Related papers: Matrix compression along isogenic blocks
Matrix-vector multiplication forms the basis of many iterative solution algorithms and as such is an important algorithm also for hierarchical matrices which are used to represent dense data in an optimized form by applying low-rank…
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…
Boundary integral equations lead to dense system matrices when discretized, yet they are data-sparse. Using the $\mathcal{H}$-matrix format, this sparsity is exploited to achieve $\mathcal{O}(N\log N)$ complexity for storage and…
The hierarchical matrix framework partitions matrices into subblocks that are either small or of low numerical rank, enabling linear storage complexity and efficient matrix-vector multiplication. This work focuses on the $H^2$-matrix format…
We provide a computational framework for approximating a class of structured matrices; here, the term structure is very general, and may refer to a regular sparsity pattern (e.g., block-banded), or be more highly structured (e.g., symmetric…
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…
We propose a new approximate factorization for solving linear systems with symmetric positive definite sparse matrices. In a nutshell the algorithm is to apply hierarchically block Gaussian elimination and additionally compress the fill-in.…
Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory…
Co-clustering simultaneously clusters rows and columns, revealing more fine-grained groups. However, existing co-clustering methods suffer from poor scalability and cannot handle large-scale data. This paper presents a novel and scalable…
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…
Matrix sketching is a recently developed data compression technique. An input matrix A is efficiently approximated with a smaller matrix B, so that B preserves most of the properties of A up to some guaranteed approximation ratio. In so…
This technical report introduces a novel approach to efficient computation in homological algebra over fields, with particular emphasis on computing the persistent homology of a filtered topological cell complex. The algorithms here…
In recent years, several algorithms, which approximate matrix decomposition, have been developed. These algorithms are based on metric conservation features for linear spaces of random projection types. We show that an i.i.d sub-Gaussian…
The Hadamard decomposition is a powerful technique for data analysis and matrix compression, which decomposes a given matrix into the element-wise product of two or more low-rank matrices. In this paper, we develop an efficient algorithm to…
We devise achievable encoding schemes for distributed source compression for computing inner products, symmetric matrix products, and more generally, square matrix products, which are a class of nonlinear transformations. To that end, our…
Random matrices now play a role in many parts of computational mathematics. To advance these applications, it is desirable to have tools that are flexible, easy to use, and powerful. Over the last 25 years, researchers have developed a…
Overparameterized models have proven to be powerful tools for solving various machine learning tasks. However, overparameterization often leads to a substantial increase in computational and memory costs, which in turn requires extensive…
Multiresolution analysis and matrix factorization are foundational tools in computer vision. In this work, we study the interface between these two distinct topics and obtain techniques to uncover hierarchical block structure in symmetric…
Theoretical studies have proven that the Hilbert space has remarkable performance in many fields of applications. Frames in tensor product of Hilbert spaces were introduced to generalize the inner product to high-order tensors. However,…
We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of…