[Paper Review] Newton's method on Gra{\ss}mann manifolds

Uwe Helmke, Knut Hüper+1 more|arXiv (Cornell University)|Sep 14, 2007

Matrix Theory and Algorithms11 references24 citations

TL;DR

This paper introduces a generalized Newton method on Graßmann and Lagrange–Graßmann manifolds using arbitrary pairs of local coordinates with equal derivatives at the base point, proving local quadratic convergence. The approach enables efficient algorithms for principal component analysis and invariant subspace computation via simplified linear matrix solves, significantly improving computational complexity over prior methods.

ABSTRACT

A general class of Newton algorithms on Gra{\\ss}mann and Lagrange-Gra{\\ss}mann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition on the choice of coordinate systems. Our result extends and unifies previous convergence results for Newton's method on a manifold. Using special choices of the coordinates, new numerical algorithms are derived for principal component analysis and invariant subspace computations with improved computational complexity properties.

Motivation & Objective

• To develop a unified framework for Newton-type optimization on Graßmann and Lagrange–Graßmann manifolds using arbitrary coordinate pairs.
• To extend existing Riemannian Newton methods by replacing fixed projections with flexible coordinate-based update schemes.
• To derive computationally efficient algorithms for eigenvalue and invariant subspace problems by exploiting coordinate choices like QR factorization.
• To establish local quadratic convergence under a mild condition on coordinate derivatives, generalizing prior results.
• To provide new numerical algorithms for Rayleigh quotient optimization and invariant subspace computation with improved complexity.

Proposed method

• Proposes a generalized Newton algorithm on Graßmann manifolds using an arbitrary pair of local coordinate systems $\mu_p, \nu_p$ with $D\mu_p(0) = D\nu_p(0)$, replacing traditional projections.
• Derives the Newton step via a pull-back/push-forward scheme based on coordinate charts, enabling intrinsic geometric computation.
• Uses Riemannian normal coordinates and $QR$-based coordinate systems to simplify the exponential map approximation, reducing computational cost.
• Applies the method to the classical Graßmann manifold $\mathrm{Gr}_{m,n}$ and the Lagrange–Graßmann manifold $\mathrm{LG}_n$, deriving explicit update rules.
• For the Rayleigh quotient on $\mathrm{LG}_n$, the algorithm reduces to solving a Lyapunov equation in each step, enabling efficient implementation.
• For invariant subspace computation, the method requires solving a structured linear matrix equation that can be solved recursively or via vectorization.

Experimental results

Research questions

• RQ1 Can Newton’s method on manifolds be generalized beyond Riemannian normal coordinates to include arbitrary coordinate pairs with matching derivatives?
• RQ2 Under what conditions does such a generalized Newton algorithm achieve local quadratic convergence on Graßmann manifolds?
• RQ3 How can coordinate choices like $QR$ factorization or normal coordinates be leveraged to reduce computational complexity in Newton-type algorithms?
• RQ4 Can the generalized Newton framework yield efficient algorithms for eigenvalue and invariant subspace problems on Graßmann manifolds?
• RQ5 What are the structural properties of the Newton step when applied to the Lagrange–Graßmann manifold, and how do they enable new numerical methods?

Key findings

• Local quadratic convergence of the generalized Newton algorithm is proven under the condition that the two coordinate systems have identical derivatives at the base point.
• The algorithm for Rayleigh quotient optimization on the Lagrange–Graßmann manifold requires solving a Lyapunov equation per iteration, enabling efficient implementation.
• The invariant subspace computation algorithm converges locally quadratically to projectors onto stable invariant subspaces of a matrix $A$.
• The Newton step for the invariant subspace problem reduces to solving a structured linear matrix equation that is uniquely solvable if the spectral components of $A_{11}$ and $A_{22}$ are disjoint.
• The proposed method generalizes and unifies previous Riemannian and affine connection-based Newton methods, subsuming earlier approaches as special cases.
• The use of $QR$-based coordinates allows for efficient approximation of the exponential map, leading to simplified and computationally favorable implementations.