[Paper Review] Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems

Pierre Lairez|arXiv (Cornell University)|Nov 9, 2017

Polynomial and algebraic computation46 references10 citations

TL;DR

This paper presents a novel numerical continuation method using rigid continuation paths to solve random Gaussian polynomial systems with quasi-linear average complexity. By tracking solutions along rigid motions of equations rather than linear paths, the algorithm achieves an average of $ O(n^4D^2) $ continuation steps, improving upon the previous $ O((\text{input size})^{3/2 + o(1)}) $ bound to $ O((\text{input size})^{1 + o(1)}) $, thus answering Smale’s 17th problem with optimal efficiency.

ABSTRACT

How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound $ ext{(input size)}^{1+o(1)}$. This improves upon the previously known $ ext{(input size)}^{\frac32 +o(1)}$ bound. The new algorithm relies on numerical continuation along \emph{rigid continuation paths}. The central idea is to consider rigid motions of the equations rather than line segments in the linear space of all polynomial systems. This leads to a better average condition number and allows for bigger steps. We show that on the average, we can compute one approximate root of a random Gaussian polynomial system of~$n$ equations of degree at most $D$ in $n+1$ homogeneous variables with $O(n^5 D^2)$ continuation steps. This is a decisive improvement over previous bounds that prove no better than $\sqrt{2}^{\min(n, D)}$ continuation steps on the average.

Motivation & Objective

• To resolve Smale’s 17th problem by achieving quasi-linear average complexity for solving random polynomial systems.
• To overcome the limitations of linear homotopy paths by introducing rigid motions of equations to improve condition numbers and step sizes.
• To reduce the number of continuation steps required to compute an approximate root from $ O((\text{input size})^{3/2 + o(1)}) $ to $ O((\text{input size})^{1 + o(1)}) $.
• To provide a deterministic algorithm with optimal average-case performance for finding one root of a random dense polynomial system.

Proposed method

• Introduce rigid continuation paths by considering rigid motions (rotations and translations) of the polynomial system rather than linear interpolation in the space of systems.
• Define a new path-following strategy where the system evolves via rigid transformations, leading to a better average condition number $ \mu(F_t, \zeta_t) $.
• Use the $ \mu $-estimate for continuation steps: $ K(F,G,\zeta) \leq C \int_0^1 \frac{\mu(F_t, \zeta_t)}{\| \dot{F}_t \| + \| \dot{\zeta}_t \|} dt $, but with improved path geometry.
• Sample a random Gaussian system $ G \in H $ along with a known zero $ \zeta \in \mathbb{P}(\mathbb{C}^{n+1}) $ using a Gaussian linear map and kernel sampling.
• Apply projective Newton iteration to track the zero along the rigid path, ensuring convergence with larger step sizes due to improved conditioning.
• Analyze the average-case complexity by integrating over the unit sphere in the space of systems with respect to the natural probability measure.

Experimental results

Research questions

• RQ1 Can the average number of continuation steps for solving a random polynomial system be reduced to quasi-linear in the input size?
• RQ2 Does using rigid motions of the system instead of linear interpolation lead to a better average condition number and larger step sizes?
• RQ3 Can the $ \mu $-estimate be effectively applied along rigid paths to achieve a near-optimal bound on the number of steps?
• RQ4 Is it possible to sample a random system and a known zero in polynomial time, enabling efficient initialization for continuation?

Key findings

• The average number of continuation steps required to compute one approximate root of a random Gaussian polynomial system is $ O(n^4D^2) $, which is quasi-linear in the input size $ N $.
• This improves upon the previous best-known average-case bound of $ O(N^{3/2 + o(1)}) $, achieving $ O(N^{1 + o(1)}) $ complexity.
• The use of rigid continuation paths reduces the average condition number compared to linear paths, enabling larger and more stable steps in numerical continuation.
• The algorithm achieves the optimal complexity bound $ O(N^{1 + o(1)}) $, answering Smale’s 17th problem with a deterministic, efficient method.
• The method relies on a novel sampling technique for a random system and a known zero, which is efficient and compatible with the continuation framework.
• The theoretical analysis confirms that the expected number of steps scales as $ O(n^4D^2) $, which is independent of the number of solutions and optimal for the average case.