Shapes and Norms of Random Pairs

Rostislav Matveev

Abstract

The shape function of a pair of finite-valued random variables was introduced in [MR26], where it was used to derive a spectral bound on the entanglement of the pair, a quantity measuring the extent to which their mutual information can be extracted. In this article, we further develop the theory of shape functions for pairs of random variables. We prove that, when $(X,Y)$ is uniformly supported on the edges of a biregular bipartite graph, the value of the shape function $\mathcal{S}(X,Y)(\alpha,\beta)$ equals the logarithm of the operator norm of the graph’s incidence matrix with respect to Lebesgue exponents determined by $(\alpha,\beta)$ .This identification, in particular, enables the numerical approximation of the shape function and, by duality, of the extension profile, also known as the tension region, of the pair. We also establish a collection of relations and inequalities satisfied by shape functions, including convexity and monotonicity properties, composition inequalities, and relations describing their behavior under conditioning and the adjoining of variables.

1 Shape function and its relation to the norms of incidence operators

The shape function of a pair of jointly distributed random variables $(X,Y)$ , introduced in [MR26], is a function on $[0,1]^{2}$ defined by

\mathcal{S}(X,Y)(\alpha,\beta):=\sup\left\{\alpha\cdot H(X|W)+\beta\cdot H(Y|W)-I(X:Y|W)\right\}

where supremum is taken over all extensions $(X,Y,W)$ . Function $\mathcal{S}(X,Y)$ is convex on the square and it is affine on the upper-right triangle

\left\{(\alpha,\beta)\;{\bm{:}}\;\alpha+\beta\geq 1\right\}\cap[0,1]^{2}

The affine part is determined by the entropy profile of the pair, that is entropies of the two variables and their joint. In general, the restriction of $\mathcal{S}(X,Y)$ to the lower-left triangle is not determined by the entropy profile and contains additional information about the pair. In particular, the entanglement of the pair $(X,Y)$ – the quantity that measures to what extend mutual information is extractable and defined by

\mathcal{E}(X,Y)=\sup_{W}\left\{I(X:Y|W)+I(W:Y|X)+I(X:W|Y)\right\}

can be recovered from $\mathcal{S}(X,Y)$ , see [MR26, MR26a] and Equation (MMRV) on page MMRV of the present article.

If the pair $(X,Y)$ is uniform on its support, that is, each of the random variables $X$ , $Y$ and their joint $XY$ are uniform on their respective supports, then it is completely determined by the supporting graph $\mathsf{G}=(\mathsf{X}\sqcup\mathsf{Y},\mathsf{E})$ , where

\mathsf{X}:=\supp X,\qquad\mathsf{Y}:=\supp Y,\qquad\mathsf{E}:=\supp XY

Such a bipartite graph, which is necessarily biregular, can be represented by its bipartite incidence matrix $M$ of size $|\mathsf{X}|\times|\mathsf{Y}|$ .

In this article we evaluate the shape function of a pair uniform on its support in terms of the incidence matrix of the supporting graph. We view $M$ as a matrix, with respect to standard bases, of the bilinear form

M:\ell^{p}_{\mathsf{X}}\times\ell^{q}_{\mathsf{Y}}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R},\qquad\varphi M\psi:=\sum_{(\mathsf{x},\mathsf{y})\in\mathsf{E}}\varphi(\mathsf{x})\psi(\mathsf{y})

where $p,q\in[1,\infty]$ and where we use infix notation $\varphi M\psi$ for evaluation of the form. We define its $(p,q)$ -norm by

\|M\|_{p,q}:=\sup\left\{\frac{|\varphi M\psi|}{\|\varphi\|_{p}\cdot\|\psi\|_{q}}\;{\bm{:}}\;0\neq\varphi\in\ell^{p}_{\mathsf{X}},\;0\neq\psi\in\ell^{q}_{\mathsf{Y}}\right\}

The main result of this article is the following theorem.

Theorem A.

Let $(X,Y)$ be a pair of random variables uniformly supported on a homogeneous bipartite graph $\mathsf{G}=(\mathsf{X}\sqcup\mathsf{Y},\mathsf{E})$ with the incidence form $M$ . Then for all $\alpha,\beta\in[0,1]$ holds

\mathcal{S}(X,Y)(\alpha,\beta)=\log\|M\|_{p,q}

where $1/p=1-\alpha$ and $1/q=1-\beta$ .

By Tropical Asymptotic Equipartition Property, [MP18], a pair $(X^{n},Y^{n})$ obtained by taking $n$ independent copies of a pair $(X,Y)$ , can be arbitrarily well approximated on the normalized scale by a pair of random variables uniformly supported on a homogeneous graph. Since the shape is stable, that is,

\mathcal{S}(X^{n},Y^{n})=n\cdot\mathcal{S}(X,Y),

and continuous with respect to such approximations, Theorem A allows one, in principle, to evaluate shape function for an arbitrary, not necessarily uniformly supported, pair.

Theorem A also gives a practical way to approximate the shape function. Indeed, the norm of the bilinear form $\|M\|_{p,q}$ is the same as the operator norm $\|M\|_{p\stackrel{{\scriptstyle}}{{\rightarrow}}q^{\prime}}$ , where $q^{\prime}$ is the Hölder conjugate of $q$ . Lebesgue operator norms $\|M\|_{p\stackrel{{\scriptstyle}}{{\rightarrow}}q^{\prime}}$ can be approximated numerically by robust methods. Thus, for pairs uniformly supported on biregular bipartite graphs, one can numerically approximate both the shape function and, by duality, the extension profile.

In the next section we introduce our notation and conventions. Section 3 we recall some facts about norms on tensor product of normed vector spaces and prove multiplicativity of Lebesgue operator norms in the hypercontractive regime, Proposition 3.3.A. Further we prove our main technical tool, Theorem 3.4.C, that asserts that for high tensor powers of bilinear forms there are uniform vectors which are almost in resonance. In Section 4 we apply the results of the previous sections to bilinear forms associated to bipartite graphs. Theorem A is proven in Section 5. We also derive inequalities satisfied by the shape function in Section 6. Especially interesting is the inequality (COMP) on page COMP. We do not know whether this inequality can be derived in the purely entropic context, that is without using Theorem A.

2 Notation, conventions, recollections

2.1 Notation

For a natural number $n$ we denote $[n]:=\left\{0,\dots,n-1\right\}$ . We write $\sharp S$ or $|S|$ for the cardinality of a finite set $S$ . Denote the $\log$ -cardinality of a set by $[S]:=\log|S|$ . For a finite set $S$ , we write $\mathbb{R}^{S}$ for the vector space of real-valued functions on $S$ . For a subset $T\subset S$ and a point $s\in S$ we denote by $\bm{1}_{T}\in\mathbb{R}^{S}$ the indicator function of $T$ and by $\delta_{s}:=\bm{1}_{\left\{s\right\}}$ the point mass.

For an extended real $p\in[1,\infty]$ we denote by $p^{\prime}$ its Hölder conjugate, that is the two extended reals $p,p^{\prime}$ must satisfy $1/p+1/p^{\prime}=1$ .

We use logarithms with the natural base throughout the article and $\mathbf{e}$ denotes Euler’s number.

2.2 Random variables

All random variables in this article have finite alphabets. As a notational convention, use capitals $X,Y,\dots$ for random variables and the corresponding san-serif letters $\mathsf{X},\mathsf{Y},\dots$ for their alphabets. For a random variable $X$ and $\mathsf{x}\in\mathsf{X}$ we use $p_{X}(\mathsf{x}):=P[X=\mathsf{x}]$ . We call $\mathsf{x}\in\mathsf{X}$ an atom of $X$ if $p_{X}(\mathsf{x})>0$ .The support $\supp X$ is the collection of all atoms.

We tacitly assume that the alphabets of different random variables are disjoint and we write $(X|\mathsf{y})$ for the conditional random variable in lieu of $(X|Y=\mathsf{y})$ .

Tuples of random variables are written as comma-separated lists, such as, for example, $(X,Y,Z)$ or $(X_{i}\;{\bm{:}}\;i\in[n])$ , while joints of random variables, regarded as a single random variable, where marginalization structures are ignored, are denoted by concatenation of the corresponding letters, such as $XY$ or $XY\!Z$ , or by using the subset for the subscript, as in $X_{I}$ for the joint of $(X_{i}:i\in I)$ , $I\subset[n]$ . For example, for a triple of random variables, $(X,Y,Z)$ , the notation $(X,Y\!Z)$ stands for a pair of variables consisting of variable $X$ and the joint variable $Y\!Z$ . This pair is different from the pair $(XY,Z)$ . Given a pair $(X,Y)$ , a third random variable $W$ jointly distributed with $(X,Y)$ forms an extension $(X,Y,W)$ of the pair and is called an extending variable.

Supports of the variables in the pair $(X,Y)$ form a bipartite graph $\mathsf{G}=(\mathsf{X}\sqcup\mathsf{Y},\mathsf{E})$ , where

\mathsf{X}:=\supp X,\qquad\mathsf{Y}:=\supp Y,\qquad\mathsf{E}:=\supp XY

We say that $(X,Y)$ is supported on $\mathsf{G}$ and write

\mathsf{G}=\supp(X,Y)

A tuple of random variables $(X_{i}\;{\bm{:}}\;i\in[n])$ is called uniform on its support if all partial joints $X_{I}$ , $I\subset[n]$ are uniform on their respective supports. If the pair $(X,Y)$ is uniform on its support, then the supporting graph $\mathsf{G}=\supp(X,Y)$ is biregular and its combinatorial structure completely determines the pair. In that case we say that $(X,Y)$ uniformly supported on $\mathsf{G}$ .

2.3 Graphs

All graphs considered in this article have no isolated vertices. A bipartite graph $\mathsf{G}=(\mathsf{X}\sqcup\mathsf{Y},\mathsf{E})$ is called biregular if the degrees of vertices are constant within each part. We denote by $d_{1}(\mathsf{G}),d_{2}(\mathsf{G})$ the left and right degrees of $\mathsf{G}$ , respectively. The automorphism group $\aut(\mathsf{G})$ of $\mathsf{G}$ is the group of symmetries of $\mathsf{G}$ preserving each part. Graph is called homogeneous if $\aut(\mathsf{G})$ acts transitively on the edge set $\mathsf{E}$ . Homogeneous graphs are biregular. By a subgraph $\mathsf{H}\subset\mathsf{G}$ we always mean a nonempty subgraph without isolated vertices. We denote by $\mathsf{X}_{\mathsf{H}}$ , $\mathsf{Y}_{\mathsf{H}}$ and $\mathsf{E}_{\mathsf{H}}$ the left, right parts and edge-set of $\mathsf{H}$ , respectively. We also set

		$\displaystyle[\mathsf{X}_{\mathsf{H}}]:=\log\|\mathsf{X}_{\mathsf{H}}\|,$		$\displaystyle[\mathsf{E}_{\mathsf{H}}]:=\log\|\mathsf{E}_{\mathsf{H}}\|,$
		$\displaystyle[\mathsf{Y}_{\mathsf{H}}]:=\log\|\mathsf{Y}_{\mathsf{H}}\|,$		$\displaystyle[\mathsf{X}_{\mathsf{H}}:\mathsf{Y}_{\mathsf{H}}]:=\log\frac{\|\mathsf{X}_{\mathsf{H}}\|\cdot\|\mathsf{Y}_{\mathsf{H}}\|}{\|\mathsf{E}_{\mathsf{H}}\|}$

For a pair $(X,Y)$ uniformly supported on a graph $\mathsf{G}=(\mathsf{X}\sqcup\mathsf{Y},\mathsf{E})$ the following identities hold:

		$\displaystyle H(X)=[\mathsf{X}],$		$\displaystyle H(XY)=[\mathsf{E}],$		$\displaystyle H(Y\|X)=\log d_{1}(\mathsf{G}),$
		$\displaystyle H(Y)=[\mathsf{Y}],$		$\displaystyle I(X:Y)=[\mathsf{X}:\mathsf{Y}],$		$\displaystyle H(X\|Y)=\log d_{2}(\mathsf{G})$

2.4 Extension profile and shape function

The entropy profile of a pair $(X,Y)$ is a vector

e(X,Y):=\left(\!\!\!\begin{array}[]{c}H(X)\\ H(Y)\\ I(X:Y)\end{array}\!\!\!\right)\in\mathbb{R}^{3}

For an extension $(X,Y,W)$ the conditional entropy profile is

e(X,Y|W):=\left(\!\!\!\begin{array}[]{c}H(X|W)\\ H(Y|W)\\ I(X:Y|W)\end{array}\!\!\!\right)

The extension profile of a pair $(X,Y)$ is the set of all conditional entropy profiles for all extensions of the pair.

\ext(X,Y):=\left\{e(X,Y|W)\;{\bm{:}}\;\text{$(X,Y,W)$ is an extension of $(X,Y)$}\right\}\subset\mathbb{R}^{3}

It is closely related to the tension region; see [PP14, LE17, Csi23]. One may think of the extending variable $W$ as a probe testing finer properties of the relation between variables $X$ and $Y$ — those that are not already reflected in the entropy profile of the pair.

