Graph neural networks are shown to be as powerful as a graph isomorphism heuristic, leading to a new approach for higher-order graph structures.
problem Understanding and distinguishing non-isomorphic graphs and their higher-order structures.
method Relating GNNs to the Weisfeiler-Leman heuristic and proposing k-dimensional GNNs. result GNNs have the same expressiveness as the Weisfeiler-Leman heuristic in distinguishing graphs and their higher-order structures.
Introduces fractional k-dimensional measure bridging fractional length and area.
problem Defining fractional measures for dimensions between 0 and n-1.
method Introduces a parameterized fractional measure σ that converges to Hausdorff measure. result Fractional measure converges to Hausdorff measure with a known constant factor.
Study growth of systoles in arithmetic manifolds, focusing on k-dimensional cases.
problem Growth of systoles in arithmetic n-manifolds along congruence coverings. method Analyzes growth of k-dimensional systoles in arithmetic n-manifolds, proving polylogarithmic and constant power bounds. result Growth of systoles for k=r oscillates between a power of a logarithm and a power function of the degree of the covering. This paper examines the category C^k_{d,n} whose morphisms are d-dimensional smooth manifolds that are properly embedded in the product of a k-dimensional cube with an (d+n-k)-dimensional Euclidean space. There are k directions to compose k-dimensional cubes, so C^k_{d,n} is a (strict) k-tuple category. The geometric r…
First variation of fractional k-dimensional measure for submanifolds
problem Computing the first variation of a fractional k-dimensional measure for submanifolds method First variation computation
result Definition of a nonlocal mean-curvature vector for embedded submanifolds
The paper studies the topology of hyperspaces of k-dimensional convex sets.
problem Topology of hyperspaces of k-dimensional closed convex sets.
method Proved that hyperspaces are Hilbert cube manifolds with fiber bundle structure over Grassmann manifold.
result Fiber of Kk,bn is homeomorphic with R2k(k+1)+2nimesQ. The k-dimensional coding schemes refer to a collection of methods that attempt to represent data using a set of representative k-dimensional vectors, and include non-negative matrix factorization, dictionary learning, sparse coding, k-means clustering and vector quantization as special cases. Previous generalizat…
In 1972, Marcel Berger defined a metric invariant that captures the `size' of k-dimensional homology of a Riemannian manifold. This invariant came to be called the k-dimensional SYSTOLE. He asked if the systoles can be constrained by the volume, in the spirit of the 1949 theorem of C. Loewner. We construct metrics, ins…
Eigen-GNN enhances GNNs by preserving graph structures.
problem Existing shallow GNNs fail to effectively preserve graph structures.
method Integrates eigenspace of graph structures into GNNs as a dimensionality reduction module.
result Eigen-GNN boosts GNNs' ability to preserve graph structures without increasing depth.
A manifold is locally \emph{k-fold symmetric}, if for any point and any k-dimensional vector subspace tangent to this point there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that k-dimensional vector subspace is minus the identity. We show that …
In hyperbolic space Hn we set a geodesic ball of radius ρ. Consider a k dimensional minimal submanifold passing through the origin of the geodesic ball with boundary lies on the boundary of that geodesic ball. We prove that its area is no less than the totally geodesic k dimensional submanifold passing through…
Let Σ be a k-dimensional minimal submanifold in the n-dimensional unit ball Bn which passes through a point y∈Bn and satisfies ∂Σ⊂∂Bn. We show that the k-dimensional area of Σ is bounded from below by ∣Bk∣(1−∣y∣2)2k. This settles a question left open by …
The paper deals with amoebas of k-dimensional algebraic varieties in the algebraic complex torus of dimension n≥2k. First, we show that the area of complex algebraic curve amoebas is finite. Moreover, we give an estimate of this area in the rational curve case in terms of the degree of the rational parametrizat…
We prove that Dranishnikov's k-dimensional resolution dk:μk→Q is a UVn−1-divider of Chigogidze's k-dimensional resolution ck. This fact implies that dk−1 preserves Z-sets. A further development of the concept of UVn−1-dividers permits us to find sufficient conditions for $d_k^{-1}(…
SHAKE-GNN scales GNNs for large graphs with multi-scale representations.
problem Scaling Graph Neural Networks (GNNs) to large graphs.
method SHAKE-GNN uses a hierarchy of Kirchhoff Forests for stochastic multi-resolution graph decompositions.
result SHAKE-GNN achieves competitive performance on large-scale graph classification benchmarks.
