New kernels for permutations improve accuracy in ranking tasks.
problem Improving ranking accuracy in permutation-based tasks.
method Introduced weighted Kendall kernel, supervised learning for weights, and higher-order permutation kernels.
result Supervised learning of weights enhances kernel performance for permutation tasks.
In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some esti…
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.
The study solves a problem in conformal geometry with applications to Q-curvature.
problem Existence of solutions to conformally invariant equations.
method Volume comparison theorems and volume rigidity theorems with respect to Q-curvature.
result Sufficient and necessary conditions for the existence of solutions to conformally invariant equations.
In this note we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we make a quick comparison with the notion of a curve presented here are other commo…
Develops a fast, accurate method for comparing networks.
problem Comparing networks with repeated observations and varying sizes/sparsity.
method A novel two-sample hypothesis testing method with theoretical guarantees.
result Outperforms existing tools in speed and accuracy, power-optimal.
New Gini indices capture more nuanced income inequality.
problem Measuring joint dispersion across multiple observations.
method Axiomatic approach to define and characterize n-th order Gini deviations.
result Higher-order Gini coefficients reveal more extreme income disparities.
Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.
problem Establishing inequalities for radial functions on hyperbolic spaces without zero boundary conditions.
method Novel approach considering both bounded and unbounded domains, focusing on weighted Sobolev and Adams-Trudinger-Moser embeddings.
result Theorems 1.2, 1.3, and 1.4 for weighted Sobolev embedding theorems, and Theorems 1.5 and 1.6 for Adams-Trudinger-Moser type embedding theorems.
Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow \cite{KZ:PAMI04} has had significant impact in computer vision \cite{BVZ:PAMI01,Kwatra:SIGGRAPH03,Rother:GrabCut04}. In this paper we address the important class of sum-of-submodular (SoS) function…
New stability conditions identified from quadratic differentials on surfaces.
problem Identifying stability conditions from quadratic differentials.
method Comparison of exchange graphs from tilting hearts and flipping mixed angulations.
result Spaces of stability conditions identified with moduli spaces of quadratic differentials.
Introduces neural point-forms for learning geometric features from noisy point clouds.
problem Learning geometric features from noisy point clouds with missing tangency information.
method Uses Laplacian-based techniques to build comparison matrices for point clouds, proving consistency under various assumptions.
result Neural point-forms provide a competitive and interpretable representation, especially beneficial for dense or manifold-like structures.
The paper establishes inequalities for convex curves and applies them to lattice point estimates.
problem Estimating the number of lattice points on convex curves.
method Developed comparison theorems for affine curves and used them to estimate areas and lattice points.
result Established inequalities for areas of inscribed triangles in terms of affine curvature and distance.
We study the geometry of jets of submanifolds with special interest in the relationship with the calculus of variations. We give a new proof of the fact that higher order jets of submanifolds are affine bundles; as a by-product we obtain a new expression for the associated vector bundles. We use Green-Vinogradov formul…
Proves limitations of higher-order optimization for convex problems.
problem Limitations of higher-order optimization methods for convex problems.
method Proves polynomial dependence on approximation guarantee and higher-order smoothness parameters.
result Nesterov's accelerated cubic regularization method is nearly tight.
Paper compares optimization methods for sparse NCP decomposition of tensors.
problem Efficiently extract meaningful nonnegative and sparse components from tensors.
method Sparse NCP decomposition with l1-norm regularization and block coordinate descent.
result Comparison of optimization methods for tensor decomposition effectiveness and speed.
Graphs improve theorem proving in higher-order logic.
problem Challenges in converting higher-order logic formulas into graph-based representations.
method Used graph neural networks (GNNs) to represent and search higher-order logic.
result GNNs outperform state-of-the-art methods in higher-order theorem proving.
Paper introduces techniques to learn higher-order programs, improving predictive accuracy and reducing learning times.
problem Expressing and learning complex programs in ILP.
method Extending meta-interpretive learning to support higher-order definitions as background knowledge.
result Learning higher-order programs reduces hypothesis space and sample complexity, improving predictive accuracy and reducing learning times.
For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann ρ-invariants, which we call higher-order signatures. The higher-order genera o…
The paper proves sampling methods using discrete-time processes and information theory.
problem Proving convergence guarantees for diffusion-based sampling methods.
method Directly works with discrete-time stochastic processes and uses information theory.
result Discrepancy between sampling and comparison processes is bounded using information theory.
A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering coefficient, which is the probability that a length-2 path is closed, i.e., induces a triangle in the network. However, higher-order cliques beyond triangles are crucia…
HONE learns higher-order network embeddings from graph data.
problem Capturing higher-order structures in network data.
method HONE framework based on network motifs, with interchangeable components.
result HONE outperforms other embedding methods by up to 75% in AUC.
Stability of capillary hypersurfaces with higher order mean curvature.
problem Stability of capillary hypersurfaces with constant higher order mean curvature.
method Generalization of classical stability theory for capillary hypersurfaces.
result Results on stability for capillary hypersurfaces with higher order mean curvature.
A new numerical framework simplifies elastic surface matching and comparison.
problem Challenging problem in surface comparison and matching in computer vision.
method Relaxing the geodesic boundary constraint using a varifold fidelity metric.
result Flexibility to deal with arbitrary topologies and sampling patterns, scalability to large meshes.
