Grothendieck inequality

Some historical data on best known lower bounds of ${\displaystyle K_{G}^{\mathbb {R} }(d)}$ is summarized in the following table.

d	Grothendieck, 1953[2]	Krivine, 1979[5]	Davie, 1984[9]	Fishburn et al., 1994[10]	Vértesi, 2008[11]	Briët et al., 2011[12]	Hua et al., 2015[13]	Diviánszky et al., 2017[14]	Designolle et al., 2023 [15]
2		${\displaystyle {\sqrt {2}}}$ ≈ 1.41421
3					1.41724		1.41758	1.4359	1.4367
4					1.44521		1.44566	1.4841
5				${\displaystyle {\frac {10}{7}}}$ ≈ 1.42857	1.46007		1.46112
6							1.47017
7						1.46286	1.47583
8						1.47586	1.47972
9						1.48608
∞	${\displaystyle {\frac {\pi }{2}}}$ ≈ 1.57079		1.67696

Some historical data on best known upper bounds of ${\displaystyle K_{G}^{\mathbb {R} }(d)}$:

d	Grothendieck, 1953[2]	Rietz, 1974[16]	Krivine, 1979[5]	Braverman et al., 2011[6]	Hirsch et al., 2016[17]	Designolle et al., 2023 [15]
2			${\displaystyle {\sqrt {2}}}$ ≈ 1.41421
3			1.5163		1.4644	1.4546
4			${\displaystyle {\frac {\pi }{2}}}$ ≈ 1.5708
8			1.6641
∞	${\displaystyle \operatorname {sinh} {\frac {\pi }{2}}}$ ≈ 2.30130	2.261	${\displaystyle {\frac {\pi }{2\ln(1+{\sqrt {2}})}}}$ ≈ 1.78221	${\displaystyle {\frac {\pi }{2\ln(1+{\sqrt {2}})}}-\varepsilon }$

Given an ${\displaystyle m\times n}$ real matrix ${\displaystyle A=(a_{ij})}$, the cut norm of ${\displaystyle A}$ is defined by

${\displaystyle \|A\|_{\square }=\max _{S\subset [m],T\subset [n]}\left|\sum _{i\in S,j\in T}a_{ij}\right|.}$

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 functions ${\displaystyle W:[0,1]^{2}\to \mathbb {R} }$ so that the cut norm of ${\displaystyle W}$ is defined by

${\displaystyle \|W\|_{\square }=\sup _{S,T\subset [0,1]}\left|\int _{S\times T}W\right|.}$

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 of a graph.

An application of the Grothendieck inequality is to give an efficient algorithm for approximating the cut norm of a given real matrix ${\displaystyle A}$; specifically, given an ${\displaystyle m\times n}$ real matrix, one can find a number ${\displaystyle \alpha }$ such that

${\displaystyle \|A\|_{\square }\leq \alpha \leq C\|A\|_{\square },}$

where ${\displaystyle C}$ is an absolute constant.[18] This approximation algorithm uses semidefinite programming.

We give a sketch of this approximation algorithm. Let ${\displaystyle B=(b_{ij})}$ be ${\displaystyle (m+1)\times (n+1)}$ matrix defined by

${\displaystyle {\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}&-\sum _{k=1}^{n}a_{1k}\\a_{21}&a_{22}&\ldots &a_{2n}&-\sum _{k=1}^{n}a_{2k}\\\vdots &\vdots &\ddots &\vdots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}&-\sum _{k=1}^{n}a_{mk}\\-\sum _{\ell =1}^{m}a_{\ell 1}&-\sum _{\ell =1}^{m}a_{\ell 2}&\ldots &-\sum _{\ell =1}^{m}a_{\ell n}&\sum _{k=1}^{n}\sum _{\ell =1}^{m}a_{\ell k}\end{pmatrix}}.}$

One can verify that ${\displaystyle \|A\|_{\square }=\|B\|_{\square }}$ by observing, if ${\displaystyle S\in [m+1],T\in [n+1]}$ form a maximizer for the cut norm of ${\displaystyle B}$, then

