论文标题
单位球上的均质多项式和杂项局部最小值
Homogeneous polynomials and spurious local minima on the unit sphere
论文作者
论文摘要
我们考虑欧几里得单位球上的学位-D形式。我们专门研究Nie在更一般的框架中获得的一般性结果。我们在F的系数中表现出均匀的多项式RES，因此，如果RES（F）= 0，那么所有满足一阶必要最佳条件的点实际上是单位球体上F的局部最小值。然后，我们仅根据（i）f，（ii）其梯度的规范和（iii）其Hessian的前两个最小特征值的范围，从而获得了所有通用度-D形式的所有局部最小值的简单而紧凑的表征。实际上，该属性还具有两倍的连续可区分函数，这些功能是均匀的。最后，我们通过使用代数几何形状中的梯度理想的特性在单位球上获得了通用度-D形式的表征，而单位球上没有虚假的局部最小值。
We consider degree-d forms on the Euclidean unit sphere. We specialize to our setting a genericity result by Nie obtained in a more general framework. We exhibit an homogeneous polynomial Res in the coefficients of f , such that if Res(f) = 0 then all points that satisfy first-and second-order necessary optimality conditions are in fact local minima of f on the unit sphere. Then we obtain obtain a simple and compact characterization of all local minima of generic degree-d forms, solely in terms of the value of (i) f , (ii) the norm of its gradient, and (iii) the first two smallest eigenvalues of its Hessian, all evaluated at the point. In fact this property also holds for twice continuous differentiable functions that are positively homogeneous. Finally we obtain a characterization of generic degree-d forms with no spurious local minimum on the unit sphere by using a property of gradient ideals in algebraic geometry.