GPT-GNN pre-trains GNNs on unlabeled graphs to improve downstream performance.
problem Training GNNs requires labeled data, which is expensive.
method Generative pre-training of GNNs on unlabeled data with self-supervision.
result GPT-GNN significantly outperforms state-of-the-art GNNs without pre-training.
LC-GNN improves GNNs for node classification by incorporating label consistency.
problem Limited performance of GNNs due to label consistency assumption not always holding.
method LC-GNN uses node pairs with the same label but unconnected to expand GNN's receptive field.
result LC-GNN outperforms traditional GNNs in semi-supervised node classification.
New insights into GNN optimization reveal skip connections and depth accelerate training.
problem Understanding and optimizing the training of Graph Neural Networks (GNNs).
method Analysis of gradient dynamics in linearized GNNs and empirical validation.
result GNNs are implicitly accelerated by skip connections, more depth, and good label distribution during training.
Survey on GNNs' power and limitations.
problem Theoretical limitations of GNNs.
method Comprehensive overview of GNNs and their variants.
result Provably powerful variants of GNNs.
Riemannian manifolds with bounded Ricci curvature have finite Uryson width.
problem Bounding Uryson width for manifolds with Ricci curvature constraints.
method Continuous map to a polyhedral space with controlled diameter.
result Riemannian manifolds with specific Ricci curvature bounds have finite Uryson width.
GNNs with random node initialization are shown to be universally expressive.
problem Limitations of standard GNNs in distinguishing graphs.
method Random node initialization (RNI) to enhance GNNs' expressive power.
result GNNs with RNI are proven to be universally expressive.
Unified framework for adaptive connection sampling in GNNs improves performance and robustness.
problem Over-smoothing and over-fitting in deep GNNs.
method Adaptive connection sampling trained jointly with GNN model parameters.
result Adaptive connection sampling mathematically equivalent to Bayesian GNNs approximation.
Hardness proven for embedding simplicial complexes in R^d, especially for k-dimensional ones.
problem Recognizing almost embeddability of k-dimensional complexes in R^d.
method NP-hardness proof using configuration spaces and preimage cycle properties.
result Embedding obstruction is incomplete for k-dimensional complexes in R^d.
New algorithms reduce communication in GNN training.
problem Higher communication costs in GNNs due to sparse connectivity.
method Parallel algorithms for sparse-dense matrix multiplication.
result Asymptotic reduction in communication compared to previous methods.
DefNet defends GNNs against adversarial attacks by identifying and mitigating vulnerabilities.
problem Vulnerability of GNNs to adversarial attacks.
method Investigates latent vulnerabilities in GNN layers, proposes dual-stage aggregation and bottleneck perceptron, and uses adversarial contrastive learning for training.
result DefNet significantly improves GNN robustness under various adversarial attacks.
New approach to k-dimensional torus differential equations.
problem Generalizing Poincaré's results for higher-order equations on a torus.
method Non-Hamiltonian approach using continuous vector functions.
result New results even in the k=1 case. ADMP-GNN dynamically adjusts message-passing layers for better graph learning performance.
problem Fixed message-passing steps in GNNs do not account for nodes' varying computational needs.
method Proposes ADMP-GNN, which dynamically adjusts the number of message-passing layers for each node.
result Improves performance on node classification tasks compared to baseline GNN models.
Adding random features to GNNs improves their performance.
problem Limitations of GNNs in distinguishing graphs and learning efficient algorithms.
method Adding random features to each node in GNNs.
result Random features enable GNNs to learn optimal algorithms for graph problems.
CI-GNN uses GNNs to diagnose psychiatric disorders by identifying causally relevant brain regions.
problem Leveraging GNNs for psychiatric diagnosis requires interpretable models to understand decision-making.
method CI-GNN integrates Granger causality into GNNs to identify causally relevant subgraphs.
result CI-GNN provides more reliable and concise explanations of psychiatric diagnoses.
This Ph.D. thesis is devoted to the constructions of Lagrangian formulation on Finsler and Kawaguchi manifolds. While Finsler geometry is a natural extension of Riemannian geometry, Kawaguchi geometry is the extension of Finsler geometry to higher order derivatives and to k-dimensional parameter space. The latter exten…
We prove that any asymptotically locally Euclidean scalar-flat Kähler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a k-dimensional quaternionic vector space by a (k−1)-torus. In order to do so, we first prove that any compact anti…
P-GNNs learn node embeddings considering node positions in graphs.
problem Capturing node positions in graph structures.
method Samples anchor nodes, computes distances, and learns weighted aggregation.
result P-GNNs outperform state-of-the-art GNNs in link prediction and community detection.
