Concept

Saddle point

A saddle point is a critical point that is a maximum along one direction and a minimum along another, like a mountain pass. On a surface it is where two pieces of a region above a height join or a loop closes, and it counts minus one towards the Euler characteristic.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

One minimum, or several. Two curves side by side with their local minima marked: a convex one with a single minimum, and a fourth-power well with 2.

Where the guarantee stops

Convexity converts every downhill method into a correct one, and its absence removes the guarantee entirely rather than degrading it. What is left is a collection of partial answers, and knowing which of them apply to a given problem is most of what non-convex optimisation is.

analysis · Convexity

A random surface and the region above height 1. A smooth Gaussian random field on a 64 by 64 torus and its excursion set above 1: 14 pieces, 0 holes, Euler characteristic 14; expected 13.8.

Pieces minus holes on a random surface

Cut a random landscape at some height and the region above it breaks into pieces with holes in them. How many pieces, and how many holes, has no formula. Their difference does: the average Euler characteristic of the region above height u is a constant times u multiplied by the bell curve at u, exactly, for every smooth Gaussian surface — because the Euler characteristic is a sum of peaks, pits and saddles, and the average number of those can be computed point by point.

topology · Euler characteristic

Named alongside it

The objects these essays reach for when they reach for this one.

Complexity Convexity Counterexample Critical point Euler characteristic Gradient Local minimum Morse theory Optimisation Random field Relaxation Torus

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