The paper connects higher order risk measures and stochastic dominance, showing their equivalence and integrating them with optimization.
problem Comparing and characterizing random outcomes in risk assessment.
method Exploring the equivalence between higher order risk measures and stochastic dominance, using stochastic optimization and expectiles as examples.
result Higher order risk measures and stochastic dominance are equivalent and can be used to characterize random outcomes.
Study compares hypergraph and graph-level models for higher-order relational learning.
problem Evaluating effectiveness of hypergraph-level vs. graph-level models in relational learning.
method Systematic evaluation of various hypergraph and graph-level architectures.
result Graph-level models applied to hypergraph expansions outperform hypergraph-level models.
H-GAT improves stock selection by capturing complex higher-order stock relations and integrating both technical and fundamental analysis.
problem Stock selection difficulty and lack of comprehensive analysis.
method Higher-order Graph Attention Network (H-GAT) that incorporates both technical and fundamental analysis.
result H-GAT outperforms existing methods in stock selection metrics.
Paper proves Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.
problem Proving relations between Reshetikhin-Turaev and re-normalized link invariants for links.
method Analyzing higher order terms of re-normalized link invariants for plumbed links.
result Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.
Combines neural networks and probabilistic graphical models for efficient higher-order inference.
problem Lack of efficient higher-order relational information in graph neural networks and probabilistic graphical models.
method Derives efficient approximate sum-product loopy belief propagation for higher-order PGMs, embeds into neural network, proposes methods for constructing higher-order factors.
result Substantially outperforms state-of-the-art k-order graph neural networks in molecular datasets.
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 consider higher-order linear-chain conditional random fields (HO-LC-CRFs) for sequence modelling, and use sum-product networks (SPNs) for representing higher-order input- and output-dependent factors. SPNs are a recently introduced class of deep models for which exact and efficient inference can be performed. By com…
The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.
problem Proving higher order trace inequalities on Poincaré-Einstein manifolds.
method Introducing conformally covariant boundary operators and using them to set up higher order Dirichlet problems.
result Sharp higher order Sobolev trace inequalities on the ball are obtained.
Higher-order tangent bundles have geometric structures compatible with their iterated bundle structure.
problem Connection towers and Sasaki metrics on higher-order tangent bundles
method Introduce the notion of a connection tower and study the geometric structures induced by such towers.
result Connection towers determine multiconnections, adapted splittings, and canonical vector bundle structures.
In this paper we derive the symplectic framework for field theories defined by higher-order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher-order system of PDEs to a constrained first-order one, the symplectic structures natura…
Advances combinatorial complexes for better modeling of hierarchical and set-type relations.
problem Lack of effective modeling for complex hierarchical and set-type relations in high-dimensional data.
method Introduces combinatorial complexes as a bridge between cell complexes and hypergraphs, emphasizing their different types of relations.
result Combining set-type and hierarchical relations in a single model can be advantageous in learning tasks.
Disproves Fedotov's conjecture on higher-order Shephard inequalities.
problem Fedotov's conjecture on higher-order Shephard inequalities.
method Using Hodge-Riemann relations for simple convex polytopes.
result Fedotov's conjecture is disproved.
We analyze oversquashing in topological message-passing using relational structures.
problem Oversquashing in topological message-passing remains understudied.
method A unifying axiomatic framework that bridges graph and topological message-passing.
result Potential to advance topological deep learning.
In this paper, we investigate the popular deep learning optimization routine, Adam, from the perspective of statistical moments. While Adam is an adaptive lower-order moment based (of the stochastic gradient) method, we propose an extension namely, HAdam, which uses higher order moments of the stochastic gradient. Our …
Topo-MLP learns network representations without message passing.
problem Lack of efficient higher-order network modeling methods.
method Proposes Topo-MLP, a simplicial neural network algorithm using MLP and HONC loss.
result Demonstrates improved robustness and efficiency in representation learning.
We present a geometric approach to the field theory with higher order anisotropic interactions. The concepts of higher order space, or locally anisotropic, space (in brief, h-space, or la-space) are introduced as general ones for various types of higher order extensions of Lagrange and Finsler geometry and higher dimen…
Recalls and refines the concept of algebraically rectifiable curves.
problem Classical notion of algebraically rectifiable plane curves.
method Provides new criteria, relates to quadratic differentials, and generalizes to higher order differentials.
result Generalization and new criteria for algebraic rectifiability.
Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this notion and other related ones to systems of higher-order differential equations …
Method provides formal guarantees for decomposing model uncertainty.
problem Decomposing model uncertainty into aleatoric and epistemic components.
method Higher-order calibration using k-snapshots.
result Formal guarantees for aleatoric uncertainty matching real-world distribution.
We show how to measure the failure of the Whitney trick in dimension 4 by constructing higher- order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previous…
LATTE tackles heterogeneous network embedding challenges with layer-stacked attention.
problem Aggregating higher-order indirect relations in heterogeneous networks.
method Layer-stacked ATTention Embedding (LATTE) that decomposes meta relations at each layer.
result LATTE achieves state-of-the-art performance on benchmark datasets.
Unified framework for higher-order network analysis.
problem Complex structure of space of networks.
method Measure-theoretic formalism, Gromov-Wasserstein distance, co-optimal transport distance.
result Unified theoretical treatment of generalized networks.
Enhances stock movement prediction using Higher Order Transformers for multimodal time-series data.
problem Predicting stock movements in financial markets with complex dynamics.
method Introduced Higher Order Transformers, extending self-attention and transformer architecture to capture complex market dynamics. Employed low-rank tensor decomposition and kernel attention to manage computational complexity. Integrated technical and fundamental analysis from historical prices and tweets.
result Demonstrated effectiveness of the method on the Stocknet dataset, improving stock movement prediction.
Self-supervised pretraining for heterogeneous hypergraphs improves graph-based tasks.
problem Capturing higher-order relations in heterogeneous hypergraphs.
method SPHH framework for self-supervised pretraining of heterogeneous HyperGNNs.
result SPHH consistently outperforms state-of-the-art baselines in various downstream tasks.
Mixes higher-order simplicial complexes for data augmentation.
problem Lack of labeled data for complex systems with multiway interactions.
method Proposes mixup mechanisms for simplicial complexes, including linear and nonlinear mixup, and a convex clustering mixup.
result Synthetic simplicial complexes interpolate between existing data based on homomorphism densities.
This paper simplifies computing higher-order U-statistics efficiently.
problem The inefficiency of computing higher-order U-statistics in practice. method Decomposition, connection to Einstein summation, and treewidth-based complexity estimate.
result A new, more efficient algorithm to compute U-statistics. Tutorials on signal processing on higher-order networks like simplicial complexes and hypergraphs.
problem Processing complex data structures with polyadic relationships.
method Introduction to simplicial complexes and hypergraphs, Fourier analysis, signal denoising, interpolation, embeddings, neural networks.
result Multi-relational operators like the Hodge Laplacian for simplicial complexes and tensor representations for hypergraphs.
New method for learning on heterogeneous graphs without meta-paths.
problem Learning on heterogeneous graphs is sensitive to meta-paths choice, leading to poor performance.
method Decompose heterogeneous graph into homogeneous relation-type graphs, combine higher-order representations, use attention mechanisms.
result Our model outperforms state-of-the-art baselines in vertex classification tasks on heterogeneous graph datasets.
Closed-form relations and approximations for SE(3) derivatives for robust numerical simulations.
problem Deriving closed-form derivatives and approximations for SE(3) for robust numerical simulations.
method Avoiding block partitioning, deriving higher-order approximations for differential, first and second derivatives, Jacobian, and Hessian.
result Compact and numerically robust closed-form relations for SE(3) derivatives.
A framework infers hyperedges and overlapping communities in hypergraphs.
problem Characterizing the structural organization of hypergraphs with higher-order interactions.
method Statistical inference to infer missing hyperedges and detect overlapping communities.
result Efficient numerical implementation and strong performance on real-world systems.
