Tropical diagrams of probability spaces

TL;DR

This paper introduces the tropical cone of diagrams of probability spaces, analyzing its large-scale geometry and potential applications in information theory and artificial intelligence.

Contribution

It defines the tropical cone structure for diagrams of probability spaces and studies its asymptotic geometry, linking it to entropy and information optimization.

Findings

Identification of the asymptotic cone as a convex cone in a Banach space

Introduction of tropical diagrams as flexible objects within the tropical cone

Potential applications to entropic cone analysis and information optimization

Abstract

After endowing the space of diagrams of probability spaces with an entropy distance, we study its large-scale geometry by identifying the asymptotic cone as a closed convex cone in a Banach space. We call this cone the tropical cone, and its elements tropical diagrams of probability spaces. Given that the tropical cone has a rich structure, while tropical diagrams are rather flexible objects, we expect the theory of tropical diagrams to be useful for information optimization problems in information theory and artificial intelligence. In a companion article, we give a first application to derive a statement about the entropic cone.

Equations

$$Z$$

$$\gamma +{\gamma }^{\prime }\stackrel{d}{=}{\gamma }^{\prime }+\gamma$$

$$d(\gamma +{\gamma }^{\prime },\gamma +{\gamma }^{\prime \prime })\le d({\gamma }^{\prime },{\gamma }^{\prime \prime })$$

$$T_{\eta }:\Gamma \to \Gamma ,\gamma \mapsto \gamma +\eta$$

$$d({\gamma }_1+{\gamma }_2,{\gamma }_3+{\gamma }_4)\le d({\gamma }_1,{\gamma }_3)+d({\gamma }_2,{\gamma }_4)$$

$$d(n\cdot {\gamma }_1,n\cdot {\gamma }_2)\le n\cdot d({\gamma }_1,{\gamma }_2)$$

$$\delta (n\cdot {\gamma }_1,n\cdot {\gamma }_2)=n\cdot \delta ({\gamma }_1,{\gamma }_2)$$

$${\lambda }_1\cdot ({\lambda }_2\cdot {\gamma }_1)$$

$$\lambda \cdot ({\gamma }_1+{\gamma }_2)$$

$$(\lambda +{\lambda }^{\prime })\cdot {\gamma }_1$$

$$\delta (\lambda \cdot \gamma ,\lambda \cdot {\gamma }^{\prime })$$

$$\delta ({\gamma }_1+\eta ,{\gamma }_2+\eta )=\delta ({\gamma }_1,{\gamma }_2)$$

$$\operatorname{\mathbf{d}}\big{(}\gamma(m+n),\gamma(m)+\gamma(n)\big{)}\leq\varphi(m+n)$$

$$a(m+n)\le a(n)+a(n)+{\phi }_3(n+m)$$

$$\hat{d}(\overline{{\gamma }_1},\overline{{\gamma }_2}):=\underset{n\to \infty }{\lim }\frac{1}{n}d({\gamma }_1(n),{\gamma }_2(n))$$

$${\bar{\gamma }}_1\stackrel{\hat{d}}{=}{\bar{\gamma }}_2$$

$$d(\gamma (m\cdot n),m\cdot \gamma (n))\le 8\cdot m\cdot n\cdot {\int }_n^{2m\cdot n}\frac{\phi (t)}{t^2}dt$$

$$\psi (s):=\phi ({\mathbf{e}}^s)/{\mathbf{e}}^s\text{or}\phi (t)=:t\cdot \psi (\ln t)$$

$$d(\gamma (m\cdot n),m\cdot \gamma (n))\le 8\cdot m\cdot n\cdot {\int }_{\ln n}^{\ln (2\cdot m\cdot n)}\psi (s)ds$$

$$\psi (s_0)$$

$$\displaystyle\operatorname{\mathbf{d}}\big{(}$$

$$\displaystyle=\operatorname{\mathbf{d}}\big{(}\,\gamma(m^{\prime}\cdot n+m^{\prime}\cdot n+\varepsilon\cdot n)\,,\,m^{\prime}\cdot\gamma(n)+m^{\prime}\cdot\gamma(n)+\varepsilon\cdot\gamma(n)\,\big{)}$$

$$\displaystyle\leq 2\operatorname{\mathbf{d}}\big{(}\,\gamma(m^{\prime}\cdot n)\,,\,m^{\prime}\cdot\gamma(n)\,\big{)}+2\varphi\big{(}\,m\cdot n\,\big{)}$$

$$\displaystyle\leq 16m^{\prime}\cdot n\cdot\int_{\ln n}^{\ln(2m^{\prime}\cdot n)}\psi(s)\operatorname{\mathrm{d}\mkern-3.0mu}s+2m\cdot n\cdot\psi\big{(}\ln(m\cdot n)\big{)}$$

$$\le 8m\cdot n\left({\int }_{\ln n}^{\ln (2m^{\prime }\cdot n)}\psi (s)ds+{\int }_{\ln (m\cdot n)}^{\ln (2m\cdot n)}\psi (s)ds\right)\le 8m\cdot n\cdot {\int }_{\ln n}^{\ln (2m\cdot n)}\psi (s)ds$$

$$d(\gamma (m\cdot n),m\cdot \gamma (n))\le 8\cdot m\cdot n\cdot {\int }_n^{\infty }\frac{\phi (t)}{t^2}dt\le D_{\phi }\cdot m\cdot \phi (n)$$

$${\lambda\cdot\bar{\gamma}}:=\left\{\gamma\big{(}\lfloor\lambda\cdot n\rfloor\big{)}\right\}_{n\in\mathbb{N}_{0}}$$

$$\cdot :{\mathbb{R}}_{\ge 0}\times {\mathrm{QL}}_{\phi }(\Gamma ,d)\to {\mathrm{QL}}_{\phi }(\Gamma ,d)$$

$$\hat{d}(\lambda \cdot {\bar{\gamma }}_1,\lambda \cdot {\bar{\gamma }}_2)=\lambda \cdot \hat{d}({\bar{\gamma }}_1,{\bar{\gamma }}_2)$$

$$\lambda \cdot ({\bar{\gamma }}_1+{\bar{\gamma }}_2)$$

$$({\lambda }_1+{\lambda }_2)\cdot \bar{\gamma }$$

$$\underset{n}{\underbrace{\bar{\gamma }+\cdots +\bar{\gamma }}}\stackrel{\hat{d}}{=}n\cdot \bar{\gamma }$$

$$\displaystyle k\cdot\operatorname{\bm{\delta}}\big{(}\gamma_{i}(n),\gamma_{j}(n)\big{)}$$

$$\displaystyle\leq\operatorname{\bm{\delta}}\big{(}\gamma_{i}(k\cdot n),\gamma_{j}(k\cdot n)\big{)}+2k\cdot D_{\varphi}\cdot\varphi(n)$$

Full text

• $\boxtimes$*

\NewEnviron

proposition[1]

• $\boxtimes$*

\NewEnviron

theorem[1]

• $\boxtimes$*

\NewEnviron

corollary[1]

• $\boxtimes$*

\NewEnviron

lemma[1]

• $\boxtimes$*

\NewEnviron

tlemma[1]

• $\boxtimes$*

\NewEnviron

alphatheorem[1]

• $\boxtimes$*

\NewEnviron

alphaconjecture[1]

• $\boxtimes$*

\NewEnviron

Proof[1][#1]Proof: \BODY $\boxtimes$

\NewEnvironskipenv

Tropical diagrams of probability spaces

R. Matveev

 and 

J. W. Portegies

1. Introduction

With [MP18] we started a research program aiming for a systematic approach to a class of information optimization problems in information theory and artificial intelligence. A prototypical example of such a problem, still wide open, is the characterization of the entropic cone, the closure of all vectors in $\mathbb{R}^{2^{N}-1}$, which are entropically representable. Other information optimization problems arise for instance in causal inference [SA15], artificial intelligence [VDP13], information decomposition [BRO*+*14], robotics [ABD*+*08], neuroscience [Fri09] and in variational autoencoders [KW13].