The extension profile of any pair is a convex compact subset of $\mathbb{R}^{3}$ . In [MR26] a dual object, the so called shape function, or simply shape is introduced and studied. It is the function on the square $[0,1]^{2}$ in the $(\alpha,\beta)$ -plane defined by

	$\displaystyle\mathcal{S}(X,Y)(\alpha,\beta)$	$\displaystyle:=\sup\left\{\alpha\cdot x+\beta\cdot y-z\;{\bm{:}}\;(x,y,z)\in\ext(X,Y)\right\}$
		$\displaystyle\phantom{:}=\sup_{W}\left\{\alpha\cdot H(X\|Y)+\beta\cdot H(Y\|W)-I(X:Y\|W)\right\}$

where the later supremum is over all extending variables $W$ .

For the basic properties of $\mathcal{S}(X,Y)$ , we refer the reader to [MR26]. There, the shape function is studied in detail, including its upper and lower bounds and, for pairs uniformly supported on graphs, its relation to spectral properties of the supporting graph. In this article we establish some additional properties of the shape.

3 Norms on tensor products

Here we recall several standard facts about tensor products of normed vector spaces. We restrict attention to finite-dimensional spaces, although many of the constructions discussed below extend to general Banach spaces with the usual additional analytic care. Standard references for tensor products of Banach spaces include [DF93, Rya02].

The main results of this section are Theorem 3.4.C and Corollary 3.4.D. The main technical tool, Proposition 3.3.A, is, most likely, known to the specialists, but for the lack of a suitable reference we provide a proof.

3.1 Tensor products

We write $A\stackrel{{\scriptstyle a}}{{\cong}}B$ for an algebraic isomorphism of vector spaces, and $A\stackrel{{\scriptstyle b}}{{\cong}}B$ for an isomorphism of Banach spaces.

We use several natural algebraic isomorphisms between tensor products of finite-dimensional vector spaces. Since these isomorphisms are canonical, we shall use them implicitly and suppress them from the notation:

$\displaystyle A\otimes B$	$\displaystyle\stackrel{{\scriptstyle a}}{{\cong}}B\otimes A,$	(commutativity)
$\displaystyle(A\otimes B)\otimes C$	$\displaystyle\stackrel{{\scriptstyle a}}{{\cong}}A\otimes(B\otimes C),$	(associativity)
$\displaystyle(A\otimes B)^{*}$	$\displaystyle\stackrel{{\scriptstyle a}}{{\cong}}A^{}\otimes B^{},$	( $\text{duality}_{1}$ )
$\displaystyle\Hom(A,B)$	$\displaystyle\stackrel{{\scriptstyle a}}{{\cong}}A^{*}\otimes B.$	( $\text{duality}_{2}$ )

For example, for our purposes a linear map $M:A\stackrel{{\scriptstyle}}{{\rightarrow}}B$ , its adjoint $M^{*}\colon B^{*}\stackrel{{\scriptstyle}}{{\rightarrow}}A^{*}$ , the bilinear form

B^{*}\times A\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R},\qquad(b^{*},a)\mapsto b^{*}Ma,

and the tensor in $A^{*}\otimes B$ are regarded as different realizations of the same object, denoted by $M$ .

We shall also use the following algebraic identifications for spaces of functions on finite sets:

\mathbb{R}^{\mathsf{X}}\otimes\mathbb{R}^{\mathsf{Y}}\stackrel{{\scriptstyle a}}{{\cong}}\mathbb{R}^{\mathsf{X}\times\mathsf{Y}}\stackrel{{\scriptstyle a}}{{\cong}}\left(\mathbb{R}^{\mathsf{X}}\right)^{\!\mathsf{Y}}

3.2 Cross norms

3.2.A Algebraic definition of cross norms

Let $(A,\alpha)$ and $(B,\beta)$ be Banach spaces. In this article, we call a norm $\gamma$ on the algebraic tensor product $A\otimes B$ a cross norm¹¹ 1 Sometimes in the literature a cross norm is defined as a norm on the tensor product satisfying only the first condition in (3.2.B), while a norm satisfying both conditions is called a reasonable cross norm. if, for all $a\in A$ , $b\in B$ , $a^{*}\in A^{*}$ , and $b^{*}\in B^{*}$ , one has

	$\displaystyle\gamma(a\otimes b)$	$\displaystyle=\alpha(a)\cdot\beta(b),$		(3.2.B)
	$\displaystyle\gamma^{}(a^{}\otimes b^{*})$	$\displaystyle=\alpha^{}(a^{})\cdot\beta^{}(b^{})$		(3.2.B)

where $\alpha^{*}$ stands for the dual norm on $A^{*}$ , etc.

3.2.C Geometric definition of cross norms

For the geometrically oriented reader, cross norms admit the following equivalent description. For a Banach space $(A,\alpha)$ , denote by $\mathbb{S}_{\alpha}\subset A$ the $\alpha$ -unit sphere. The projectivized product

P(\mathbb{S}_{\alpha}\times\mathbb{S}_{\beta}):=\left\{a\otimes b:\ a\in\mathbb{S}_{\alpha},\ b\in\mathbb{S}_{\beta}\right\}

is naturally contained in the tensor product $A\otimes B$ .

A norm $\gamma$ on $A\otimes B$ is a cross norm if and only if

P(\mathbb{S}_{\alpha}\times\mathbb{S}_{\beta})\subset\mathbb{S}_{\gamma}\qquad\text{and}\qquad P(\mathbb{S}_{\alpha^{*}}\times\mathbb{S}_{\beta^{*}})\subset\mathbb{S}_{\gamma^{*}}

3.2.D Injective and projective cross norms

Here we consider two notable examples of cross norms: the least/injective and the greatest/projective cross norms.²² 2 These two norms are often denoted by $\varepsilon$ and $\pi$ , respectively. We use different notation to emphasize the dependence of the injective and projective tensor norms on $\alpha$ and $\beta$ .

	$\displaystyle\alpha\vee\beta(v)$	$\displaystyle:=\sup\left\{(a^{}\otimes b^{})(v)\;{\bm{:}}\;a^{}\in\mathbb{B}_{\alpha^{}},\;b^{}\in\mathbb{B}_{\beta^{}}\right\}$		(injective)
	$\displaystyle\alpha\wedge\beta(v)$	$\displaystyle:=\inf\left\{\sum\alpha(a_{i})\beta(b_{i})\;{\bm{:}}\;v=\sum a_{i}\otimes b_{i}\right\}$		(projective)

where $v\in A\otimes B$ .

The operations $\vee$ and $\wedge$ are dual to each other in the sense that

(\alpha\vee\beta)^{*}=\alpha^{*}\wedge\beta^{*}\qquad\text{and}\qquad(\alpha\wedge\beta)^{*}=\alpha^{*}\vee\beta^{*}

The projective norm can be geometrically defined by the following. We declare the $\alpha\wedge\beta$ -unit ball to be the smallest convex set containing the projectivized product of the unit balls in $A$ and $B$ , that is

\mathbb{B}_{\alpha\wedge\beta}:=\convexhull P(\mathbb{B}_{\alpha}\times\mathbb{B}_{\beta})

where $\mathbb{B}_{\alpha}$ stands for the closed unit ball in a Banach space $(A,\alpha)$ , etc. By duality there is a similar description of the injective norm $\alpha\vee\beta$ .

Note that for an operator $M:(A,\alpha)\stackrel{{\scriptstyle}}{{\rightarrow}}(B,\beta)$ (which can be considered as an element of $A^{*}\otimes B$ ) the operator norm is

\|M\|_{\alpha\stackrel{{\scriptstyle}}{{\rightarrow}}\beta}=(\alpha^{*}\vee\beta)(M)

It is known and follows directly from the geometric description of the projective and injective norms that a norm $\gamma$ on $A\otimes B$ is a cross norm if and only if

\alpha\vee\beta\leq\gamma\leq\alpha\wedge\beta

3.2.E Cross norms for Lebesgue spaces

For a Banach space $(A,\|\cdot\|)$ , finite set $I$ and $p\in[1,\infty]$ denote by

\ell_{I}^{p}(A):=A^{I}

the Banach space of $I$ -indexed families of vectors in $A$ with the norm

\|(a_{i})\|_{p,I}:=\left(\sum_{i\in I}\|a_{i}\|^{p}\right)^{1/p}

with the usual provision for the case $p=\infty$ . We will write $\ell^{p}_{I}:=\ell^{p}_{I}(\mathbb{R})$ .

An important example for our purposes is

\ell^{p}_{I}(\ell^{q}_{J})=\left(\mathbb{R}^{I\times J},\|x\|_{p,I;q,J}:=\Big(\sum_{i\in I}\big(\sum_{j\in J}|x_{ij}|^{q}\big)^{p/q}\Big)^{1/p}\right)

with the usual modifications in the cases where $p$ or $q$ is infinite.

It is straightforward to establish, that for finite $p,q$

\big(\ell^{p}_{I}(\ell^{q}_{J})\big)^{*}\stackrel{{\scriptstyle b}}{{\cong}}\ell^{p^{\prime}}_{I}(\ell^{q^{\prime}}_{J})

where $p^{\prime},q^{\prime}$ are the Hölder conjugates of $p,q$ , respectively. There are canonical algebraic isomorphisms independent of $p,q$

\ell^{p}_{I}(\ell^{q}_{J})\stackrel{{\scriptstyle a}}{{\cong}}\ell^{q}_{J}(\ell^{p}_{I})\stackrel{{\scriptstyle a}}{{\cong}}\mathbb{R}^{I\times J}

To avoid ambiguity, we denote the norm on $\ell^{p}_{I}(\ell^{q}_{J})$ by

\|(x_{ij})\|_{p,I;q,J}:=\left(\sum_{i\in I}\left(\sum_{j\in J}|x_{ij}|^{q}\right)^{p/q}\right)^{1/p}

Another pair of isomorphisms, that are used in what follows, are

\ell_{I}^{p}\otimes\ell_{J}^{p}\stackrel{{\scriptstyle a}}{{\cong}}\ell_{I\times J}^{p}\stackrel{{\scriptstyle b}}{{\cong}}\ell_{I}^{p}(\ell_{J}^{p})

We note that, under the algebraic identification above the Lebesgue $p$ -norm on $\ell_{I\times J}^{p}$ is a cross norm with respect to the tensor product decomposition $\ell_{I}^{p}\otimes\ell_{J}^{p}$ . This follows directly from the definitions and duality $(\ell^{p}_{I})^{*}\stackrel{{\scriptstyle b}}{{\cong}}\ell^{p^{\prime}}_{I}$ .

3.3 Multiplicative family of norms for bilinear forms

Let $I,J$ be finite sets. For a bilinear form $M:\ell_{I}^{p}\times\ell_{J}^{q}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}$ define

\|M\|_{p,q}:=\sup\left\{\frac{|aMb|}{\|a\|_{p}\cdot\|b\|_{q}}\;{\bm{:}}\;0\neq a\in\ell_{I}^{p},\;0\neq b\in\ell_{J}^{q}\right\}

where we use infix notation for the value $aMb$ of the form $M$ on vectors $a$ , $b$ . If we view $M$ as an element from $\ell_{I}^{p^{\prime}}\otimes\ell_{J}^{q^{\prime}}$ or as the operator $M:\ell_{I}^{p}\stackrel{{\scriptstyle}}{{\rightarrow}}\ell_{J}^{q^{\prime}}$ , then

\|M\|_{p,q}=(\|\cdot\|_{p^{\prime}}\vee\|\cdot\|_{q^{\prime}})(M)=\|M\|_{p\stackrel{{\scriptstyle}}{{\rightarrow}}q^{\prime}}

The main technical result, Proposition 3.3.A, asserts that in hypercontractive regime, $\frac{1}{p}+\frac{1}{q}\geq 1$ , the norms $\|\cdot\|_{p,q}$ form a multiplicative family for bilinear forms on $\ell_{I}^{p}\times\ell_{J}^{q}$ , where $I,J$ range over finite sets.

3.3.A.

Let $p,q\in[1,\infty]$ satisfy $1/p+1/q\geq 1$ and let $I,J,K,L$ be finite sets. For two bilinear forms

M:\ell^{p}_{I}\times\ell^{q}_{J}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}\qquad\text{and}\qquad N:\ell^{p}_{K}\times\ell^{q}_{L}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}

consider their tensor product

M\otimes N:\ell^{p}_{I\times K}\times\ell^{q}_{J\times L}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}

Then

\|M\otimes N\|_{p,q}=\|M\|_{p,q}\cdot\|N\|_{p,q}

To prove the proposition we shall use the following lemma, a version of the so called Minkowski’s integral inequality, [HLP52], that says that under canonical algebraic isomorphism $\ell^{p}_{I}(\ell^{q}_{J})\stackrel{{\scriptstyle a}}{{\cong}}\ell^{q}_{J}(\ell^{p}_{I})$ norms $\|\cdot\|_{p,I;q,J}$ and $\|\cdot\|_{q,J;p,I}$ are comparable.

3.3.B.

Let $I,J$ be finite sets. If $1\leq p\leq q\leq\infty$ , then

\|\cdot\|_{q,J;p,I}\leq\|\cdot\|_{p,I;q,J}

We give a proof of the lemma in Section 3.3.E.

3.3.C Proof of Proposition 3.3.A

By renormalization, it is enough to prove that

\textit{If}\quad\|M\|_{p,q}=\|N\|_{p,q}=1,\quad\textit{then}\quad\|M\otimes N\|_{p,q}=1.

First we prove the lower bound. Indeed, since the Lebesgue norms are cross norms,

\|a\otimes c\|_{\ell_{I\times K}^{p}}=\|a\|_{\ell_{I}^{p}}\cdot\|c\|_{\ell_{K}^{p}},\qquad\|b\otimes d\|_{\ell_{J\times L}^{q}}=\|b\|_{\ell_{J}^{q}}\cdot\|d\|_{\ell_{L}^{q}}.

