Novel multigraph network improves chemical classification tasks.
problem Learning from variable graphs with multiple relationships.
method Proposed a multigraph network using Chebyshev GCNs to handle variable graphs and learned edges.
result Achieved competitive results on chemical classification benchmarks.
Proposes a new method to optimize graph neural network architectures on heterogeneous information networks.
problem Weaknesses in instability and inflexibility of existing graph neural architecture search methods.
method Partial Message Meta Multigraph search (PMMM) using a differentiable framework to search for a meaningful meta multigraph.
result Significantly more stable and effective than state-of-the-art heterogeneous GNNs.
L-GCNs learn from complex multigraphs, improving node classification performance.
problem Learning from complex multigraphs with rich edge labels.
method Latent-Graph Convolutional Networks (L-GCNs) that propagate information to a latent adjacency tensor.
result L-GCNs improve node classification performance, especially with nonlinear interactions.
New model captures sparse, evolving multigraph structures.
problem Understanding sparse, evolving multigraph structures in dynamic interaction data.
method Dynamic nonparametric Bayesian model combining sparsity and clustering.
result Improved held-out likelihood and predictive performance.
The study proves planes are the only complete uniformly elliptic Weingarten multigraphs.
problem Proving planes are the only complete uniformly elliptic Weingarten multigraphs.
method Proving planes are the only complete multigraphs with quasiconformal Gauss map and bounded second fundamental form.
result Proves planes are the only complete uniformly elliptic Weingarten multigraphs.
Classifies multigraphs for torus actions on 6D manifolds with isolated fixed points.
problem Classifying torus actions on 6D manifolds with isolated fixed points.
method Associate multigraphs to fixed point data, study operations, and prove classification.
result Classifies multigraphs for 6D manifolds by converting them into the empty graph.
MGMC method handles missing data in medical datasets for accurate disease classification.
problem Handling missing data in incomplete medical datasets for accurate disease classification.
method Multigraph Geometric Matrix Completion (MGMC) using multiple graph convolutional networks.
result MGMC achieves superior classification and imputation performance compared to state-of-the-art approaches.
New models capture heterogeneous network density, improving community detection.
problem Empirical networks are often globally sparse but locally dense.
method Latent Poisson models generating hidden multigraphs.
result These models improve community detection in sparse networks.
A typical census of 3-manifolds contains all manifolds (under various constraints) that can be triangulated with at most n tetrahedra. Al- though censuses are useful resources for mathematicians, constructing them is difficult: the best algorithms to date have not gone beyond n = 12. The underlying algorithms essential…
Study circle actions on unitary manifolds with discrete fixed points.
problem Understanding circle actions on compact unitary manifolds with discrete fixed points.
method Prove relationships between weights at fixed points and derive results regarding the first equivariant Chern class and Hirzebruch χy-genus. result Derive a multigraph encoding fixed point data, leading to new insights into unitary S1-manifolds. We study the spectrum of the Laplace operator of a complete minimal properly immersed hypersurface M in Rn+1. (1) Under a volume growth condition on extrinsic balls and a condition on the unit normal at infinity, we prove that M has only essential spectrum consisting of the half line [0,+∞). This is t…
The paper explores holonomy, zeta functions, and cohomology in foliated manifolds with stratified boundaries.
problem Understanding symmetries and cohomology in foliated manifolds with stratified boundaries.
method Developed a novel formalism for the Gamma-set and defined an Ihara zeta function to encode symmetries. Investigated the relationship between holonomy and zeta functions, and analyzed how the twist map impacts cohomology.
result Conjectured a duality between holonomy fixed points and the poles of the Ihara zeta function, extending to twisted cohomology classes.
In this paper, we study a circle action on a compact oriented manifold with a discrete fixed point set. The fixed point data consists of the weights of the S1-representations at the fixed points. We prove various results and properties of the action, in terms of the fixed point data. We show that the manifold can be…
Dynamic model clusters interactions over time, improving prediction.
problem Sparse, evolving interaction graphs with temporal dynamics.
method Structured, nonparametric edge-exchangeable model for dynamic clustering.
result Improved predictive performance compared to static models.
In this paper, we present a novel way to summarize the structure of large graphs, based on non-parametric estimation of edge density in directed multigraphs. Following coclustering approach, we use a clustering of the vertices, with a piecewise constant estimation of the density of the edges across the clusters, and ad…
Constructs polyhedral chains with prescribed tangent plane distributions.
problem Constructing polyhedral chains with specific tangent plane distributions.
method Explicit construction of polyhedral chains that approximate prescribed measures on Grassmannian.
result Polyconvexity is equivalent to quasiconvexity of associated Q-integrands under certain conditions.
We proposed a probabilistic approach to joint modeling of participants' reliability and humans' regularity in crowdsourced affective studies. Reliability measures how likely a subject will respond to a question seriously; and regularity measures how often a human will agree with other seriously-entered responses coming…
We prove that any complete surface with constant mean curvature in a homogeneous space E(κ,τ) which is transversal to the vertical Killing vector field is, in fact, a vertical graph. As a consequence we get that any orientable, parabolic, complete, immersed surface with constant mean curvature H in E(κ,τ) (different fr…
GRCN improves GCNs by predicting missing edges and revising weights.
problem Sub-optimal solutions due to incomplete and noisy real-world graphs.
method Introduces a GCN-based graph revision module for predicting missing edges and revising weights.
result GRCN consistently outperforms strong baseline methods, especially on incomplete graphs.
Proves generic existence of spectral networks for many cases.
problem Existence of spectral networks for a broad range of spectral data.
method Generic existence proof for spectral networks.
result Proves existence for a large class of spectral data.
