The sparseness of g-convex functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912040178679808 |
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| author | Wang, Yu Ye, Ke |
| author_facet | Wang, Yu Ye, Ke |
| contents | The g-convexity of functions on manifolds is a generalization of the convexity of functions on Rn. It plays an essential role in both differential geometry and non-convex optimization theory. This paper is concerned with g-convex smooth functions on manifolds. We establish criteria for the existence of a Riemannian metric (or connection) with respect to which a given function is g-convex. Using these criteria, we obtain three sparseness results for g-convex functions: (1) The set of g-convex functions on a compact manifold is nowhere dense in the space of smooth functions. (2) Most polynomials on Rn that is g-convex with respect to some geodesically complete connection has at most one critical point. (3) The density of g-convex univariate (resp. quadratic, monomial, additively separable) polynomials asymptotically decreases to zero |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14434 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The sparseness of g-convex functions Wang, Yu Ye, Ke Differential Geometry Optimization and Control The g-convexity of functions on manifolds is a generalization of the convexity of functions on Rn. It plays an essential role in both differential geometry and non-convex optimization theory. This paper is concerned with g-convex smooth functions on manifolds. We establish criteria for the existence of a Riemannian metric (or connection) with respect to which a given function is g-convex. Using these criteria, we obtain three sparseness results for g-convex functions: (1) The set of g-convex functions on a compact manifold is nowhere dense in the space of smooth functions. (2) Most polynomials on Rn that is g-convex with respect to some geodesically complete connection has at most one critical point. (3) The density of g-convex univariate (resp. quadratic, monomial, additively separable) polynomials asymptotically decreases to zero |
| title | The sparseness of g-convex functions |
| topic | Differential Geometry Optimization and Control |
| url | https://arxiv.org/abs/2409.14434 |