${\displaystyle S^{*}={\begin{cases}S,&{\text{if }}m+1\not \in S,\\{[m]}\setminus S,&{\text{otherwise}},\end{cases}}\qquad T^{*}={\begin{cases}T,&{\text{if }}n+1\not \in T,\\{[n]}\setminus S,&{\text{otherwise}},\end{cases}}\qquad }$

form a maximizer for the cut norm of ${\displaystyle A}$. Next, one can verify that ${\displaystyle \|B\|_{\square }=\|B\|_{\infty \to 1}/4}$, where

${\displaystyle \|B\|_{\infty \to 1}=\max \left\{\sum _{i=1}^{m+1}\sum _{j=1}^{n+1}b_{ij}\varepsilon _{i}\delta _{j}:\varepsilon _{1},\ldots ,\varepsilon _{m+1}\in \{-1,1\},\delta _{1},\ldots ,\delta _{n+1}\in \{-1,1\}\right\}.}$[19]

Although not important in this proof, ${\displaystyle \|B\|_{\infty \to 1}}$ can be interpreted to be the norm of ${\displaystyle B}$ when viewed as a linear operator from ${\displaystyle \ell _{\infty }^{m}}$ to ${\displaystyle \ell _{1}^{m}}$.

Now it suffices to design an efficient algorithm for approximating ${\displaystyle \|A\|_{\infty \to 1}}$. We consider the following semidefinite program:

${\displaystyle {\text{SDP}}(A)=\max \left\{\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}\left\langle x_{i},y_{j}\right\rangle :x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n}\in S^{n+m-1}\right\}.}$

Then ${\displaystyle {\text{SDP}}(A)\geq \|A\|_{\infty \to 1}}$. The Grothedieck inequality implies that ${\displaystyle {\text{SDP}}(A)\leq K_{G}^{\mathbb {R} }\|A\|_{\infty \to 1}}$. Many algorithms (such as interior-point methods, first-order methods, the bundle method, the augmented Lagrangian method) are known to output the value of a semidefinite program up to an additive error ${\displaystyle \varepsilon }$ in time that is polynomial in the program description size and ${\displaystyle \log(1/\varepsilon )}$.[20] Therefore, one can output ${\displaystyle \alpha ={\text{SDP}}(B)}$ which satisfies

${\displaystyle \|A\|_{\square }\leq \alpha \leq C\|A\|_{\square }\qquad {\text{with}}\qquad C=K_{G}^{\mathbb {R} }.}$

Szemerédi's regularity lemma is a useful tool in graph theory, asserting (informally) that any graph can be partitioned into a controlled number of pieces that interact with each other in a pseudorandom way. Another application of the Grothendieck inequality is to produce a partition of the vertex set that satisfies the conclusion of Szemerédi's regularity lemma, via the cut norm estimation algorithm, in time that is polynomial in the upper bound of Szemerédi's regular partition size but independent of the number of vertices in the graph.[19]

It turns out that the main "bottleneck" of constructing a Szemeredi's regular partition in polynomial time is to determine in polynomial time whether or not a given pair ${\displaystyle (X,Y)}$ is close to being ${\displaystyle \varepsilon }$-regular, meaning that for all ${\displaystyle S\subset X,T\subset Y}$ with ${\displaystyle |S|\geq \varepsilon |X|,|T|\geq \varepsilon |Y|}$, we have

${\displaystyle \left|{\frac {e(S,T)}{|S||T|}}-{\frac {e(X,Y)}{|X||Y|}}\right|\leq \varepsilon ,}$

where ${\displaystyle e(X',Y')=|\{(u,v)\in X'\times Y':uv\in E\}|}$ for all ${\displaystyle X',Y'\subset V}$ and ${\displaystyle V,E}$ are the vertex and edge sets of the graph, respectively. To that end, we construct an ${\displaystyle n\times n}$ matrix ${\displaystyle A=(a_{xy})_{(x,y)\in X\times Y}}$, where ${\displaystyle n=|V|}$, defined by

