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

Laplacian regularized stratified models (LRSM) are models that utilize the explicit or implicit network structure of the sub-problems as defined by the categorical features called strata (e.g., age, region, time, forecast horizon, etc.),…

Multi-view learning has progressed rapidly in recent years. Although many previous studies assume that each instance appears in all views, it is common in real-world applications for instances to be missing from some views, resulting in…

Graphs are a powerful tool for representing and analyzing unstructured, non-Euclidean data ubiquitous in the healthcare domain. Two prominent examples are molecule property prediction and brain connectome analysis. Importantly, recent works…

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…

Laplacian eigenvectors capture natural community structures on graphs and are widely used in spectral clustering and manifold learning. The use of Laplacian eigenvectors as embeddings for the purpose of multiscale graph comparison has…

Despite the strong abilities, large language models (LLMs) still suffer from hallucinations and reliance on outdated knowledge, raising concerns in knowledge-intensive tasks. Graph-based retrieval-augmented generation (GRAG) enriches LLMs…

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…

Naive Retrieval-Augmented Generation (RAG) focuses on individual documents during retrieval and, as a result, falls short in handling networked documents which are very popular in many applications such as citation graphs, social media, and…

The computation of stationary distributions of Markov chains is an important task in the simulation of stochastic models. The linear systems arising in such applications involve non-symmetric M-matrices, making algebraic multigrid methods a…

Graph algorithms can be expressed in terms of linear algebra. GraphBLAS is a library of low-level building blocks for such algorithms that targets algorithm developers. LAGraph builds on top of the GraphBLAS to target users of graph…

Retrieval-augmented generation (RAG) empowers large language models to access external and private corpus, enabling factually consistent responses in specific domains. By exploiting the inherent structure of the corpus, graph-based RAG…

In this paper, we study FPGA based pipelined and superscalar design of two variants of conjugate gradient methods for solving Laplacian equation on a discrete grid; the first version corresponds to the original conjugate gradient algorithm,…

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…

In this paper, we consider a classical form of optimal algebraic multigrid (AMG) interpolation that directly minimizes the two-grid convergence rate and compare it with the so-called ideal form that minimizes a certain weak approximation…

Isogeometric analysis (IgA) offers enhanced approximation capabilities for the discretization of elliptic boundary-value problems, yet it results in large, sparse, and increasingly ill-conditioned linear systems due to higher…

Large linear systems with sparse, non-symmetric matrices arise in the modeling of Markov chains or in the discretization of convection-diffusion problems. Due to their potential to solve sparse linear systems with an effort that is linear…

In recent contributions, algebraic multigrid methods have been designed and studied from the viewpoint of the spectral complementarity. In this note we focus our efforts on specific applications and, more precisely, on large linear systems…

Machine learning methods have greatly changed science, engineering, finance, business, and other fields. Despite the tremendous accomplishments of machine learning and deep learning methods, many challenges still remain. In particular, the…

Retrieval-Augmented Generation (RAG) improves factuality by grounding LLMs in external knowledge, yet conventional centralized RAG requires aggregating distributed data, raising privacy risks and incurring high retrieval latency and cost.…

Multigrid methods despite being known to be asymptotically optimal algorithms, depend on the careful selection of their individual components for efficiency. Also, they are mostly restricted to standard cycle types like V-, F-, and…