Why Machines Learn by Anil Ananthaswamy

Author:Anil Ananthaswamy [Ananthaswamy, Anil]
Language: eng
Format: epub
Publisher: Penguin Publishing Group
Published: 2024-07-16T00:00:00+00:00

FLIP-FLOP

Some simple math connects phenomena as seemingly diverse as the process that gives us window glass, the magnetization of materials, and the workings of some types of neural networks, at least the artificial kind.

Let’s start with window glass. One method for making such glass is to start with the raw materials—usually silica (sand), soda ash, and limestone, with silica being the primary component. The mixture is melted to form molten glass and then poured into a “float bath.” The bath gives plate glass its flatness and helps cool the molten material from temperatures of over 1,000°C down to about 600°C. This flat material is further “annealed,” a process that releases any accumulated stresses in the glass. The key, for our purposes, is that the resulting glass is neither a solid with an ordered crystalline structure nor a liquid. Instead, it’s an amorphous solid where the material’s atoms and molecules don’t conform to the regularity of a crystal lattice.

There’s an interesting analog in magnetism. Certain materials, for example, are ferromagnetic, a state in which the magnetic moments of the material’s atoms (or ions) are all aligned, generating a net magnetism. A ferromagnet is analogous to a solid with a definite crystalline structure. However, if the magnetic moments of the atoms, or ions, are randomly oriented, the material has no permanent magnetism—analogous to the structure of glass. Each individual magnetic moment is the outcome of the spin of an elementary particle in the material. Hence, materials with disordered magnetic moments are called spin glasses.

In the early 1920s, the German physicist Wilhelm Lenz and his graduate student Ernst Ising developed a simple model of such materials. It came to be called the Ising model. For his doctoral thesis, Ising analyzed a one-dimensional case of magnetic moments. The engendering spins can be either up (+1) or down (-1). In the model, any given spin state is influenced only by its immediate neighbors. For example, if one spin state is -1, but both its neighbors are +1, then the spin will flip directions. It’s clear that such a system will have some dynamics, because as each spin state reacts to its nearest neighbors, the effects of spin flips will ripple back and forth through the system. If all the spins taken together constitute the system’s state, then the system traverses a state space, going from one state to another, possibly settling into some stable state or continually oscillating. Ising showed that a 1D system cannot be ferromagnetic (meaning, the spins will never all align in one direction). He even argued—erroneously, it turned out—that state transitions from disorderly to orderly would not happen even in the three-dimensional case.

In 1936, Rudolf Ernst Peierls, a German physicist who left Germany during the Nazi era and became a British citizen, rigorously studied the model for the 2D case. (It was Peierls who attributed the model to Ising, giving it its name.) “For sufficiently low temperatures the Ising model in two dimensions shows ferromagnetism and the same holds a fortiori also for the three-dimensional model,” Peierls wrote.

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