相关论文: Mixed precision HODLR matrices

Hierarchical Matrix (H-matrix) is an approximation technique which splits a target dense matrix into multiple submatrices, and where a selected portion of submatrices are low-rank approximated. The technique substantially reduces both time…

Hierarchical matrix approximations have gained significant traction in the machine learning and scientific community as they exploit available low-rank structures in kernel methods to compress the kernel matrix. The resulting compressed…

We study low-rank matrix regression in settings where matrix-valued predictors and scalar responses are observed across multiple individuals. Rather than assuming a fully homogeneous coefficient matrices across individuals, we accommodate…

Obtaining lightweight and accurate approximations of Hessian applies in inverse problems governed by partial differential equations (PDEs) is an essential task to make both deterministic and Bayesian statistical large-scale inverse problems…

Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. $\mathcal{H}^2$-matrices refine this representation following the ideas of fast multipole…

The discretization of non-local operators, e.g., solution operators of partial differential equations or integral operators, leads to large densely populated matrices. $\mathcal{H}^2$-matrices take advantage of local low-rank structures in…

Hierarchical matrices provide a powerful representation for significantly reducing the computational complexity associated with dense kernel matrices. For general kernel functions, interpolation-based methods are widely used for the…

Toeplitz matrices are abundant in computational mathematics, and there is a rich literature on the development of fast and superfast algorithms for solving linear systems involving such matrices. Any Toeplitz matrix can be transformed into…

Hierarchical matrices can be used to construct efficient preconditioners for partial differential and integral equations by taking advantage of low-rank structures in triangular factorizations and inverses of the corresponding stiffness…

Using lower precision in algorithms can be beneficial in terms of reducing both computation and communication costs. Motivated by this, we aim to further the state-of-the-art in developing and analyzing mixed precision variants of iterative…

Low rank matrix approximations appear in a number of scientific computing applications. We consider the Nystr\"{o}m method for approximating a positive semidefinite matrix $A$. In the case that $A$ is very large or its entries can only be…

Low-rank modeling plays a pivotal role in signal processing and machine learning, with applications ranging from collaborative filtering, video surveillance, medical imaging, to dimensionality reduction and adaptive filtering. Many modern…

This paper proposes an efficient algorithm (HOLRR) to handle regression tasks where the outputs have a tensor structure. We formulate the regression problem as the minimization of a least square criterion under a multilinear rank…

We present a mixed-precision benchmark called HPL-MxP that uses both a lower-precision LU factorization with a non-stationary iterative refinement based on GMRES. We evaluate the numerical stability of one of the methods of generating the…

Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner. However, modern matrices can involve billions of elements, making their storage and…

Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…

We study the robust matrix completion problem for the low-rank Hankel matrix, which detects the sparse corruptions caused by extreme outliers while we try to recover the original Hankel matrix from the partial observation. In this paper, we…

Low rank approximation is an important tool used in many applications of signal processing and machine learning. Recently, randomized sketching algorithms were proposed to effectively construct low rank approximations and obtain approximate…

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 examine a class of embeddings based on structured random matrices with orthogonal rows which can be applied in many machine learning applications including dimensionality reduction and kernel approximation. For both the…