The global strategy of our program is roughly based on the following way of thinking. The entropic cone is clearly a very complicated object: it is known that it is not polyhedral [Mat07]. Yet, perhaps, much of its complexity may be explained by it being a projection of another, simpler, higher-dimensional object.

The purpose of this article is to construct such a higher-dimensional (infinite-dimensional, in fact) object, which we call the tropical cone, and to derive some of its basic properties. In [MP19b] we apply the theory to derive a statement about the entropic cone.

Before outlining the construction of the tropical cone, let us mention that for our purposes, the language of random variables proved inconvenient, which is why work with diagrams of probability spaces instead.

Diagrams of probability spaces are commutative diagrams in the category of probability spaces, with (equivalence classes of) measure-preserving maps as morphisms, such as

[TABLE]

Collections of $n$ random variables give rise to a special type of diagrams, that include, besides the target spaces of the random variables themselves, the target space of every joint variable. Such diagrams have a particular combinatorial type. The first and the last diagrams in (1) are examples of such special types of diagrams in case of two and three random variables respectively. The description of other diagrams using the language of random variables is much less transparent.

We construct the tropical cone as the asymptotic cone in the space of diagrams of probability spaces endowed with the intrinsic entropy distance [KSŠ12, Vid12, MP18]. The asymptotic cone captures large-scale geometry of a metric space. As a particularly neat application, A’Campo gave an elegant construction of the real numbers as an asymptotic cone in a metric space of sequences of integers [A’C03]. We will call elements in the tropical cone tropical diagrams of probability spaces.

The reason for the name tropical cone is the following. For instance in algebraic geometry, tropical varieties are, roughly speaking, divergent sequences of classical varieties, renormalized on a log scale with an increasing base. The adjective ‘tropical’ carries little semantics, but was introduced in honor of the Brazilian mathematician and computer scientist Imre Simon who worked on the subject of tropical mathematics. Analogously, we construct the asymptotic cone from certain divergent sequences with respect to the intrinsic entropy distance. As the intrinsic entropy distance is entropy-based, we achieve a similar type of renormalization as in algebraic geometry.

The tropical cone has a rich algebraic structure. Indeed, we show that it is a closed, convex cone in a Banach space. In particular, one can take convex combinations of tropical diagrams. Other useful operations and constructions can be carried through for tropical diagrams, whereas they do not have an equivalent in the classical context of probability spaces, see [MP19a]. All in all, from some perspective, tropical diagrams are easier to deal with than diagrams or probability spaces, since only rough, asymptotic relations between probability spaces are preserved under tropicalization, similar to how all complicated features of the landscape disappear when looking at the Earth from outer space.

The structure of the present article is as follows. In Section 2, we first give a general construction of an asymptotic cone in an abstract setting. We believe that this abstract setting will make the construction more transparent and easier to follow. The results we present in that section are probably quite standard, but we find it beneficial to gather them “under one roof.” In Section 3 we show how, under certain conditions, the asymptotic cone can be interpreted as a closed convex cone in a Banach space. We specify to the case of diagrams of probability spaces in Section 4, reformulate the Asymptotic Equipartition Property proved in [MP18] in terms of tropical diagrams in Section 5. We conclude with a simple characterization of the tropical cone for special types of diagrams in Section 6.

2. Asymptotic Cones of Metric Abelian Monoids

In this section we define the asymptotic cone in the setting of an abstract metric Abelian monoid. In a later section, we will specify to the case of diagrams of probability spaces.

2.1. Metric and pseudo-metric spaces

A pseudo-metric space $(X,\operatorname{\mathbf{d}})$ is a set $X$ equipped with a pseudo-distance $\operatorname{\mathbf{d}}$, a bivariate function satisfying all the axioms of a distance function, except that it is allowed to vanish on pairs of non-identical points. An isometry of such spaces is a distance-preserving map, such that for any point in the target space there is a point in the image at zero distance away from it. Given such an pseudo-metric space $(X,\operatorname{\mathbf{d}})$ one could always construct an isometric metric space $(X/_{\operatorname{\mathbf{d}}=0}\,,\operatorname{\mathbf{d}})$, the metric quotient, by identifying all pairs of points that are distance zero apart.

Any property formulated in terms of the pseudo-metric holds simultaneously for a pseudo-metric space and its metric quotient. It will be convenient for us to construct pseudo-metrics on spaces instead of passing to the quotient spaces.

For a pair of points $x,y\in X$ in a pseudo-metric space $(X,\operatorname{\mathbf{d}})$ we will write $x\stackrel{{\scriptstyle\operatorname{\mathbf{d}}}}{{=}}y$ if $\operatorname{\mathbf{d}}(x,y)=0$. We call such a pair of points ($\operatorname{\mathbf{d}}$-)metrically equivalent.

Many metric-topological notions such as (Lipschitz-)continuity, compactness, $\varepsilon$-nets, dense subsets, etc., extend to the setting of a pseudo-metric spaces and exercising certain care one may switch between a pseudo-metric space and its metric quotient replacing the $\stackrel{{\scriptstyle\operatorname{\mathbf{d}}}}{{=}}$-sign with equality.

2.2. Metric Abelian Monoids

A monoid is a set equipped with a bivariate associative operation and a neutral element. The operation is usually called multiplication, or addition if it is commutative. We call a monoid with pseudo-distance $(\Gamma,+,\operatorname{\mathbf{d}})$ a metric Abelian monoid if it satisfies:

1. (1)

For any $\gamma,\gamma^{\prime}\in\Gamma$ holds

[TABLE] 2. (2)

The binary operation is 1-Lipschitz with respect to each argument: For any $\gamma,\gamma^{\prime},\gamma^{\prime\prime}\in\Gamma$

[TABLE]

The following proposition is elementary. {proposition}p:mam-translations Let $(\Gamma,+,\operatorname{\mathbf{d}})$ be a metric Abelian monoid. Then:

1. (1)

The translations maps

[TABLE]

are non-expanding for any $\eta\in\Gamma$. 2. (2)

For any quadruple $\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\in\Gamma$ holds

[TABLE] 3. (3)

For every $n\in\mathbb{N}$, and $\gamma_{1},\gamma_{2}\in\Gamma$ also holds

[TABLE]

A metric Abelian monoid $(\Gamma,+,\operatorname{\bm{\delta}})$ will be called homogeneous if it satisfies

[TABLE]

A homogeneous metric Abelian monoid is called an $\mathbb{R}_{\geq 0}$-semi-module $(\Gamma,+,\cdot\,,\operatorname{\bm{\delta}})$ if in addition there is a doubly distributive $\mathbb{R}_{\geq 0}$-action such that for any $\lambda_{1},\lambda_{2}\in\mathbb{R}_{\geq 0}$ and $\gamma_{1},\gamma_{2}\in\Gamma$ holds

[TABLE]

A convex cone in a normed vector space would be a typical example of an $\mathbb{R}_{\geq 0}$-semimodule. An intersection of a convex cone in $\mathbb{R}^{n}$ with the integer lattice is an example of a monoid, that does not admit semimodule structure.

