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Average conditioning of underdetermined polynomial systems

Federico Carrasco

TL;DR

This work analyzes the average conditioning of random underdetermined polynomial systems and derives closed-form moment formulas for the relative and absolute Frobenius condition numbers. Using a geometric framework with the solution variety and a double-fibration technique, the authors connect these moments to moments of random matrices, yielding explicit expressions in terms of Gamma functions and Grassmannian volumes. The main contributions include exact formulas for the $\alpha$-th moment of the relative normalized Frobenius condition number and a transfer principle linking the polynomial-case averages to corresponding random-matrix expectations, with concrete results such as the $\mathbb E(\hat\mu^{r,2})$ and $\mathbb E(\mu^{r,2}_{Av})$ values. These results advance probabilistic conditioning analysis for underdetermined systems and illuminate how average behavior scales with problem dimensions, providing insights relevant to the numerical performance of Newton-type methods.

Abstract

This article study the average conditioning for a random underdetermined polynomial system. The expected value of the moments of the condition number are compared to the moments of the condition number of random matrices. An expression for these moments is given by studying the kernel finding problem for random matrices. Furthermore, the second moment of the Frobenius condition number is computed.

Average conditioning of underdetermined polynomial systems

TL;DR

This work analyzes the average conditioning of random underdetermined polynomial systems and derives closed-form moment formulas for the relative and absolute Frobenius condition numbers. Using a geometric framework with the solution variety and a double-fibration technique, the authors connect these moments to moments of random matrices, yielding explicit expressions in terms of Gamma functions and Grassmannian volumes. The main contributions include exact formulas for the -th moment of the relative normalized Frobenius condition number and a transfer principle linking the polynomial-case averages to corresponding random-matrix expectations, with concrete results such as the and values. These results advance probabilistic conditioning analysis for underdetermined systems and illuminate how average behavior scales with problem dimensions, providing insights relevant to the numerical performance of Newton-type methods.

Abstract

This article study the average conditioning for a random underdetermined polynomial system. The expected value of the moments of the condition number are compared to the moments of the condition number of random matrices. An expression for these moments is given by studying the kernel finding problem for random matrices. Furthermore, the second moment of the Frobenius condition number is computed.
Paper Structure (11 sections, 10 theorems, 72 equations)

This paper contains 11 sections, 10 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Condition Number
  4. Main Result
  5. Proof of Main Theorem
  6. Geometric framework
  7. Double-fibration technique
  8. The linear solution variety
  9. The solution variety for the finding kernel problem
  10. Proof of Theorem \ref{['relavg']}
  11. Some useful computations

Key Result

Theorem 1

The expected value, with respect to the standard Gaussian distribution, of the $2$-nd moment of the relative Frobenius condition number $\frac{\mu^{r,2}_{F,Av}(h)}{\|h\|^2}$ satisfies:

Theorems & Definitions (25)

  • Theorem 1: Main Theorem
  • Remark 2
  • Proposition 1
  • Remark 3
  • Theorem : BCSS*p. 241 The smooth coarea formula
  • Proposition 2
  • proof
  • Proposition 3
  • Remark 4
  • Remark 5
  • ...and 15 more