This paper explains spectral clustering and its equivalence to PCA, breaking it into fully connected and multi-connected cases.
problem Understanding the mathematics behind spectral clustering and its equivalence to PCA.
method Dividing spectral clustering into two categories based on graph connectivity and proving the equivalence to PCA.
result Spectral clustering and PCA are equivalent, with specific proofs for fully connected and multi-connected graphs.
Defines spectral varieties for non-simply connected manifolds and constructs conformal invariants.
problem Analyzing spectra of magnetic Laplacians on non-simply connected manifolds.
method Definition of spectral varieties and construction of conformal invariants.
result New conformal invariants of immersions of surfaces into 3- and 4-dimensional spaces.
The paper connects group extensions, cochains, and spectral sequences.
problem Understanding the relationship between group extensions and spectral sequences.
method Using connection cochains, the paper derives a formula for the extension class.
result A formula clarifies the relation among connection cochains, extension classes, and the LHS spectral sequence.
The paper studies spectral convergence of connections on vector and principal bundles.
problem Continuity of eigenvalues of connection Laplacians on vector bundles.
method Introducing a new topology on metric measure spaces with isometric G-actions to analyze convergence of G-connections. result Established spectral convergence of connections on vector and principal bundles.
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.
Optimizes projections for binary data separation using spectral connectivity.
problem Maximizing separability of binary partitions in unlabelled datasets.
method Minimizes the second eigenvalue of the graph Laplacian, using spectral connectivity.
result Optimal projections converge to the maximum margin hyperplane as scaling parameter approaches zero.
We explain that spectral networks are a unifying framework that incorporates both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate systems on moduli spaces of flat SL(2,C) connections, in the following sense. Given a spectral network W on a punctured Riemann surface C, we explain the process of "abe…
The paper proves conditions for Kähler and Riemannian manifolds to be simply connected.
problem Conditions for Kähler and Riemannian manifolds to be simply connected.
method Spectral positivity assumptions for Kähler manifolds and a specific spectral positivity assumption for Riemannian manifolds.
result Compact Kähler manifolds and Riemannian manifolds under the specified spectral positivity assumptions are simply connected.
New spectral functionals related to noncommutative residue and torsion Dirac operators.
problem Spectral functionals and Dirac operators with torsion.
method Noncommutative residue and Dirac operators with torsion.
result Extension of spectral functionals to noncommutative realm with torsion.
New spectral triples for higher-rank graphs linked to wavelet decompositions.
problem Creating spectral triples for higher-rank graph C∗-algebras. method Generalizing spectral triples from Cuntz-Krieger algebras to higher-rank graph C∗-algebras and connecting them to wavelet decompositions. result Wavelet decompositions describe eigenspaces of Dirac operators in these spectral triples.
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.
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…
This paper provides theoretical guarantees for spectral clustering using graph cuts.
problem Lack of performance guarantees for spectral clustering.
method Convex relaxation of graph cuts, spectral proximity condition, algebraic connectivity, inter-cluster connectivity.
result Deterministic bounds for successful spectral clustering are derived.
Proposes CRG_IMSC for better clustering of multi-view data.
problem Lack of effective connectivity in clustering results.
method Directly obtains clustering result with nonnegative constraint; constructs connectivity matrix based on spectral clustering result; uses multiplicative update algorithm.
result Improves clustering performance on benchmark datasets.
The paper defines surface area for graphs and derives spectral estimates.
problem Understanding connectivity measures and spectral properties of graphs.
method Introducing surface area concepts related to inverse degree and deriving spectral bounds.
result An upper bound on the second eigenvalue for planar graphs.
Defines curvature for spectral triples and applies to θ-deformations.
problem Defining curvature for noncommutative spectral triples.
method Using Levi-Civita connection, defines curvature tensors and derives Weitzenbock formula.
result Riemann and Ricci tensors transform naturally under θ-deformation, while scalar curvature is invariant.
Study Higgs bundles using cameral and spectral data.
problem Classify connected components of Higgs bundle moduli spaces.
method Analyze cameral and spectral data to deduce moduli space structure.
result Toledo invariant classifies Hitchin map fibers.
Study connects spectral clustering to maximum margin and level set estimation.
problem Connecting spectral clustering to maximum margin and level set estimation.
method Obtained bounds on eigenvectors of graph Laplacian matrices in terms of cluster separation and connectivity. Showed sensitivity mitigation by removing outliers and estimating level sets.
result Spectral clustering converges to maximum margin clustering as scaling parameter approaches zero.
Paper develops a method for estimating spectral density matrices in high-dimensional time series.
problem Estimating spectral density matrices in high-dimensional time series.
method Thresholded versions of averaged periodograms for regularized estimation.
result Consistent estimation of spectral density matrices possible under high-dimensional regime.
The paper proves spectral clustering consistency for point cloud data.
problem Consistency of spectral clustering for point cloud data.
method Variational convergence approach to graph Laplacians.
result Sharp conditions for spectral convergence with respect to sample size.
Study of meromorphic connections and their spectral duals in gl3(C).
problem Exploring ℏ-deformed meromorphic connections and their spectral duals. method Using apparent singularities and their dual partners as Darboux coordinates, the Hamiltonian evolutions and reductions are derived.
result Spectral duality extends to Hamiltonian evolutions, tau-functions, and Hermitian matrix models on both sides.
New non-Kähler manifolds constructed with specific properties.
problem Creating non-Kähler manifolds with cohomological properties.
method Four constructions using quotient singularities and spectral sequences.
result Disproved a conjecture by Popovici.
