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. Bayesian framework integrates spectral deconvolution with expert reasoning for robust peak estimation.
problem Challenges in extracting meaningful peaks from noisy or complex spectra.
method Bayesian spectral deconvolution coupled with a physical-property regression layer.
result Recovery of weak peaks in poly(lactic acid) IR spectra related to degradation rates.
Study approximates sub-Riemannian structures with Riemannian metrics and analyzes spectral convergence.
problem Approximating sub-Riemannian structures for analysis.
method Constructing Riemannian metrics tailored to sub-Riemannian structures and studying spectral convergence.
result Riemannian volumes converge to Popp's volume and spectral convergence of Laplace operators is studied.
Spectral Adaptive Conformal Prediction for Structured Non-Exchangeable Data
problem Improving prediction intervals for non-exchangeable time-indexed datasets
method Spectral adaptive conformal prediction
result Improves on fixed spectral weighting while monitoring uncertainty changes
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. New spectral clustering method handles discrete covariates for better community detection.
problem Community detection in networks with discrete covariates.
method Spectral algorithm that separates latent network structure from observed covariates.
result Achieves perfect clustering with high probability in large, sparse networks.
Spheres' spectral structure converges to Gaussian space's as dimensions grow.
problem Understanding spectral convergence between high-dimensional spheres and Gaussian spaces.
method Proving spectral convergence using projections and eigenvalues.
result Spectral structure on high-dimensional spheres converges to Gaussian space's as dimensions increase.
Optimal spectral method found for inhomogeneous spiked Wigner model.
problem Structured noise in learning scenarios.
method Random matrix theory and spectral analysis.
result Optimal threshold for phase transition in block-structured Wigner model.
LASE improves local network structure visualization by targeting locally low-dimensional regions.
problem Global spectral embedding fails to capture local geometric features in sparse, transitive networks.
method Local Adjacency Spectral Embedding (LASE) using weighted spectral decomposition.
result LASE reveals locally low-dimensional structure, improving local reconstruction and visualization.
Spectral images captured by satellites and radio-telescopes are analyzed to obtain information about geological compositions distributions, distant asters as well as undersea terrain. Spectral images usually contain tens to hundreds of continuous narrow spectral bands and are widely used in various fields. But the vast…
New spectral clustering method using LASSO regularization for robust graph partitioning.
problem Lack of theoretical guarantees for spectral clustering on general graph models.
method 1-spectral clustering on a new random model with LASSO regularization.
result Effective and robust to small noise perturbations, validated by simulations and real data.
Model structures on multicomplexes help study complex geometry.
problem Understanding homotopy types of complex manifolds.
method Model category structures on N-multicomplexes with weak equivalences induced by quasi-isomorphisms. result Establishes a basis for studying almost and generalized complex manifolds.
Introduces internal Lagrangians for differential equations and connects them to presymplectic structures.
problem Understanding the geometry of differential equations and their solutions.
method Develops a spectral sequence related to internal Lagrangians and investigates connections to presymplectic structures.
result Interprets a term in Vinogradov's spectral sequence for gauge theories.
The study reveals the spectral structure of attention layers and its implications for generalization.
problem Understanding the spectral structure and generalization of trained attention layers.
method Empirical risk minimization in a single-head tied-attention layer, using random matrix theory, spin-glass theory, and approximate message passing.
result Exact high-dimensional characterization of training and test error, interpolation and recovery thresholds, and spectrum of the key and query matrices.
The paper constructs toric vector bundles using spectral networks and non-abelianization.
problem Understanding how holomorphic vector bundles arise from spectral networks and non-abelianization.
method Constructing toric vector bundles on complete toric surfaces via spectral networks and non-abelianization.
result The moduli space of rank 2 toric vector bundles over toric surfaces admits an A-type X-cluster structure. This paper proposes a spectral clustering algorithm for hyperbolic spaces, improving efficiency over Euclidean methods.
problem Inefficient clustering in Euclidean spaces for complex data structures.
method Developed a spectral clustering algorithm using hyperbolic similarity matrices.
result The algorithm converges at least as fast as Euclidean spectral clustering and performs better on complex datasets.
Rewiring GNNs to optimize community and feature alignment improves their performance.
problem Improving GNNs' performance by addressing over-squashing and generalization issues.
method Three rewiring strategies: ComMa, FeaSt, and ComFy, targeting community structure, node labels, and their alignment.
result Rewiring strategies enhance GNNs' performance by optimizing label-community alignment.
Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the s…
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…
New spectral invariants distinguish Joyce orbifolds from other manifolds.
problem Distinguishing Joyce orbifolds from other G2-structures. method Introducing and computing two new spectral invariants for Joyce orbifolds.
result These invariants are more effective than existing invariants for distinguishing Joyce orbifolds.
Szabó recently introduced a combinatorially-defined spectral sequence in Khovanov homology. After reviewing its construction and explaining our methodology for computing it, we present results of computations of the spectral sequence. Based on these computations, we make a number of conjectures concerning the structure…
New methods rank players using covariates and comparisons, outperforming existing algorithms.
problem Ranking players based on incomplete and noisy pairwise comparisons.
method Three spectral ranking methods incorporating player covariates.
result Proposed methods outperform existing algorithms in simulations.
