Groups with specific properties have similar cubulations and coarse median structures.
problem Understanding the structure of certain groups through cubical coarsening.
method Analyzing right-angled Artin and Coxeter groups, focusing on automorphisms and cubulations.
result Automorphisms of specific groups preserve coarse median structures and have nice fixed subgroups.
New graph coarsening method preserves GNN message-passing signals.
problem Preserving GNN message-passing on coarsened graphs.
method Proposed a new oriented message-passing operation for coarsened graphs.
result Improved node classification results compared to naive methods.
Unified framework for graph coarsening using node features and graph matrices.
problem Dimensionality reduction of large graphs while preserving node features.
method Optimization-based framework that unifies graph learning and dimensionality reduction.
result The learned coarsened graph is ε-similar to the original graph, where ε is a small positive number.
We develop a method to summarize causal models with cycles in cubic time.
problem Cycles in high-dimensional causal models limit applicability of existing methods.
method We relax the acyclicity assumption in LiNG models and develop a low-dimensional DAG summary.
result Our method allows recovery of a low-dimensional DAG from high-dimensional data with cycles.
Paper introduces a taxonomy of reduction matrices for more efficient graph coarsening.
problem Efficiently reducing graph size while preserving important information.
method Introduces a more general notion of reduction matrix, not necessarily the pseudo-inverse of the lifting matrix.
result Reducing the Restricted Spectral Approximation (RSA) by modifying the reduction matrix.
Efficiently computes embeddings for large graphs using coarsening.
problem Inefficient computation of graph embeddings for large-scale graphs.
method Graph coarsening based on Schur complements and Gaussian elimination.
result Efficiently computed embeddings on coarsened graph match Schur complement embeddings in expectation.
How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to depend on standard graph-theoretic properties, such as the degree and eigenvalue di…
A new unsupervised method learns graph hierarchies using optimal transport.
problem Learning meaningful graph hierarchies without labeled data.
method Differentiable coarsening and optimal transport.
result OTCoarsening produces meaningful coarse graphs and competitive performance.
This study improves graph coarsening methods by preserving graph spectrum and distances.
problem Solving large-scale graph problems by working on a smaller graph.
method Developed a geometric approach using Gromov--Wasserstein distance to minimize the difference between graph distances and their coarsened versions.
result Minimizing the difference between graph distances and their coarsened versions can be achieved using the weighted kernel K-means method. Study intersection homotopy groups in coarsenings of CS sets.
problem Understanding invariance of intersection homotopy groups in coarsenings of CS sets.
method Introduced a general perversity and its pushforward, established invariance theorems for intersection homotopy groups in coarsenings of CS sets.
result Found invariance theorems for intersection homotopy groups in coarsenings of CS sets.
Paper improves GNN inference speed and memory usage.
problem Scalability issues in GNNs during inference.
method Graph coarsening techniques for faster inference.
result Significant reduction in inference time and memory usage.
New estimators improve efficiency in two-phase designs with coarsened data.
problem Efficient estimation in two-phase designs with incomplete data.
method Developed new estimators within the TMLE framework.
result New estimators are asymptotically equivalent and more efficient.
A novel algorithm for unsupervised graph representation learning combining coarsening and mutual information maximization.
problem Current limitations in unsupervised graph representation learning, especially in embedding new graphs and considering both micro- and macro-structures.
method Combines coarsening with mutual information maximization to produce high-quality embeddings.
result The algorithm produces high-quality embeddings that are competitive with state-of-the-art methods.
We consider the problem of statistical inference for ranking data, specifically rank aggregation, under the assumption that samples are incomplete in the sense of not comprising all choice alternatives. In contrast to most existing methods, we explicitly model the process of turning a full ranking into an incomplete on…
New method reduces spatial graphs while preserving their topological features.
problem Finding a smaller spatial graph with the same structure.
method Topological spatial graph coarsening approach based on triangle-aware graph filtration.
result Significant reduction in graph size while preserving topological information.