New methods for faster ranking and link prediction using higher-order motifs.
problem Real-time ranking and link prediction in applications like web search.
method Higher-order ranking and link prediction methods based on closing higher-order network motifs.
result The methods are faster and more efficient than existing methods based on closing triangles.
The paper improves CR Sobolev inequalities and classifies minimizers.
problem Higher-order CR Sobolev inequalities on the CR sphere.
method Improvement through vanishing higher order moments of the volume element.
result New direct proof of minimizers' classification and existence of minimizers in C2k(N). In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We present the corresponding Tulczyjew triple for this higher order situation and derive in this framework the phase equations fro…
Sharp convergence analysis for nonconvex regression models.
problem Nonconvex optimization in regression models with normally distributed covariates.
method Gaussian comparison theorems for analyzing iterative algorithms.
result Sharp global convergence rates for various statistical models.
A new method predicts higher-order interactions in evolving graphs using simplicial complexes.
problem Predicting higher-order interactions in dynamic graphs with theoretical guarantees.
method Capturing higher-order interactions as simplices, modeling neighborhoods with face-vectors, and developing a nonparametric kernel estimator.
result Our method outperforms existing higher-order prediction methods and is theoretically consistent.
The paper glosses different forms of an introducing of higher order tangent-like functors, especially functors derived from higher order nonholonomic tangent functors. A special attention is devoted to higher order osculating bundles: their identification with higher order tangent bundles is demonstrated as the main re…
Novel higher-order group synchronization for noisy local measurements on hypergraphs.
problem Synchronizing higher-order local measurements on hyperedges to global estimates on nodes.
method Message passing algorithm for global synchronization of higher-order measurements.
result Higher-order method outperforms standard pairwise synchronization methods in certain applications.
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.
Study higher-order interactions in networks, proposing link prediction as a new benchmark.
problem Understanding higher-order interactions in complex systems.
method Temporal analysis of 19 datasets, focusing on interactions involving more than two nodes.
result Higher-order interactions are consistent across different systems, with tie strength and edge density influencing their presence.
In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beiträge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the synthetic approach to jet bundles in terms of infinitesimal spaces Dⁿ's. In our succeeding [Synthetic differential geo…
We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss t…
A new GCN model learns higher-order neighbors without explicit adjacency matrix computation.
problem GCN's performance drops for deeper structures due to limited neighborhood information.
method Assumes higher-order neighbors are similar to first-order neighbors, learns weights through Lasso to minimize feature loss.
result HWGCN achieves state-of-the-art results on various datasets.
Study on bias-variance trade-off in hierarchical models with higher-order interactions.
problem Understanding the bias-variance trade-off in hierarchical probabilistic models with higher-order interactions.
method Proposed an efficient inference algorithm using Gibbs sampling and annealed importance sampling for log-linear higher-order Boltzmann machine.
result Higher-order interactions produce less variance for smaller sample size and comparable error with hidden layers.
We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type (r,s) in a vector space of signature (p,q). We then use these examples to establish some results concerning higher order Osserman and highe…
After defining cohomologically higher order BRST and anti-BRST operators for a compact simple algebra {\cal G}, the associated higher order Laplacians are introduced and the corresponding supersymmetry algebra Σ is analysed. These operators act on the states generated by a set of fermionic ghost fields transforming u…
HONEM learns embeddings for higher-order networks, improving performance in various tasks.
problem Existing methods fail to capture non-Markovian higher-order dependencies in networks.
method HONEM is a higher-order network embedding method designed for HON, capturing non-Markovian dependencies.
result HONEM outperforms other methods in node classification, network reconstruction, link prediction, and visualization.
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.
Given a closed, oriented, connected 3-manifold, M, we define higher-order linking forms on the higher-order Alexander modules of M. These higher-order linking forms generalize similar linking forms for knots previously studied by the author, which were themselves generalizations of the classical Blanchfield linking for…
The paper studies higher order Dirichlet-to-Neumann maps on graphs and their eigenvalues.
problem Analyzing eigenvalues of higher order Dirichlet-to-Neumann maps on graphs.
method Introducing and studying higher order Dirichlet-to-Neumann maps on graphs, deriving estimates on eigenvalues.
result Raulot-Savo-type estimates on the eigenvalues of the DtN maps.
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.
New estimator stabilizes higher-order influence functions for stable statistical inference.
problem Numerical instability in estimating inverse population Gram matrix.
method Proposes a new stabilized higher-order estimator without sample splitting.
result Stabilized estimator exhibits more stable performance and similar statistical guarantees.
This work explores higher-order algebroids via vector bundle comorphisms.
problem Generalizing concepts of higher-order tangent bundles and Lie algebroids.
method Introduces a vector bundle comorphism approach to describe higher-order algebroids.
result Establishes a one-to-one correspondence between higher-order Lie algebroids and specific algebraic structures.
New estimator stabilizes higher-order influence functions for bilinear forms.
problem Stability issues in estimating bilinear forms using higher-order influence functions.
method Proposes a new stabilized higher-order estimator for a class of bilinear forms without sample splitting.
result New estimator exhibits more stable finite-sample performance compared to the empirical higher-order estimator.
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…
Enhanced Adam uses higher-order moments for better performance.
problem Improving the performance of Adam optimization algorithm.
method Proposes HAdam, an extension of Adam using higher-order moments of the stochastic gradient.
result Higher-order moments of the stochastic gradient can lead to better performance than vanilla Adam.