The main purpose of the present work is to investigate statistical manifolds endowed with almost product structures. We prove that structure a para-Kähler-like manifold constant curvature in Kurose's sense Hessian structure. also derive properties submersions which are compatible results illustrated by several nontrivial examples.

We solve the Dirichlet problem for $k$-Hessian equations on compact complex manifolds with boundary, given existence of a subsolution. Our method is based second order priori estimate solution boundary particular gradient scale. The scale allows us to apply blow-up argument obtain control all necessary norms solution.

We prove that generalised Monge-Ampère equations (a family of which includes the inverse Hessian like J-equation, as well equation) on projective manifolds have smooth solutions if certain intersection numbers are positive. As corollaries our work, we improve a result Chen (albeit in case) existence to and conjecture Székelyhidi case solvability equations. The key new ingredient improving Chen’...

Let $(X,\omega)$ be a compact K\ahler manifold of dimension $n$ and fix $1\leq m\leq n$. We prove that the total mass complex Hessian measure $\omega$-$m$-subharmonic functions is non-decreasing with respect to singularity type. then solve equations prescribed singularity, Hodge index type inequality for positive currents.

Abstract We study Hessian estimators for functions defined over an $n$-dimensional complete analytic Riemannian manifold. introduce new stochastic zeroth-order using $O (1)$ function evaluations. show that, real-valued $f$, our estimator achieves a bias bound of order $ O ( \gamma \delta ^2 ) $, where depends on both the Levi–Civita connection and $\delta is finite difference step size. To best...

We discuss the use of the Newton method in the computation of max(p 7→ Ep [f ]), where p belongs to a statistical exponential family on a finite state space. In a number of papers, the authors have applied first order search methods based on information geometry. Second order methods have been widely used in optimization on manifolds, e.g., matrix manifolds, but appear to be new in statistical ...

In this paper, we extend the DC Calculus introduced by Perelman on finite dimensional Alexandrov spaces with curvature bounded below. Among other things, our results allow us to define the Hessian and the Laplacian of DC functions (including distance functions as a particular instance) as a measure-valued tensor and a Radon measure respectively. We show that these objects share various properti...