- Person
- Leonid Kantorovich
- Lived
- 1912–1986
- Field
- Mathematical economics
- From
- Soviet
The story
Among the economic problems Leonid Kantorovich identified in the late 1930s was the distribution of transport flows: how to move goods from many sources to many destinations at the lowest possible cost. In his 1939 booklet The Mathematical Method of Production Planning and Organization he sketched methods for the rational routing of transported goods, alongside his new technique of linear programming.
The question had a long history. In 1781 the French mathematician Gaspard Monge, who studied geometry in the service of military fortification, had asked how to move earth dug from one place to build up ramparts elsewhere with the least effort, and he turned this into a problem of pure geometry. Kantorovich took the question much further. He treated it as an infinite-dimensional optimisation problem, and his formulation became known as the Monge-Kantorovich transportation problem.
His analysis led to what is now called the Kantorovich-Rubinstein metric. It measures the distance between two probability distributions as the minimum cost of turning one into the other, like the effort needed to reshape one pile of earth into another. The metric was first defined in Kantorovich's 1939 work, but in 1970 the Russian mathematician Roland Dobrushin named it after Leonid Vaserstein, who had used it in 1969, and it is still widely called the Wasserstein distance. Some scholars encourage the names Kantorovich metric or Kantorovich distance.
The metric is used in probability theory, particularly in the study of the weak convergence of probability measures. In computer science the same idea is known as the earth mover's distance: it was applied to images in 1989, named in 1994, and is widely used in content-based image retrieval to compare the colour histograms of digital images.
Today the metric is also used in machine learning: researchers have used it to improve generative adversarial networks, the systems that learn to create realistic images, and related measures help evaluate them. What began as a practical question about shipping goods in Soviet industry now helps computers compare pictures and data around the world.
Why it matters
Kantorovich's transport ideas underpin tools used today in probability, statistics and computing to compare distributions and images.
About Leonid Kantorovich
Leningrad mathematician who invented linear programming in 1939 while solving a plywood factory’s puzzle, and the only Soviet economist to win the Nobel Prize, in 1975.
More about Leonid Kantorovich
- Leonid Kantorovich showed why rational prices matterKantorovich's book The Best Use of Economic Resources, drafted during the war and published in 1959, linked efficient planning to a rational system of prices.
- Leonid Kantorovich pioneered parallel computing ideasIn the late 1940s Leonid Kantorovich organised parallel calculations on punched-card machines and proposed computer designs based on large-block processing.
- Leonid Kantorovich linked pure maths to practical computingLeonid Kantorovich's paper Functional Analysis and Applied Mathematics showed how abstract mathematics could power numerical methods, earning a State Prize in 1949.
- Leonid Kantorovich invented linear programming in 1939Asked in 1938 to help a Leningrad plywood trust get more from its machines, Leonid Kantorovich devised the method now known as linear programming.
- Leonid Kantorovich's methods cut waste in steel sheetsFrom 1948 to 1950, work guided by Leonid Kantorovich at the Leningrad Carriage-Building Works used linear programming to cut steel sheets with less waste.
- Leonid Kantorovich trained a new kind of economistIn Novosibirsk from 1960, Leonid Kantorovich built a laboratory and school that trained a generation of mathematician-economists.
- Leonid Kantorovich kept the Road of Life trucks on the iceDuring the Siege of Leningrad, Leonid Kantorovich calculated safe spacing for supply trucks crossing frozen Lake Ladoga on the Road of Life.
Sources
- Leonid Kantorovich Wikipedia
- Wasserstein metric Wikipedia
- Leonid Vital'evich Kantorovich MacTutor, University of St Andrews
- Earth mover's distance Wikipedia
Another act of tikkun olam. This is part of tikkun olam, the Jewish idea of repairing the world, and every one of us can add to it.