UM-GNN improves GNN robustness against poisoning attacks.
problem Vulnerability of GNNs to poisoning attacks.
method UM-GNN uses epistemic uncertainties from message passing to build a surrogate predictor.
result UM-GNN achieves significantly improved robustness against poisoning attacks.
TF-GNN simplifies graph neural networks in TensorFlow.
problem Handling rich heterogeneous graph data in machine learning.
method A scalable library with a Keras message passing API.
result Enables low-code solutions for broader developers.
Paper tackles fairness issues in GNNs by proposing ELEGANT for certification.
problem Fairness issues in GNN predictions due to graph data perturbations.
method Proposes ELEGANT framework for certifying fairness of any GNN without assumptions or re-training.
result The fairness of any GNN backbone is impossible to be corrupted under certain perturbation budgets.
We show that for closed orientable manifolds the k-dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree k that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles…
Simplified NAS for GNN architectures improves efficiency and expressiveness.
problem Efficient and effective discovery of optimal GNN architectures.
method SNAG framework with a novel search space and reinforcement learning.
result SNAG framework outperforms human-designed and existing NAS methods.
A new method for training GNNs without a teacher model.
problem Training over-parameterized GNN models is difficult and inefficient.
method GNN Self-Distillation (GNN-SD) with NDR and ADR.
result Improves GNN performance with less training cost and better generalization.
GNNs generalize CNNs for graph data, showing equivariance and stability.
problem Processing signals on graphs.
method Graph convolutional filters, nonlinearities, stacked layers.
result GNNs converge to graphon neural networks under graph convergence.
Researchers explore statistical perspectives to understand GNN generalization.
problem Limited mathematical understanding of GNN performance.
method Three broad frameworks: learning theory, asymptotics, and random graph models.
result Various theoretical results and open questions identified.
Tail-GNNs improve protein function prediction using relational reinforcement.
problem Predicting hierarchical protein functions from sequence data.
method Combining Tail-GNNs with dilated convolutional networks for multi-task learning.
result Significant improvement in F_1 score for protein function prediction.
GraphNorm accelerates GNN training by adapting InstanceNorm, improving convergence and generalization.
problem Improving convergence and generalization of Graph Neural Networks (GNNs).
method Adapting InstanceNorm to GNNs, proposing GraphNorm with a learnable shift.
result GNNs with GraphNorm converge faster and achieve better performance on benchmarks.
Boost GNNs for node classification by incorporating label dependencies.
problem Current GNNs lack expressiveness and fail to capture label dependencies.
method Proposes a collective learning framework combining collective classification and self-supervised learning.
result Consistent, significant improvement in node classification accuracy across various GNNs.
AGNN automates GNN architecture search, achieving best performance.
problem Finding optimal GNN architectures is laborious and requires human expertise.
method AGNN uses reinforcement learning to search for optimal GNN architectures within a predefined space, with a novel parameter sharing strategy.
result AGNN identifies optimal GNN architectures achieving best performance.
SGQuant reduces GNN memory usage without significant accuracy loss.
problem High memory consumption in GNNs limits their applicability on memory-constrained devices.
method Proposes a specialized GNN quantization scheme (SGQuant) with a quantization algorithm, fine-tuning scheme, and multi-granularity strategy.
result SGQuant reduces GNN memory footprint from 4.25x to 31.9x with minimal accuracy loss.
New research limits what GNNs can compute and generalizes their performance.
problem Limits of GNNs in computing graph properties and generalization bounds.
method Novel graph-theoretic formalism and data-dependent generalization bounds.
result Proves GNNs can't compute certain graph properties and provides tighter generalization bounds.
PA-GNN enhances GNN robustness against poisoning attacks using clean graph knowledge.
problem Improving robustness of GNNs against poisoning attacks.
method PA-GNN uses a penalized aggregation mechanism and meta-optimization to transfer robustness from clean graphs.
result PA-GNN significantly improves GNN robustness against poisoning attacks on real-world graphs.
Alt-GNNs improve travel mode choice modeling by integrating graph neural networks with GEV models.
problem Capturing alternative dependence in discrete choice models with predefined, symmetric, and uniform dependence.
method Introducing Alternative Graph Neural Networks (Alt-GNNs) that embed alternative dependence within a unified framework.
result Alt-GNNs significantly improve predictive performance over benchmark models in travel mode choice datasets.