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On Smale's 17th problem over the reals

Andrea Montanari, Eliran Subag

Abstract

We consider the problem of efficiently solving a system of $n$ non-linear equations in ${\mathbb R}^d$. Addressing Smale's 17th problem stated in 1998, we consider a setting whereby the $n$ equations are random homogeneous polynomials of arbitrary degrees. In the complex case and for $n= d-1$, Beltrán and Pardo proved the existence of an efficient randomized algorithm and Lairez recently showed it can be de-randomized to produce a deterministic efficient algorithm. Here we consider the real setting, to which previously developed methods do not apply. We describe a polynomial time algorithm that finds solutions (with high probability) for $n= d -O(\sqrt{d\log d})$ if the maximal degree is bounded by $d^2$ and for $n=d-1$ if the maximal degree is larger than $d^2$.

On Smale's 17th problem over the reals

Abstract

We consider the problem of efficiently solving a system of non-linear equations in . Addressing Smale's 17th problem stated in 1998, we consider a setting whereby the equations are random homogeneous polynomials of arbitrary degrees. In the complex case and for , Beltrán and Pardo proved the existence of an efficient randomized algorithm and Lairez recently showed it can be de-randomized to produce a deterministic efficient algorithm. Here we consider the real setting, to which previously developed methods do not apply. We describe a polynomial time algorithm that finds solutions (with high probability) for if the maximal degree is bounded by and for if the maximal degree is larger than .
Paper Structure (24 sections, 31 theorems, 251 equations, 3 algorithms)

This paper contains 24 sections, 31 theorems, 251 equations, 3 algorithms.

Table of Contents

  1. Introduction and main result
  2. Approximate solutions
  3. Optimization
  4. Hessian descent: large $d$ and moderate $p_{\max}\le d^2$
  5. Multi-Scale Search: large $d$ and $p_{\max}\ge d^2$ or bounded $p_{\max}\le p_0$, $d\leq d_0$
  6. Smale's 17th problem in the complex case
  7. Preliminaries
  8. Bounds on the Hessian
  9. Minimal singular value at solutions
  10. Upper bound on volume in expectation
  11. Lower bound on the volume of a neighborhood
  12. Proof of Proposition \ref{['prop:sigmamin_MSS']}
  13. Proof of Proposition \ref{['prop:sigmamin']}
  14. Good events
  15. Sufficient conditions for approximate solutions
  16. ...and 9 more sections

Key Result

Theorem 1

There exist absolute constants $A,C,\delta_0$ such that, for any $\delta\in (0,\delta_0)$ and any $\delta'>0$ the following holds.The only assumption on $\delta_0$ is that it is sufficiently small so that the condition $Ce^{-d/C}<\delta<\delta_0$ implies that $d$ is large enough so that $\lfloor d- Then, there exists an algorithm which solves $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ with p

Theorems & Definitions (58)

  • Theorem 1
  • Theorem 2: Hessian Descent
  • Theorem 3: Multi-Scale Search
  • Definition 1.1: Approximate solution
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 48 more