James Mercer FRS (15 January 1883 – 21 February 1932) was a

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, born in

Bootle Bootle (pronounced ) is a town in the Metropolitan Borough of Sefton, Merseyside, England, which had a population of 51,394 in 2011; the wider Parliamentary constituency had a population of 98,449. Historically part of Lancashire, Bootle's ...

, close to

Liverpool Liverpool is a city and metropolitan borough in Merseyside, England. With a population of in 2019, it is the 10th largest English district by population and its metropolitan area is the fifth largest in the United Kingdom, with a populat ...

, England. He was educated at

University of Manchester , mottoeng = Knowledge, Wisdom, Humanity , established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria Univ ...

, and then

University of Cambridge The University of Cambridge is a public collegiate research university in Cambridge, England. Founded in 1209 and granted a royal charter by Henry III in 1231, Cambridge is the world's third oldest surviving university and one of its most pr ...

. He became a Fellow, saw active service at the Battle of Jutland in World War I and, after decades of ill health, died in London. He proved

Mercer's theorem In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in , is one of the most no ...

, which states that

positive-definite kernel In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving ...

s can be expressed as a

dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...

in a high-dimensional

space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...

. This theorem is the basis of the

kernel trick In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). The general task of pattern analysis is to find and study general types of relations (for example ...

( applied by Aizerman), which allows linear algorithms to be easily converted into non-linear algorithms.

References

1883 births 1932 deaths 19th-century British mathematicians 20th-century British mathematicians Mathematical analysts People from Bootle Alumni of the University of Manchester Senior Wranglers Scientists from Liverpool Fellows of the Royal Society {{UK-mathematician-stub