Spectral clustering identifies node groups in time-varying networks.
problem Identifying evolving community structures in dynamic networks.
method Estimate edge probabilities using a kernel-type procedure and apply spectral clustering.
result The method is computationally efficient and robust to varying membership rates.
Functorial correspondence found between logarithmic connections and abelian connections.
problem Establishing a relationship between logarithmic sl2-connections and abelian connections. method Constructing inverse functors and a canonical cocycle.
result Functorial correspondence proved between logarithmic sl2-connections and abelian connections. 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.
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.
Spectral sequence connects knot homologies via algebraic geometry.
problem Connecting algebraic and geometric knot homologies.
method Bigraded spectral sequence from gl(0)-homology to knot Floer homology.
result Constructs a Bockstein-type spectral sequence.
Curvature defined for Hilbert modules and Kasparov modules.
problem Defining and studying curvature in Hilbert modules and Kasparov modules.
method Introduced curvature for densely defined universal connections on Hilbert C∗-modules relative to spectral triples. result Curvature only depends on the represented form of the universal connection modulo junk forms.
Reduces SL(3) pre-buildings from spectral curves.
problem Constructing pre-buildings from spectral curves.
method Reduction steps from spectral curves to pre-buildings.
result Core of pre-building constructed from spectral curve differential.
Study connects Morin singularities to sphere homotopy groups.
problem Computing stable homotopy groups of spheres.
method Apply Morin singularities to sphere homotopy groups.
result Differentials in spectral sequence linked to sphere homotopy groups.
New spectral functionals for Dirac operators with inner fluctuations computed.
problem Spectral functionals and Dirac operators with inner fluctuations.
method Extension of spectral functionals for Dirac operators with inner fluctuations.
result Computed spectral Einstein functional for Dirac operator with inner fluctuations on even-dimensional spin manifolds.
Interpolates mean shift and spectral clustering on graphs.
problem Data clustering algorithms.
method Fokker-Planck equations on data graphs.
result New theoretical insights on diffusion maps and mean shift dynamics.
Study on spectral stability of Riemannian coverings.
problem Stability of eigenvalues in Riemannian coverings.
method Analysis of Laplacian eigenvalues under finite coverings.
result Necessary conditions for spectral stability or instability.
Estimates the number of connected components in a graph from a sampled subgraph.
problem Inferring the number of connected components in a larger graph from a sampled subgraph.
method A highly redundant and large-dimensional representation of the subgraph using counts of network motifs, leading to a novel estimator for the number of connected components.
result Improves upon competing algorithms for graphs with spectral gaps bounded away from zero.
This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
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.
Spectral sequence connects knot homologies to quotient knots.
problem Distinguishing knots using homology.
method Construct spectral sequence relating Khovanov homology to quotient knots.
result Khovanov homology distinguishes certain slice disks.
New interpretation of Mayer-Vietoris sequence using überhomology.
problem Understanding the Mayer-Vietoris spectral sequence.
method Identifying the second page of the Mayer-Vietoris spectral sequence with überhomology.
result Combinatorial interpretation of the second page of the Mayer-Vietoris sequence.
The paper introduces a trilinear functional to recover torsion in spectral triples.
problem Recovering torsion in noncommutative spectral triples.
method Introduces a trilinear functional for spectral triples and demonstrates its application to recover torsion.
result The trilinear functional recovers the torsion of the linear connection in canonical spectral triples.
Connected sum of manifolds preserves Ricci lower bounds.
problem Proving connected sum of manifolds with spectral Ricci lower bounds.
method Geometric construction resembling Gromov-Lawson tunnel, focusing on γ>n−2n−1. result Connected sum M#N also admits a metric satisfying the Ricci lower bound condition. We establish a splitting formula for the spectral flow of the odd signature operator on a closed 3-manifold M coupled to a path of SU(2) connections, provided M = S cup X, where S is the solid torus. It describes the spectral flow on M in terms of the spectral flow on S, the spectral flow on X (with certain Atiyah-Pato…
Formula for spectral flow connects manifold properties to index theorem.
problem Establishing a formula for spectral flow on manifolds.
method Reduction to Atiyah-Patodi-Singer index theorem for manifolds with boundary.
result Formula for spectral flow expressed in manifold and connection properties.
The paper analyzes spectral properties of connection Laplacian on tori, proving convergence to real torus.
problem Spectral analysis of connection Laplacian on tori.
method Employing parallel orthonormal basis in pullback bundle, examining eigenvalues of connection Laplacian on real and discrete tori.
result Eigenvalues of connection Laplacian on discrete tori converge to those on real torus, with unique twist in torsion matrix.
Paper introduces a new multilinear functional for spectral triples and computes its properties.
problem Computing properties of spectral triples and their associated Hodge operators.
method Introduces a new multilinear functional for spectral triples and computes its properties using noncommutative residue and perturbed de-Rham Hodge operators.
result Recover two forms, torsion of the linear connection, and four forms by the noncommutative residue and perturbed de-Rham Hodge Dirac triple.
Study Berry connections for 2d GLSMs, linking to cohomology theories.
problem Quantise ground states of 2d (2,2) GLSMs on a circle. method Relate periodic monopole solutions to difference modules and vector bundles with filtrations.
result Derive novel difference equations for brane amplitudes and vortex partition functions.
Corners can be identified by a drum's sound spectrum.
problem Determining the presence of corners in a drum's shape from its sound.
method Proving spectral invariance of corners in domains with Lipschitz, piecewise smooth boundaries.
result Corners are uniquely determined by a drum's spectrum among domains with fixed genus.
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. Study on spectral clustering phase transitions in noisy networks.
problem Detecting communities in noisy networks.
method Proved phase transitions using Erdos-Renyi random noise model.
result Critical external edge connection probability for community detectability.