In recent years, graph neural networks (GNNs) have emerged as a powerful neural architecture to learn vector representations of nodes and graphs in a supervised, end-to-end fashion. Up to now, GNNs have only been evaluated empirically -- showing promising results. The following work investigates GNNs from a theoretical…
The set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk-Livingston-Wang's argument over the Gassner repres…
Flat surfaces that correspond to k-differentials on compact Riemann surfaces are of finite area provided there is no pole of order k or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat structures defined by a k-differential with at least one pole of order at least $k…
New method finds open subsets with trivial holonomy for certain geometries.
problem Finding open subsets with trivial holonomy for Cartan geometries.
method Analyzing the behavior of isotropies in model geometries to generalize properties of isolated higher-order fixed points.
result Existence of open subsets with trivial holonomy for Cartan geometries with certain isotropies.
TopoNTK kernel captures higher-order interactions in simplicial complexes.
problem Graph neural networks miss higher-order interactions in relational systems.
method Introduces TopoNTK, an infinite-width kernel for simplicial message passing.
result TopoNTK captures topology invisible to graph kernels, improving expressivity and interpretability.
The paper solves a 25-year-old problem about maximal growth distributions on manifolds.
problem Existence and classification of maximal growth distributions on smooth manifolds.
method Higher order convex integration and new criteria for ampleness of differential relations.
result Positive answer to the open question about parallelizable manifolds admitting maximal growth distributions.
Generative model for hypergraph clustering improves detection of higher-order structure.
problem Detecting clusters in complex relational systems modeled as hypergraphs.
method Poisson degree-corrected hypergraph stochastic blockmodel (DCHSBM) and Louvain-type algorithms.
result AON hypergraph Louvain algorithm efficiently detects higher-order structure in large hypergraphs.
We prove a lower bound for the k-th Steklov eigenvalues in terms of an isoperimetric constant called the k-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…
Study privacy vs. utility in estimating network parameters with aggregated data.
problem Privacy-preserving estimation of network parameters from aggregated node degrees.
method β model, local and central differential privacy, minimax lower bounds, simple estimators.
result Achieved minimax-optimal risk bounds for parameter estimation under privacy constraints.
In this paper we present a recurrent relation for counting meaningful compositions of the higher-order differential operations on the space Rn (n=3,4,...) and extract the non-trivial compositions of order higher than two.
Diffusion models learn simple statistics before complex ones, revealing a sample complexity exponent.
problem Understanding the learning dynamics of diffusion models.
method Empirical observations and theoretical analysis of diffusion models and denoisers.
result Diffusion models learn simple statistics (pair-wise correlations) at linear sample complexity, while higher-order statistics (e.g., fourth cumulant) require cubic sample complexity.
We study higher-order conservation laws of the non-linearizable elliptic Poisson equation ∂z∂zˉ∂2u=−f(u) as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for diffe…
We give two applications of the 2-Engel relation, classically studied in finite and Lie groups, to the 4-dimensional topological surgery conjecture. The A-B slice problem, a reformulation of the surgery conjecture for free groups, is shown to admit a homotopy solution. We also exhibit a new collection of universal surg…
Paper characterizes equilibrium strategies for stochastic control with higher-order moments.
problem Stochastic control problems with higher-order moments.
method Novel characterization of time-consistent control problems, deriving equilibrium conditions via BSDEs.
result Derives sufficient and necessary conditions for an open-loop Nash equilibrium control (ONEC) in a novel way.
Derives formulae for general permutation equivariant layers and presents a second order graph variational encoder.
problem Tackles the limitation of previous equivariant neural networks by considering permutations of matrices.
method Derives formulae for general permutation equivariant layers, including matrix permutations. Presents a second order graph variational encoder.
result Latent distribution of equivariant generative models must be exchangeable.
In this paper, we present a Lichnerowicz type estimate and (higher order) Buser type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential. These results relate eigenvalues, magnetic fields, Ricci curvature, and Cheeger type constants.
The paper extends Gaussian processes to model complex interactions in cellular complexes.
problem Capturing topological inductive biases in machine learning models.
method Proposes Gaussian processes on cellular complexes, introducing novel kernels.
result Derives two novel kernels for modeling interactions between cells.
BScNets expands graph learning to higher-order interactions.
problem Link prediction on graphs with higher-order interactions.
method Proposes a new SNN using block Hodge-Laplacian and block simplicial complexes.
result BScNets outperforms state-of-the-art models in link prediction.