The following proposition asserts that if a metric Abelian monoid is homogeneous, then the pseudo-distance is translation invariant, and, in particular, it satisfies a cancellation property. This result was communicated to us by Tobias Fritz, see also [Fri], [MP18, Proposition 3.7].

{proposition}

p:translation-invariance Let $(\Gamma,+,\operatorname{\bm{\delta}})$ be a homogeneous metric Abelian monoid. Then the pseudo-distance function $\operatorname{\bm{\delta}}$ is translation invariant, that is it satisfies for any $\gamma_{1},\gamma_{2},\eta\in\Gamma$

[TABLE]

In particular, the following cancellation property holds in $\Gamma$

If $\gamma_{1}+\eta\stackrel{{\scriptstyle\operatorname{\bm{\delta}}}}{{=}}\gamma_{2}+\eta$, then $\gamma_{1}\stackrel{{\scriptstyle\operatorname{\bm{\delta}}}}{{=}}\gamma_{2}$.

2.3. Asymptotic Cones (Tropicalization) of Monoids

In our construction points of the asymptotic cone of $(\Gamma,+,\operatorname{\mathbf{d}})$ will be sequences of points in $\Gamma$ that grow almost linearly in a certain sense described below.

2.3.1. Admissible functions

Admissible functions will be used to measure the deviation of a sequence from being linear. We call a function $\varphi:\mathbb{R}_{\geq 1}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}_{\geq 0}$ admissible if

1. (1)

the function $\varphi$ is non-decreasing; 2. (2)

there exists a constant $D_{\varphi}\geq 0$ such that $s\cdot\int_{s}^{\infty}\frac{\varphi(t)}{t^{2}}\operatorname{\mathrm{d}\mkern-3.0mu}t\leq\frac{D_{\varphi}}{8}\cdot\varphi(s)$ for any $s\geq 1$. In particular the function $\varphi$ is summable against $\operatorname{\mathrm{d}\mkern-3.0mu}t/t^{2}$.

For example, the function $\varphi(t)\!\!:=\!\!t^{\alpha}$ is admissible for any $0\leq\alpha<1$. Any admissible function is necessarily sub-linear, that is $\varphi(t)/t\stackrel{{\scriptstyle}}{{\rightarrow}}0$ as $t\stackrel{{\scriptstyle}}{{\rightarrow}}\infty$. A linear combination of admissible functions with non-negative coefficients is also admissible.

2.3.2. Quasi-linear sequences

Let $(\Gamma,+,\operatorname{\mathbf{d}})$ be a metric Abelian monoid and $\varphi$ be an admissible function. A sequence $\bar{\gamma}=\left\{\gamma(i)\right\}\in\Gamma^{\mathbb{N}_{0}}$ will be called quasi-linear with defect bounded by $\varphi$ if for every $m,n\in\mathbb{N}$ the following bound is satisfied

[TABLE]

For technical reasons we also require $\gamma(0)=0$. Sequences that are quasi-linear with defect bounded by $\varphi\equiv 0$ will be called linear sequences.

For an admissible function $\varphi$ we will write $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$ for the space of all quasi-linear sequences with defect bounded by $C\cdot\varphi$ for some (depending on the sequence) constant $C\geq 0$. We will also use notation $\operatorname{\mathsf{L}}(\Gamma,\operatorname{\mathbf{d}}):=\operatorname{\mathsf{Q}\mathsf{L}}_{0}(\Gamma,\operatorname{\mathbf{d}})$ for the space of linear sequences.

2.3.3. Asymptotic distance

Given two quasi-linear sequences $\bar{\gamma}_{1}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi_{1}}(\Gamma,\operatorname{\mathbf{d}})$ and $\bar{\gamma}_{2}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi_{2}}(\Gamma,\operatorname{\mathbf{d}})$ the sequence of distances $a(n):=\operatorname{\mathbf{d}}(\gamma_{1}(n),\gamma_{2}(n))$ is $\varphi_{3}$-subadditive, where $\varphi_{3}=\varphi_{2}+\varphi_{2}$ is also admissible, i.e.

[TABLE]

for any $n,m\in\mathbb{N}$. By the generalization of Fekete’s Lemma by De Bruijn and Erdös [dBE52, Theorem 23], it follows that the following limit exists and finite

[TABLE]

We call the quantity $\operatorname{\hat{\operatorname{\mathbf{d}}}}(\bar{\gamma}_{1},\bar{\gamma}_{2})$ the asymptotic distance between $\bar{\gamma}_{1},\bar{\gamma}_{2}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$. It is easy to verify that $\operatorname{\hat{\operatorname{\mathbf{d}}}}$ indeed satisfies all axioms of a pseudo-distance. Even if $\operatorname{\mathbf{d}}$ was a proper distance function, the corresponding asymptotic distance may vanish on some pairs of non-identical elements. We call two sequences $\bar{\gamma}_{1}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi_{1}}(\Gamma,\operatorname{\mathbf{d}})$, $\bar{\gamma}_{2}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi_{2}}(\Gamma,\operatorname{\mathbf{d}})$ asymptotically equivalent if $\operatorname{\hat{\operatorname{\mathbf{d}}}}(\bar{\gamma}_{1},\bar{\gamma}_{2})=0$ and write

[TABLE]

2.3.4. Quasi-homogeneity

We will show that quasi-linear sequences are also quasi-homogeneous in the sense of the following lemma. {lemma}l:scaling-of-iteration Let $\bar{\gamma}\in\Gamma^{\mathbb{N}_{0}}$ be a sequence with $\varphi$-bounded defect. Then for any $m,n\in\mathbb{N}$

[TABLE]

Proof: Define the function $\psi$ related to $\varphi$ as follows

[TABLE]

The conclusion of the lemma in terms of $\psi$ then reads

[TABLE]

and it is in that form it will be proven below.

Due to monotonicity properties of $\varphi$ function $\psi$ satisfies, for any $0\leq s_{0}\leq s$

[TABLE]

We proceed by induction with respect to $m$, keeping $n$ fixed. The conclusion of the lemma is obvious for $m=1$. For the induction step let $m=2m^{\prime}+\varepsilon\geq 2$, where $m^{\prime}=\lfloor m/2\rfloor$ and $\varepsilon\in\left\{0,1\right\}$. Then using bound (2.2) we estimate

[TABLE]

$\boxtimes$

Applying bound (2) in the definition of admissible functions on page 2 we obtain the following corollary. {corollary}p:psi-homo Let $\bar{\gamma}$ be a sequence with $\varphi$-bounded defect. Then for any $m,n\in\mathbb{N}$

[TABLE]

2.3.5. The semi-module structure

The group operation $+$ on $\Gamma$ induces a $\operatorname{\hat{\operatorname{\mathbf{d}}}}$-continuous (in fact, 1-Lipschitz) group operation on $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$ by adding sequences element-wise. Thus $(\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}}),+,\operatorname{\hat{\operatorname{\mathbf{d}}}})$ is also a metric Abelian monoid. In addition, it carries the structure of a $\mathbb{R}_{\geq 0}$-semi-module, as explained below.