Improved graph clustering with modularity and coarsening for attributes and communities.
problem Inaccurate community detection and computational inefficiency in graph clustering.
method Integrates coarsening and modularity maximization, using a loss function with log-determinant, smoothness, and modularity components.
result Superior clustering outcomes, proven consistent under DC-SBM, and efficient algorithm integration with GNNs and VGAEs.
PASCO speeds up graph clustering for large graphs.
problem Efficiently clustering large graphs with many communities.
method Overlay method combining coarsening and parallel clustering.
result PASCO accelerates clustering with improved efficiency and quality.
New method for valid prediction intervals with coarsened data.
problem Handling missing data and censored outcomes in training samples.
method Multiply robust conformal risk control with semiparametric theory.
result Stronger coverage properties under covariate shift.
NDP improves GNN efficiency by coarsening graphs without losing structure.
problem Efficiently summarize graph data for deep learning models.
method Node Decimation Pooling (NDP) reduces graph density while preserving topology.
result NDP achieves comparable performance to state-of-the-art pooling methods but with improved efficiency.
CoSimGNN improves graph similarity computation for large graphs.
problem Efficiently computing graph similarity scores for large graphs.
method Embedding-coarsening-matching framework with adaptive pooling and fine-grained interactions.
result CoSimGNN achieves best performance in graph similarity computation.
Paper uses Shapley values to identify key confounders in product funnel data.
problem Identifying important confounders in product funnel data.
method Applies Shapley values for scalable coarsened exact matching.
result Shapley values provide robust importance-ranking for confounders.
New methods for handling confounding in observational studies.
problem Handling confounding variables in observational studies.
method Generalized coarsened procedures for clustering confounding variables, followed by estimation of treatment effects and variance.
result Developed a general asymptotic framework for the average causal effect estimator and variance formulae.
Study recovers community structure from coarse graph measurements.
problem Community recovery from low-resolution graph measurements.
method Formalized coarsening process of graph measurements, developed conditions for perfect recovery.
result Simple and closed-form asymptotic conditions for perfect recovery of coarse graph communities.
This thesis explores GNNs, categorizing them into local and global approaches.
problem Understanding the convergence of global GNNs and connecting local and global approaches.
method Categorization of GNNs into local and global, study of Invariant Graph Networks, connecting local and global approaches, and using local MPNN for graph coarsening.
result Established a connection between local and global GNN approaches.
New algorithm tackles self-selection bias in estimating linear regressors.
problem Estimating k linear regressors with self-selection bias in d dimensions. method First local convergence algorithm for self-selection, reducing to coarsening problem.
result Improves running time of previous algorithms by a poly(d, k, 1/ε) factor.
Paper creates transparent, safe synthetic data from coarsened margins.
problem Creating synthetic data that maintains original relationships and is safe from disclosure.
method Defining and curating margins, applying SDC, coarsening counts, and using IPF algorithm.
result Synthetic data derived from safe, coarsened margins maintains original relationships.
GraphZoom improves graph embedding accuracy and scalability.
problem Node attribute noise and scalability issues in graph embedding models.
method GraphZoom combines graph fusion and multi-level coarsening to improve accuracy and scalability.
result GraphZoom significantly increases classification accuracy and speeds up the embedding process.
Efficient memory layer improves graph neural networks for graph classification and regression.
problem Efficiently learning node representations and graph coarsening for arbitrary graph topology.
method Introduces a memory layer for GNNs that learns node representations and graph coarsening, and two new networks: MemGNN and GMN.
result Proposed models achieve state-of-the-art results in graph classification and regression benchmarks.
{\em Riemannian cubics} are curves in a manifold M that satisfy a variational condition appropriate for interpolation problems. When M is the rotation group SO(3), Riemannian cubics are track-summands of {\em Riemannian cubic splines}, used for motion planning of rigid bodies. Partial integrability results are know…
Computes the monodromy of cubic surfaces branching over smooth cubic curves.
problem Understanding the monodromy of cubic surfaces.
method Computational approach using the relationship between inflection points and lines on cubic surfaces.
result The monodromy map is surjective onto the centralizer of the image of a generator of the deck group.
