We derive estimates of the Hessian of two smooth functions defined on Grassmannian manifold. Based on it, we can derive curvature estimates for minimal submanifolds in Euclidean space via Gauss map as [24]. In this way, the result for Bernstein type theorem done by Jost and the first author could be improved.

An unbiased low-variance gradient estimator, termed GO gradient, was proposed recently for expectation-based objectives E_q_γ(y) [f(y)], where the random variable (RV) y may be drawn from a stochastic computation graph (SCG) with continuous (non-reparameterizable) internal nodes and continuous/discrete leaves. Based on we present [f(y)] an Hessian named Hessian, which contains deterministic as ...

We introduce an evolution equation which deforms metrics on 3-manifolds with sectional curvature of one sign. Given a closed 3-manifold with an initial metric with negative sectional curvature, we conjecture that this flow will exist for all time and converge to a hyperbolic metric after a normalization. We shall establish a monotonicity formula in support of this conjecture. Note that in contr...

Minimum mode following algorithms are widely used for saddle point searching in chemical and material systems. Common to these algorithms is a component to find the minimum curvature mode of the second derivative, or Hessian matrix. Several methods, including Lanczos, dimer, Rayleigh-Ritz minimization, shifted power iteration, and locally optimal block preconditioned conjugate gradient, have be...

We propose a block-diagonal approximation of the positive-curvature Hessian (BDA-PCH) matrix to measure curvature. Our proposed BDAPCH matrix is memory efficient and can be applied to any fully-connected neural networks where the activation and criterion functions are twice differentiable. Particularly, our BDA-PCH matrix can handle non-convex criterion functions. We devise an efficient scheme ...

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R3, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove a general local existence result for a large class of degenerate Monge-Ampère...

The Armijo and Goldstein step-size rules are modified to allow steps along a curvilinear path of the form x(a) = x + as + a2d, where x is the current estimate of the minimum, s is a descent direction and d is a nonascent direction of negative curvature. By using directions of negative curvature when they exist, we are able to prove, under fairly mild assumptions, that the sequences of iterates ...