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

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.

Figures (13)

• Figure 1: Geometric comparison between the Euclidean and affine normals
• Figure 2: Normal–aligned frame at $x_k$ illustrating three typical constructions of $d_k$. The analytic affine normal $d_{\mathrm{AN}}(x_k)$ is represented in the frame $\{e_1,\dots,e_n,e_{n+1}\}$ with its $(n+1)$-st component normalized to $-1$. Case 1: the affine normal is already a descent direction ($\langle\nabla f(x_k), d_{\mathrm{AN}}(x_k)\rangle<0$), hence $d_k=d_{\mathrm{AN}}(x_k)$. Case 2: the affine normal points uphill ($\langle\nabla f(x_k), d_{\mathrm{AN}}(x_k)\rangle>0$), so we flip the sign and set $d_k=-d_{\mathrm{AN}}(x_k)$. Case 3: the affine normal is orthogonal to the gradient ($\langle\nabla f(x_k), d_{\mathrm{AN}}(x_k)\rangle=0$); in this degenerate case we revert to the steepest–descent direction $d_k=-\nabla f(x_k)/\|\nabla f(x_k)\|$.
• Figure 3: YAND on the well-conditioned quadratic \ref{['eq:quad-well']} with three different line-search strategies. Each panel shows (from left to right) the YAND trajectory on level sets, the function value $f(x_k)-f^\star$ (log scale), and the gradient norm $\|\nabla f(x_k)\|_2$ (log scale).
• Figure 4: Optimization trajectories in the original $x$-coordinates for $f_\gamma(x)=\tfrac{1}{2}(x_1^2+\gamma^2 x_2^2)$ with $\gamma=1,10^2,10^4$. As $\gamma$ increases, the level sets become increasingly elongated. YAND-Exact and Newton remain essentially one-step methods, while the behavior of gradient descent depends more strongly on the step-size rule. In particular, GD-Fixed becomes substantially slower as the anisotropy increases, whereas GD-Exact remains convergent on this diagonal quadratic.
• Figure 5: YAND trajectories after normalization $y=B_\gamma x$ for $\gamma=1,10^2,10^4$. After mapping to the intrinsic coordinates of $\phi(y)=\tfrac{1}{2}\|y\|^2$, the trajectories collapse onto nearly identical paths, illustrating the affine invariance predicted by the theory.

Theorems & Definitions (76)

• Remark 2.1: Moment viewpoint
• Theorem 2.2: Consistency of slice-centroid and analytical affine normal under convexity
• proof
• Lemma 2.3: Fourth-moment tensor on the sphere
• Theorem 3.1: Affine normal coincides with the Newton direction on strictly convex quadratics
• Corollary 3.2: One-step convergence with exact line search
• proof
• Theorem 4.1: Strict descent at elliptic points
• proof
• Remark 4.2: Geometric meaning