Sub-Riemannian cubics are a generalisation of Riemannian cubics to a sub-Riemannian manifold. Cubics are curves which minimise the integral of the norm squared of the covariant acceleration. Sub-Riemannian cubics are cubics which are restricted to move in a horizontal subspace of the tangent space. When the sub-Riemann…
New framework estimates target functions from incomplete data.
problem Estimating target functions from partially observed data.
method IF-learning framework using influence functions.
result Two learning algorithms developed for estimation.
In rank aggregation (RA), a collection of preferences from different users are summarized into a total order under the assumption of homogeneity of users. Model misspecification in RA arises since the homogeneity assumption fails to be satisfied in the complex real-world situation. Existing robust RAs usually resort to…
The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain t…
New method reduces confidence interval sizes for causal inference.
problem Inaccurate propensity scores and extreme scores cause large confidence intervals.
method Data-dependent Coarse IPW (CIPW) estimators.
result Robust CIPW estimators reduce confidence interval sizes to ε+1/√n.
This paper refines homotopy theory for cubical sets and uniform spaces.
problem Classical homotopy theory limitations in cubical sets and uniform spaces.
method Develops a uniform-theoretic refinement for cubical sets and uniform spaces, lifting to a full and faithful embedding.
result Lifts classical homotopy categories to new uniform homotopy categories, generalizing cohomology theories.
According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and to obtain a pure deformati…
The paper studies conformally flat cubic metrics with isotropic curvature, finding they must be Minkowski.
problem Understanding conformal properties of cubic metrics with isotropic scalar curvature.
method Analyzing the conformal flatness and isotropic scalar curvature of cubic metrics.
result Cubic metrics with weakly isotropic scalar curvature must be Minkowski metrics.
Cubic fourfolds have K-stability and admit Kähler-Einstein metrics.
problem Understanding K-stability and Kähler-Einstein metrics for cubic fourfolds.
method Local volume estimates and Ambro-Kawamata's non-vanishing theorem.
result All smooth cubic fourfolds admit Kähler-Einstein metrics.
The study shows that certain cubical presentations lead to aspherical spaces.
problem Understanding the asphericity of cubical presentations in 2D.
method Analyzing the second homotopy group of coned-off spaces associated with cubical presentations.
result The coned-off space is aspherical under specific conditions.
New cubic forms linked to η-invariants and mod 2 indices.
problem Anomaly cancellation and modularity in 12-manifolds.
method Combination of Witten classes and affine E8 character. result Relates cubic forms to η-invariants and mod 2 indices.
It is shown that there exist non-singular cubic surfaces in CP^3 containing 5 twistor lines. This is the maximum number of twistor fibres that a non-singular cubic can contain. Cubic surfaces in CP^3 with 5 twistor lines are classified up to transformations preserving the conformal structure of S^4.
Dynamical systems with large state-spaces are often expensive to thoroughly explore experimentally. Coarse-graining methods aim to define simpler systems which are more amenable to analysis and exploration; most current methods, however, focus on a priori state aggregation based on similarities in transition rates, whi…
We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kahler-Einstein metric on two singular cubic surfaces.
Solves infinite family of cubic polynomial problems.
problem Infinite family of twisted polynomial problems.
method Using Dehn twists and 9-adic expansions.
result Result of twisting depends on 9-adic expansion.
Graph convolutional networks (GCNs) have been successfully applied in node classification tasks of network mining. However, most of these models based on neighborhood aggregation are usually shallow and lack the "graph pooling" mechanism, which prevents the model from obtaining adequate global information. In order to …
Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with norm…
A natural family of affine cubic surfaces arises from SL(2)-characters of the 4-holed sphere and the 1-holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic s…