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We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or…
We propose two novel techniques for overcoming load-imbalance encountered when implementing so-called look-ahead mechanisms in relevant dense matrix factorizations for the solution of linear systems. Both techniques target the scenario…
In the current era of vast data and transparent machine learning, it is essential for techniques to operate at a large scale while providing a clear mathematical comprehension of the internal workings of the method. Although there already…
Triangular factorizations are an important tool for solving integral equations and partial differential equations with hierarchical matrices ($\mathcal{H}$-matrices). Experiments show that using an $\mathcal{H}$-matrix LR factorization to…
Finding the largest cardinality feasible subset of an infeasible set of linear constraints is the Maximum Feasible Subsystem problem (MAX FS). Solving this problem is crucial in a wide range of applications such as machine learning and…
Large models and enormous data are essential driving forces of the unprecedented successes achieved by modern algorithms, especially in scientific computing and machine learning. Nevertheless, the growing dimensionality and model…
Persistent homology is a leading tool in topological data analysis (TDA). Many problems in TDA can be solved via homological -- and indeed, linear -- algebra. However, matrices in this domain are typically large, with rows and columns…
Recent SVD-free matrix factorization formulations have enabled rank minimization for systems with millions of rows and columns, paving the way for matrix completion in extremely large-scale applications, such as seismic data interpolation.…
We propose an algorithm for clustering high dimensional data. If $P$ features for $N$ objects are represented in an $N\times P$ matrix ${\bf X}$, where $N\ll P$, the method is based on exploiting the cluster-dependent structure of the…
Incomplete factorizations have long been popular general-purpose algebraic preconditioners for solving large sparse linear systems of equations. Guaranteeing the factorization is breakdown free while computing a high quality preconditioner…
This paper introduces the hierarchical interpolative factorization for integral equations (HIF-IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU…
The high-dimensional rank lasso (hdr lasso) model is an efficient approach to deal with high-dimensional data analysis. It was proposed as a tuning-free robust approach for the high-dimensional regression and was demonstrated to enjoy…
Supervised cross-modal hashing has gained increasing research interest on large-scale retrieval task owning to its satisfactory performance and efficiency. However, it still has some challenging issues to be further studied: 1) most of them…
Low-rank methods have shown success in accelerating simulations of a collisionless plasma described by the Vlasov equation, but still rely on computationally costly linear algebra every time step. We propose a data-driven factorization…
In this article, we prove new theorems bounding the rank of different sub-matrices arising from these kernel functions. Bounds like these are often useful for analyzing the complexity of various hierarchical matrix algorithms. We also plot…
We consider the linear time-dependent Schr\"odinger equation with a time-dependent smooth potential on an unbounded domain. A Galerkin spectral method with a tensor-product Hermite basis is used as a discretization in space. Discretizing…
In the age of big data and interpretable machine learning, approaches need to work at scale and at the same time allow for a clear mathematical understanding of the method's inner workings. While there exist inherently interpretable…
Semidefinite programs (SDPs) and their solvers are powerful tools with many applications in machine learning and data science. Designing scalable SDP solvers is challenging because by standard the positive semidefinite decision variable is…
This paper presents a generalization of the "weighted least-squares" (WLS), named "weighted pairing least-squares" (WPLS), which uses a rectangular weight matrix and is suitable for data alignment problems. Two fast solving methods,…
Integral equations are commonly encountered when solving complex physical problems. Their discretization leads to a dense kernel matrix that is block or hierarchically low-rank. This paper proposes a new way to build a low-rank…