DeepAI AI Chat
Log In Sign Up

A note on the variation of geometric functionals

by   Nir Sochen, et al.

Calculus of Variation combined with Differential Geometry as tools of modelling and solving problems in image processing and computer vision were introduced in the late 80's and the 90s of the 20th century. The beginning of an extensive work in these directions was marked by works such as Geodesic Active Contours (GAC), the Beltrami framework, level set method of Osher and Sethian the works of Charpiat et al. and the works by Chan and Vese to name just a few. In many cases the optimization of these functional are done by the gradient descent method via the calculation of the Euler-Lagrange equations. Straightforward use of the resulted EL equations in the gradient descent scheme leads to non-geometric and in some cases non sensical equations. It is costumary to modify these EL equations or even the functional itself in order to obtain geometric and/or sensical equations. The aim of this note is to point to the correct way to derive the EL and the gradient descent equations such that the resulted gradient descent equation is geometric and makes sense.


page 1

page 2

page 3

page 4


Stochastic Gradient Descent for Semilinear Elliptic Equations with Uncertainties

Randomness is ubiquitous in modern engineering. The uncertainty is often...

A Geometric Approach of Gradient Descent Algorithms in Neural Networks

In this article we present a geometric framework to analyze convergence ...

Gradient Descent Can Take Exponential Time to Escape Saddle Points

Although gradient descent (GD) almost always escapes saddle points asymp...

Training Two-Layer ReLU Networks with Gradient Descent is Inconsistent

We prove that two-layer (Leaky)ReLU networks initialized by e.g. the wid...

A Note on the Convergence of Mirrored Stein Variational Gradient Descent under (L_0,L_1)-Smoothness Condition

In this note, we establish a descent lemma for the population limit Mirr...

Two almost-circles, and two real ones

Implicit locus equations in GeoGebra allow the user to do experiments wi...

Revisiting the acceleration phenomenon via high-resolution differential equations

Nesterov's accelerated gradient descent (NAG) is one of the milestones i...