[Paper Review] Probabilistic Condition Number Estimates for Real Polynomial Systems II: Structure and Smoothed Analysis.

Alperen A. Ergür, Grigoris Paouris+1 more|arXiv (Cornell University)|Sep 10, 2018

Polynomial and algebraic computation28 references4 citations

TL;DR

This paper provides explicit probabilistic estimates for condition numbers of structured real polynomial systems, analyzing their sensitivity to coefficient perturbations. It extends these results to a smoothed analysis framework, offering theoretical guarantees on numerical stability for random structured systems.

ABSTRACT

We consider the sensitivity of real zeros of structured polynomial systems to perturbations of their coefficients. In particular, we provide explicit estimates for condition numbers of structured random real polynomial systems, and extend these estimates to smoothed analysis setting.

Motivation & Objective

• To analyze the sensitivity of real zeros in structured polynomial systems to coefficient perturbations.
• To derive explicit probabilistic estimates for condition numbers in structured polynomial systems.
• To extend condition number analysis to the smoothed analysis setting for improved robustness guarantees.
• To provide theoretical foundations for understanding numerical stability in random structured polynomial systems.

Proposed method

• The authors use probabilistic techniques to estimate condition numbers for structured real polynomial systems.
• They model coefficient perturbations as random variables with bounded variance to assess sensitivity.
• The analysis incorporates structural constraints inherent in the polynomial systems, such as symmetry or sparsity.
• Smoothed analysis is applied by adding small random noise to coefficients, enabling robustness quantification.
• Key estimates are derived using concentration inequalities and tail bounds on condition number distributions.
• The framework allows for explicit bounds on the probability that condition numbers exceed certain thresholds.

Experimental results

Research questions

• RQ1 How do condition numbers of structured real polynomial systems behave under random coefficient perturbations?
• RQ2 What is the probability that the condition number exceeds a given threshold in structured systems?
• RQ3 How does smoothed analysis refine traditional condition number estimates for structured systems?
• RQ4 What structural properties of polynomial systems influence their sensitivity to perturbations?
• RQ5 Can explicit probabilistic bounds be derived for condition numbers in the smoothed analysis setting?

Key findings

• The paper derives explicit upper bounds on the probability that the condition number of a structured real polynomial system exceeds a given threshold.
• These bounds depend on the system's structure and the variance of coefficient perturbations.
• The smoothed analysis framework yields tighter and more robust condition number estimates compared to worst-case analysis.
• The results show that structured systems exhibit favorable probabilistic stability under small random perturbations.
• The derived estimates are applicable to a broad class of structured polynomial systems, including symmetric and sparse configurations.
• The framework enables quantification of numerical stability in randomized algorithms for solving structured polynomial systems.