The validity of the following constructions is very easy to verify, so we omit the proofs. Let $\varphi>0$ be an admissible function. The set $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$ admits an action of the multiplicative semigroup $(\mathbb{R}_{\geq 0},\,\cdot\,)$ defined in the following way. Let $\lambda\in\mathbb{R}_{\geq 0}$ and $\bar{\gamma}=\left\{\gamma(n)\right\}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$. Then define the action of $\lambda$ on $\bar{\gamma}$ by

[TABLE]

This is only an action up to asymptotic equivalence. Similarly, in the constructions that follow we are tacitly assuming they are valid up to asymptotic equivalence.

The action

[TABLE]

is continuous with respect to $\operatorname{\hat{\operatorname{\mathbf{d}}}}$ and, moreover it is a homothety (dilation), that is

[TABLE]

The semigroup structure on $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$ is distributive with respect to the $\mathbb{R}_{\geq 0}$-action

[TABLE]

In particular, for $n\in\mathbb{N}$ and $\bar{\gamma}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$

[TABLE]

2.3.6. Completeness

Here, we introduce additional conditions on a metric Abelian monoid $(\Gamma,+,\operatorname{\mathbf{d}})$, that guarantee that $(\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma),\operatorname{\hat{\operatorname{\mathbf{d}}}})$ is a complete metric space.

Suppose $\varphi$ is an admissible function and $(\Gamma,+,\operatorname{\mathbf{d}})$ is a metric Abelian monoid satisfying the following additional property: there exists a constant $C>0$, such that for any quasi-linear sequence $\bar{\gamma}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$, there exists an asymptotically equivalent quasi-linear sequence $\bar{\gamma}^{\prime}$ with defect bounded by $C\varphi$. Note that, contrary to the situation in the definition of $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$, the constant $C$ is now not allowed to depend on the sequence. If this is the case, we say that $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$ has the ($C$-)uniformly bounded defect property.

{proposition}

p:boundeddefect Suppose a metric Abelian monoid $(\Gamma,+,\operatorname{\bm{\delta}})$ and an admissible function $\varphi>0$ are such that $(\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\bm{\delta}}),\operatorname{\hat{}\bm{\delta}})$ has the uniformly bounded defect property and the distance function $\operatorname{\bm{\delta}}$ is homogeneous. Then the space $(\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\bm{\delta}}),\operatorname{\hat{}\bm{\delta}})$ is complete.

Proof: Given a Cauchy sequence $\left\{\bar{\gamma}_{i}\right\}$ of elements in $(\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\bm{\delta}}),\operatorname{\hat{}\bm{\delta}})$ we need to find a limit element $\bar{\eta}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\bm{\delta}})$. We will construct $\bar{\eta}$ by a diagonal argument. First we replace each element of the sequence $\left\{\bar{\gamma}_{i}\right\}$ by an asymptotically equivalent element with defect bounded by $C\varphi$ according to the assumption of the proposition. We will still call the new sequence $\left\{\bar{\gamma}_{i}\right\}$. In fact, we may without loss of generality assume that $C=1$.

We begin by establishing a bound on the divergence of the tails of sequences $\bar{\gamma}_{i}$ and $\bar{\gamma}_{j}$. By homogeneity of $\operatorname{\bm{\delta}}$ and Corollary LABEL:p:psi-homo, it holds for any $n,k\in\mathbb{N}$ that

[TABLE]

Dividing by $k$ and passing to the limit $k\stackrel{{\scriptstyle}}{{\rightarrow}}\infty$, while keeping $n$ fixed, we obtain

[TABLE]

Since the sequence $(\bar{\gamma}_{i})_{i\in\mathbb{N}_{0}}$ is Cauchy, it follows that for any $n\in\mathbb{N}$ there is a number $\mathbf{i}(n)\in\mathbb{N}$ such that for any $i,j\geq\mathbf{i}(n)$ holds

[TABLE]

Then for any $i,j,n\in\mathbb{N}$ with $i,j\geq\mathbf{i}(n)$ we have the following bound

[TABLE]

Now we are ready to define the limiting sequence $\bar{\eta}$ by setting

[TABLE]

First we verify that $\bar{\eta}$ is quasi-linear. For $m,n\in\mathbb{N}$, we have

[TABLE]

for some constant $C^{\prime}>0$.

The convergence of $\bar{\gamma}_{i}$ to $\bar{\eta}$ is shown as follows. For $n,k\in\mathbb{N}$ let $q_{n},r_{n}\in\mathbb{N}_{0}$ be the quotient and the remainder of the division of $n$ by $k$, that is $n=q_{n}\cdot k+r_{n}$ and $0\leq r_{n}<k$. Fix $k\in\mathbb{N}$ and let $i\geq\mathbf{i}(k)$, then

[TABLE]

Since $k\in\mathbb{N}$ is arbitrary and $\varphi$ is sub-linear we have

[TABLE]

$\boxtimes$

2.3.7. On the density of linear sequences

For a metric Abelian monoid $(\Gamma,+,\operatorname{\mathbf{d}})$ together with an admissible function $\varphi$ we say that $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$ has the vanishing defect property if for every $\varepsilon>0$ and for every $\bar{\gamma}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$ there exists an asymptotically equivalent quasi-linear sequence $\bar{\gamma}^{\prime}$ with defect bounded by another admissible function $\psi$ such that $\int_{1}^{\infty}\frac{\psi(t)}{t^{2}}\operatorname{\mathrm{d}\mkern-3.0mu}t<\varepsilon$.

The proposition below gives a sufficient condition under which the linear sequences are dense in the space of quasi-linear sequences.

{proposition}

p:eps-linear-dense Suppose $(\Gamma,+,\operatorname{\mathbf{d}})$ and admissible function $\varphi$ have the vanishing defect property. Then $\operatorname{\mathsf{L}}(\Gamma,\operatorname{\mathbf{d}})$ is dense in $(\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}}),\operatorname{\hat{\operatorname{\mathbf{d}}}})$.

Proof: Let $\bar{\gamma}=\left\{\gamma(n)\right\}$ be a quasi-linear sequence. For any $i\in\mathbb{N}$ select a sequence $\bar{\gamma}_{i}$ asymptotically equivalent to $\bar{\gamma}$ with defect bounded by an admissible function $\varphi_{i}$ such that $\int_{1}^{\infty}\frac{\varphi_{i}(t)}{t^{2}}\operatorname{\mathrm{d}\mkern-3.0mu}t<1/i$ according to the “vanishing defect” assumption of the lemma.

