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Turing meets Moore-Penrose: Computing the Pseudoinverse on Turing Machines

Holger Boche, Adalbert Fono, Gitta Kutyniok

TL;DR

The paper studies whether the Moore-Penrose pseudoinverse $A^agger$ can be computed effectively on digital machines modeled as Turing machines. Using computable analysis and notions of Borel-Turing and Banach-Mazur computability, it proves a universal finite-error algorithm does not exist for arbitrary matrices, while demonstrating effective algorithms for restricted input sets such as full-rank matrices or matrices with entries bounded away from zero. It then analyzes iterative schemes, showing that convergence alone does not yield effective computability without suitable initialization or stopping criteria, and it fully characterizes the computability breakdown via rank-based input properties. The results delineate fundamental limits of digital linear-algebra computation and point to practical guidance: focus on benign input domains where provable computability guarantees can be realized, and acknowledge inherent non-computability for general inputs.

Abstract

The pseudoinverse of a matrix, a generalized notion of the inverse, is of fundamental importance in linear algebra and, thereby, in many different fields. Despite its proven existence, an algorithmic approach is typically necessary to obtain the pseudoinverse in practical applications. Therefore, we analyze if and to what degree the pseudoinverse can be computed on perfect digital hardware platforms modeled as Turing machines. For this, we utilize the notion of an effective algorithm that describes a provably correct computation: upon an input of any error parameter, the algorithm provides an approximation within the given error bound with respect to the unknown solution. We prove that a universal effective algorithm for computing the pseudoinverse of any matrix with a finite error bound does not exist on Turing machines. However, for specific classes of matrices, we show that provably correct algorithms exist and obtain a characterization of the properties of the input set, leading to the effective computability breakdown.

Turing meets Moore-Penrose: Computing the Pseudoinverse on Turing Machines

TL;DR

The paper studies whether the Moore-Penrose pseudoinverse can be computed effectively on digital machines modeled as Turing machines. Using computable analysis and notions of Borel-Turing and Banach-Mazur computability, it proves a universal finite-error algorithm does not exist for arbitrary matrices, while demonstrating effective algorithms for restricted input sets such as full-rank matrices or matrices with entries bounded away from zero. It then analyzes iterative schemes, showing that convergence alone does not yield effective computability without suitable initialization or stopping criteria, and it fully characterizes the computability breakdown via rank-based input properties. The results delineate fundamental limits of digital linear-algebra computation and point to practical guidance: focus on benign input domains where provable computability guarantees can be realized, and acknowledge inherent non-computability for general inputs.

Abstract

The pseudoinverse of a matrix, a generalized notion of the inverse, is of fundamental importance in linear algebra and, thereby, in many different fields. Despite its proven existence, an algorithmic approach is typically necessary to obtain the pseudoinverse in practical applications. Therefore, we analyze if and to what degree the pseudoinverse can be computed on perfect digital hardware platforms modeled as Turing machines. For this, we utilize the notion of an effective algorithm that describes a provably correct computation: upon an input of any error parameter, the algorithm provides an approximation within the given error bound with respect to the unknown solution. We prove that a universal effective algorithm for computing the pseudoinverse of any matrix with a finite error bound does not exist on Turing machines. However, for specific classes of matrices, we show that provably correct algorithms exist and obtain a characterization of the properties of the input set, leading to the effective computability breakdown.
Paper Structure (10 sections, 9 theorems, 36 equations)

This paper contains 10 sections, 9 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Contributions
  4. Outline
  5. Digital Computations
  6. Computable Analysis
  7. Main Results
  8. Banach-Mazur Non-Computability of the Pseudoinverse
  9. Non-Effectiveness of Iterative Processes computing the Pseudoinverse
  10. Full description of computability breakdown

Key Result

Theorem 6

For any function $f$, denote a function with the same domain and codomain by $\hat{f}$. Then, for $m,n \geq 2$, the functions I.-VI. are not algorithmically approximable, i.e., any function $\hat{h}$ is not Banach-Mazur computable if

Theorems & Definitions (24)

  • Definition 1
  • Remark 2
  • Definition 3
  • Remark 4
  • Remark 5
  • Theorem 6
  • Remark 7
  • Corollary 8
  • Proposition 9
  • Remark 10
  • ...and 14 more