光滑DC优化中的有效曲率维数
Effective curvature dimension in smooth DC optimization
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中文总结 AI 辅助
本文提出光滑DC分解的有效曲率维数，证明其最小可达值等于一致控制Hessian的正半定矩阵的最小秩，并揭示该维数决定算法中子问题求解数量，为优化提供新视角。
中文摘要 AI 辅助
我们引入了光滑DC分解的有效曲率维数，定义为其第一个凸分量的Hessian值域的跨度维数。我们证明了在所有光滑DC分解上可达到的最小有效维数等于一个正半定矩阵的最小秩，该矩阵一致地控制DC目标函数的Hessian。因此，最优维数总是由二次凸化器实现。我们进一步通过广义Schur补准则刻画了可行的无曲率子空间，揭示了由交叉曲率引起的障碍。作为有效维数框架的算法推论，我们证明了全局求解的下层模型子问题的数量取决于有效维数而非环境维数。最后，对于二次第一分量，我们确定了在内部覆盖数下均匀逼近所需的最小切向支撑数，并在非退化条件下获得了匹配的维数指数。
英文摘要
We introduce the effective curvature dimension of a smooth DC decomposition, defined as the dimension of the span of the Hessian ranges of its first convex component. We prove that the smallest attainable effective dimension over all smooth DC decompositions equals the minimum rank of a positive semidefinite matrix that uniformly majorizes the Hessian of the DC objective. Hence, an optimal dimension is always attained by a quadratic convexifier. We further characterize feasible curvature-free subspaces through generalized Schur-complement criteria, revealing an obstruction caused by cross-curvature.As an algorithmic consequence of the effective dimension framework, we show that the number of globally solved lower model subproblems depends on the effective dimension rather than the ambient dimension. Finally, for quadratic first components, we identify the minimum number of tangent supports required for uniform approximation with an internal covering number and obtain a matching dimensional exponent under a nondegeneracy condition.
发表机构
- Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。