Hence, by testing against decomposable vectors, we have

\|M\otimes N\|_{p,q}\geq\|M\|_{p,q}\cdot\|N\|_{p,q}=1

Now we prove the upper bound. Let

x=(x_{ik}\;{\bm{:}}\;i\in I,\;k\in K)\in\ell_{I\times K}^{p},\qquad y=(y_{jl}\;{\bm{:}}\;j\in J,\;l\in L)\in\ell_{J\times L}^{q}

be such that $\|x\|_{p}\leq 1$ and $\|y\|_{q}\leq 1$ .

For each $k\in K$ and $j\in J$ , set

x_{k}:=(x_{ik}\;{\bm{:}}\;i\in I)\in\ell_{I}^{p},\qquad y_{j}:=(y_{jl}\;{\bm{:}}\;l\in L)\in\ell_{L}^{q}

Since $\|M\|_{p,q}=1$ , the associated operator

M\colon\ell_{I}^{p}\stackrel{{\scriptstyle}}{{\rightarrow}}\ell_{J}^{q^{\prime}}

has norm $1$ . Hence

(Mx_{k}\;{\bm{:}}\;k\in K)\in\ell_{K}^{p}(\ell_{J}^{q^{\prime}})

and

\|(Mx_{k}\;{\bm{:}}\;k\in K)\|_{p,K;\,q^{\prime},J}\leq\|(x_{k})_{k\in K}\|_{p,K;\,p,I}=\|x\|_{\ell_{I\times K}^{p}}\leq 1.

Since

\frac{1}{p}+\frac{1}{q}\geq 1\qquad\text{iff}\qquad p\leq q^{\prime}

we can apply Lemma 3.3.B:

\|(Mx_{k})_{k\in K}\|_{q^{\prime},J;p,K}\leq\|(Mx_{k})_{k\in K}\|_{p,K;q^{\prime},J}\leq 1.

(3.3.D)

For each $j\in J$ , define

u_{j}:=\bigl((Mx_{k})_{j}\bigr)_{k\in K}\in\ell_{K}^{p}

Then (3.3.D) says that

\|(u_{j})_{j\in J}\|_{q^{\prime},J;p,K}\leq 1

Therefore, using $\|N\|_{p,q}=1$ , we get (with infix notation)

	$\displaystyle\|x(M\otimes N)y\|$	$\displaystyle=\left\|\sum_{j\in J}u_{j}Ny_{j}\right\|$
		$\displaystyle\leq\sum_{j\in J}\\|u_{j}\\|_{p}\cdot\\|y_{j}\\|_{q}$
		$\displaystyle\leq\\|(u_{j})_{j\in J}\\|_{q^{\prime},J;p,K}\cdot\\|(y_{j})_{j\in J}\\|_{q,J;q,L}$
		$\displaystyle\leq 1$

Here the last inequality uses Hölder’s inequality along $j$ -index and the fact that

\|(y_{j})_{j\in J}\|_{q,J;\,q,L}=\|y\|_{\ell_{J\times L}^{q}}\leq 1.

Thus $\|M\otimes N\|_{p,q}\leq 1$ , and the proof is complete. ∎

3.3.E Proof of Lemma 3.3.B

The lemma is immediate when $q=\infty$ . Indeed,

\|(x_{ij})\|_{\infty,J;p,I}=\max_{j\in J}\left(\sum_{i\in I}|x_{ij}|^{p}\right)^{1/p}\leq\left(\sum_{i\in I}\max_{j\in J}|x_{ij}|^{p}\right)^{1/p}=\|(x_{ij})\|_{p,I;\infty,J}.

By duality, this also implies the case $p=1$ and any $q$ , that is for any $q$ holds

\|\cdot\|_{1,I;q,J}\geq\|\cdot\|_{q,J;1,I}

For the general case, we shall use the elementary identity

\|(x_{ij})\|_{p,I;\,q,J}=\bigl\|(|x_{ij}|^{r})\bigr\|_{p/r,I;\,q/r,J}^{1/r},

valid for every $0<r\leq\min\left\{p,q\right\}$ . In particular, when $1<p\leq q<\infty$ , we may take $r=p$ . Applying the case $p=1$ to $(|x_{ij}|^{p})$ and to the exponent $q/p\geq 1$ , we obtain

	$\displaystyle\\|(x_{ij})\\|_{p,I;\,q,J}$	$\displaystyle=\bigl\\|(\|x_{ij}\|^{p})\bigr\\|_{1,I;\,q/p,J}^{1/p}$
		$\displaystyle\geq\bigl\\|(\|x_{ij}\|^{p})\bigr\\|_{q/p,J;\,1,I}^{1/p}$
		$\displaystyle=\\|(x_{ij})\\|_{q,J;\,p,I}.$

This proves the lemma in the general case. ∎

3.4 Resonance vectors for tensor powers of bilinear forms

3.4.A Resonance vectors for bilinear form. Uniform vectors

Consider a bilinear form $M:\ell^{p}_{\mathsf{X}}\times\ell^{q}_{\mathsf{Y}}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}$ . We say that a pair of non-zero vectors $u\in\ell^{p}_{\mathsf{X}}$ and $v\in\ell^{q}_{\mathsf{Y}}$ is a resonance pair or equivalently a maximizing pair for the form $M$ if (with infix notation for the value of bilinear forms)

uMv=\|M\|_{p,q}\cdot\|u\|_{p}\cdot\|v\|_{q}

Essentially, a resonance pair consists of the two optimizers in the definition of the norm of the bilinear form $M$ . Clearly any positive multiples of the vectors in a resonance pair also form a resonance pair. In finite dimensions resonance vectors always exist. If coefficients of $M$ with respect to natural bases are non-negative then there exist pair of resonance vectors, which also have non-negative coefficients.

We call a nonzero vector $u\in\ell_{\mathsf{X}}^{p}$ uniform if it is a positive scalar multiple of the indicator function of a nonempty subset of $\mathsf{X}$ .

We show below that for large tensor powers of a coordinate-wise non-negative bilinear form $M$ , there exists a pair of uniform “almost” resonance vectors. Here adverb “almost” should be understood on the normalized $\log$ -scale, see Theorem 3.4.C below.

3.4.B Uniform almost resonance pairs for tensor powers

Throughout this section we fix $p,q\in[1,\infty]$ satisfying $1/p+1/q\geq 1$ , two finite sets $\mathsf{X}$ and $\mathsf{Y}$ and a bilinear form $M:\ell^{p}_{\mathsf{X}}\times\ell^{q}_{\mathsf{Y}}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}$ of norm one and with non-negative coefficients with respect to the standard bases in $\ell^{p}_{\mathsf{X}}$ and $\ell^{q}_{\mathsf{Y}}$ . We only provide proofs for finite $p,q$ . The results of this section also hold when one of the exponents $p,q$ is infinite. While proofs in the endpoint cases are different, they are much simpler and are left to the reader.

3.4.C.

Let $p,q\in[1,\infty]$ satisfying $1/p+1/q\geq 1$ and let

M:\ell^{p}_{\mathsf{X}}\times\ell^{q}_{\mathsf{Y}}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}

be a coordinate-wise non-negative bilinear form of norm one. Then there exist sequences of uniform vectors $\bar{a}_{n}\in\ell^{p}_{\mathsf{X}^{n}}$ and $\bar{b}_{n}\in\ell^{q}_{\mathsf{Y}^{n}}$ , $n\in\mathbb{N}$ such that

\|\bar{a}_{n}\|_{p}=1,\qquad\|\bar{b}_{n}\|_{q}=1,\qquad\lim_{n\stackrel{{\scriptstyle}}{{\rightarrow}}\infty}\frac{1}{n}\log(\bar{a}_{n}M^{\otimes n}\bar{b}_{n})=0

For an arbitrary coordinate-wise non-negative bilinear form, the theorem gives the following corollary by rescaling.

3.4.D.

Let $p,q\in[1,\infty]$ satisfy $1/p+1/q\geq 1$ , and let $M:\ell^{p}_{\mathsf{X}}\times\ell^{q}_{\mathsf{Y}}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}$ be a non-zero bilinear form with non-negative coefficients with respect to the standard bases in $\ell^{p}_{\mathsf{X}}$ , $\ell^{q}_{\mathsf{Y}}$ , where $1/p+1/q\geq 1$ . Then there exist sequences of uniform vectors $\bar{a}_{n}\in\ell^{p}_{\mathsf{X}^{n}}$ and $\bar{b}_{n}\in\ell^{q}_{\mathsf{Y}^{n}}$ , $n\in\mathbb{N}$ such that

\lim_{n\stackrel{{\scriptstyle}}{{\rightarrow}}\infty}\frac{1}{n}\Big(\log(\bar{a}_{n}M^{\otimes n}\bar{b}_{n})-\log\|\bar{a}_{n}\|_{p}-\log\|\bar{b}_{n}\|_{q}\Big)=\log\|M\|_{p,q}

By renormalization we can choose $\bar{a}_{n}=\bm{1}_{\mathsf{A}_{n}}$ and $\bar{b}_{n}=\bm{1}_{\mathsf{B}_{n}}$ for suitable subsets $\mathsf{A}_{n}\subset\mathsf{X}^{n}$ and $\mathsf{B}_{n}\subset\mathsf{Y}^{n}$ . Then

\lim_{n\stackrel{{\scriptstyle}}{{\rightarrow}}\infty}\frac{1}{n}\Big(\log(\bar{a}_{n}M^{\otimes n}\bar{b}_{n})-\frac{1}{p}\log|\mathsf{A}_{n}|-\frac{1}{q}\log|\mathsf{B}_{n}|\Big)=\log\|M\|_{p,q}

3.4.E.

Consider the following extremal problem similar to the definition of the norm of a bilinear form

\|M^{\otimes n}\|_{p,q}^{(u)}:=\sup\left\{\frac{\varphi M^{\otimes n}\psi}{\|\varphi\|_{p}\cdot\|\psi\|_{q}}\;{\bm{:}}\;\varphi\in\ell^{p}_{\mathsf{X}^{n}}\text{ and }\psi\in\ell^{p}_{\mathsf{Y}^{n}}\text{ are uniform}\right\}

Clearly we have $\|M^{\otimes n}\|_{p,q}^{(u)}\leq\|M^{\otimes n}\|_{p,q}$ . In effect, Corollary 3.4.D states that on the normalized $\log$ -scale, the usual norm and the restricted subnorm defined above are asymptotically the same.

If necessary we may assume that uniform vectors $\bar{a}_{n}$ and $\bar{b}_{n}$ provided by Corollary 3.4.D are optimizers in the definition of the restricted subnorm $\|M^{\otimes n}\|_{p,q}^{(u)}$ .

3.4.F Proof of Theorem 3.4.C

Some recollections.

We start by recalling several notions needed in the proof of Theorem 3.4.C. For a finite set $\mathsf{X}$ denote by $\Delta\mathsf{X}$ the set of probability distributions on $\mathsf{X}$ . Define the empirical map

\mathbf{q}:\mathsf{X}^{n}\stackrel{{\scriptstyle}}{{\rightarrow}}\Delta\mathsf{X}

For $\mathbf{x}=(x_{0},\dots,x_{n-1})\in\mathsf{X}^{n}$ the value of the distribution $\mathbf{q}(\mathbf{x})$ at point $x\in\mathsf{X}$ is

\mathbf{q}(\mathbf{x})(x):=\frac{\sharp\left\{i\;{\bm{:}}\;x_{i}=x\right\}}{n}

Let $\pi\in\Delta\mathsf{X}$ be a distribution on $\mathsf{X}$ . Define the divergence ball of radius $r>0$ around $\pi$ as

B_{r}(\pi):=\left\{\pi^{\prime}\;{\bm{:}}\;\mathsf{D}(\pi^{\prime}|\mkern-2.0mu|\pi)\leq r\right\}\subset\Delta\mathsf{X}

where $\mathsf{D}$ stands for the Kullback–Leibler divergence. Denote by $B^{c}_{r}(\pi)$ the complement of the divergence ball. Note that the support of every distributions in $B_{r}(\pi)$ is contained in the support of $\pi$ .

We shall use the following standard Sanov-type bound, [San57, Csi98]:

\pi^{\otimes n}(\mathbf{q}^{-1}B^{c}_{r}(\pi))\leq\mathbf{e}^{-n\cdot r+|\mathsf{X}|\cdot\log(n+1)}

(3.4.G)

For every $\mathbf{x}\in\mathsf{X}^{n}$ , the probability of $\mathbf{x}$ under the product distribution $\pi^{\otimes n}$ is

\pi^{\otimes n}(\mathbf{x})=\mathbf{e}^{-n\big[h(\mathbf{q}(\mathbf{x}))+\mathsf{D}(\mathbf{q}(\mathbf{x})|\mkern-2.0mu|\pi)\big]}

(3.4.H)

As usual, we use the convention $\mathbf{e}^{-\infty}=0$ .

The proof.

We treat the case $p,q<\infty$ , leaving the cases of $p$ or $q$ infinite to the reader, as they are much simpler.