Let G be a finite group. Noncommutative geometry of unital G-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical o…
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.
Researchers prove spectral uniqueness of complex/quaternionic structures on manifolds.
problem Spectral uniqueness of complex/quaternionic structures on manifolds.
method Explicit expression for smallest positive eigenvalue of Laplace-Beltrami operator.
result Irreducible symmetric spaces are spectrally unique within families of homogeneous metrics.
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.
A holomorphic Poisson structure induces a deformation of the complex structure as Hitchin's generalized geometry. Its associated cohomology naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is the Dolbeault cohomology with coefficients in the exterior a…
Through the theory of Lie bi-algebroids and generalized complex structures, one could define a cohomology theory naturally associated to a holomorphic Poisson structure. It is known that it is the hypercohomology of a bi-complex such that one of the two operators is the classical ∂-operator. Another…
Spectral analysis detects structural changes in financial networks.
problem Detecting structural transitions in financial networks to assess systemic risk.
method Ensemble properties of spectral radius of random graph models calibrated on real-world evolving networks.
result The spectral deviation captures ongoing topological changes in financial networks.
Study spectral settings of generalized Laplacians on homogeneous spaces.
problem Understanding the spectral properties of generalized Laplacians on compact homogeneous spaces.
method Investigates the generic spectral configuration of operators on G-invariant metrics on M=G/K. result The spectral setting depends on G-isometries and hidden symmetries. 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.
FoSR adds edges to graphs to prevent oversquashing and oversmoothing in GNNs.
problem Oversquashing and oversmoothing in graph neural networks (GNNs).
method First-order spectral rewiring to add edges based on spectral expansion, combined with a relational architecture.
result Our algorithm outperforms existing graph rewiring methods in graph classification tasks.
Paper improves robust spectral clustering for noisy data.
problem Noisy data and heavy-tailed entries hinder traditional clustering methods.
method Robust spectral clustering with rank statistics for latent structure recovery.
result Provable recovery of latent block structure in large data matrices.
Researchers find spectral gaps in quantum flag manifolds using twisted operators.
problem Finding spectral gaps in quantum flag manifolds.
method Tensoring Laplace and Dolbeault-Dirac operators with negative Hermitian holomorphic modules.
result Twisting Dirac and Laplace operators by negative line bundles produces a spectral gap for q close to 1.
Proves a conjecture for a specific group using spectral sequences and homology.
problem Proves the Gromov-Lawson-Rosenberg Conjecture for the group Z/4xZ/4.
method Used the Adams spectral sequence and detection theorems to compute connective real k-homology.
result Determines differentials of the Adams spectral sequence and studies the cap structure of relevant sub-hopf algebras.
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 present Spectral Inference Networks, a framework for learning eigenfunctions of linear operators by stochastic optimization. Spectral Inference Networks generalize Slow Feature Analysis to generic symmetric operators, and are closely related to Variational Monte Carlo methods from computational physics. As such, the…
A fast spectral algorithm detects community structure in evolving graphs.
problem Detecting community structure in time-evolving sparse graphs.
method Extension of the Bethe-Hessian matrix for spectral community detection.
result The algorithm reaches the optimal detectability threshold and outperforms other methods.
We define an integer graded symplectic Floer cohomology and a Fintushel-Stern type spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopes. The Z-graded symplectic Floer cohomology is an integral lifting of the usual Z_Sigma(L)-graded Floer-Oh cohomology. We prove the Kunneth…
We show that there is a well-defined cap-product structure on the Fintushel-Stern spectral sequence. Hence we obtain the induced cap-product structure on the ${\BZ}_8$-graded instanton Floer homology. The cap-product structure provides an essentially new property of the instanton Floer homology, from a topological poin…
Study on spectral points of Inoue surfaces with Tricerri metric.
problem Identifying spectral points on Inoue surfaces.
method Analyzing chiral Dirac operators twisted by flat C∗-connections. result No spectral points inside the annulus α−1/4<∣z∣<α1/4, with spectral points on boundary. Spectral mixture (SM) kernels comprise a powerful class of generalized kernels for Gaussian processes (GPs) to describe complex patterns. This paper introduces model compression and time- and phase (TP) modulated dependency structures to the original (SM) kernel for improved generalization of GPs. Specifically, by adop…
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.
The notion of a Kähler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kähler structure has a canonically associated triple satisf…
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.
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.
Researchers describe a spectral sequence for knots in 3D space.
problem Understanding the Sinha spectral sequence for knots in R^3.
method Explicit description using Fox Neuwirth chain complexes and multicomplex structure.
result A non-trivial third page differential found, contradicting the rational case.
The paper explores spectral sequences of complex manifolds with special metrics.
problem Understanding spectral sequences of compact complex manifolds with special metrics.
method Investigation of Frölicher spectral sequences and special metrics (balanced, SKT, Gauduchon) on manifolds.
result Found compact manifolds where spectral sequences do not degenerate at the second page, providing counterexamples and new families.