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Generalized Wedderburn Rank Reduction

Oskar Kędzierski

Abstract

We generalize the Wedderburn rank reduction formula by replacing the inverse with the Moore--Penrose pseudoinverse. In particular, this allows one to remove the non--singularity of a certain matrix from assumptions. The results implies in a straightforward way Nystroem, CUR decompositions, meta-factorization, and a result of Ameli, Shadden. We investigate which properties of the matrix are inherited by the generalized Wedderburn reduction. Reductions leading to the best low-rank approximation are explicitly described in terms of singular vectors. We give a self--contained calculation of the range and the nullspace of the projection $A(BA)^+B$ and prove that any projection can be expressed in this way.

Generalized Wedderburn Rank Reduction

Abstract

We generalize the Wedderburn rank reduction formula by replacing the inverse with the Moore--Penrose pseudoinverse. In particular, this allows one to remove the non--singularity of a certain matrix from assumptions. The results implies in a straightforward way Nystroem, CUR decompositions, meta-factorization, and a result of Ameli, Shadden. We investigate which properties of the matrix are inherited by the generalized Wedderburn reduction. Reductions leading to the best low-rank approximation are explicitly described in terms of singular vectors. We give a self--contained calculation of the range and the nullspace of the projection and prove that any projection can be expressed in this way.
Paper Structure (9 sections, 32 theorems, 136 equations)

This paper contains 9 sections, 32 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Projection Formula
  5. Generalized Wedderburn Rank Reduction
  6. Wedderburn Decomposition
  7. Best Rank Reduction via Generalized Wedderburn Rank Reduction
  8. Properties Inherited by Generalized Wedderburn Rank Reduction
  9. Examples

Key Result

Lemma 1.1

Let $A\in\mathbb{K}^{m\times n}$ be any matrix. Let $x\in\mathbb{K}^n,\ y\in\mathbb{K}^m$. Assume that Then In general, if $X\in\mathbb{K}^{n\times k},\ Y\in\mathbb{K}^{m\times k}$ and $M\in\mathbb{K}^{k\times k}$ is given by is invertible then

Theorems & Definitions (64)

  • Lemma 1.1: Wedderburn rank reduction formula
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 4.1
  • proof
  • ...and 54 more