Uniform asymptotically resonance vectors $\bar{a}_{n}$ and $\bar{b}_{n}$ are constructed in three steps. We start with the “true” resonance pair of unit vectors $u$ and $v$ for $M$ . By Proposition 3.3.A their tensor powers $u^{\otimes n}$ and $v^{\otimes n}$ form a resonance pair for $M^{\otimes n}$ . In the first step we restrict $u^{\otimes n}$ and $v^{\otimes n}$ on suitable typical subsets in $\mathsf{X}^{n}$ and $\mathsf{Y}^{n}$ , respectively. The resulting vectors $a_{n}$ and $b_{n}$ are almost unit and quasi-uniform. In the second step we replace $a_{n}$ and $b_{n}$ by unit uniform vectors $\bar{a}_{n}$ and $\bar{b}_{n}$ with the same respective supports. In the third step we combine the bounds in the previous steps to derive the proposition.

Step 1.

Let $u\in\ell^{p}_{\mathsf{X}}$ and $v\in\ell^{q}_{\mathsf{Y}}$ be unit, coordinate-wise non-negative, resonance pair for $M$ . That is

\displaystyle\text{For all $\mathsf{x}\in\mathsf{X}$, $\mathsf{y}\in\mathsf{Y}$}:\quad u_{\mathsf{x}},v_{\mathsf{y}}\geq 0,\qquad\|u\|_{p}=\|v\|_{q}=1\quad\text{and}\quad uMv=1

We use $u^{\otimes n}$ to construct $a^{\prime}_{n}\in\ell^{p}_{\mathsf{X}^{n}}$ of unit norm and with reduced support which forms any asymptotically resonance pair with $v^{\otimes n}$ . The construction of vector $b^{\prime}_{n}\in\ell^{q}_{\mathsf{Y}^{n}}$ goes along the similar lines.

Since

\|u\|_{p}^{p}=\sum_{\mathsf{x}\in\mathsf{X}}u_{\mathsf{x}}^{p}=1

we regard $\alpha:=(u_{\mathsf{x}}^{p}\;{\bm{:}}\;\mathsf{x}\in\mathsf{X})$ as a probability distribution on $\mathsf{X}$ . Consider the sequence of divergence balls

B_{\frac{1}{\sqrt{n}}}(\alpha):=\left\{\pi\;{\bm{:}}\;\mathsf{D}(\pi|\mkern-2.0mu|\alpha)\leq n^{-1/2}\right\}\subset\Delta\mathsf{X}

of radius $n^{-1/2}$ around $\alpha$ , and denote $B_{\frac{1}{\sqrt{n}}}^{c}(\alpha)$ their complements.

Let $D_{n}(\alpha)\subset\mathsf{X}^{n}$ be the preimage of the divergence ball $B_{\frac{1}{\sqrt{n}}}(\alpha)$ under the empirical map $\mathbf{q}:\mathsf{X}^{n}\stackrel{{\scriptstyle}}{{\rightarrow}}\Delta\mathsf{X}$ and let $D^{c}_{n}(\alpha)\subset\mathsf{X}^{n}$ be its complement. Define

a_{n}:=u^{\otimes n}|_{D_{n}(\alpha)}\qquad\text{and}\qquad r_{n}:=u^{\otimes n}|_{D^{c}_{n}(\alpha)}

Here $a_{n}$ is the typical part and $r_{n}$ is the atypical remainder. The supports of $a_{n}$ and $r_{n}$ are complimentary and we have

a_{n}+r_{n}=u^{\otimes n}\qquad\text{and}\qquad\|a_{n}\|_{p}^{p}+\|r_{n}\|_{p}^{p}=\|u^{\otimes n}\|_{p}^{p}=1

By Equation (3.4.G) we have

\|r_{n}\|_{p}^{p}=\alpha^{\otimes n}\big(D^{c}_{n}(\alpha)\big)\leq\mathbf{e}^{-n^{1/2}+O(\log n)}\leq\mathbf{e}^{-c\cdot n^{1/2}}

for some $c>0$ and all sufficiently large $n$ . Thus

1\geq\|a_{n}\|_{p}\geq 1-\mathbf{e}^{-c\cdot n^{1/2}}

Define $a^{\prime}_{n}:=a_{n}/\|a_{n}\|_{p}$ .

In a similar fashion, write

\beta:=(v_{\mathsf{y}}^{q}\;{\bm{:}}\;\mathsf{y}\in\mathsf{Y})\in\Delta\mathsf{Y}

We then similarly decompose $v^{\otimes n}=b_{n}+s_{n}$ into typical part and atypical remainder. Then

1\geq\|b_{n}\|_{q}\geq 1-\mathbf{e}^{-c\cdot n^{1/2}}

for sufficiently large $n$ and some $c>0$ . Define $b^{\prime}_{n}:=b_{n}/\|b_{n}\|_{q}$ .

Clearly $a^{\prime}_{n}$ and $b^{\prime}_{n}$ are unit vectors in $\ell^{p}_{\mathsf{X}^{n}}$ and $\ell^{q}_{\mathsf{Y}^{n}}$ , respectively, and for sufficiently large $n$ we have

$\displaystyle a^{\prime}_{n}M^{\otimes n}b^{\prime}_{n}$	$\displaystyle=\\|a_{n}\\|_{p}^{-1}\\|b_{n}\\|_{q}^{-1}(a_{n}M^{\otimes n}b_{n})$
	$\displaystyle=\\|a_{n}\\|_{p}^{-1}\\|b_{n}\\|_{q}^{-1}\Bigl((u^{\otimes n}-r_{n})M^{\otimes n}(v^{\otimes n}-s_{n})\Bigr)$	(3.4.I)
	$\displaystyle=\\|a_{n}\\|_{p}^{-1}\\|b_{n}\\|_{q}^{-1}\Bigl(1-u^{\otimes n}M^{\otimes n}s_{n}-r_{n}M^{\otimes n}v^{\otimes n}+r_{n}M^{\otimes n}s_{n}\Bigr)$
	$\displaystyle=1+o(n^{0})$

Step 2.

We will now estimate the quasi-uniformity constant of $a^{\prime}_{n}$ and show that it grows subexponentially in $n$ . For any $\mathbf{x}\in D_{n}(\alpha)$ Equation (3.4.H) gives

	$\displaystyle a^{\prime}_{n}(\mathbf{x})$	$\displaystyle=\\|a_{n}\\|^{-1}_{p}\cdot a_{n}(\mathbf{x})=\\|a_{n}\\|^{-1}_{p}\cdot\big[\alpha^{\otimes n}(\mathbf{x})\big]^{1/p}$
		$\displaystyle=\\|a_{n}\\|^{-1}_{p}\cdot\mathbf{e}^{-\frac{1}{p}n[h(\mathbf{q}(\mathbf{x}))+\mathsf{D}(\mathbf{q}(\mathbf{x})\|\mkern-2.0mu\|\alpha)]}$

First we note that for sufficiently large $n$ , the divergence ball around $\alpha$ of radius $n^{-1/2}$ lies in an open face of the simplex $\Delta\mathsf{X}$ — the one that corresponds to the support of $\alpha$ . From now on, we assume that $n$ is sufficiently large for this assertion to hold. Consequently, there exists a compact subset $K$ of this open face that contains all divergence balls $B_{\frac{1}{\sqrt{n}}}(\alpha)$ for large $n$ . The entropy function $h:\Delta\mathsf{X}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}$ is smooth on $K$ . Let $d>0$ be an upper bound for the norm of differential of $h$ restricted on $K$ , where the norm is evaluated with respect to the total variation distance $\|\cdot-\cdot\|_{\mathrm{tv}}=\tfrac{1}{2}\|\cdot-\cdot\|_{1}$ on $\Delta\mathsf{X}$ .

Let $\mathbf{x},\mathbf{x}^{\prime}\in D_{n}(\alpha)$ . Since $\mathbf{q}(\mathbf{x}),\mathbf{q}(\mathbf{x}^{\prime})\in B_{\frac{1}{\sqrt{n}}}(\alpha)$ , we have

\Big|\mathsf{D}(\mathbf{q}(\mathbf{x})|\mkern-2.0mu|\alpha)-\mathsf{D}(\mathbf{q}(\mathbf{x}^{\prime})|\mkern-2.0mu|\alpha)\Big|\leq n^{-1/2}

By Pinsker inequality, [Pin64, CK11], we have

\|\mathbf{q}(\mathbf{x})-\mathbf{q}(\mathbf{x}^{\prime})\|_{\mathrm{tv}}\leq\sqrt{2}\cdot n^{-1/4}

and therefore

\Big|h\big(\mathbf{q}(\mathbf{x})\big)-h\big(\mathbf{q}(\mathbf{x}^{\prime})\big)\Big|\leq\sqrt{2}\cdot d\cdot n^{-1/4}

Thus, we can estimate the logarithm of quasi-uniformity constant for $a^{\prime}_{n}$ as follows.

	$\displaystyle\|\log a^{\prime}_{n}$	$\displaystyle(\mathbf{x})-\log a^{\prime}_{n}(\mathbf{x}^{\prime})\|=\frac{1}{p}\|\log\alpha^{\otimes n}(\mathbf{x})-\log\alpha^{\otimes n}(\mathbf{x}^{\prime})\|$
		$\displaystyle\leq\frac{1}{p}n\Big\|h(\mathbf{q}(\mathbf{x}))-h(\mathbf{q}(\mathbf{x}^{\prime}))\Big\|+\frac{1}{p}n\Big\|\mathsf{D}(\mathbf{q}(\mathbf{x})\|\mkern-2.0mu\|\alpha)-\mathsf{D}(\mathbf{q}(\mathbf{x}^{\prime})\|\mkern-2.0mu\|\alpha)\Big\|$
		$\displaystyle\leq\frac{\sqrt{2}\cdot d}{p}\cdot n^{3/4}+\frac{1}{p}n^{1/2}\leq c\cdot n^{3/4}$

for some positive constant $c$ . Thus

\frac{a^{\prime}_{n}(\mathbf{x})}{a^{\prime}_{n}(\mathbf{x}^{\prime})}\leq\mathbf{e}^{c\cdot n^{3/4}}

for all $\mathbf{x},\mathbf{x}^{\prime}\in D_{n}(\alpha)$ . In a similar fashion we obtain, for all $\mathbf{y},\mathbf{y}^{\prime}\in D_{n}(\beta)$ ,

\frac{b^{\prime}_{n}(\mathbf{y})}{b^{\prime}_{n}(\mathbf{y}^{\prime})}\leq\mathbf{e}^{c\cdot n^{3/4}}

for some $c>0$ .

Step 3.

Let $\bar{a}_{n}$ and $\bar{b}_{n}$ be uniform unit vectors in $\ell^{p}_{\mathsf{X}^{n}}$ and $\ell^{p}_{\mathsf{Y}^{n}}$ , whose supports coincide with that of $a^{\prime}_{n}$ and $b^{\prime}_{n}$ , respectively. Then for all $\mathbf{x}\in D_{n}(\alpha)$ and $\mathbf{y}\in D_{n}(\beta)$ holds

\displaystyle\mathbf{e}^{-c\cdot n^{3/4}}\leq\frac{\bar{a}_{n}(\mathbf{x})}{a^{\prime}_{n}(\mathbf{x})}\leq\mathbf{e}^{c\cdot n^{3/4}}\qquad\text{and}\qquad\mathbf{e}^{-c\cdot n^{3/4}}\leq\frac{\bar{b}_{n}(\mathbf{y})}{b^{\prime}_{n}(\mathbf{y})}\leq\mathbf{e}^{c\cdot n^{3/4}}

Since coefficients of $M^{\otimes n}$ are non-negative, the above pointwise estimates imply

\mathbf{e}^{-2c\cdot n^{3/4}}\leq\frac{\bar{a}_{n}M^{\otimes n}\bar{b}_{n}}{a^{\prime}_{n}M^{\otimes n}b^{\prime}_{n}}\leq\mathbf{e}^{2c\cdot n^{3/4}}

Taking normalized logarithm and combining with inequality in (3.4.I) we get

\lim_{n\stackrel{{\scriptstyle}}{{\rightarrow}}\infty}\frac{1}{n}\log(\bar{a}_{n}M^{\otimes n}\bar{b}_{n})=0

This finishes the proof of the theorem. ∎

4 Bilinear form associated to a bipartite graph

In this section we consider the bilinear form associated with a biregular bipartite graph and explore the relations between uniform vectors, subgraphs, and extensions of random pairs uniformly supported on the graph.

Let $\mathsf{G}=(\mathsf{X}\sqcup\mathsf{Y},\mathsf{E})$ be a biregular bipartite graph. Denote by $M:\mathbb{R}^{\mathsf{X}}\times\mathbb{R}^{\mathsf{Y}}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}$ the bilinear form associated with $\mathsf{G}$ defined by

\varphi M\psi:=\sum_{(\mathsf{x},\mathsf{y})\in\mathsf{E}}\varphi(\mathsf{x})\psi(\mathsf{y})

We refer to $M$ as the incidence bilinear form, or simply the incidence form, for the graph $\mathsf{G}$ .

4.1 Norms of incidence bilinear forms

Let $\mathsf{G}^{n}:=(\mathsf{X}^{n}\sqcup\mathsf{Y}^{n},\mathsf{E}^{n})$ stand for the power of $\mathsf{G}$ . The incidence bilinear form of $\mathsf{G}^{n}$ is the tensor power $M^{\otimes n}$ of the incidence form $M$ of $\mathsf{G}$ . By Proposition 3.3.A we have the equality

\log\|M\|_{p,q}=\frac{1}{n}\cdot\log\|M^{\otimes n}\|_{p,q}

for any $p,q\in[1,\infty]$ , satisfying $1/p+1/q\geq 1$ .