${\displaystyle a_{xy}={\begin{cases}1-{\frac {e(X,Y)}{|X||Y|}},&{\text{if }}xy\in E,\\-{\frac {e(X,Y)}{|X||Y|}},&{\text{otherwise}}.\end{cases}}}$

Then for all ${\displaystyle S\subset X,T\subset Y}$,

${\displaystyle \left|\sum _{x\in S,y\in T}a_{xy}\right|=|S||T|\left|{\frac {e(S,T)}{|S||T|}}-{\frac {e(X,Y)}{|X||Y|}}\right|.}$

Hence, if ${\displaystyle (X,Y)}$ is not ${\displaystyle \varepsilon }$-regular, then ${\displaystyle \|A\|_{\square }\geq \varepsilon ^{3}n^{2}}$. It follows that using the cut norm approximation algorithm together with the rounding technique, one can find in polynomial time ${\displaystyle S\subset X,T\subset Y}$ such that

${\displaystyle \min \left\{n|S|,n|T|,n^{2}\left|{\frac {e(S,T)}{|S||T|}}-{\frac {e(X,Y)}{|X||Y|}}\right|\right\}\geq \left|\sum _{x\in S,y\in T}a_{xy}\right|\geq {\frac {1}{K_{G}^{\mathbb {R} }}}\varepsilon ^{3}n^{2}\geq {\frac {1}{2}}\varepsilon ^{3}n^{2}.}$

Then the algorithm for producing a Szemerédi's regular partition follows from the constructive argument of Alon et al.[21]

The Grothendieck inequality of a graph states that for each ${\displaystyle n\in \mathbb {N} }$ and for each graph ${\displaystyle G=(\{1,\ldots ,n\},E)}$ without self loops, there exists a universal constant ${\displaystyle K>0}$ such that every ${\displaystyle n\times n}$ matrix ${\displaystyle A=(a_{ij})}$ satisfies that

${\displaystyle \max _{x_{1},\ldots ,x_{n}\in S^{n-1}}\sum _{ij\in E}a_{ij}\left\langle x_{i},x_{j}\right\rangle \leq K\max _{\varepsilon _{1},\ldots ,\varepsilon _{n}\in \{-1,1\}}\sum _{ij\in E}a_{ij}\varepsilon _{1}\varepsilon _{n}.}$[22]

The Grothendieck constant of a graph ${\displaystyle G}$, denoted ${\displaystyle K(G)}$, is defined to be the smallest constant ${\displaystyle K}$ that satisfies the above property.

The Grothendieck inequality of a graph is an extension of the Grothendieck inequality because the former inequality is the special case of the latter inequality when ${\displaystyle G}$ is a bipartite graph with two copies of ${\displaystyle \{1,\ldots ,n\}}$ as its bipartition classes. Thus,

${\displaystyle K_{G}=\sup _{n\in \mathbb {N} }\{K(G):G{\text{ is an }}n{\text{-vertex bipartite graph}}\}.}$

For ${\displaystyle G=K_{n}}$, the ${\displaystyle n}$-vertex complete graph, the Grothendieck inequality of ${\displaystyle G}$ becomes

${\displaystyle \max _{x_{1},\ldots ,x_{n}\in S^{n-1}}\sum _{i,j\in \{1,\ldots ,n\},i\neq j}a_{ij}\left\langle x_{i},x_{j}\right\rangle \leq K(K_{n})\max _{\varepsilon _{1},\ldots ,\varepsilon _{n}\in \{-1,1\}}\sum _{i,j\in \{1,\ldots ,n\},i\neq j}a_{ij}\varepsilon _{i}\varepsilon _{j}.}$

It turns out that ${\displaystyle K(K_{n})\asymp \log n}$. On one hand, we have ${\displaystyle K(K_{n})\lesssim \log n}$.[23][24][25] Indeed, the following inequality is true for any ${\displaystyle n\times n}$ matrix ${\displaystyle A=(a_{ij})}$, which implies that ${\displaystyle K(K_{n})\lesssim \log n}$ by the Cauchy-Schwarz inequality:[22]

