds.
problem Quantization of symplectic manifolds with non-singular Lagrangian fibrations.
method Using spectral convergence of metric measure spaces, the authors develop a new geometric quantization approach.
result Spectral and quantum Hilbert space convergence results for Kähler and almost Kähler quantizations.
SNN architecture shows gradient descent converges to regularized solution in matrix sensing problems.
problem Understanding implicit regularization in neural networks for matrix sensing.
method Developed Spectral Neural Networks (SNN) for matrix learning problems, rigorously demonstrating implicit regularization.
result Gradient descent converges to the solution of a regularized learning problem in matrix sensing problems.
The paper studies magnetic field effects on surface eigenvalues and spectral properties.
problem Understanding magnetic effects on surface eigenvalues and spectral properties.
method Provided precise spectral asymptotics expansion for the magnetic Dirichlet-to-Neumann map on surfaces.
result The spectrum of the magnetic Dirichlet-to-Neumann map uniquely determines the number and length of boundary components, parallel transport, and magnetic flux.
Study geometric properties and spectral estimates on warped products.
problem Investigate Ricci curvature and spectral estimates in warped products.
method Establish integral inequalities and sufficient conditions for geometric properties.
result Sufficient conditions for intersection of warped products with totally geodesic hypersurfaces.
We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (call…
Graph pruning improves neural network performance by addressing squashing and smoothing issues.
problem Over-squashing and over-smoothing in Graph Neural Networks.
method Proposes edge deletions to simultaneously address over-squashing and over-smoothing, optimizing spectral gap.
result Edge deletions improve generalization and distinguishability of nodes of different classes.
Incompatible operations affect Khovanov homology and spectral sequences.
problem Incompatibility between operations and spectral sequences.
method Observation of obstructions to integral lifting and spectrification.
result Lipshitz-Sarkar Steenrod operations are incompatible with Szabo's spectral sequence.
Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral triple in terms of σ-connections on finitely generated projective modules. This ma…
Study on model collapse in regression models, proposing a mitigation strategy.
problem Model collapse in recursive training of regression models.
method Analytic formulae and adaptive regularization strategy.
result Modified scaling laws and mitigation strategy for model collapse.
Paper connects neural networks to Gaussian processes for understanding double-descent.
problem Understanding the double-descent phenomenon in neural networks.
method Uses techniques from random matrix theory and Gaussian processes.
result Establishes a connection between NNGP and random matrix theory for neural networks.
New insights into why neural networks generalize well.
problem Understanding why neural networks generalize well despite heavy-tailed weight distributions.
method Developed a simple model to analyze the emergence of heavy-tailed empirical spectral densities (ESDs) in two-layer neural networks without gradient noise.
result Learning rates play a crucial role in shaping the ESDs of two-layer neural networks, leading to better generalization.
Despite excellent progress in recent years, mode collapse remains a major unsolved problem in generative adversarial networks (GANs).In this paper, we present spectral regularization for GANs (SR-GANs), a new and robust method for combating the mode collapse problem in GANs. Theoretical analysis shows that the optimal …
Large learning rates improve kernel method performance.
problem Improving generalization in kernel methods with large learning rates.
method Analyzing the spectral decomposition of the solution to a quadratic objective in a separable Hilbert space.
result Large learning rates influence the spectral decomposition of the solution, leading to better generalization.
It is shown that the topological phenomenon "zero in the continuous spectrum", discovered by S.P.Novikov and M.A.Shubin, can be explained in terms of a homology theory on the category of finite polyhedra with values in certain abelian category. This approach implies homotopy invariance of the Novikov-Shubin invariants.…
How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to depend on standard graph-theoretic properties, such as the degree and eigenvalue di…
We obtain a Bonnet-Myers theorem under a spectral condition: a closed Riemannian manifold (Mn,g) for which the lowest eigenvalue of the Ricci tensor ρ is such that the Schrödinger operator (n−2)Δ+ρ is positive has finite fundamental group. As a continuation of our earlier results, we obtain isoperimetric inequa…
Study on spectral stability of an embedded annulus under curve shortening and Ricci flows.
problem Spectral stability of Dirichlet eigenvalues on an evolving annulus.
method Variational formulas, Rellich-type identities, and harmonic capacity methods.
result Established quantitative bounds comparing the spectrum of the evolving annulus with a flat cylinder.