Generalized Pseudoskeleton Decompositions

@article{Hamm2022GeneralizedPD,
  title={Generalized Pseudoskeleton Decompositions},
  author={Keaton Hamm},
  journal={ArXiv},
  year={2022},
  volume={abs/2206.14905}
}
A BSTRACT . We characterize some variations of pseudoskeleton (also called CUR) de- compositions for matrices andtensorsover arbitraryfields. These characterizations extend previous results to arbitrary fields and to decompositions which use generalized inverses of the constituent matrices, in contrast to Moore–Penrose pseudoinverses in prior works which are specific to real or complex valued matrices, and are significantly more structured. 

References

SHOWING 1-10 OF 34 REFERENCES

A Theory of Pseudoskeleton Approximations

Pseudo-Skeleton Approximations by Matrices of Maximal Volume

1. M. M. Gekhtman and I. V. Stankevich, Spectral Theory of Some Nonclassical Differential Operators [in Russian], Textbook, Dagestan State Univ., Makhachkala (1985). 2. S. Albeverio and F. Gestezi,

Computation of generalized inverses of tensors via t‐product

Generalized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory of generalized inverses of tensors

Efficient algorithms for cur and interpolative matrix decompositions

The manuscript describes efficient algorithms for the computation of the CUR and ID decompositions. The methods used are based on simple modifications to the classical truncated pivoted QR

Multilinear operators for higher-order decompositions

  • T. Kolda
  • Mathematics, Computer Science
  • 2006
TLDR
Two new multilinear operators are proposed for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions and one of them is shorthand for performing an n-mode matrix multiplication for every mode of a given tensor.

LU and CR Elimination

TLDR
This paper creates a column-row rank-revealing factorization A = CR, with the first r independent columns of A in C and the r nonzero rows of rref (A) in R, to reimagine the start of a linear algebra course.

On best approximate solutions of linear matrix equations

  • R. Penrose
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1956
In an earlier paper (4) it was shown how to define for any matrix a unique generalization of the inverse of a non-singular matrix. The purpose of the present note is to give a further application

Generalizing the column–row matrix decomposition to multi-way arrays

Relative-Error CUR Matrix Decompositions

TLDR
These two algorithms are the first polynomial time algorithms for such low-rank matrix approximations that come with relative-error guarantees; previously, in some cases, it was not even known whether such matrix decompositions exist.

Revisiting the Nystrom Method for Improved Large-scale Machine Learning

TLDR
An empirical evaluation of the performance quality and running time of sampling and projection methods on a diverse suite of SPSD matrices and a suite of worst-case theoretical bounds for both random sampling and random projection methods are complemented.