We introduce three spectral sequences which give some expressions of colored Jones polynomials. Each spectral sequence contains a Khovanov-type homology groups. Two of them are derived from a bicomplex of the colored Jones polynomial. The other is the spectral sequence that deduces a colored Rasmussen invariant of link…
Paper revisits graph-CNNs using Laplace-Beltrami spectral filters and polynomials.
problem Improving spectral graph convolutional neural networks (graph-CNNs).
method Developed Laplace-Beltrami CNN (LB-CNN) by replacing graph Laplacian with LB operator and approximating spectral filters using Chebyshev, Laguerre, and Hermite polynomials.
result Classification accuracy of LB-CNN is not dependent on the type of polynomials or operators.
Random covers of hyperbolic surfaces have a spectral gap with polynomial rate.
problem Finding spectral gaps in random covers of hyperbolic surfaces.
method Applying recent work on spectral gaps to uniformly random covers of closed hyperbolic surfaces.
result Uniformly random degree-n covers of a closed hyperbolic surface have no new Laplacian eigenvalues below a specific threshold with high probability.
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.
PolyNSD improves Neural Sheaf Diffusion with polynomial operators and spectral rescaling.
problem Limitations of common Neural Sheaf Diffusion implementations, including scalability and stability issues.
method Introduces Polynomial Neural Sheaf Diffusion (PolyNSD) with a degree-K polynomial propagation operator and spectral rescaling.
result PolyNSD achieves state-of-the-art results on both homophilic and heterophilic benchmarks with reduced runtime and memory requirements.
Study polynomial cubic differentials on Riemann surfaces using spectral networks.
problem Characterize polynomial cubic differentials with saddle connections or critical tripods.
method Introduced spectral core, refined classical core concept, and applied Gaiotto-Moore-Neitzke's algorithm.
result Completely characterized polynomial cubic differentials up to degree 3, including wall-and-chamber structure.
Study shows link polynomial evaluations from Heegaard Floer theory.
problem Link polynomial evaluations from Heegaard Floer theory.
method Definition of Euler characteristic for fractionally-graded complexes based on roots of unity.
result Equality of Alexander polynomial evaluations and sl(n) polynomial evaluations at certain roots of unity. FNOs learn solution operators of dissipative equations efficiently via spectral methods.
problem Learning and approximation of solution operators for dissipative equations.
method Introducing spectral methods and deriving FNO approximation bounds and sample complexity guarantees.
result Polynomial sample complexity guarantees for FNOs learning solution operators of dissipative equations.
Random surfaces have a strong spectral gap with polynomial rate.
problem Understanding spectral gaps in random hyperbolic surfaces.
method Adapting polynomial method for random matrices to Laplacian on surfaces.
result Laplacian spectral gap at least 1/4 - O(1/g^c) for large g.
Study on polynomial growth functions and forms on gradient Ricci solitons.
problem Estimating dimensions of polynomial growth holomorphic functions and forms.
method Relating to spectral data of the f-Laplacian, proving estimates under curvature assumptions. result Sharp dimension estimates and almost sharp frequency estimates for polynomial growth holomorphic functions.
Proves polynomial error rate for equidistribution of unipotent flows.
problem Equidistribution of orbits of unipotent subgroups in arithmetic quotients.
method Uses Margulis function, incidence geometry tools, and spectral gap.
result Polynomial error rate for equidistribution theorems.
Let EkF(D) be the spectral sequence induced by the oriented cube of resolutions on knot Floer homology. We prove that E2F(D) is a triply graded link invariant whose graded Euler characteristic is the HOMFLY-PT polynomial and that the higher pages are link invariants. By construction, the spectral sequen…
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.
Topological recursion recovers a specific partition function for colored knots.
problem Recovering the extended Ooguri-Vafa partition function for colored HOMFLY-PT polynomials of torus knots.
method Applying topological recursion to the spectral curve of colored HOMFLY-PT polynomials of torus knots.
result Topological recursion reproduces the n-point functions of the extended Ooguri-Vafa partition function.
