Gradient descent
No. 176, 1976
(Calculus of variations and geometry)
India ink and pencil on paper, 31.5x44 cm.
A long beam protrudes into space, pierced by two slender pins on which hangs a tattered cloth. In the background, great tubular structures jut into the sky, melting away at bottom while masked in part from view by a tenuous haze. The theme behind this image involves an operation of the geometric calculus of variations, namely the process of lowering a cycle along the integral path of a vector field. As it slides downward, the cycle eventually catches on the critical points of some function. In this image the pins represent the critical points, while the beam represents the body of a manifold. The further study of such cycles under gradient descent falls under finite-dimensional Morse theory and its diverse generalizations to the infinite dimensional case.
