Related papers: Consistent Computation of First- and Second-Order …
The analysis of gradient descent-type methods typically relies on the Lipschitz continuity of the objective gradient. This generally requires an expensive hyperparameter tuning process to appropriately calibrate a stepsize for a given…
We propose a new approach for controlling the characteristics of certain mesh faces during optimization of high-order curved meshes. The practical goals are tangential relaxation along initially aligned curved boundaries and internal…
In this paper we propose a Newton method for shape functions defined on an image set generated by the (Micheletti) metric group. We review basic properties of the metric group and a quotient associated with the metric group and a fixed…
We develop Policy Gradient with Second-Order Momentum (PG-SOM), a lightweight second-order optimisation scheme for reinforcement-learning policies. PG-SOM augments the classical REINFORCE update with two exponentially weighted statistics: a…
Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of…
Higher-order tensor methods were recently proposed for minimizing smooth convex and nonconvex functions. Higher-order algorithms accelerate the convergence of the classical first-order methods thanks to the higher-order derivatives used in…
Due to the effectiveness of second-order algorithms in solving classical optimization problems, designing second-order optimizers to train deep neural networks (DNNs) has attracted much research interest in recent years. However, because of…
We introduce a smoothing algorithm for triangle, quadrilateral, tetrahedral and hexahedral meshes whose centerpiece is a simple geometric triangle transformation. The first part focuses on the mathematical properties of the element…
Gradient methods are widely used in optimization problems. In practice, while the smoothness parameter can be estimated utilizing techniques such as backtracking, estimating the strong convexity parameter remains a challenge; moreover, even…
Neuroscientific studies of drawing-like movements usually analyze neural representation of either geometric (eg. direction, shape) or temporal (eg. speed) features of trajectories rather than trajectory's representation as a whole. This…
We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end, we extend the variational time…
We embark in a program of studying the problem of better approximating surfaces by triangulations(triangular meshes) by considering the approximating triangulations as finite metric spaces and the target smooth surface as their…
An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow, powers of mean and inverse mean curvature flow, etc. Error estimates are proven for semi- and full…
We give a condition under which the findings of the paper cited above work well and determine the surfaces that were not considered before. In this paper, we show that a parallel mean curvature surface of a general type in a complex…
Sketching is a dimensionality reduction technique where one compresses a matrix by linear combinations that are chosen at random. A line of work has shown how to sketch the Hessian to speed up each iteration in a second order method, but…
Consider the following distributed optimization scenario. A worker has access to training data that it uses to compute the gradients while a server decides when to stop iterative computation based on its target accuracy or delay…
An expression for the first variation of the area functional of the second fundamental form is given for a hypersurface in a semi-Riemannian space. The concept of the "mean curvature of the second fundamental form" is then introduced. Some…
A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as…
We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic geometries are the study of a three-dimensional space equipped with a rank 2 metric of…
In this paper, we present the detailed mathematical derivation of the gradient and Hessian matrix for the Vora-Value based colorimetric filter optimization. We make a full recapitulation of the steps involved in differentiating the…