Related papers: Lean Algebraic Multigrid (LAMG): Fast Graph Laplac…

Algebraic multigrid (AMG) is conventionally applied in a black-box fashion, agnostic to the underlying geometry. In this work, we propose that using geometric information -- when available -- to assist with setting up the AMG hierarchy is…

Embedding parameterized optimization problems as layers into machine learning architectures serves as a powerful inductive bias. Training such architectures with stochastic gradient descent requires care, as degenerate derivatives of the…

Graph Neural Networks (GNNs) have emerged as a dominant paradigm for graph classification. Specifically, most existing GNNs mainly rely on the message passing strategy between neighbor nodes, where the expressivity is limited by the…

This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies, the least squares method for…

Enabling effective and efficient machine learning (ML) over large-scale graph data (e.g., graphs with billions of edges) can have a great impact on both industrial and scientific applications. However, existing efforts to advance…

Fully realizing the potential of multigrid solvers often requires custom algorithms for a given application model, discretizations and even regimes of interest, despite considerable effort from the applied math community to develop fully…

Recent advances in Large Language Models have demonstrated their capabilities across a variety of tasks. However, automatically extracting implicit knowledge from natural language remains a significant challenge, as machines lack active…

We numerically analyze the possibility of turning off post-smoothing (relaxation) in geometric multigrid when used as a preconditioner in conjugate gradient linear and eigenvalue solvers for the 3D Laplacian. The geometric Semicoarsening…

We introduce Algorithm MGB (Multi Grid Barrier) for solving highly nonlinear convex Euler-Lagrange equations. This class of problems includes many highly nonlinear partial differential equations, such as $p$-Laplacians. We prove that, if…

A large number of real-world graphs or networks are inherently heterogeneous, involving a diversity of node types and relation types. Heterogeneous graph embedding is to embed rich structural and semantic information of a heterogeneous…

Signed graphs are equipped with both positive and negative edge weights, encoding pairwise correlations as well as anti-correlations in data. A balanced signed graph has no cycles of odd number of negative edges. Laplacian of a balanced…

We consider effective preconditioners for solving Laplacians of general weighted graphs. Theoretically, spectral sparsifiers (SSs) provide preconditioners of optimal computational complexity. However, they are not easy to use for real-world…

Graph Neural Networks (GNNs) are powerful techniques in representation learning for graphs and have been increasingly deployed in a multitude of different applications that involve node- and graph-wise tasks. Most existing studies solve…

Large Language Models (LLMs) demonstrate strong reasoning abilities but face limitations such as hallucinations and outdated knowledge. Knowledge Graph (KG)-based Retrieval-Augmented Generation (RAG) addresses these issues by grounding LLM…

Many real world graphs, such as the graphs of molecules, exhibit structure at multiple different scales, but most existing kernels between graphs are either purely local or purely global in character. In contrast, by building a hierarchy of…

It is challenging for generative models to learn a distribution over graphs because of the lack of permutation invariance: nodes may be ordered arbitrarily across graphs, and standard graph alignment is combinatorial and notoriously…

Generic deep learning (DL) networks for image restoration like denoising and interpolation lack mathematical interpretability, require voluminous training data to tune a large parameter set, and are fragile in the face of covariate shift.…

In this paper, we discuss the convergence of an Algebraic MultiGrid (AMG) method for general symmetric positive-definite matrices. The method relies on an aggregation algorithm, named \emph{coarsening based on compatible weighted matching},…

Delaunay and Gabriel graphs are widely studied geometric proximity structures. Motivated by applications in wireless routing, relaxed versions of these graphs known as \emph{Locally Delaunay Graphs} ($LDGs$) and \emph{Locally Gabriel…

This paper introduces layout-aware graph modeling for multimodal RAG. Different from traditional RAG methods that mostly deal with flat text chunks, the proposed method takes into account the relationship of multimodalities by using a graph…