A new polynomial invariant for strongly involutive links.
problem Characterizing strongly involutive links using polynomial invariants.
method Introducing a two-variable polynomial invariant \(P^e\) with equivariant skein relations.
result Specialisation of \(P^e\) recovers the graded Euler characteristic of a spectral sequence.
We derive the Do and Norbury recursion formula for the one-loop mean of an irregular spectral curve from a variant of replica method by Brezín and Hikami. We express this recursion in special times in which all terms W1(g) of the genus expansion of the one-loop mean are polynomials. We find a generalization of th…
Explains Khovanov homology and its applications.
problem Understanding Khovanov homology and its applications.
method Expository lecture notes covering Jones polynomial, Khovanov homology, cobordism category, spectral sequences, and skein lasagna modules.
result Explains the Jones polynomial, Khovanov homology, and their applications.
Polynomial density theorem for specific subgroup orbits in quotient spaces.
problem Effective density of orbits in arithmetic quotients of SL2(C) and SL2(R)imesSL2(R). method Use of Margulis function, incidence geometry tools, and spectral gap of ambient space.
result Proved effective density theorems with polynomial error rate.
Proposes a Gaussian process for graph signals using adaptive spectral kernels.
problem Predicting signals on graph nodes with various structures.
method Spectral kernel learning approach that incorporates a polynomial function in the graph spectral domain.
result The model accurately recovers ground truth spectral filters and outperforms in real-world graph data.
Developed a new homology theory for graph chromatic polynomials.
problem Categorification of chromatic polynomials.
method Introduced a new homology theory HLee(G) and developed a spectral sequence. result Spectral sequence converges to HLee(G), supporting H∗(G). We encode the variation structure of a quasihomogeneous polynomial with an isolated singularity as introduced by Nemethi in a set of spectral flows of the signature operator on the Milnor bundle by varying global elliptic boundary conditions in a specific way using the quasihomogeneous circle action on the Brieskorn la…
We investigate solutions of the elliptic sinh-Gordon equation of spectral genus g<3. These solutions are parametrized by complex matrix-valued polynomials called potentials. On the space of these potentials there act two commuting flows. The orbits of these flows are called Polynomial Killing fields and are double peri…
The paper certifies projective rigidity for once-punctured torus bundles using twisted Alexander polynomials.
problem Certifying infinitesimal projective rigidity for hyperbolic once-punctured torus bundles.
method Using twisted Alexander polynomials of representations associated with the holonomy.
result The induced action on the tangent space of the character variety matches the group theoretic action.
We study minimal annuli in S2×R of finite type by relating them to harmonic maps C→S2 of finite type. We rephrase an iteration by Pinkall-Sterling in terms of polynomial Killing fields. We discuss spectral curves, spectral data and the geometry of the isospectral set…
Study connects spectral properties to frame flows on curved manifolds.
problem Spectral properties and frame flows on curved manifolds.
method Link between spectral properties, frame flows, and polynomial maps between spheres.
result Ergodicity of frame flows on low-rank bundles.
Analysis of DPPs and k-DPPs via spectral decomposition reveals identifiable parameters and non-identifiability gaps.
problem Identifying parameters of DPPs and k-DPPs through spectral decomposition.
method Spectral decomposition of the covariance matrix, analysis of invariances, and counting arguments.
result Identifiability of parameters changes fundamentally for k-DPPs, with specific invariances and non-identifiability gaps.
We give a polynomial-time algorithm for learning latent-state linear dynamical systems without system identification, and without assumptions on the spectral radius of the system's transition matrix. The algorithm extends the recently introduced technique of spectral filtering, previously applied only to systems with a…
We give a method of decomposing bundle-valued polynomials compatible with the action of the Lie group Spin(n), where important tools are Spin(n)-equivariant operators and their spectral decompositions. In particular, the top irreducible component is realized as an intersection of kernels of these operators.
