EM算法 EM(Expectation Maximization Algorithm),期望最大值算法推理时,用到了这些数学知识: (1)正/负定矩阵 (2)[严格]凹函数 (3)[严格]凸函数 (4)[最大]似然函数 (5)Hessian矩阵 (6)Jensen不等式 1.正/负定矩阵 positive definite matrix,正定矩阵; negative definite matrix... 查看原文 EM算法 。
在数学中,海森矩阵(Hessianmatrix或Hessian)是一个自变量为向量的实值函数的二阶偏导数组成的方块矩阵,此函数如下: 如果f所有的二阶导数都存在,那么f的海森矩阵即: H(f)ij(x)=DiDjf(x) 其中,即 (也有人把海森定义为以上矩阵的行列式)海森矩阵被应用于牛顿法解决的大规模优化问题。
This note points out that the positive semidefiniteness of the discrete Hessian matrix does not imply nor is implied by convex extensibility of discrete functions.doi:10.15807/jorsj.55.48Moriguchi, SatokoMurota, KazuoJournal of the Operations Research Society of Japan...
The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplyinga symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval uponwhich the parametric Hessian is positive semidefinite. The algorithm is ...
where Hk∈Rn×n is a symmetric positive-definite matrix that approximates the exact Hessian ∇2f(xk) in some manner. The basic premise of secant methods for unconstrained optimization is that at each iteration, we use curvature information measured from the most recent point, xk−1, to upd...
are positive semidefinite, and so is \({\hat{\textbf{h}}}(\textbf{x})\) . thus we can use the matrix chernoff inequality (theorem 1.1 [ 58 ]), which provides a sharper guarantee: $$\begin{aligned}&\mathbb {p}\big (\vert {\hat{\textbf{h}}}(\textbf{x})-\textbf{h}(\...
It is easy to see that every elementwise nonnegative matrix is copositive and every positive semidefinite matrix is copositive. Therefore, (4.1)Ccop⊇C++Cpsd and (4.2)Ccp=Ccop∗⊆(C++Cpsd)∗=C+∗∩Cpsd∗=C+∩Cpsd. Diananda [20] shows that the inclusions in (4.1), (4.2) are ...
Hessian matrixsemidefinite programmingFor functions defined on integer lattice points, discrete versions of the Hessian matrix have been considered in various contexts. In discrete convex analysis, for example, certain combinatorial properties of the discrete Hessian matrices are known to characterize M-...
was introduced in [2]. In line with (HBF), it contains afixedpositive friction coefficient. The introduction of the Hessian-driven damping makes it possible to neutralize the transversal oscillations likely to occur with (HBF), as observed in [2] in the case of the Rosenbrook function. The...
The underestimation procedure is based on Gerschgorin's circle theorem, which gives bounds on the eigenvalues of a matrix in terms of its elements (Gerschgorin, 1931). The underestimator is convex if its Hessian is positive semidefinite everywhere on D. Therefore Gerschgorin's theorem is extended ...