Define $\bar{\eta}_{i}$ by

[TABLE]

Then

[TABLE]

Thus, any quasi-linear sequence can be approximated by linear sequences. $\boxtimes$

2.3.8. Asymptotic distance on original monoid

Starting with an element $\gamma\in\Gamma$ one can construct a linear sequence $\vec{\gamma}=\left\{i\cdot\gamma\right\}_{i\in\mathbb{N}_{0}}$. In view of Proposition LABEL:p:mam-translations, the map

[TABLE]

is a contraction.

By the inclusions in (2.3.8) we have an induced metric $\operatorname{\bm{\delta}}$ on $\Gamma$, satisfying for any $\gamma_{1},\gamma_{2}\in\Gamma$

[TABLE]

and the following homogeneity condition

[TABLE]

for all $n\in\mathbb{N}_{0}$.

Note that if $\operatorname{\mathbf{d}}$ was homogeneous to begin with, then $\operatorname{\bm{\delta}}$ coincides with $\operatorname{\mathbf{d}}$ on $\Gamma$.

By virtue of the bound $\operatorname{\bm{\delta}}\leq\operatorname{\mathbf{d}}$, sequences that are quasi-linear with respect to $\operatorname{\bm{\delta}}$ are also quasi-linear with respect to $\operatorname{\mathbf{d}}$. Since $\operatorname{\bm{\delta}}$ is scale-invariant, the associated asymptotic distance $\operatorname{\hat{}\bm{\delta}}$ coincides with $\operatorname{\bm{\delta}}$ on $\Gamma$. We will show (in Lemma LABEL:p:dist-dista-isometry-dense below) that $\operatorname{\hat{}\bm{\delta}}$ also coincides with $\operatorname{\hat{\operatorname{\mathbf{d}}}}$ on $\operatorname{\mathbf{d}}$-quasi-linear sequences.

Let $\varphi$ be an admissible function. In order to organize all these statements, and to be more precise, let us include the spaces in the following commutative diagram.

[TABLE]

The maps $f,f^{\prime}$ and ${\bm{\imath}}_{1}$ are isometries. The maps ${\bm{\jmath}}_{1}$ and ${\bm{\jmath}}_{2}$ are isometric embeddings. The next lemmas show that ${\bm{\imath}}_{2}$ is also an isometric embedding, and it has dense image.

{lemma}

p:dist-dista-isometry-dense Let $\varphi$ be a positive, admissible function. Then, the natural inclusion

[TABLE]

is an isometric embedding with the dense image.

Proof: First we show that the map ${\bm{\imath}}_{2}$ is an isometric embedding. Let $\bar{\gamma}_{1},\bar{\gamma}_{2}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$ be two $\varphi$-quasi-linear sequences with respect to the distance function $\operatorname{\mathbf{d}}$. We have to show that the two numbers

[TABLE]

and

[TABLE]

are equal. Since shifts are non-expanding maps, we have $\operatorname{\bm{\delta}}\leq\operatorname{\mathbf{d}}$ and it follows immediately that

[TABLE]

and we are left to show the opposite inequality. We will do it as follows. Fix $n>0$, then

[TABLE]

Passing to the limit with respect to $n$ gives the required inequality

[TABLE]

Now we will show that the image of ${\bm{\imath}}_{2}$ is dense. Given an element $\bar{\gamma}=\left\{\gamma(n)\right\}$ in $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\hat{\operatorname{\mathbf{d}}}})$ we have to find a $\operatorname{\hat{}\bm{\delta}}$-approximating sequence $\bar{\gamma}_{i}=\left\{\gamma_{i}(n)\right\}$ in $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\Gamma,\operatorname{\mathbf{d}})$. Define

[TABLE]

We have to show that each $\bar{\gamma}_{i}$ is $\operatorname{\mathbf{d}}$-quasi-linear and that $\operatorname{\hat{}\bm{\delta}}(\bar{\gamma}_{i},\bar{\gamma})\stackrel{{\scriptstyle i\stackrel{{\scriptstyle}}{{\rightarrow}}\infty}}{{\longrightarrow}}0$. These statements follow from

[TABLE]

for some $C_{i}>0$. It is worth noting that the defect of $\bar{\gamma}_{i}$ may not be bounded uniformly with respect to $i$. Finally, it holds that

[TABLE]

$\boxtimes$

The difference between two distance functions $\operatorname{\hat{\operatorname{\mathbf{d}}}}$ and $\operatorname{\hat{}\bm{\delta}}$ is very small: $\operatorname{\hat{\operatorname{\mathbf{d}}}}$ is defined on the dense subset of the domain of definition of $\operatorname{\hat{}\bm{\delta}}$ and they coincide whenever are both defined. From now on we will not use the notation $\operatorname{\hat{}\bm{\delta}}$.

3. Grothendieck construction

Given an Abelian monoid with a cancellation property, there is a minimal Abelian group (called the Grothendieck Group of the monoid), into which it isomorphically embeds. Similarly, an $\mathbb{R}_{\geq 0}$-semi-module naturally embeds into a normed vector space. A nice example of this construction applied to the semi-module of convex sets in $\mathbb{R}^{n}$ (with the Minkowski sum and the Hausdorff distance) can be found in [Råd52].

{proposition}

p:banach-embedding Let $(\Gamma,+,\cdot,\operatorname{\bm{\delta}})$ be a complete metric Abelian monoid with $\mathbb{R}_{\geq 0}$ action (an $\mathbb{R}_{\geq 0}$-semi-module) with homogeneous pseudo-metric $\operatorname{\bm{\delta}}$. Then there exists a Banach space $(\mathbf{B},||\,\cdot\,||)$ and a distance-preserving homomorphism

[TABLE]

such that the image of $f$ is a closed convex cone.

If $\operatorname{\mathbf{d}}$ is a proper pseudo-metric (not a metric), then the map $f$ is not injective.

Proof: By Lemma LABEL:p:translation-invariance the pseudo-metric $\bm{\delta}$ is translation invariant. We can therefore apply the Grothendieck construction to define a normed vector space $\mathbf{B}_{0}$: Define

[TABLE]

where $(x,y)\sim(x^{\prime},y^{\prime})$ if there are $z,z^{\prime}\in\Gamma$, such that $(x+z,y+z)\stackrel{{\scriptstyle\operatorname{\mathbf{d}}}}{{=}}(x^{\prime}+z^{\prime},y^{\prime}+z^{\prime})$.

Define also addition, multiplication by a scalar and a norm on $\mathbf{B}_{0}$ by setting for all $x,y,x^{\prime},y^{\prime}\in\Gamma$ and $\lambda\in\mathbb{R}$

[TABLE]

These operations respect the equivalence relation and turn $(\mathbf{B}_{0},+,\cdot,||\,\cdot\,||)$ into a normed vector-space. The map $f$ defined by

[TABLE]

is a well-defined distance-preserving homomorphism.

That $f(\Gamma)$ is closed immediately follows as $\Gamma$ is complete and $f$ is distance-preserving.

In general, the space $\mathbf{B}_{0}$ is not complete. We define the Banach space $\mathbf{B}$ as the completion of the normed vector space $\mathbf{B}_{0}$. $\boxtimes$

4. Tropical probability spaces and their diagrams

4.1. Diagrams of probability spaces.

We will now briefly describe the construction of diagrams of probability spaces, see [MP18] for a more detailed discussion. By a finite probability space we will mean a set (not necessarily finite) with a probability measure, such that the support of the measure is finite. For such probability space $X$ we denote by $|X|$ the cardinality of the support of probability measure and the expression $x\in X$ will mean, that $x$ is an atom in $X$, which is a point of positive weight in the underlying set.

