Yau's Affine Normal Descent: Algorithmic Framework and Convergence Analysis
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2026
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| _version_ | 1866913000484503552 |
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| author | Niu, Yi-Shuai Sheshmani, Artan Yau, Shing-Tung |
| author_facet | Niu, Yi-Shuai Sheshmani, Artan Yau, Shing-Tung |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28448 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Yau's Affine Normal Descent: Algorithmic Framework and Convergence Analysis Niu, Yi-Shuai Sheshmani, Artan Yau, Shing-Tung Optimization and Control Machine Learning Numerical Analysis Differential Geometry 90C15, 90C30, 49M37, 53A15 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. |
| title | Yau's Affine Normal Descent: Algorithmic Framework and Convergence Analysis |
| topic | Optimization and Control Machine Learning Numerical Analysis Differential Geometry 90C15, 90C30, 49M37, 53A15 |
| url | https://arxiv.org/abs/2603.28448 |