Extends Einstein-Hilbert action to higher-order spectral triples.
problem No specific problem stated; focuses on extending action.
method Introduced two second-order spectral triples and computed their Einstein-Hilbert actions.
result Demonstrated applicability of the theoretical framework.
Calculates knot X-torsion order using spectral sequences.
problem Calculating the X-torsion order of knots. method Using the reduced Bar-Natan--Lee--Turner spectral sequence.
result Example of X-torsion order 4. The C-spectral sequence was introduced by Vinogradov in the late Seventies as a fundamental tool for the study of algebro-geometric properties of jet spaces and differential equations. A spectral sequence arise from the contact filtration of the modules of forms on jet spaces of a fibring (or on a differential equation…
New framework detects directional influence in multivariate time series.
problem Detecting directional influence in multivariate time series.
method Order-constrained spectral non-invariance.
result Unique diagnostic functional for directional influence.
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.
Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.
problem Estimating spectral gaps for higher-order operators on Cartan-Hadamard manifolds.
method Symmetrization-free proofs based on general functional inequalities.
result Solves a sharp asymptotic problem from Cheng and Yang and answers a question from Kristály.
Introduces new spectral triples for parabolic geometry.
problem Anisotropies and varying orders in parabolic geometry.
method Tangled spectral triples incorporating directional Dirac operators.
result Higher order spectral triples for hypoelliptic complexes and nilpotent group algebras.
Study non-squeezing phenomena in contact geometry using specific capacities.
problem Detect and quantify non-squeezing in contact geometry.
method Defined and computed two contact capacities, using spectral selectors and Givental's non-linear Maslov index.
result Discovered and quantified non-squeezing phenomena in lens spaces and strongly order able closed prequantizations.
The study proves properties of spectral selectors for contact manifolds and applies them to contact big fibers and geodesics.
problem Properties of spectral selectors for contact manifolds.
method Algebraic properties of spectral selectors for strongly orderable contact manifolds.
result Established contact big fiber theorem and constructed norms on contactomorphism group universal cover.
Paper proposes efficient methods for high-order clustering in tensor block models.
problem High-order clustering of multiway datasets in neuroimaging, genomics, etc.
method Tensor block model and computationally efficient algorithms (HLloyd, HSC)
result Achieves high-order exact clustering with statistical optimality and computational efficiency.
Let Y be a compact, oriented 3-manifold with a contact form a. For any Dirac operator D, we study the asymptotic behavior of the spectral flow between D and D+cl(-ira) as r very large. If a is the Thurston-Winkelnkemper contact form whose monodromy is the product of Dehn twists along disjoint circles, we prove that the…
Study spectral analysis on lens spaces, proving isospectral lens spaces with prime order fundamental groups.
problem Spectral analysis of the Kohn Laplacian on lens spaces.
method Analog of Weyl's law and isospectral lens spaces with prime order fundamental groups.
result Two 3D lens spaces with prime order fundamental groups are isospectral with respect to the Kohn Laplacian if and only if they are CR isometric.
Introduce Collapsed Effective Operators for higher-order structures.
problem Existing spectral operators decompose topology into separate ranks, leaving practitioners to fuse information back to vertices.
method Introduce Collapsed Effective Operators via Schur complementation of a graded Laplacian.
result Preserves positive semi-definiteness, lowers system energy under higher-order connectivity.
We extend the Heegaard Floer homological definition of spectral order for closed contact 3-manifolds due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact 3-manifolds with convex boundary. We show that the order of a codimension zero contact submanifold bounds the order of the ambient manifold from above. As the…
Clustering is fundamental for gaining insights from complex networks, and spectral clustering (SC) is a popular approach. Conventional SC focuses on second-order structures (e.g., edges connecting two nodes) without direct consideration of higher-order structures (e.g., triangles and cliques). This has motivated SC ext…
Solves the Wiegold problem by showing free products of left-orderable groups have normal rank > 1.
problem Wiegold problem about groups of normal rank > 1
method Topological argument and intricate construction of left-orders
result Free products of nontrivial left-orderable groups have normal rank > 1
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'.
Higher-order motif structures and multi-vertex interactions are becoming increasingly important in studies that aim to improve our understanding of functionalities and evolution patterns of networks. To elucidate the role of higher-order structures in community detection problems over complex networks, we introduce the…
Enhances clustering performance with a novel high-order Laplacian matrix.
problem Limited representation capability and insufficient information exploitation in multi-view spectral clustering.
method Proposes a multi-view spectral clustering algorithm that learns a high-order optimal neighborhood Laplacian matrix.
result Improves clustering performance through enhanced representation capacity of the learned optimal Laplacian matrix.
RP-GFRFT unifies fractional order and rotation control for graph signals.
problem Lack of rotation-based spectral control in GFRFT and zero-angle degeneracy in AGFT.
method Rotation-parameterized graph fractional Fourier transform (RP-GFRFT) with degeneracy preserving rotation matrix.
result RP-GFRFT improves spectral filtering performance over existing methods.
Calculates spectral flow bounds for reducible solutions to Vafa-Witten equations.
problem Bounding spectral flow between diverging reducible solutions.
method Localization and excision techniques to calculate spectral flow.
result Bounds on spectral flow are given for reducible solutions.