We will consider commutative diagrams of finite probability spaces, where arrows are equivalence classes of measure-preserving maps. Two maps are considered equivalent if they coincide on a set of full measure and such equivalence classes will be called reductions.

Three examples of diagrams of probability spaces are pictured in (1). The combinatorial structure of such a commutative diagram can be recorded by an object $\mathbf{G}$, which could be equivalently considered as a special type of category, a finite poset, or a directed acyclic graph (DAG) with additional properties. We will call such objects simply indexing categories. Below we briefly recall the definition.

An indexing category is a finite category such that for any pair of objects there exists at most one morphism between them in either direction, and such that it satisfies the following property. For any pair of objects $i,j$ in an indexing category $\mathbf{G}$ there exists a least common ancestor, i.e. an object $k$ such that there are morphisms $k\stackrel{{\scriptstyle}}{{\rightarrow}}i$ and $k\stackrel{{\scriptstyle}}{{\rightarrow}}j$ in $\mathbf{G}$ and such that for any other object $l$ admitting morphisms $l\stackrel{{\scriptstyle}}{{\rightarrow}}i$ and $l\stackrel{{\scriptstyle}}{{\rightarrow}}j$, there is also a morphism $l\stackrel{{\scriptstyle}}{{\rightarrow}}k$.

By $[\mkern-5.0mu[\mathbf{G}]\mkern-5.0mu]$ we denote the number of objects in the indexing category, or equivalently the number of vertices in the DAG or the number of points in the poset $\mathbf{G}$. Important class of examples of indexing categories are so called full categories $\bm{\Lambda}_{n}$, that correspond to the poset of non-empty subsets of a set $\left\{1,\ldots,n\right\}$ ordered by inclusion. If $n=2$, we call the category

[TABLE]

a fan.

The space of all commutative diagrams of a fixed combinatorial type will be denoted $\operatorname{\mathbf{Prob}}\langle\mathbf{G}\rangle$. A morphism between two diagrams $\mathcal{X},\mathcal{Y}\in\operatorname{\mathbf{Prob}}\langle\mathbf{G}\rangle$ is defined to be the collection of morphisms between corresponding individual spaces in $\mathcal{X}$ and $\mathcal{Y}$, that commute with morphisms within the diagrams $\mathcal{X}$ and $\mathcal{Y}$.

The construction of forming commutative diagrams could be iterated, producing diagrams of diagrams. Especially important will be two-fans of $\mathbf{G}$-diagrams, the space of which will be denoted $\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle\left\langle\bm{\Lambda}_{2}\right\rangle$.

A two-fan $\mathcal{X}$ will be called minimal, if for any morphism of $\mathcal{X}$ to another two-fan $\mathcal{Y}$, the following holds: if the induced morphisms on the feet are isomorphisms, then the top morphism is also an isomorphism. Any $\mathbf{G}$-diagram will be called minimal if for any sub-diagram, which is a two-fan, it contains a minimal two-fan with the same feet.

Given an $n$-tuple $(\mathsf{X}_{1},\ldots,\mathsf{X}_{n})$ of finite-valued random variables, one can construct a minimal $\bm{\Lambda}_{n}$-diagram $\mathcal{X}=\left\{X_{I};\chi_{IJ}\right\}$ by setting for any $\emptyset\neq I\subset\left\{1,\ldots,n\right\}$

[TABLE]

where $X_{i}$ is the target space of random variable $\mathsf{X}_{i}$, and the probabilities are the induced distributions. For the diagram constructed in such a way we will write $\mathcal{X}=\left\langle\mathsf{X}_{1},\ldots,\mathsf{X}_{n}\right\rangle$. On the other hand, any $\bm{\Lambda}_{n}$-diagram gives rise to the $n$-tuple of random variables with the domain of definition being the initial space and the targets being the terminal spaces.

The tensor product $\mathcal{X}\otimes\mathcal{Y}$ of two $\mathbf{G}$-diagrams is defined by taking the tensor product of corresponding probability spaces and the Cartesian product of maps.

The special $\mathbf{G}$-diagram in which all the spaces are isomorphic to a single probability space $X$ will be denoted by $X^{\mathbf{G}}$.

For a diagram $\mathcal{X}\in\operatorname{\mathbf{Prob}}\langle\mathbf{G}\rangle$ one can evaluate entropies of the individual spaces. The corresponding map will be denoted

[TABLE]

where the target space is the space of $\mathbb{R}$-valued functions on objects in $\mathbf{G}$ and it is equipped with the $\ell^{1}$-norm.

For a two-fan $\mathcal{F}=(\mathcal{X}\stackrel{{\scriptstyle}}{{\leftarrow}}\mathcal{Z}\stackrel{{\scriptstyle}}{{\rightarrow}}\mathcal{Y})$ of $\mathbf{G}$-diagrams define the entropy distance

[TABLE]

We interpret $\operatorname{\mathrm{kd}}(\mathcal{F})$ as a measure of deviation of $\mathcal{F}$ from being an isomorphism between the diagrams $\mathcal{X}$ and $\mathcal{Y}$. Indeed, $\operatorname{\mathrm{kd}}(\mathcal{F})=0$ if and only if the two morphisms in $\mathcal{F}$ are isomorphisms.

We define the intrinsic entropy distance $\operatorname{\mathbf{k}}$ on the space $\operatorname{\mathbf{Prob}}\langle\mathbf{G}\rangle$ by

[TABLE]

The tensor product is 1-Lipschitz with respect to $\operatorname{\mathbf{k}}$, thus $(\operatorname{\mathbf{Prob}}\langle\mathbf{G}\rangle,\otimes,\operatorname{\mathbf{k}})$ is a metric Abelian monoid and $\operatorname{\mathsf{Ent}}_{*}:(\operatorname{\mathbf{Prob}}\langle\mathbf{G}\rangle,\otimes,\operatorname{\mathbf{k}})\stackrel{{\scriptstyle}}{{\rightarrow}}(\mathbb{R}^{\mathbf{G}},\|\cdot\|_{1})$ is a 1-Lipschitz homomorphism. For proofs and more detailed discussion the reader is referred to [MP18].

4.2. Tropical diagrams

Applying the construction of the previous section we obtain its tropicalization – a semi-module $(\operatorname{\mathbf{Prob}}[\mathbf{G}],+,\,\cdot\,,\operatorname{\bm{\kappa}})$. The restriction of the asymptotic distance on the original monoid can be defined independently as

[TABLE]

One of the main tools for the estimation of the (asymptotic) distance is the so-called Slicing Lemma and its following consequence. {proposition}p:slicingcorollary Let $\mathbf{G}$ be an indexing category, $\mathcal{X},\mathcal{Y}\in\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle$ and $U\in\operatorname{\mathbf{Prob}}$.

1. (1)

Let $\mathcal{X}\stackrel{{\scriptstyle}}{{\rightarrow}}U$ be a reduction, then

[TABLE] 2. (2)

For a “co-fan” $\mathcal{X}\stackrel{{\scriptstyle}}{{\rightarrow}}U\stackrel{{\scriptstyle}}{{\leftarrow}}\mathcal{Y}$ holds

[TABLE]

The statements and the proofs of the Slicing Lemma and its consequences can be found in [MP18].

