In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Grothendieck inequality states that there is a universal constant
with the following property. If ''M''
''ij'' is an ''n'' × ''n'' (
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
)
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
with
:
for all (real or complex) numbers ''s''
''i'', ''t''
''j'' of absolute value at most 1, then
:
for all
vectors ''S''
''i'', ''T''
''j'' in the
unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
''B''(''H'') of a (real or complex)
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''H'', the constant
being independent of ''n''. For a fixed Hilbert space of dimension ''d'', the smallest constant that satisfies this property for all ''n'' × ''n'' matrices is called a Grothendieck constant and denoted
. In fact, there are two Grothendieck constants,
and
, depending on whether one works with real or complex numbers, respectively.
The Grothendieck inequality and Grothendieck constants are named after
Alexander Grothendieck, who proved the existence of the constants in a paper published in 1953.
Motivation and the operator formulation
Let
be an
matrix. Then
defines a linear operator between the normed spaces
and
for
. The
-norm of
is the quantity
If
, we denote the norm by
.
One can consider the following question: For what value of
and
is
maximized? Since
is linear, then it suffices to consider
such that
contains as many points as possible, and also
such that
is as large as possible. By comparing
for
, one sees that
for all
.
One way to compute
is by solving the following quadratic
integer program
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objectiv ...
:
To see this, note that
, and taking the maximum over
gives
. Then taking the maximum over
gives
by the convexity of
and by the triangle inequality. This quadratic integer program can be relaxed to the following
semidefinite program:
It is known that exactly computing
for
is
NP-hard
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
, while exacting computing
is
NP-hard
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
for
.
One can then ask the following natural question: How well does an optimal solution to the
semidefinite program approximate
? The Grothendieck inequality provides an answer to this question: There exists a fixed constant
such that, for any
, for any
matrix
, and for any Hilbert space
,
Bounds on the constants
The sequences
and
are easily seen to be increasing, and Grothendieck's result states that they are
bounded,
[.] so they have
limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
.
Grothendieck proved that
where
is defined to be
.
[.] improved the result by proving that
, conjecturing that the upper bound is tight. However, this conjecture was disproved by .
[.]
Grothendieck constant of order ''d''
Boris Tsirelson
Boris Semyonovich Tsirelson (May 4, 1950 – January 21, 2020) ( he, בוריס סמיונוביץ' צירלסון, russian: Борис Семёнович Цирельсон) was a Russian–Israeli mathematician and Professor of Mathematic ...
showed that the Grothendieck constants
play an essential role in the problem of
quantum nonlocality: the
Tsirelson bound of any full correlation bipartite
Bell inequality for a quantum system of dimension ''d'' is upperbounded by
.
Lower bounds
Some historical data on best known lower bounds of
is summarized in the following table.
Upper bounds
Some historical data on best known upper bounds of
:
Applications
Cut norm estimation
Given an
real matrix
, the cut norm of
is defined by
The notion of cut norm is essential in designing efficient approximation algorithms for dense graphs and matrices. More generally, the definition of cut norm can be generalized for symmetric
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
functions
so that the cut norm of
is defined by
This generalized definition of cut norm is crucial in the study of the space of
graphons, and the two definitions of cut norm can be linked via the
adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simp ...
of a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
.
An application of the Grothendieck inequality is to give an efficient algorithm for approximating the cut norm of a given real matrix
; specifically, given an
real matrix, one can find a number
such that
where
is an absolute constant. This approximation algorithm uses
semidefinite programming Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize)
over the intersection of the cone of positive ...
.
We give a sketch of this approximation algorithm. Let
be
matrix defined by
One can verify that
by observing, if