Recently a majorization method for optimizing partition functions of log-linear models was proposed alongside a novel quadratic variational upper-bound. In the batch setting, it outperformed state-of-the-art firstand second-order optimization methods on various learning tasks. We propose a stochastic version of this bound majorization method as well as a low-rank modification for highdimensiona...

Here (u) denotes the inverse of the Hessian matrix uij = ∂u ∂xi∂xj . We call this PDE Abreu’s equation since the expression S(u) appears in [1], in the study of the differential geometry of toric varieties. In this paper we will be primarily interested in the case when A is a constant. Solutions to Equation 1 then correspond to certain Kahler metrics of constant scalar curvature, as we will rec...

Let M denote a compact real hyperbolic manifold with dimensionm 2: 5 and sectional curvature K = I , and let 1: be an exotic sphere ofdimension m. Given any small number t5 > 0 , we show that there is a finitecovering space M of M satisfying the following properties: the connectedsum M#1: is not diffeomorphic to M, but it is homeomorphic to M; M#1:supports a Riemannian metri...

The effectiveness of Newton’s method for finding an unconstrained minimizer of a strictly convex twice continuously differentiable function has prompted the proposal of various modified Newton methods for the nonconvex case. Line search modified Newton methods utilize a linear combination of a descent direction and a direction of negative curvature. If these directions are sufficient in a certa...

We study the effects of entanglement and control parameters on symmetry energy landscape optimization performance variational quantum circuit. Through a systematic analysis Hessian spectrum, we characterize local geometry at random point along an trajectory. argue that decreasing entangling capability increasing number circuit have same qualitative effect eigenspectrum. Both low-entangling abun...

hadamard (or complete $cat(0)$) spaces are complete, non-positive curvature, metric spaces. here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. our results extend the standard non-linear ergodic theorems for non-expansive maps on real hilbert spaces, to non-expansive maps on had...

We prove that the manifold Mn of minimal radial curvature Kmin o ≥ 1 is homeomorphic to the sphere Sn if its radius or volume is larger than half the radius or volume of the round sphere of constant curvature 1. These results are optimal and give a complete generalization of the corresponding results for manifolds of sectional curvature bounded from below. 0. Introduction and results 0.1. A sta...

If π : M → B is a Riemannian Submersion and M has positive sectional curvature, O’Neill’s Horizontal Curvature Equation shows that B must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian submersions from...

We develop a 2nd-order optimization method based on the “Hessian-free” approach, and apply it to training deep auto-encoders. Without using pre-training, we obtain results superior to those reported by Hinton & Salakhutdinov (2006) on the same tasks they considered. Our method is practical, easy to use, scales nicely to very large datasets, and isn’t limited in applicability to autoencoders, or...

This paper describes a node relocation algorithm based on nonlinear optimization which delivers excellent results for both unstructured and structured plane triangle meshes over convex as well as non-convex domains with high curvature. The local optimization scheme is a damped Newton’s method in which the gradient and Hessian of the objective function are evaluated exactly. The algorithm has be...