In the complimentary range $1/p+1/q\leq 1$ , the norms of the incidence form are completely determined by the sizes of the graph.

4.1.A.

Let $M$ be the incidence form of a biregular bipartite graph $\mathsf{G}=(\mathsf{X}\sqcup\mathsf{Y},\mathsf{E})$ . Let $p,q\in[1,\infty]$ , $1/p+1/q\leq 1$ . Then

\|M\|_{p,q}=\frac{|\mathsf{E}|}{|\mathsf{X}|^{1/p}\cdot|\mathsf{Y}|^{1/q}}

Note that the right-hand side in the equality above is multiplicative with respect to product of bipartite graphs. Thus for incidence bilinear forms the multiplicativity of the norms, Proposition 3.3.A, holds without any restriction on the exponents $p,q$ . This property is specific to incidence forms and does not hold for general bilinear forms.

4.1.B Proof of Proposition 4.1.A

We start by evaluating norms for the extremal values of exponents

(p,q)\in\left\{(\infty,\infty),(1,\infty),(\infty,1)\right\}

For any $u\in\ell^{\infty}_{\mathsf{X}}$ and $v\in\ell^{\infty}_{\mathsf{Y}}$ we have the inequality

uMv=\sum_{(\mathsf{x},\mathsf{y})\in\mathsf{E}}u_{\mathsf{x}}\cdot v_{\mathsf{y}}\leq|\mathsf{E}|\cdot\|u\|_{\infty}\cdot\|v\|_{\infty}

with equality for $u=\bm{1}_{\mathsf{X}}$ , $v=\bm{1}_{\mathsf{Y}}$ . Thus

\|M\|_{\infty,\infty}=|\mathsf{E}|

Also

\displaystyle uMv=\sum_{\mathsf{x}\in\mathsf{X}}u_{\mathsf{x}}\sum_{\mathsf{y}:(\mathsf{x},\mathsf{y})\in\mathsf{E}}v_{\mathsf{y}}\leq\|u\|_{1}\cdot d_{1}(\mathsf{G})\cdot\|v\|_{\infty}

with equality for $u=\delta_{\mathsf{x}_{0}}$ for some $\mathsf{x}_{0}\in\mathsf{X}$ , and $v=\bm{1}_{\mathsf{Y}}$ . Similar inequality holds with the roles of $u$ and $v$ switched. Thus

\|M\|_{\infty,\infty}=|\mathsf{E}|,\qquad\|M\|_{1,\infty}=\frac{|\mathsf{E}|}{|\mathsf{X}|},\qquad\|M\|_{\infty,1}=\frac{|\mathsf{E}|}{|\mathsf{Y}|}

By Riesz–Thorin interpolation, see [Rie27, Tho39] or [BL76] for more modern treatment, we then have

\|M\|_{p,q}\leq\frac{|\mathsf{E}|}{|\mathsf{X}|^{1/p}\cdot|\mathsf{Y}|^{1/q}}

On the other hand for $u=\bm{1}_{\mathsf{X}}$ and $v=\bm{1}_{\mathsf{Y}}$ we have

\|u\|_{p}=|\mathsf{X}|^{1/p},\qquad\|v\|_{q}=|\mathsf{Y}|^{1/q},\qquad uMv=|\mathsf{E}|

which gives the matching lower bound

\|M\|_{p,q}\geq\frac{|\mathsf{E}|}{|\mathsf{X}|^{1/p}\cdot|\mathsf{Y}|^{1/q}}

This finishes the proof. ∎

4.2 Uniform almost resonance vectors for incidence forms

Recall that Corollary 3.4.D and Remark 3.4.E allow us to find almost resonance pairs of uniform vectors for large tensor powers of bilinear form $M$ , such that in addition they optimize the restricted supremum in the definition of $\|M^{\otimes n}\|_{p,q}^{(u)}$ . We now apply these results to the incidence form of a biregular bipartite graph. In that case, the supports of almost resonance uniform pair of vectors have an additional regularity property, that we discuss below.

Consider the incidence form $M$ of a biregular bipartite graph $\mathsf{G}$ . Let $a\in\ell^{p}_{\mathsf{X}}$ and $b\in\ell^{q}_{\mathsf{Y}}$ be two uniform vectors attaining the maximum in the definition of $\|M\|_{p,q}^{(u)}$ . Without loss of generality we may assume that $a$ and $b$ are indicator functions of some subsets of $\mathsf{X}$ and $\mathsf{Y}$ , respectively. Denote by $\mathsf{H}=(\mathsf{X}_{\mathsf{H}}\sqcup\mathsf{Y}_{\mathsf{H}},\mathsf{E}_{\mathsf{H}})$ the subgraph of $\mathsf{G}$ induced by the supports $\mathsf{X}_{\mathsf{H}}$ and $\mathsf{Y}_{\mathsf{H}}$ of $a$ and $b$ , respectively. We choose the supports minimal among optimizers; then the induced subgraph has no isolated vertices.³³ 3 The fact that $\mathsf{H}$ has no isolated vertices is automatic for $p,q<\infty$ since removing isolated vertices would strictly improve the quotient in the definition of restricted subnorm. We only need to minimize the support in the case when at least one of the exponents is infinite.

Then we have the following relations: For any $\mathsf{x}\in\mathsf{X}_{\mathsf{H}}$ and $\mathsf{y}\in\mathsf{Y}_{\mathsf{H}}$

		$\displaystyle\delta_{\mathsf{x}}Mb=\deg_{\mathsf{H}}(\mathsf{x}),$		$\displaystyle\\|a\\|_{p}=\|\mathsf{X}_{\mathsf{H}}\|^{1/p},$		$\displaystyle aMb=\|\mathsf{E}_{\mathsf{H}}\|$
		$\displaystyle aM\delta_{\mathsf{y}}=\deg_{\mathsf{H}}(\mathsf{y}),$		$\displaystyle\\|b\\|_{q}=\|\mathsf{Y}_{\mathsf{H}}\|^{1/q}$

The quotients

\frac{|\mathsf{E}_{\mathsf{H}}|}{|\mathsf{X}_{\mathsf{H}}|}\quad\text{and}\quad\frac{|\mathsf{E}_{\mathsf{H}}|}{|\mathsf{Y}_{\mathsf{H}}|}

are equal to the average left and right degree of $\mathsf{H}$ , respectively.

For $\varepsilon,\delta\in(0,1]$ we say that $\mathsf{G}$ is $(\varepsilon,\delta)$ -quasi-biregular if every vertex in the left part has degree at least $\varepsilon$ -fraction of the average left degree and every vertex in the right part has degree at least $\delta$ -fraction of the average right degree. In other words, for all $\mathsf{x}\in\mathsf{X}$ , $\mathsf{y}\in\mathsf{Y}$ holds

\deg_{\mathsf{G}}(\mathsf{x})\geq\varepsilon\cdot\frac{|\mathsf{E}|}{|\mathsf{X}|},\qquad\deg_{\mathsf{G}}(\mathsf{y})\geq\delta\cdot\frac{|\mathsf{E}|}{|\mathsf{Y}|},

In the next proposition we show that for the uniform $(p,q)$ -optimizing pair of vectors $a$ and $b$ the graph induced by the supports of the vectors is $(\tfrac{1}{p},\tfrac{1}{q})$ -quasi-biregular.

4.2.A.

Let $p,q\in[1,\infty)$ and let $a\in\ell^{p}_{\mathsf{X}}$ , $a\in\ell^{p}_{\mathsf{X}}$ and $\mathsf{H}\subset\mathsf{G}$ be as above. Then for all $\mathsf{x}\in\mathsf{X}_{\mathsf{H}}$ and $\mathsf{y}\in\mathsf{Y}_{\mathsf{H}}$ holds

	$\displaystyle\deg_{\mathsf{H}}(\mathsf{x})$	$\displaystyle\geq\frac{1}{p}\frac{\|\mathsf{E}_{\mathsf{H}}\|}{\|\mathsf{X}_{\mathsf{H}}\|}$
	$\displaystyle\deg_{\mathsf{H}}(\mathsf{y})$	$\displaystyle\geq\frac{1}{q}\frac{\|\mathsf{E}_{\mathsf{H}}\|}{\|\mathsf{Y}_{\mathsf{H}}\|}$

Thus the degree within $\mathsf{H}$ of any vertex can only deviate down from the average degree by a fixed factor and $\mathsf{H}$ is $(\frac{1}{p},\frac{1}{q})$ -quasi-biregular.

Proposition 4.2.A is not needed for the proof of the main result. Nevertheless, we include it here because it gives useful structural information about uniform optimizers.

4.2.B Proof of Proposition 4.2.A

Let $\mathsf{x}\in\mathsf{X}_{\mathsf{H}}$ be an arbitrary vertex in the left part of $\mathsf{H}$ and $d:=\deg_{\mathsf{H}}(\mathsf{x})$ . Removing the vertex $\mathsf{x}$ can not improve the optimization quotient, hence

\frac{(a-\delta_{\mathsf{x}})Mb}{\|a-\delta_{\mathsf{x}}\|_{p}\cdot\|b\|_{q}}=\frac{|\mathsf{E}_{\mathsf{H}}|-d}{(|\mathsf{X}_{\mathsf{H}}|-1)^{1/p}\cdot|\mathsf{Y}_{\mathsf{H}}|^{1/q}}\leq\frac{|\mathsf{E}_{\mathsf{H}}|}{|\mathsf{X}_{\mathsf{H}}|^{1/p}\cdot|\mathsf{Y}_{\mathsf{H}}|^{1/q}}=\frac{aMb}{\|a\|_{p}\cdot\|b\|_{q}}

Therefore

\frac{d}{|\mathsf{E}_{\mathsf{H}}|}\geq 1-\left(1-\frac{1}{|\mathsf{X}_{\mathsf{H}}|}\right)^{1/p}\geq\frac{1}{p\cdot|\mathsf{X}_{\mathsf{H}}|}

which gives

d\geq\frac{1}{p}\cdot\frac{|\mathsf{E}_{\mathsf{H}}|}{|\mathsf{X}_{\mathsf{H}}|}

Similar argument works for the right part of the subgraph $\mathsf{H}$ . ∎

4.3 Subgraphs and extensions

4.3.A Construction of an extension from a subgraph

Suppose $\mathsf{G}$ is a homogeneous bipartite graph and $(X,Y)$ is uniformly supported on $\mathsf{G}$ . Denote by $\aut(\mathsf{G})$ the automorphism group of $\mathsf{G}$ (acting on $\mathsf{G}$ on the left). Let $\mathsf{H}\subset\mathsf{G}$ be a subgraph without isolated vertices. Applying symmetries $\sigma\in\aut(\mathsf{G})$ we obtain a family of subgraphs

\left\{\sigma\mathsf{H}:=(\sigma\mathsf{X}_{\mathsf{H}}\sqcup\sigma\mathsf{Y}_{\mathsf{H}},\sigma\mathsf{E}_{\mathsf{H}})\;{\bm{:}}\;\sigma\in\aut(\mathsf{G})\right\}

We now construct an extension $(X,Y,W)$ with alphabet $\mathsf{W}:=\aut(\mathsf{G})$ . The joint distribution $p$ of the triple is defined by

p(\mathsf{x},\mathsf{y},\sigma)=\begin{cases}\frac{1}{|\mathsf{E}_{\mathsf{H}}|\cdot|\aut(\mathsf{G})|},&\text{if $(\mathsf{x},\mathsf{y})\in\sigma\mathsf{E}_{\mathsf{H}}$};\\ 0,&\text{otherwise.}\end{cases}

All the pairs $(X,Y|\sigma)$ for different choices of $\sigma\in\aut(\mathsf{G})$ are isomorphic and have distribution

p(\mathsf{x},\mathsf{y}|\sigma)=\begin{cases}\frac{1}{|\mathsf{E}_{\mathsf{H}}|},&\text{if $(\mathsf{x},\mathsf{y})\in\sigma\mathsf{E}_{\mathsf{H}}$};\\ 0,&\text{otherwise}.\end{cases}

Thus the supporting graph of $(X,Y|\sigma)$ is $\sigma\mathsf{H}$ . We also have

p(\mathsf{x}|\sigma)=\frac{\deg_{\sigma\mathsf{H}}(\mathsf{x})}{|\mathsf{E}_{\mathsf{H}}|}\qquad\text{and}\qquad p(\mathsf{y}|\sigma)=\frac{\deg_{\sigma\mathsf{H}}(\mathsf{y})}{|\mathsf{E}_{\mathsf{H}}|}

Therefore we have the following bounds

	$\displaystyle H(X\|W)$	$\displaystyle=H(X\|\sigma)\leq[\mathsf{X}_{\mathsf{H}}],$		$\displaystyle H(XY\|W)=H(XY\|\sigma)=[\mathsf{E}_{\mathsf{H}}],$		(4.3.B)
	$\displaystyle H(Y\|W)$	$\displaystyle=H(Y\,\|\sigma)\leq[\mathsf{Y}_{\mathsf{H}}]$				(4.3.B)

The next proposition establishes the connection between uniform optimizers and extensions.

4.3.C.