${\displaystyle \max _{x_{1},\ldots ,x_{n}\in S^{n-1}}\sum _{i,j\in \{1,\ldots ,n\},i\neq j}a_{ij}\left\langle x_{i},x_{j}\right\rangle \leq \log \left({\frac {\sum _{i\in \{1,\ldots ,n\}}\sum _{j\in \{1,\ldots ,n\}\setminus \{i\}}|a_{ij}|}{\sqrt {\sum _{i\in \{1,\ldots ,n\}}\sum _{j\in \{1,\ldots ,n\}\setminus \{i\}}a_{ij}^{2}}}}\right)\max _{\varepsilon _{1},\ldots ,\varepsilon _{n}\in \{-1,1\}}\sum _{i,j\in \{1,\ldots ,n\},i\neq j}a_{ij}\varepsilon _{1}\varepsilon _{n}.}$

On the other hand, the matching lower bound ${\displaystyle K(K_{n})\gtrsim \log n}$ is due to Alon, Makarychev, Makarychev and Naor in 2006.[22]

The Grothendieck inequality ${\displaystyle K(G)}$ of a graph ${\displaystyle G}$ depends upon the structure of ${\displaystyle G}$. It is known that

${\displaystyle \log \omega \lesssim K(G)\lesssim \log \vartheta ,}$[22]

and

${\displaystyle K(G)\leq {\frac {\pi }{2\log \left({\frac {1+{\sqrt {(\vartheta -1)^{2}+1}}}{\vartheta -1}}\right)}},}$[26]

where ${\displaystyle \omega }$ is the clique number of ${\displaystyle G}$, i.e., the largest ${\displaystyle k\in \{2,\ldots ,n\}}$ such that there exists ${\displaystyle S\subset \{1,\ldots ,n\}}$ with ${\displaystyle |S|=k}$ such that ${\displaystyle ij\in E}$ for all distinct ${\displaystyle i,j\in S}$, and

${\displaystyle \vartheta =\min \left\{\max _{i\in \{1,\ldots ,n\}}{\frac {1}{\langle x_{i},y\rangle }}:x_{1},\ldots ,x_{n},y\in S^{n},\left\langle x_{i},x_{j}\right\rangle =0\;\forall ij\in E\right\}.}$

The parameter ${\displaystyle \vartheta }$ is known as the Lovász theta function of the complement of ${\displaystyle G}$.[27][28][22]

In the application of the Grothendieck inequality for approximating the cut norm, we have seen that the Grothendieck inequality answers the following question: How well does an optimal solution to the semidefinite program ${\displaystyle {\text{SDP}}(A)}$ approximate ${\displaystyle \|A\|_{\infty \to 1}}$, which can be viewed as an optimization problem over the unit cube? More generally, we can ask similar questions over convex bodies other than the unit cube.

For instance, the following inequality is due to Naor and Schechtman[29] and independently due to Guruswami et al:[30] For every ${\displaystyle n\times n}$ matrix ${\displaystyle A=(a_{ij})}$ and every ${\displaystyle p\geq 2}$,

${\displaystyle \max _{x_{1},\ldots ,x_{n}\in \mathbb {R} ^{n},\sum _{k=1}^{n}\|x_{k}\|_{2}^{p}\leq 1}\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}\left\langle x_{i},x_{j}\right\rangle \leq \gamma _{p}^{2}\max _{t_{1},\ldots ,t_{n}\in \mathbb {R} ,\sum _{k=1}^{n}|t_{k}|^{p}\leq 1}\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}t_{i}t_{j},}$

where

${\displaystyle \gamma _{p}={\sqrt {2}}\left({\frac {\Gamma ((p+1)/2)}{\sqrt {\pi }}}\right)^{1/p}.}$

The constant ${\displaystyle \gamma _{p}^{2}}$ is sharp in the inequality. Stirling's formula implies that ${\displaystyle \gamma _{p}^{2}=p/e+O(1)}$ as ${\displaystyle p\to \infty }$.