Affine Normal Directions via Log-Determinant Geometry: Scalable Computation under Sparse Polynomial Structure
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| Format: | Preprint |
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2026
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| _version_ | 1866910094800715776 |
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| author | Niu, Yi-Shuai Sheshmani, Artan Yau, Shing-Tung |
| author_facet | Niu, Yi-Shuai Sheshmani, Artan Yau, Shing-Tung |
| contents | Affine normal directions provide intrinsic affine-invariant descent directions derived from the geometry of level sets. Their practical use, however, has long been hindered by the need to evaluate third-order derivatives and invert tangent Hessians, which becomes computationally prohibitive in high dimensions. In this paper, we show that affine normal computation admits an exact reduction to second-order structure: the classical third-order contraction term is precisely the gradient of the log-determinant of the tangent Hessian. This identity replaces explicit third-order tensor contraction by a matrix-free formulation based on tangent linear solves, Hessian-vector products, and log-determinant gradient evaluation. Building on this reduction, we develop exact and stochastic matrix-free procedures for affine normal evaluation. For sparse polynomial objectives, the algebraic closure of derivatives further yields efficient sparse kernels for gradients, Hessian-vector products, and directional third-order contractions, leading to scalable implementations whose cost is governed by the sparsity structure of the polynomial representation. We establish end-to-end complexity bounds showing near-linear scaling with respect to the relevant sparsity scale under fixed stochastic and Krylov budgets. Numerical experiments confirm that the proposed MF-LogDet formulation reproduces the original autodifferentiation-based affine normal direction to near machine precision, delivers substantial runtime improvements in moderate and high dimensions, and exhibits empirical near-linear scaling in both dimension and sparsity. These results provide a practical computational route for affine normal evaluation and reveal a new connection between affine differential geometry, log-determinant curvature, and large-scale structured optimization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_01163 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Affine Normal Directions via Log-Determinant Geometry: Scalable Computation under Sparse Polynomial Structure Niu, Yi-Shuai Sheshmani, Artan Yau, Shing-Tung Optimization and Control Numerical Analysis Algebraic Geometry Differential Geometry 90C30, 90C26, 65F30, 65Y20, 53A15 Affine normal directions provide intrinsic affine-invariant descent directions derived from the geometry of level sets. Their practical use, however, has long been hindered by the need to evaluate third-order derivatives and invert tangent Hessians, which becomes computationally prohibitive in high dimensions. In this paper, we show that affine normal computation admits an exact reduction to second-order structure: the classical third-order contraction term is precisely the gradient of the log-determinant of the tangent Hessian. This identity replaces explicit third-order tensor contraction by a matrix-free formulation based on tangent linear solves, Hessian-vector products, and log-determinant gradient evaluation. Building on this reduction, we develop exact and stochastic matrix-free procedures for affine normal evaluation. For sparse polynomial objectives, the algebraic closure of derivatives further yields efficient sparse kernels for gradients, Hessian-vector products, and directional third-order contractions, leading to scalable implementations whose cost is governed by the sparsity structure of the polynomial representation. We establish end-to-end complexity bounds showing near-linear scaling with respect to the relevant sparsity scale under fixed stochastic and Krylov budgets. Numerical experiments confirm that the proposed MF-LogDet formulation reproduces the original autodifferentiation-based affine normal direction to near machine precision, delivers substantial runtime improvements in moderate and high dimensions, and exhibits empirical near-linear scaling in both dimension and sparsity. These results provide a practical computational route for affine normal evaluation and reveal a new connection between affine differential geometry, log-determinant curvature, and large-scale structured optimization. |
| title | Affine Normal Directions via Log-Determinant Geometry: Scalable Computation under Sparse Polynomial Structure |
| topic | Optimization and Control Numerical Analysis Algebraic Geometry Differential Geometry 90C30, 90C26, 65F30, 65Y20, 53A15 |
| url | https://arxiv.org/abs/2604.01163 |