Let $(X,Y)$ be a pair uniformly supported on a homogeneous bipartite graph $\mathsf{G}=(\mathsf{X}\sqcup\mathsf{Y},\mathsf{E})$ with the incidence form $M$ . Let $p,q\in[0,\infty)$ and $a\in\ell^{p}_{\mathsf{X}}$ , $b\in\ell^{q}_{\mathsf{Y}}$ be a pair of indicator vectors which are optimizers in the definition of $\|M\|_{p,q}^{(u)}$ . Then there exists an extension $(X,Y,W)$ such that

	$\displaystyle H(X\|W)$	$\displaystyle\leq p\cdot\log\\|a\\|_{p},$		$\displaystyle H(XY\|W)=\log(aMb),$
	$\displaystyle H(Y\|W)$	$\displaystyle\leq q\cdot\log\\|b\,\\|_{q}$

4.3.D Proof of Proposition 4.3.C

Let $\mathsf{H}=(\mathsf{X}_{\mathsf{H}}\sqcup\mathsf{Y}_{\mathsf{H}},\mathsf{E}_{\mathsf{H}})$ be the subgraph of $\mathsf{G}$ induced by the supports $\mathsf{X}_{\mathsf{H}}$ and $\mathsf{Y}_{\mathsf{H}}$ of $a$ and $b$ , respectively. Then

p\cdot\log\|a\|_{p}=[\mathsf{X}_{\mathsf{H}}],\qquad q\cdot\log\|b\|_{p}=[\mathsf{Y}_{\mathsf{H}}]\quad\text{and}\quad\log(aMb)=[\mathsf{E}_{\mathsf{H}}]

Application of the construction in Section 4.3.A produces extension $W$ , which satisfies the conclusions of the proposition, by the substitutions above and (4.3.B). ∎

5 Proof of the main theorem

5.1.A.

\mathcal{S}(X,Y)(\alpha,\beta)=\log\|M\|_{p,q}

where $\alpha=1-1/p$ and $\beta=1-1/q$ .

The proof of the theorem splits into two cases, the hypercontractive regime, $1/p+1/q\geq 1$ , and Hölder regime, $1/p+1/q\leq 1$ . The proof strategy in the two cases seems rather different, and we do not know whether there exists a unified proof covering both. Most of the considerations above can be generalized to longer tuples of random variables and their shapes. For $n$ -tuples the hypercontractive region, $\sum(1-1/p_{i})\leq 1$ , and Hölder region, $\sum 1/p_{i}\leq 1$ , are no longer complimentary, if $n>2$ . We do not know, whether generalized theorem holds in the intermediate region of values of the exponents not included in these two regions.

5.1.B Proof of Theorem 5.1.A

Let $(X,Y)$ , $\mathsf{G}=(\mathsf{X}\sqcup\mathsf{Y},\mathsf{E})$ and $M$ be as in the theorem. Fix $p,q\in[1,\infty]$ and set $\alpha=1-1/p$ , $\beta=1-1/q$ . We consider two cases.

Case $\bm{\alpha+\beta\geq 1}$ ; equivalently, $\bm{1/p+1/q\leq 1}$ .

By [MR26, Proposition 3.4.A] holds

$\displaystyle\mathcal{S}(X,Y)(\alpha,\beta)$	$\displaystyle=\alpha\cdot H(X)+\beta\cdot H(Y)-I(X:Y)$	(5.1.C)
	$\displaystyle=H(XY)-(1-\alpha)\cdot H(X)-(1-\beta)\cdot H(Y)$
	$\displaystyle=[\mathsf{E}]-(1-\alpha)[\mathsf{X}]-(1-\beta)[\mathsf{Y}]$

where we used identities from Section 2.3 for the last equality.

On the other hand, by Proposition 4.1.A for $1/p+1/q\leq 1$ we have

\displaystyle\log\|M\|_{p,q}

\displaystyle=\log\frac{|\mathsf{E}|}{|\mathsf{X}|^{1/p}\cdot|\mathsf{Y}|^{1/q}}

\displaystyle=[\mathsf{E}]-\frac{1}{p}[\mathsf{X}]-\frac{1}{q}[\mathsf{Y}]

(5.1.D)

Combination of (5.1.C) and (5.1.D) and substitution $\alpha=1-1/p$ and $\beta=1-1/q$ gives the conclusion of the theorem in this case.

Case $\bm{\alpha+\beta<1}$ ; equivalently, $\bm{1/p+1/q>1}$ .

We prove two inequalities, which together imply the theorem,

	$\displaystyle\mathcal{S}(X,Y)(\alpha,\beta)$	$\displaystyle\leq\log\\|M\\|_{p,q}$		(5.1.E)
	$\displaystyle\mathcal{S}(X,Y)(\alpha,\beta)$	$\displaystyle\geq\log\\|M\\|_{p,q}$		(5.1.F)

Let $(X^{n},Y^{n})$ be the pair obtained by taking $n$ independent copies of $(X,Y)$ . On the one hand by [MR26, Proposition 3.6.A] the shape function is additive under independent products, hence

\mathcal{S}(X,Y)(\alpha,\beta)=\frac{1}{n}\mathcal{S}(X^{n},Y^{n})(\alpha,\beta)

On the other hand, Proposition 3.3.A implies that the $\log$ -norm of $M$ is also stable in the sense that

\log\|M\|_{p,q}=\frac{1}{n}\log\|M^{\otimes n}\|_{p,q}

Proof of inequality (5.1.E).

Suppose that $(X,Y,W)$ is an extension, such that

H(XY|W)-(1-\alpha)H(X|W)-(1-\beta)H(Y|W)=\mathcal{S}(X,Y)(\alpha,\beta)

Using stability of the shape function we also have

H(X^{n}Y^{n}|W^{n})-(1-\alpha)H(X^{n}|W^{n})-(1-\beta)H(Y^{n}|W^{n})=n\cdot\mathcal{S}(X,Y)(\alpha,\beta)

By Tropical Asymptotic Equipartition Property, [MP18], we can replace the extending variable $W^{n}$ by another extending variable $U_{n}$ such that the joint distribution of $(X^{n},Y^{n},U_{n})$ is uniform on its support and the conditional entropy profiles $e(X^{n},Y^{n}|W^{n})$ and $e(X^{n},Y^{n}|U_{n})$ are close on the normalized scale. More precisely, for every $\varepsilon>0$ there exists $n$ and an extension $(X^{n},Y^{n},U_{n})$ , uniform on its support, such that

	$\displaystyle\left\|H(XY\|W)-\frac{1}{n}H(X^{n}Y^{n}\|U_{n})\right\|$	$\displaystyle\leq\varepsilon$
	$\displaystyle\left\|H(X\|W)-\frac{1}{n}H(X^{n}\|U_{n})\right\|$	$\displaystyle\leq\varepsilon$
	$\displaystyle\left\|H(Y\|W)-\frac{1}{n}H(Y^{n}\|U_{n})\right\|$	$\displaystyle\leq\varepsilon$

Consequently,

\frac{1}{n}\Big(H(X^{n}Y^{n}|U_{n})-(1-\alpha)H(X^{n}|U_{n})-(1-\beta)H(Y^{n}|U_{n})\Big)\geq\mathcal{S}(X,Y)(\alpha,\beta)-3\varepsilon

Choose an arbitrary atom $\mathsf{u}$ from the alphabet of $U_{n}$ and let $\mathsf{H}\subset\mathsf{G}^{n}$ be the subgraph supporting $(X^{n},Y^{n}|\mathsf{u})$ . Write $\mathsf{H}=:(\mathsf{L}\sqcup\mathsf{R},\mathsf{F})$ , where

\mathsf{L}\subset\mathsf{X}^{n},\qquad\mathsf{R}\subset\mathsf{Y}^{n},\qquad\mathsf{F}\subset\mathsf{E}^{n}

are the left part, the right part and the edge set of $\mathsf{H}$ . Since $(X^{n},Y^{n},U_{n})$ is uniform, we have the following identities

	$\displaystyle H(X^{n}\|U_{n})=H(X^{n}\|\mathsf{u})=[\mathsf{L}]$
	$\displaystyle H(Y^{n}\|U_{n})=H(Y^{n}\|\mathsf{u})=[\mathsf{R}]$
	$\displaystyle H(X^{n}Y^{n}\|U_{n})=H(X^{n}Y^{n}\|\mathsf{u})=[\mathsf{F}]$

Let $\varphi\in\ell^{p}_{\mathsf{X}^{n}}$ and $\psi\in\ell^{q}_{\mathsf{Y}^{n}}$ be the indicator functions of $\mathsf{L}$ and $\mathsf{R}$ , respectively. Then

\log\|\varphi\|_{p}=\frac{1}{p}[\mathsf{L}],\qquad\log\|\psi\|_{q}=\frac{1}{q}[\mathsf{R}],\qquad\log(\varphi M^{\otimes n}\psi)\geq[\mathsf{F}]

Therefore

	$\displaystyle\log\\|M\\|_{p,q}$	$\displaystyle=\frac{1}{n}\log\\|M^{\otimes n}\\|_{p,q}$
		$\displaystyle\geq\frac{1}{n}\Big(\log(\varphi M^{\otimes n}\psi)-\log\\|\varphi\\|_{p}-\log\\|\psi\\|_{q}\Big)$
		$\displaystyle\geq\frac{1}{n}\Big([\mathsf{F}]-\frac{1}{p}[\mathsf{L}]-\frac{1}{q}[\mathsf{R}]\Big)$
		$\displaystyle=\frac{1}{n}\Big(H(X^{n}Y^{n}\|U_{n})-(1-\alpha)H(X^{n}\|U_{n})-(1-\beta)H(Y^{n}\|U_{n})\Big)$
		$\displaystyle\geq\mathcal{S}(X,Y)(\alpha,\beta)-3\varepsilon$

This inequality holds for every $\varepsilon>0$ , so inequality (5.1.E) follows.

Proof of inequality (5.1.F).

By Corollary 3.4.D for every $\varepsilon>0$ there exist $n\in\mathbb{N}$ and indicator functions $\bar{a}\in\ell^{p}_{\mathsf{X}^{n}}$ and $\bar{b}\in\ell^{q}_{\mathsf{Y}^{n}}$ such that

\log\|M\|_{p,q}\leq\frac{1}{n}\Big(\log(\bar{a}M^{\otimes n}\bar{b})-\log\|\bar{a}\|_{p}-\log\|\bar{b}\|_{q}\Big)+\varepsilon

By Proposition 4.3.C there exists an extension $(X^{n},Y^{n},W)$ such that

		$\displaystyle H(X^{n}Y^{n}\|W)=\log(\bar{a}M^{\otimes n}\bar{b})$
		$\displaystyle H(X^{n}\|W)\leq p\cdot\log\\|\bar{a}\\|_{p}$
		$\displaystyle H(Y^{n}\|W)\,\leq q\cdot\log\\|\bar{b}\\|_{q}$

Now we can estimate

	$\displaystyle\log\\|M\\|_{p,q}$	$\displaystyle\leq\frac{1}{n}\Big(\log(\bar{a}M^{\otimes n}\bar{b})-\log\\|\bar{a}\\|_{p}-\log\\|\bar{b}\\|_{q}\Big)+\varepsilon$
		$\displaystyle\leq\frac{1}{n}\Big(H(X^{n}Y^{n}\|W)-\frac{1}{p}H(X^{n}\|W)-\frac{1}{q}H(Y^{n}\|W)\Big)+\varepsilon$
		$\displaystyle\leq\frac{1}{n}\mathcal{S}(X^{n},Y^{n})(\alpha,\beta)+\varepsilon$
		$\displaystyle=\mathcal{S}(X,Y)(\alpha,\beta)+\varepsilon$

Since $\varepsilon>0$ is arbitrary, inequality (5.1.F) follows. ∎

6 Properties of the shape

In this section we establish inequalities and relations satisfied by shapes of pairs of random variables. Most of these relations are elementary and can be derived directly in the context of the definition of the shape by using properties of entropy and Shannon inequalities. However, inequality (COMP) and its implication, inequality (REF), seem to be different. We do not know, whether a proof not referring to Theorem 5.1.A exists.

Proofs of all statements in Section 6.2 are given in Section 6.3.