Polynomial error equidistribution for SL2 groups.
problem Equidistribution of orbits in arithmetic quotients.
method Margulis function, incidence geometry, spectral gap.
result Polynomial error rate for equidistribution.
This work analyzes how different layers in deep neural networks contribute to generalization error.
problem Understanding the role of each layer in deep neural networks for generalization.
method Spectral analysis, Neural Tangent Kernel, Hermite polynomials, Spherical Harmonics.
result Initial layers in deep neural networks have a larger bias towards high-frequency functions.
Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. We develop a method solving the gossip problem that depends only on the spectral dimension of the netwo…
Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky.
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.
A new method for creating simpler models from complex ones.
problem Creating accurate approximations of complex models at reduced costs.
method Sequential adaptive surrogate modeling based on locally spectral expansions.
result Stochastic spectral embedding (SSE) shows good approximation capabilities and scalability.
Unified framework explains why overfitting is benign in interpolating learning.
problem Understanding why overfitting is benign in highly overparameterized models.
method Spectral-transport stability framework.
result Sharp benign-overfitting criterion and explicit phase-transition rates.
Spectral graph sparsification preserves geometry of GNN embeddings.
problem Maintaining geometric properties of graph neural network embeddings during sparsification.
method Proving spectral sparsification preserves squared pairwise distances, class means, and covariance structure in embedding space.
result Spectral sparsification preserves the geometry of learned embeddings in GNNs.
New bounds for KRR condition number reveal overfitting phenomena.
problem Characterizing overfitting in KRR with varying kernel spectral decay.
method Derived new bounds for kernel matrices, enhanced test error bounds, and identified feature independence role.
result Identified tempered and catastrophic overfitting phenomena.
By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the i…
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…
Popular graph neural networks implement convolution operations on graphs based on polynomial spectral filters. In this paper, we propose a novel graph convolutional layer inspired by the auto-regressive moving average (ARMA) filter that, compared to polynomial ones, provides a more flexible frequency response, is more …
The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are ab…
Study reveals an equivalence principle for the spectrum of random inner-product kernel matrices in polynomial scaling.
problem Understanding the spectrum of random kernel matrices in polynomial scaling regimes.
method Investigates random matrices with nonlinear kernel functions applied to inner products of uniformly distributed vectors.
result The spectrum of the random kernel matrix is asymptotically equivalent to a simpler matrix model through free additive convolution.
We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy. Our algorithms can decompose a 4-tensor with n-dimensional orthonormal components in the…
We consider the problem of training input-output recurrent neural networks (RNN) for sequence labeling tasks. We propose a novel spectral approach for learning the network parameters. It is based on decomposition of the cross-moment tensor between the output and a non-linear transformation of the input, based on score …
This paper improves neural network learning by escaping the NTK regime and efficiently learning sparse polynomials.
problem Learning sparse polynomials efficiently using neural networks.
method Spectral analysis of NTK, identifying 'good' directions, and constructing a regularizer.
result Gradient descent on a two-layer neural network can learn sparse polynomials efficiently, improving over the NTK and QuadNTK.
The paper introduces a quantum state system to count perfect matchings in graphs.
problem Counting perfect matchings in graphs using quantum state systems.
method Topological quantum field theory (TQFT) and spectral sequences.
result The filtered n-color vertex homology for n=2 is generated by perfect matchings. We consider a fundamental algorithmic question in spectral graph theory: Compute a spectral sparsifier of random-walk matrix-polynomial Lα(G)=D−∑r=1dαrD(D−1A)r where A is the adjacency matrix of a weighted, undirected graph, D is the diagonal matrix of weighted degrees, and α=(α1...αd) are nonn…
Paper studies community detection in censored hypergraphs using information theory.
problem Community detection in censored hypergraphs with missing values.
method Information-theoretic approach, polynomial-time algorithm, spectral algorithm with refinement.
result Derives information-theoretic threshold for exact recovery of community structure.