Researchers study spectral asymmetry using pseudodifferential projections on the massless Dirac operator.
problem Understanding spectral asymmetry for the massless Dirac operator.
method Constructing a negative order pseudodifferential asymmetry operator from spectral projections.
result Computed the principal symbol of the asymmetry operator, accounting for gauge invariance.
Introduces a new spectral geometry framework with dissipative data.
problem Deforming spectral triples with dissipative Lindblad operators.
method Lindblad-deformed spectral geometry framework with heat-kernel asymptotics.
result First nontrivial dissipative effect appears at order gamma^4.
New method clusters weighted directed networks using motifs.
problem Clustering directed networks fails to consider higher-order structure and edge weights.
method Motif-based weighted spectral clustering with new matrix formulae.
result Scalable and effective clustering on large graphs and real-world data.
SASE improves attributed graph clustering for large graphs with linear time and space complexity.
problem Challenges in clustering large attributed graphs due to high computational and memory costs.
method SASE combines node features smoothing, scalable spectral clustering, and adaptive order selection.
result SASE achieves a 6.9% improvement in ACC and a 5.87x speedup on the ArXiv dataset.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n-th order spectral variations of singular values. In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit f…
A formula is given in terms of secondary characteristic classes for the leading order contribution to the spectral flow for a path of twisted Dirac operators on an odd dimensional, Riemannian manifold when the twisting is done by a path of unitary connections with large curvature.
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.
The study of higher-order homology embeddings for manifold topology.
problem Understanding the structure of higher-order homology embeddings to disclose geometric or topological information.
method Analysis of the null space of the k-th order Laplacian and proposing an algorithm to factorize the homology embedding. result The proposed spectral loop detection algorithm is more efficient and effective on various data types.
Study proves spectral determination of triangles and quadrilaterals, with restrictions on higher-order polygons.
problem Determining the geometry of convex polygons from their Steklov spectra.
method Analysis of characteristic polynomial and spectral properties of Steklov spectrum.
result Almost all triangles and certain quadrilaterals are uniquely determined by their Steklov spectra.
Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and tot…
Nonparametric models are versatile, albeit computationally expensive, tool for modeling mixture models. In this paper, we introduce spectral methods for the two most popular nonparametric models: the Indian Buffet Process (IBP) and the Hierarchical Dirichlet Process (HDP). We show that using spectral methods for the in…
The paper studies elliptic operators on manifolds with boundary.
problem Characterizing boundary conditions for elliptic operators on manifolds.
method Using Calderón projectors and mixed order Sobolev spaces, the paper describes the space of boundary values and characterizes Fredholm and regular realisations.
result Characterization of boundary conditions for elliptic operators leading to Fredholm and regular realisations.
Given a dataset and an existing clustering as input, alternative clustering aims to find an alternative partition. One of the state-of-the-art approaches is Kernel Dimension Alternative Clustering (KDAC). We propose a novel Iterative Spectral Method (ISM) that greatly improves the scalability of KDAC. Our algorithm is …
We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as $\l…
Spectral flow connects manifold geometry to rigidity criteria.
problem Tackling rigidity of simply-connected closed manifolds.
method Spectral deformation flow and invariant-based approach.
result Spherical profile is the unique manifold-compatible asymptotic realization.
Given a link in the three-sphere, Ozsváth and Szabó showed that there is a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double cover. The aim of this paper is to explicitly calculate this spectral sequence in terms of bordered Floer homolo…
Deterministic bounds for tensor singular values and vectors, differing from matrix cases.
problem Spectral learning of higher-order orthogonally decomposable tensors.
method Deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors.
result Perturbation affects each essential singular value/vector in isolation, independent of multiplicity and distance from other singular values.
A spectral approach to building the exterior calculus in manifold learning problems is developed. The spectral approach is shown to converge to the true exterior calculus in the limit of large data. Simultaneously, the spectral approach decouples the memory requirements from the amount of data points and ambient space …
For a second order operator on a compact manifold satisfying the strong Hörmander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian s…
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.
One of the longstanding problems in spectral graph clustering (SGC) is the so-called model order selection problem: automated selection of the correct number of clusters. This is equivalent to the problem of finding the number of connected components or communities in an undirected graph. In this paper, we propose AMOS…
Researchers calculate exact moduli for type II flux backgrounds using spectral sequences.
problem Determining exact moduli of type II flux backgrounds in string theory.
method Using techniques from generalised geometry, they count infinitesimal deformations via a spectral sequence.
result The spectral sequence reproduces naïve expectations and shows all obstructions vanish, impacting the tadpole conjecture.
Let Y be a compact, oriented 3-manifold with a contact form a and a metric ds2. Suppose that F→Y is a principal bundle with structure group U(2)=SU(2)×±1S1 such that F/S1 is the principal SO(3) bundle of orthonormal frames for TY. A unitary connection A0 on the Hermitian line bundle $…
Develops a new approach to spectral asymmetry using microlocal analysis.
problem Spectral asymmetry on 3-manifolds.
method Constructs an asymmetry operator using microlocal analysis.
result The asymmetry operator generalizes the eta invariant and contains spectral asymmetry information.
We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous fa…
New map constructed from equivariant spectra for manifold study.
problem Understanding equivariant parametrized h-cobordism in non-manifold settings.
method Constructed a map from suspension G-spectrum to equivariant A-theory spectrum, compatible with tom Dieck splitting formulas.
result Fiber of constructed map is wedge of stable h-cobordism spectra.