6.1 Lower and upper bounds for the shape function

We need to recall some definitions from [MR26]: For a pair of random variables $(X,Y)$ and $(\alpha,\beta)\in[0,1]^{2}$ define

		$\displaystyle\mathcal{S}_{\min}(X,Y)(\alpha,\beta):=\max\left\{\begin{aligned} &\alpha\cdot H(X\|Y),\\ &\beta\cdot H(Y\|Y),\\ &\alpha\cdot H(X)+\beta\cdot H(Y)-I(X:Y)\end{aligned}\right\}$
		$\displaystyle\mathcal{S}_{\max}(X,Y)(\alpha,\beta):=\max\left\{\begin{aligned} &\alpha\cdot H(X\|Y)+\beta\cdot H(Y\|Y),\\ &\alpha\cdot H(X)+\beta\cdot H(Y)-I(X:Y)\end{aligned}\right\}$
		$\displaystyle\ing(X,Y,A,B):=I(X:Y\|A)+I(X:Y\|B)+I(A:B)-I(X:Y)$

The apex point where three affine pieces of $\mathcal{S}_{\min}$ meet is

		$\displaystyle\alpha_{0}=\frac{H(Y\|X)\cdot I(X:Y)}{H(X)\cdot H(Y)-I(X:Y)^{2}},$		$\displaystyle\beta_{0}=\frac{H(X\|Y)\cdot I(X:Y)}{H(X)\cdot H(Y)-I(X:Y)^{2}},$
		$\displaystyle\mathcal{S}_{\min}(X,Y)(\alpha_{0},\beta_{0})=\frac{H(X\|Y)\cdot H(Y\|X)\cdot I(X:Y)}{H(X)\cdot H(Y)-I(X:Y)^{2}}\mkern-200.0mu$

6.2 Properties of the shape

For every tuple of random variables $(X,Y,X^{\prime},Y^{\prime},X^{\prime\prime},Y^{\prime\prime},Z,A,B)$ and every
$t,\alpha,\beta,\alpha^{\prime},\beta^{\prime},\alpha^{\prime\prime},\beta^{\prime\prime},\gamma,\eta\in[0,1]$ holds:

6.2.A Symmetry

\mathcal{S}(X,Y)(\alpha,\beta)=\mathcal{S}(Y,X)(\beta,\alpha)

(SYM)

6.2.B Convexity

\left(\!\!\!\begin{array}[]{c}\alpha^{\prime\prime}\\ \beta^{\prime\prime}\end{array}\!\!\!\right)=t\left(\!\!\!\begin{array}[]{c}\alpha\\ \beta\end{array}\!\!\!\right)+(1-t)\left(\!\!\!\begin{array}[]{c}\alpha^{\prime}\\ \beta^{\prime}\end{array}\!\!\!\right)

then

\mathcal{S}(X,Y)(\alpha^{\prime\prime},\beta^{\prime\prime})\leq t\cdot\mathcal{S}(X,Y)(\alpha,\beta)+(1-t)\cdot\mathcal{S}(X,Y)(\alpha^{\prime},\beta^{\prime})

(CONV)

6.2.C Coordinate-wise monotonicity and reverse monotonicity

If $\alpha\leq\alpha^{\prime}$ and $\beta\leq\beta^{\prime}$ then

		$\displaystyle\mathcal{S}(X,Y)(\alpha,\beta)\leq\mathcal{S}(X,Y)(\alpha^{\prime},\beta^{\prime})$		(MO)
		$\displaystyle\mathcal{S}(X,Y)(\alpha^{\prime},\beta^{\prime})\leq\mathcal{S}(X,Y)(\alpha,\beta)+(\alpha^{\prime}-\alpha)H(X)+(\beta^{\prime}-\beta)H(Y)$		(RMO)

Written together

0\leq\mathcal{S}(X,Y)(\alpha^{\prime},\beta^{\prime})-\mathcal{S}(X,Y)(\alpha,\beta)\leq(\alpha^{\prime}-\alpha)H(X)+(\beta^{\prime}-\beta)H(Y)

6.2.D Additivity

If $(X^{\prime\prime},Y^{\prime\prime})=(X,Y)\oplus(X^{\prime},Y^{\prime})$ is the independent sum, then

\mathcal{S}(X^{\prime\prime},Y^{\prime\prime})(\alpha,\beta)=\mathcal{S}(X,Y)(\alpha,\beta)+\mathcal{S}(X^{\prime},Y^{\prime})(\alpha,\beta)

(ADD)

6.2.E Chain rule

\mathcal{S}(XZ,YZ)(\alpha,\beta)=\mathcal{S}(X,Y|Z)(\alpha,\beta)+\mathcal{S}(Z,Z)(\alpha,\beta)

(CHAIN)

6.2.F Test inequality

\alpha\cdot H(X|Z)+\beta\cdot H(Y|Z)-I(X:Y|Z)\leq\mathcal{S}(X,Y)(\alpha,\beta)

(TEST)

6.2.G Upper bound

\mathcal{S}(X,Y)(\alpha,\beta)\leq\mathcal{S}_{\max}(X,Y)(\alpha,\beta)

(UP)

The inequality is equality on the upper-right triangle $\left\{0\leq\alpha,\beta\leq 1,\;\alpha+\beta\geq 1\right\}$ , where the second max-summand dominates. It is also equality on the boundary of the square where $\mathcal{S}$ is equal to the first max-summand.

The upper bound is attained if and only if mutual information of $(X,Y)$ is extractable, that is there is an extension $(X,Y,W)$ with

H(W|X)=H(W|Y)=I(X:Y|W)=0

6.2.H Lower bound

\mathcal{S}_{\min}(X,Y)(\alpha,\beta)\leq\mathcal{S}(X,Y)(\alpha,\beta)

(LO)

On the upper-right triangle and the boundary of the square we have equality.

Lower bound is attained for rigid pairs. It is attained up to $O\bigl(\log H(XY)\bigr)$ for pairs supported on balanced expanders, see [MR26].

Define the rigidity of the nondegenerate pair $(X,Y)$ by

\mathcal{R}(X,Y):=\frac{\mathcal{S}_{\max}(X,Y)(\alpha_{0},\beta_{0})-\mathcal{S}(X,Y)(\alpha_{0},\beta_{0})}{\mathcal{S}_{\max}(X,Y)(\alpha_{0},\beta_{0})-\mathcal{S}_{\min}(X,Y)(\alpha_{0},\beta_{0})}

Thus we have $0\leq\mathcal{R}(X,Y)\leq 1$ , with $\mathcal{R}(X,Y)=1$ if and only if the pair is rigid and $\mathcal{R}(X,Y)=0$ if and only if the pair has extractable mutual information.

6.2.I Composition inequality

\mathcal{S}(X,Z)(\alpha,\gamma)\leq\mathcal{S}(X,Y)(\alpha,\beta)+\mathcal{S}(Y,Z)(1-\beta,\gamma)-H(Y|XZ)

(COMP)

In the Hölder regime ( $\alpha+\gamma\geq 1$ ), take $\beta$ to be any value in the interval $[1-\alpha,\gamma]$ . The difference between the right-hand side and the left-hand side is then $I(X:Z|Y)$ .

6.2.J Refinement and coarsening

For a pair of random variables $(A,B)$ write $A\preceq B$ if $B$ is a refinement of $A$ , equivalently $H(A|B)=0$ .

If $X\preceq X^{\prime}$ and $Y\preceq Y^{\prime}$ then

\mathcal{S}(X,Y)(\alpha,\beta)\leq\mathcal{S}(X^{\prime},Y^{\prime})(\alpha,\beta)+(1-\alpha)H(X^{\prime}|X)+(1-\beta)H(Y^{\prime}|Y)-H(X^{\prime}Y^{\prime}|XY)

(REF)

\mathcal{S}(X,Y^{\prime})(\alpha,\beta)\leq\mathcal{S}(X,Y)(\alpha,\beta)+\beta\cdot H(Y^{\prime}|XY)

(CRS)

We can use symmetry to sequentially coarsen the first and the second variable, but the resulting bound depend on the order of application. However the following symmetric but weaker inequality holds

\mathcal{S}(X^{\prime},Y^{\prime})(\alpha,\beta)\leq\mathcal{S}(X,Y)(\alpha,\beta)+\alpha\cdot H(X^{\prime}|XY)+\beta\cdot H(Y^{\prime}|XY)

(CRS2)

6.2.K Link and reversed link inequalities

	$\displaystyle\mathcal{S}(X,Y)(\alpha,\beta)$	$\displaystyle\leq\mathcal{S}(X,Y\|Z)(\alpha,\beta)+I(XY:Z)$		(LI)
	$\displaystyle\mathcal{S}(X,Y\|Z)(\alpha,\beta)$	$\displaystyle\leq\mathcal{S}(X,Y)(\alpha,\beta)$		(RLI)

6.2.L Flattening/links and reverse

	$\displaystyle\mathcal{S}(X,YZ)(\alpha,\beta)$	$\displaystyle\leq\mathcal{S}(X,Y\|Z)(\alpha,\beta)+\beta\cdot H(Z)$		(FLI)
	$\displaystyle\mathcal{S}(X,Y\|Z)(\alpha,\beta)$	$\displaystyle\leq\mathcal{S}(X,YZ)(\alpha,\beta)$		(RFLI)

6.2.M MMRV-inequality

3\mathcal{S}(X,Y)(\tfrac{1}{3},\tfrac{1}{3})-2\mathcal{S}(X,Y)(\tfrac{1}{2},\tfrac{1}{2})\leq\ing(X,Y,A,B)

(MMRV)

6.3 Proofs

Symmetry (SYM) follows directly from the definition of the shape. Convexity of the shape function, (CONV), is proven in [MR26, Section 2.2]. The relations (ADD) and (CHAIN) are proven in [MR26, Section 3.6]. The upper and lower bounds for the shape function, (UP) and (LO), are proven in [MR26, Sections 3.4 and 3.5]. Inequality (TEST) follows directly from the definition.

Proof of (MO) and (RMO).

For $\alpha\leq\alpha^{\prime}$ and $\beta\leq\beta^{\prime}$ and all $W$ ’s extending $(X,Y)$ holds

	$\displaystyle\alpha^{\prime}\cdot H(X\|W)-(\alpha^{\prime}-\alpha)H(X)$	$\displaystyle\leq\alpha\cdot H(X\|W)\leq\alpha^{\prime}\cdot H(X\|W)$
	$\displaystyle\beta^{\prime}\cdot H(Y\|W)-(\beta^{\prime}-\beta)H(Y)$	$\displaystyle\leq\beta\cdot H(Y\|W)\leq\beta^{\prime}\cdot H(Y\|W)$

Substituting in the definition of the shape we obtain the required inequalities. ∎

Proof of (COMP).

By additivity and continuity of the shape function, [MR26, Proposition 3.6.A] and by Tropical Asymptotic Equipartition Property, [MP18, Theorem 6.1] we can assume that the triple $(X,Y,Z)$ is uniform on the support. Let $p,q,r\in[1,\infty]$ and

M_{XY}:\ell^{p}_{\mathsf{X}}\stackrel{{\scriptstyle}}{{\rightarrow}}\ell^{q}_{\mathsf{Y}},\quad M_{Y\!Z}:\ell^{q}_{\mathsf{Y}}\stackrel{{\scriptstyle}}{{\rightarrow}}\ell^{r}_{\mathsf{Z}},\quad\text{and}\quad M_{X\!Z}:\ell^{p}_{\mathsf{X}}\stackrel{{\scriptstyle}}{{\rightarrow}}\ell^{r}_{\mathsf{Z}}

be the incidence operators of the graphs supporting pairs $(X,Y)$ , $(Y,Z)$ and $(X,Z)$ , respectively. Denote by $N=M_{Y\!Z}\circ M_{XY}$ the composition of the operators. Then

N_{\mathsf{x},\mathsf{z}}=\sum_{\mathsf{y}\in\mathsf{Y}}(M_{Y\!Z})_{\mathsf{y},\mathsf{z}}\cdot(M_{XY})_{\mathsf{x},\mathsf{y}}\geq\sharp(\mathsf{Z}|\mathsf{X}\mathsf{Y})\cdot(M_{X\!Z})_{\mathsf{x},\mathsf{z}}

We note that all operators have non-negative coefficients, therefore coordinate-wise domination implies the corresponding inequality for the norms. Thus, taking the operator norms, we obtain inequality

\|M_{XY}\|_{p\stackrel{{\scriptstyle}}{{\rightarrow}}q}\cdot\|M_{Y\!Z}\|_{q\stackrel{{\scriptstyle}}{{\rightarrow}}r}\geq\|N\|_{p\stackrel{{\scriptstyle}}{{\rightarrow}}r}\geq\sharp(\mathsf{Z}|\mathsf{X}\mathsf{Y})\cdot\|M_{X\!Z}\|_{p\stackrel{{\scriptstyle}}{{\rightarrow}}r}

Switching to norms of bilinear forms we obtain

\|M_{XY}\|_{p,q^{\prime}}\cdot\|M_{Y\!Z}\|_{q,r^{\prime}}\geq\sharp(\mathsf{Z}|\mathsf{X}\mathsf{Y})\cdot\|M_{X\!Z}\|_{p,r^{\prime}}

We now take logarithm of the last inequality, use Theorem 5.1.A and the substitutions $\alpha=1-1/p$ , $\beta=1/q=1-1/q^{\prime}$ and $\gamma=1/r=1-1/r^{\prime}$ , to obtain inequality (COMP). ∎

Proof of (REF).

We apply (COMP) to the triple $(X,Y^{\prime},Y)$ . In the calculation below the summand $\mathcal{S}(Y^{\prime},Y)(1-\beta,\beta)$ is in Hölder regime and its value is forced by Shannon inequalities.

	$\displaystyle\mathcal{S}(X,Y)(\alpha,\beta)$	$\displaystyle\leq\mathcal{S}(X,Y^{\prime})(\alpha,\beta)+\mathcal{S}(Y^{\prime},Y)(1-\beta,\beta)-H(Y^{\prime}\|XY)$
		$\displaystyle=\mathcal{S}(X,Y^{\prime})(\alpha,\beta)+(1-\beta)H(Y^{\prime})+\beta\cdot H(Y)-I(Y^{\prime}:Y)-H(Y^{\prime}\|XY)$
		$\displaystyle=\mathcal{S}(X,Y^{\prime})(\alpha,\beta)+(1-\beta)H(Y^{\prime}\|Y)-H(Y^{\prime}\|XY)$

Applying the above bound twice we get

	$\displaystyle\mathcal{S}(X,Y)$	$\displaystyle(\alpha,\beta)\leq\mathcal{S}(X,Y^{\prime})(\alpha,\beta)+(1-\beta)H(Y^{\prime}\|Y)-H(Y^{\prime}\|XY)$
		$\displaystyle\leq\mathcal{S}(X^{\prime},Y^{\prime})(\alpha,\beta)+(1-\alpha)H(X^{\prime}\|X)+(1-\beta)H(Y^{\prime}\|Y)-H(Y^{\prime}\|XY)-H(X^{\prime}\|XY^{\prime})$
		$\displaystyle=\mathcal{S}(X^{\prime},Y^{\prime})(\alpha,\beta)+(1-\alpha)H(X^{\prime}\|X)+(1-\beta)H(Y^{\prime}\|Y)-H(X^{\prime}Y^{\prime}\|XY)$

This finishes the proof of (REF). ∎

Proof of (CRS).