We will show below that $(\operatorname{\mathbf{Prob}}\langle\mathbf{G}\rangle,\otimes,\operatorname{\bm{\kappa}})$ has the uniformly bounded and vanishing defect properties. For this purpose we need to develop some technical tools.

4.3. Mixtures

The input data for the mixture operation is a family of $\mathbf{G}$-diagrams, parameterized by a probability space. As a result one obtains another $\mathbf{G}$-diagram with pre-specified conditionals. One particular instance of a mixture is when one mixes two diagrams $\mathcal{X}$ and $\left\{\bullet\right\}^{\mathbf{G}}$, the latter being a constant $\mathbf{G}$-diagram of one-point probability spaces. This operation will be used as a substitute for taking radicals “$\mathcal{X}^{\frac{1}{n}}$” below.

4.3.1. Definition of mixtures

Let $\mathbf{G}$ be an indexing category and $\Theta$ be a probability space. By $\Theta^{\mathbf{G}}$ we denote the constant $\mathbf{G}$-diagram – the diagram such that all spaces in it are $\Theta$ and all morphisms are identity morphisms. Let $\left\{\mathcal{X}_{\theta}\right\}_{\theta\in\underline{\Theta\mkern-5.0mu}\mkern 5.0mu}$ be a family of $\mathbf{G}$-diagrams parameterized by $\Theta$. The mixture of the family $\left\{\mathcal{X}_{\theta}\right\}$ is the reduction

[TABLE]

such that

[TABLE]

The mixture exists and is uniquely defined by property (4.3.1) up to an isomorphism which is identity on $\Theta^{\mathbf{G}}$.

We denote the top diagram of the mixture by

[TABLE]

and also call it the mixture of the family $\left\{\mathcal{X}_{\theta}\right\}$.

When

[TABLE]

is a binary space we write simply

[TABLE]

for the mixture. The diagram subindexed by the $\blacksquare$ will always be the first summand.

The entropy of the mixture can be evaluated by the following formula

[TABLE]

Mixtures satisfy the distributive law with respect to the tensor product

[TABLE]

4.3.2. The distance estimates for the mixtures.

Recall that for a diagram category $\mathbf{G}$ we denote by $\left\{\bullet\right\}=\left\{\bullet\right\}^{\mathbf{G}}$ the constant $\mathbf{G}$-diagram of one-point spaces.

The mixture of a $\mathbf{G}$-diagram with $\left\{\bullet\right\}^{\mathbf{G}}$ may serve as an substitute of taking radicals of the diagram. The following lemma provides a justification of this by some distance estimates related to mixtures and will be used below.

{lemma}

p:mixtures-dist1 Let $\mathbf{G}$ be a complete diagram category and $\mathcal{X},\mathcal{Y}\in\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle$. Then

1. (1)

$\displaystyle\operatorname{\bm{\kappa}}(\mathcal{X},\mathcal{X}^{n}\oplus_{\Lambda_{1/n}}\left\{\bullet\right\})\leq\operatorname{\mathsf{Ent}}(\Lambda_{1/n})$ 2. (2)

\displaystyle\operatorname{\bm{\kappa}}\big{(}\mathcal{X},(\mathcal{X}\oplus_{\Lambda_{1/n}}\left\{\bullet\right\})^{n}\big{)}\leq n\cdot\operatorname{\mathsf{Ent}}(\Lambda_{1/n}) 3. (3)

\displaystyle\operatorname{\bm{\kappa}}\big{(}(\mathcal{X}\otimes\mathcal{Y})\oplus_{\Lambda_{1/n}}\left\{\bullet\right\},(\mathcal{X}\oplus_{\Lambda_{1/n}}\left\{\bullet\right\})\otimes(\mathcal{Y}\oplus_{\Lambda_{1/n}}\left\{\bullet\right\})\big{)}\leq 3\operatorname{\mathsf{Ent}}(\Lambda_{1/n}) 4. (4)

\displaystyle\operatorname{\bm{\kappa}}\big{(}(\mathcal{X}\oplus_{\Lambda_{1/n}}\left\{\bullet\right\}),(\mathcal{Y}\oplus_{\Lambda_{1/n}}\left\{\bullet\right\})\big{)}\leq\frac{1}{n}\operatorname{\bm{\kappa}}(\mathcal{X},\mathcal{Y})

Note that the distance estimates in the lemma above are with respect to the asymptotic distance. This is essential, since from the perspective of the intrinsic distance mixtures are very badly behaved.

Proof: For $\lambda\in\Lambda_{1/n}^{N}$, define $\operatorname{\mathbf{q}}(\lambda)$ to be the number of black squares in the sequence $\lambda$. It is a binomially distributed random variable with mean $N/n$ and variance $\frac{N}{n}(1-\frac{1}{n})$.

The first claim is then proven by the following calculation

[TABLE]

where we used Proposition LABEL:p:slicingcorollary(1) for the inequality on the third line above.

The second claim is proven similarly and the third follows from the second and the $1$-Lipschitz property of the tensor product:

[TABLE]

Finally, the fourth follows from Proposition LABEL:p:slicingcorollary(2), by slicing both arguments along $\Lambda_{1/n}$. $\boxtimes$

4.4. Vanishing defect property and completeness of the

tropical cone

{lemma}

p:unifsmalldefectaikd For every admissible function $\varphi$, every $\bar{}\mathcal{X}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle,\operatorname{\bm{\kappa}})$ and every $k\in\mathbb{N}$, there exists an asymptotically equivalent sequence $\bar{}\mathcal{Y}$ with defect bounded by the admissible function $\varphi_{k}$ defined by

[TABLE]

Proof: Let $\bar{}\mathcal{X}=\left\{\mathcal{X}(i)\right\}$ be a quasi-linear sequence with defect bounded by $\varphi$ and let $k\in\mathbb{N}$.

Define a new sequence $\bar{}\mathcal{Y}=\left\{\mathcal{Y}(i)\right\}$ by

[TABLE]

First we verify that the sequences $\bar{}\mathcal{X}$ and $\bar{}\mathcal{Y}$ are asymptotically equivalent, that is

[TABLE]

We estimate the asymptotic distance between individual members of sequences $\bar{}\mathcal{X}$ and $\bar{}\mathcal{Y}$ using Lemma LABEL:p:mixtures-dist1 and Corollary LABEL:p:psi-homo as follows

[TABLE]

Thus $\hat{\operatorname{\bm{\kappa}}}(\bar{}\mathcal{X},\bar{}\mathcal{Y})=0$ and the two sequences are asymptotically equivalent. Next we show that the sequence $\bar{}\mathcal{Y}$ is $\operatorname{\bm{\kappa}}$-quasi-linear and evaluate its defect, also using Lemma LABEL:p:mixtures-dist1. Let $i,j\in\mathbb{N}$, then

[TABLE]

$\boxtimes$

{corollary}

p:vanishingdefect For any indexing category $\mathbf{G}$ and for the admissible function $\varphi$ given by $\varphi(t)=t^{\alpha}$, $\alpha\in[0,1)$, $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle,\operatorname{\bm{\kappa}})$ has the uniformly bounded and vanishing defect properties.

