Yau's Affine Normal Descent: Algorithmic Framework and Convergence Analysis

2026-03-30

AI Summary

This work addresses the poor adaptability to anisotropic curvature and sensitivity to affine transformations in unconstrained smooth optimization by proposing a novel algorithm grounded in affine differential geometry. The method introduces, for the first time in optimization, the equiaffine normal vector of level-set hypersurfaces to construct a volume-preserving, affine-invariant search direction that aligns with the Newton direction for strictly convex quadratic problems, thereby achieving one-step convergence. Under standard smoothness assumptions, the algorithm is globally convergent; it exhibits linear convergence under strong convexity or the Polyak–Łojasiewicz condition, and attains quadratic local convergence near nondegenerate minima. Numerical experiments confirm its robustness and efficiency under anisotropic scaling.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We propose Yau's Affine Normal Descent (YAND), a geometric framework for smooth unconstrained optimization in which search directions are defined by the equi-affine normal of level-set hypersurfaces. The resulting directions are invariant under volume-preserving affine transformations and intrinsically adapt to anisotropic curvature. Using the analytic representation of the affine normal from affine differential geometry, we establish its equivalence with the classical slice-centroid construction under convexity. For strictly convex quadratic objectives, affine-normal directions are collinear with Newton directions, implying one-step convergence under exact line search. For general smooth (possibly nonconvex) objectives, we characterize precisely when affine-normal directions yield strict descent and develop a line-search-based YAND. We establish global convergence under standard smoothness assumptions, linear convergence under strong convexity and Polyak-Lojasiewicz conditions, and quadratic local convergence near nondegenerate minimizers. We further show that affine-normal directions are robust under affine scalings, remaining insensitive to arbitrarily ill-conditioned transformations. Numerical experiments illustrate the geometric behavior of the method and its robustness under strong anisotropic scaling.

Problem

• affine invariance
• anisotropic curvature
• ill-conditioned scaling
• unconstrained optimization
• search direction

Innovation

• Affine Normal Descent
• Equi-affine Invariance
• Anisotropic Curvature Adaptation
• Geometric Optimization
• Robustness to Ill-conditioning