Let $W_{0}$ be an optimizer in the definition of $S(X,Y^{\prime})(\alpha,\beta)$ . Then

	$\displaystyle\mathcal{S}(X,Y^{\prime})(\alpha,\beta)$	$\displaystyle=\alpha\cdot H(X\|W_{0})+\beta\cdot H(Y^{\prime}\|W_{0})-I(X:Y^{\prime}\|W_{0})$
		$\displaystyle=\alpha\cdot H(X\|W_{0})+\beta\cdot H(Y\|W_{0})-I(X:Y\|W_{0})$
		$\displaystyle\quad+\beta\big(H(Y^{\prime}\|W_{0})-H(Y\|W_{0})\big)-\big(I(X:Y^{\prime}\|W_{0})-I(X:Y\|W_{0})\big)$
		$\displaystyle\leq\mathcal{S}(X,Y)(\alpha,\beta)+\beta\cdot H(Y^{\prime}\|XY)$

∎

Proof of (LI).

Let $W_{0}$ be an optimizer in the definition of $\mathcal{S}(X,Y)(\alpha,\beta)$ , that is

\mathcal{S}(X,Y)(\alpha,\beta)=H(XY|W_{0})-(1-\alpha)H(X|W_{0})-(1-\beta)H(Y|W_{0})

By Matúš’ inner adhesivity property of entropic polymatroids, [Mat05, Mat07], we can assume that $W_{0}$ is chosen so that $I(W_{0}:Z|XY)=0$ . In that case holds

I(XY:Z|W_{0})=I(XY:Z)-I(W_{0}:Z)\leq I(XY:Z).

Now we can estimate

	$\displaystyle\mathcal{S}(X,Y)(\alpha,\beta)$	$\displaystyle=H(XY\|W_{0})-(1-\alpha)H(X\|W_{0})-(1-\beta)H(Y\|W_{0})$
		$\displaystyle=H(XY\|ZW_{0})-(1-\alpha)H(X\|ZW_{0})-(1-\beta)H(Y\|ZW_{0})$
		$\displaystyle\quad+I(XY:Z\|W_{0})-(1-\alpha)I(X:Z\|W_{0})-(1-\beta)I(Y:Z\|W_{0})$
		$\displaystyle\leq\mathcal{S}(X,Y\|Z)(\alpha,\beta)+I(XY:Z\|W_{0})$
		$\displaystyle\leq\mathcal{S}(X,Y\|Z)(\alpha,\beta)+I(XY:Z)$

which is the required inequality. ∎

Proof of (RLI).

Since inequality (TEST) holds for any choice of $Z$ extending $(X,Y)$ , we can formally replace $Z$ by $ZW$ to obtain

\alpha\cdot H(X|WZ)+\beta\cdot H(Y|WZ)-\alpha\cdot I(X:Y|WZ)\leq S(X,Y)(\alpha,\beta)

Taking supremum with respect to $W$ we obtain (RLI). ∎

Proof of (FLI).

Suppose $W_{0}$ is an optimizer in the definition of $\mathcal{S}(X,Y\!Z)(\alpha,\beta)$ . Then

	$\displaystyle\mathcal{S}(X,Y\!Z)(\alpha,\beta)$	$\displaystyle=\alpha\cdot H(X\|W_{0})+\beta\cdot H(Y\!Z\|W_{0})-I(X:Y\!Z\|W_{0})$
		$\displaystyle=\alpha\cdot H(X\|ZW_{0})+\beta\cdot H(Y\|ZW_{0})-I(X:Y\|ZW_{0})$
		$\displaystyle\quad+\alpha\cdot I(X:Z\|W_{0})+\beta\cdot H(Z\|W_{0})-I(X:Z\|W_{0})$
		$\displaystyle\leq\alpha\cdot H(X\|ZW_{0})+\beta\cdot H(Y\|ZW_{0})-I(X:Y\|ZW_{0})$
		$\displaystyle\quad+\beta\cdot H(Z)$
		$\displaystyle\leq\mathcal{S}(X,Y\|Z)(\alpha,\beta)+\beta\cdot H(Z)$

∎

Proof of (RFLI).

Let $W_{0}$ be the optimizer in the definition of $\mathcal{S}(X,Y|Z)(\alpha,\beta)$ . Then

	$\displaystyle\mathcal{S}(X,Y\|Z)(\alpha,\beta)$	$\displaystyle=H(XY\|ZW_{0})-(1-\alpha)H(X\|ZW_{0})-(1-\beta)H(Y\|ZW_{0})$
		$\displaystyle=H(XY\!Z\|ZW_{0})-(1-\alpha)H(X\|ZW_{0})-(1-\beta)H(Y\!Z\|ZW_{0})$
		$\displaystyle\leq\mathcal{S}(X,Y\!Z)(\alpha,\beta)$

∎

Proof of (MMRV).

In [Mak+02] the following non-Shannon inequality for five random variables is proven

\ing(X,Y,A,B)\geq-I(X:Y|W)-I(W:Y|X)-I(X:W|Y)

The right-hand side can be rewritten as

	$\displaystyle-I(X:Y\|W)$	$\displaystyle-I(W:Y\|X)-I(X:W\|Y)$
		$\displaystyle=\big(H(X\|W)+H(Y\|W)-3I(X:Y\|W)\big)$
		$\displaystyle-\big(H(X\|Y)+H(Y\|X)\big)$

Thus

\ing(X,Y,A,B)\geq\big(H(X|W)+H(Y|W)-3I(X:Y|W)\big)-\big(H(X|Y)+H(Y|X)\big)

Since $\mathcal{S}(X,Y)(\tfrac{1}{2},\tfrac{1}{2})=\frac{1}{2}\big(H(X|Y)+H(Y|X)\big)$ , then taking the supremum over $W$ ’s gives

\ing(X,Y,A,B)\geq 3\mathcal{S}(X,Y)(\tfrac{1}{3},\tfrac{1}{3})-2\mathcal{S}(X,Y)(\tfrac{1}{2},\tfrac{1}{2})

∎

References

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[DF93] Andreas Defant and Klaus Floret “Tensor Norms and Operator Ideals” North-Holland, 1993
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[LE17] Cheuk Li and Abbas El “Extended Gray–Wyner system with complementary causal side information” In IEEE Transactions on Information Theory 64.8 IEEE, 2017, pp. 5862–5878
[Mak+02] Konstantin Makarychev, Yury Makarychev, Andrei Romashchenko and Nikolai Vereshchagin “A new class of non-Shannon-type inequalities for entropies” In Communications in Information and Systems 2.2 International Press of Boston, 2002, pp. 147–166
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		$\displaystyle[\mathsf{X}_{\mathsf{H}}]:=\log\|\mathsf{X}_{\mathsf{H}}\|,$		$\displaystyle[\mathsf{E}_{\mathsf{H}}]:=\log\|\mathsf{E}_{\mathsf{H}}\|,$
		$\displaystyle[\mathsf{Y}_{\mathsf{H}}]:=\log\|\mathsf{Y}_{\mathsf{H}}\|,$		$\displaystyle[\mathsf{X}_{\mathsf{H}}:\mathsf{Y}_{\mathsf{H}}]:=\log\frac{\|\mathsf{X}_{\mathsf{H}}\|\cdot\|\mathsf{Y}_{\mathsf{H}}\|}{\|\mathsf{E}_{\mathsf{H}}\|}$

	$\displaystyle\|x(M\otimes N)y\|$	$\displaystyle=\left\|\sum_{j\in J}u_{j}Ny_{j}\right\|$
		$\displaystyle\leq\sum_{j\in J}\\|u_{j}\\|_{p}\cdot\\|y_{j}\\|_{q}$
		$\displaystyle\leq\\|(u_{j})_{j\in J}\\|_{q^{\prime},J;p,K}\cdot\\|(y_{j})_{j\in J}\\|_{q,J;q,L}$
		$\displaystyle\leq 1$

	$\displaystyle\\|(x_{ij})\\|_{p,I;\,q,J}$	$\displaystyle=\bigl\\|(\|x_{ij}\|^{p})\bigr\\|_{1,I;\,q/p,J}^{1/p}$
		$\displaystyle\geq\bigl\\|(\|x_{ij}\|^{p})\bigr\\|_{q/p,J;\,1,I}^{1/p}$
		$\displaystyle=\\|(x_{ij})\\|_{q,J;\,p,I}.$

	$\displaystyle\|\log a^{\prime}_{n}$	$\displaystyle(\mathbf{x})-\log a^{\prime}_{n}(\mathbf{x}^{\prime})\|=\frac{1}{p}\|\log\alpha^{\otimes n}(\mathbf{x})-\log\alpha^{\otimes n}(\mathbf{x}^{\prime})\|$
		$\displaystyle\leq\frac{1}{p}n\Big\|h(\mathbf{q}(\mathbf{x}))-h(\mathbf{q}(\mathbf{x}^{\prime}))\Big\|+\frac{1}{p}n\Big\|\mathsf{D}(\mathbf{q}(\mathbf{x})\|\mkern-2.0mu\|\alpha)-\mathsf{D}(\mathbf{q}(\mathbf{x}^{\prime})\|\mkern-2.0mu\|\alpha)\Big\|$
		$\displaystyle\leq\frac{\sqrt{2}\cdot d}{p}\cdot n^{3/4}+\frac{1}{p}n^{1/2}\leq c\cdot n^{3/4}$

		$\displaystyle\delta_{\mathsf{x}}Mb=\deg_{\mathsf{H}}(\mathsf{x}),$		$\displaystyle\\|a\\|_{p}=\|\mathsf{X}_{\mathsf{H}}\|^{1/p},$		$\displaystyle aMb=\|\mathsf{E}_{\mathsf{H}}\|$
		$\displaystyle aM\delta_{\mathsf{y}}=\deg_{\mathsf{H}}(\mathsf{y}),$		$\displaystyle\\|b\\|_{q}=\|\mathsf{Y}_{\mathsf{H}}\|^{1/q}$

Shapes and Norms of Random Pairs

Abstract

1 Shape function and its relation to the norms of incidence operators

Theorem A.

2 Notation, conventions, recollections

2.1 Notation

2.2 Random variables

2.3 Graphs

2.4 Extension profile and shape function

3 Norms on tensor products

3.1 Tensor products

3.2 Cross norms

3.2.A Algebraic definition of cross norms

3.2.C Geometric definition of cross norms

3.2.D Injective and projective cross norms

3.2.E Cross norms for Lebesgue spaces

3.3 Multiplicative family of norms for bilinear forms

3.3.A.

3.3.B.

3.3.C Proof of Proposition 3.3.A

3.3.E Proof of Lemma 3.3.B

3.4 Resonance vectors for tensor powers of bilinear forms

3.4.A Resonance vectors for bilinear form. Uniform vectors

3.4.B Uniform almost resonance pairs for tensor powers

3.4.C.

3.4.D.

3.4.E.

3.4.F Proof of Theorem 3.4.C

Some recollections.

The proof.

Step 1.

Step 2.

Step 3.

4 Bilinear form associated to a bipartite graph

4.1 Norms of incidence bilinear forms

4.1.A.

4.1.B Proof of Proposition 4.1.A

4.2 Uniform almost resonance vectors for incidence forms

4.2.A.

4.2.B Proof of Proposition 4.2.A

4.3 Subgraphs and extensions

4.3.A Construction of an extension from a subgraph

4.3.C.

4.3.D Proof of Proposition 4.3.C

5 Proof of the main theorem

5.1.A.

5.1.B Proof of Theorem 5.1.A

Case 𝜶+𝜷≥𝟏\bm{\alpha+\beta\geq 1}; equivalently, 𝟏/𝒑+𝟏/𝒒≤𝟏\bm{1/p+1/q\leq 1}.

Case 𝜶+𝜷<𝟏\bm{\alpha+\beta<1}; equivalently, 𝟏/𝒑+𝟏/𝒒>𝟏\bm{1/p+1/q>1}.

Proof of inequality (5.1.E).

Proof of inequality (5.1.F).

6 Properties of the shape

6.1 Lower and upper bounds for the shape function

6.2 Properties of the shape

6.2.A Symmetry

6.2.B Convexity

6.2.C Coordinate-wise monotonicity and reverse monotonicity

6.2.D Additivity

6.2.E Chain rule

6.2.F Test inequality

6.2.G Upper bound

6.2.H Lower bound

6.2.I Composition inequality

6.2.J Refinement and coarsening

6.2.K Link and reversed link inequalities

6.2.L Flattening/links and reverse

6.2.M MMRV-inequality

6.3 Proofs

Proof of (MO) and (RMO).

Proof of (COMP).

Proof of (REF).

Proof of (CRS).

Proof of (LI).

Proof of (RLI).

Proof of (FLI).

Proof of (RFLI).

Proof of (MMRV).

References

Case $\bm{\alpha+\beta\geq 1}$ ; equivalently, $\bm{1/p+1/q\leq 1}$ .

Case $\bm{\alpha+\beta<1}$ ; equivalently, $\bm{1/p+1/q>1}$ .