Study shows how many crossings arise in curves on surfaces.
problem Understanding crossings in curves on surfaces.
method Proved a minimum crossing number growth rate for m curves. result Effective bounds with optimal growth rates on every orientable surface.
Fatgraphs are multigraphs enriched with a cyclic order of the edges incident to a vertex. This paper presents algorithms to: (1) generate the set of all fatgraphs having a given genus and number of boundary cycles; (2) compute automorphisms of any given fatgraph; (3) compute the homology of the fatgraph complex. The al…
Given a graph where vertices represent alternatives and arcs represent pairwise comparison data, the statistical ranking problem is to find a potential function, defined on the vertices, such that the gradient of the potential function agrees with the pairwise comparisons. Our goal in this paper is to develop a method …
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…
The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.
problem Understanding spectral networks in symplectic topology and their role in Lagrangian fillings.
method Analytic results on adiabatic degeneration of Floer trajectories and explicit computation of continuation strips.
result Established equivalence between Family Floer functor and non-abelianization functor for Lagrangian fillings with spectral networks.
Spectral decoupling improves neural network generalization in medical imaging.
problem Poor generalization of neural networks trained on medical imaging data.
method Spectral decoupling, a regularization technique that encourages learning more features.
result Spectral decoupling increases network robustness and performance on external datasets.
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. Exact spectral norm regularization improves neural network generalization.
problem Improving neural network generalization while protecting against noise.
method Exact spectral norm regularization of the Jacobian.
result Improved generalization performance compared to previous methods.
ARFF reduces spectral bias in SGD-trained neural networks.
problem Spectral bias in two-layer neural networks.
method Comparison of SGD and ARFF on spectral bias and robustness.
result ARFF yields a closer to zero spectral bias compared to SGD.
Researchers calculate spectral dimension of complex networks using renormalization group theory.
problem Understanding diffusion properties in complex systems.
method Renormalization group theory applied to graph Laplacians of simplicial complexes.
result Spectral dimension decreases with randomness in topological structure.
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.
This paper proves a generalization bound for complex-valued neural networks scaling with spectral complexity.
problem Ensuring the performance of complex-valued neural networks on unseen data.
method Theoretical derivation using Maurey Sparsification Lemma and Dudley Entropy Integral, empirical validation on various datasets.
result The spectral complexity of weight matrices is a significant factor in the generalization ability of complex-valued neural networks.
New spectral clustering method for multi-layer networks improves accuracy.
problem Detecting community structure in multi-layer networks.
method Integrative spectral clustering based on adaptive layer aggregation.
result Our methods minimize mis-clustering error and outperform existing methods.
Introduces Spectral Graph Network combining spatial and spectral message passing.
problem Relational reasoning in graph structured data.
method Applies message passing to both spatial and spectral domains of a graph.
result Promotes efficient training with fewer iterations and robustness to edge dropout.
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.
This paper improves spectral embedding for multipartite networks, revealing latent subspaces and providing consistent node representations.
problem Improving spectral embedding for multipartite networks to better represent node types.
method Developed a follow-on step to spectral embedding that recovers node representations in their intrinsic rather than ambient dimension, proving consistency under a specific model.
result Node representations in multipartite networks lie near type-specific subspaces, and the proposed method recovers these representations consistently.
Paper optimizes Laplacian regularization for sparse network clustering.
problem Improving spectral clustering in sparse networks.
method Formally determines optimal Laplacian regularization.
result Proper regularization is closely tied to state-of-the-art techniques.
Paper studies randomized spectral clustering for large-scale networks.
problem Computational challenges in large-scale network community detection.
method Randomized sketching algorithms for spectral clustering.
result Theoretical bounds for approximation, misclassification, and link probability estimation.
New spectral clustering method for graphs with uneven node degrees.
problem Challenges in community detection for graphs with heterogeneous degree distributions.
method Spectral clustering on spherical coordinates with degree correction.
result Improved performance in representing computer networks.
The paper studies almost complex torus manifolds using graphs and Hirzebruch genera, proving properties of their fixed points and cohomology.
problem Understanding the fixed points and cohomology of almost complex torus manifolds.
method Using directed labeled multigraphs and Hirzebruch genera to encode and analyze the manifolds.
result Almost complex torus manifolds have positive Todd genus and at least n+1 fixed points.
Proof of wall-crossing formula using spectral networks.
problem Proving the Kontsevich-Soibelman wall-crossing formula.
method Path-lifting rules for spectral networks, convergence justification.
result Definition and justification of path lifting rules for spectral networks.
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.
Unified view of spectral networks linking geometry and gauge theory.
problem Understanding BPS states in gauge theories.
method Unified geometric and physical approaches, focusing on spectral networks.
result Spectral networks provide a framework for determining BPS spectra.
Improved spectral-based GCN for directed graphs.
problem Cannot directly work on directed graphs.
method Redefined Laplacians to improve propagation model.
result Outperforms state-of-the-art methods on directed graph datasets.
New approach connects 3D Chern-Simons theory to spectral networks.
problem Understanding Chern-Simons invariants in 3D manifolds.
method Constructing equivalences between bundles and spectral networks.
result New formulas for Chern-Simons invariants of 3D manifolds.
Dual regularized graph Laplacian improves spectral clustering for community detection.
problem Detecting clusters in networks with improved spectral clustering methods.
method Proposes dual regularized graph Laplacian for three spectral clustering approaches.
result Theoretical analysis shows DRSC and DRSLIM yield stable consistent community detection.
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 portfolio theory links neural networks to wealth dynamics via SGD weight matrices.
problem Understanding wealth dynamics from neural network training.
method Direct identification of weight matrices as portfolio allocation matrices, linking SGD forces to portfolio dynamics.
result Spectral properties of SGD weight matrices transition between additive and multiplicative regimes, influencing wealth dynamics.