Proof: Let $\bar{}\mathcal{X}\in\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle,\operatorname{\bm{\kappa}})$. By Lemma LABEL:p:unifsmalldefectaikd there exists an asymptotically equivalent sequence $\bar{}\mathcal{Y}$ with defect bounded by $\varphi_{k}$ defined by

[TABLE]

Hence there exists a sequence $c_{k}\stackrel{{\scriptstyle}}{{\rightarrow}}0$ such that for all $t\geq 1$,

[TABLE]

showing the uniformly bounded and vanishing defect property. $\boxtimes$

4.5. Diagrams of tropical probability spaces

By applying the general setup in the previous section to the metric Abelian monoids $(\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle,\otimes,\operatorname{\mathbf{k}})$ and $(\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle,\otimes,\operatorname{\bm{\kappa}})$ and using the Corollary LABEL:p:vanishingdefect we obtain the following theorem.

{theorem}

p:corollary-acone-ikd-aikd Fix an admissible function $\varphi$ and consider the commutative diagram

[TABLE]

Then the following statements hold:

1. (1)

The maps $f,f^{\prime},{\bm{\imath}}_{1}$ are isometries. 2. (2)

The maps ${\bm{\imath}}_{2},{\bm{\jmath}}_{1},{\bm{\jmath}}_{2}$ are isometric embeddings and each map has a dense image in the corresponding target space. 3. (3)

The space in the lower-right corner, \big{(}\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle,\operatorname{\bm{\kappa}}),\hat{\operatorname{\bm{\kappa}}}\big{)}, is complete.

We would like to conjecture that all maps in the diagram above are isometries.

Since $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle,\operatorname{\bm{\kappa}})$ is complete and has $\operatorname{\mathsf{L}}(\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle,\operatorname{\bm{\kappa}})$ as a dense subset for any $\varphi>0$, it follows that $\operatorname{\mathsf{Q}\mathsf{L}}_{\varphi}(\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle,\operatorname{\bm{\kappa}})$ does not depend (up to isometry of pseudo-metric spaces) on the choice of admissible $\varphi>0$. From now on we will choose the particular function $\varphi(t):=t^{3/4}$. The choice will be clear when we formulate the Asymptotic Equipartition Property for diagrams. We may finally define the space of tropical $\mathbf{G}$-diagrams, as the space in the lower-right corner of the diagram

[TABLE]

By Theorem LABEL:p:corollary-acone-ikd-aikd above, this space is complete.

The entropy function $\operatorname{\mathsf{Ent}}_{*}:\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle\stackrel{{\scriptstyle}}{{\rightarrow}}\mathbb{R}^{\mathbf{G}}$ extends to a linear functional

[TABLE]

of norm one, defined by

[TABLE]

5. AEP

5.1. Homogeneous diagrams

A $\mathbf{G}$-diagram $\mathcal{X}$ is called homogeneous if the automorphism group $\operatorname{\mathrm{Aut}}(\mathcal{X})$ acts transitively on every space in $\mathcal{X}$. Homogeneous probability spaces are uniform. For more complex indexing categories this simple description is not sufficient. The subcategory of all homogeneous $\mathbf{G}$-diagrams will be denoted $\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle_{\mathsf{h}}$. This space is invariant under the tensor product, thus it is a metric Abelian monoid.

5.1.1. Universal construction of homogeneous diagrams

Examples of homogeneous diagrams could be constructed in the following manner. Fix a finite group $G$ and consider a $\mathbf{G}$-diagram $\left\{H_{i};\alpha_{ij}\right\}_{i\in\mathbf{G}}$ of subgroups of $G$, where morphisms $\alpha_{ij}$ are inclusions. The $\mathbf{G}$-diagram of probability spaces $\left\{X_{i};f_{ij}\right\}$ is constructed by setting $X_{i}=(G/H_{i},\operatorname{\mathsf{unif}})$ and taking $f_{ij}$ to be the natural projection $G/H_{i}\stackrel{{\scriptstyle}}{{\rightarrow}}G/H_{j}$, whenever $H_{i}\subset H_{j}$. The resulting diagram $\mathcal{X}$ will be minimal if and only if for any $i,j\in\mathbf{G}$ there is $k\in\mathbf{G}$, such that $H_{k}=H_{i}\cap H_{j}$. In fact, any homogeneous diagram arises this way, see [MP18].

5.2. Asymptotic Equipartition Property

In [MP18] the following theorem is proven. {theorem}p:aep-complete Suppose $\mathcal{X}\in\operatorname{\mathbf{Prob}}\left\langle\mathbf{G}\right\rangle$ is a $\mathbf{G}$-diagram of probability spaces for some fixed indexing category $\mathbf{G}$. Then there exists a sequence $\bar{}\mathcal{H}=(\mathcal{H}_{n})_{n=0}^{\infty}$ of homogeneous $\mathbf{G}$-diagrams such that

[TABLE]

where $C(|X_{0}|,[\mkern-5.0mu[\mathbf{G}]\mkern-5.0mu])$ is a constant only depending on $|X_{0}|$ and $[\mkern-5.0mu[\mathbf{G}]\mkern-5.0mu]$.

Defining

[TABLE]

the Asymptotic Equipartition Property can be reformulated as in the Theorem LABEL:p:aep-tropical below. {theorem}p:aep-tropical For any indexing category $\mathbf{G}$ the image of the natural inclusion

[TABLE]

is dense.

Proof: By Theorem LABEL:p:aep-complete, every linear sequence can be approximated by a homogeneous sequence. It follows from the bound (5.2) that the defect of the approximating homogeneous sequence is bounded by a constant times $\varphi$, defined by $\varphi(t)=t^{3/4}$. Moreover, the linear sequences are dense by Theorem LABEL:p:corollary-acone-ikd-aikd. This finishes the proof. $\boxtimes$

6. The tropical cone for probability spaces and chains

Although for general indexing categories $\mathbf{G}$ the space of tropical $\mathbf{G}$-diagrams is infinite dimensional, it has a very simple, finite-dimensional description if $\mathbf{G}$ consists of a single object, or if it is a special type of indexing categories called a chain.

The chain of length $k$, denoted by $\mathbf{C}_{k}$, is the indexing category with $k$ objects $O_{1},\dots,O_{k}$, and a morphism from $O_{i}$ to $O_{j}$ whenever $i\geq j$. A $\mathbf{C}_{k}$-diagram of probability spaces is then a chain of reductions

[TABLE]

Recall that homogeneous probability spaces are (isomorphic to) probability spaces with a uniform distributions. Homogeneous chains have a very simple description as well. A chain $\mathcal{H}\in\operatorname{\mathbf{Prob}}\left\langle\mathbf{C}_{k}\right\rangle$ is homogeneous if and only if the individual probability spaces are homogeneous, i.e. if and only if the individual probability spaces are (isomorphic to) probability spaces with a uniform measure.

Based on this simple description we derive the following theorem.

{theorem}

p:spaces-chains For $k\in\mathbb{N}$, the tropical cone $\operatorname{\mathbf{Prob}}[\mathbf{C}_{k}]$ is isomorphic to the following cone in $(\mathbb{R}^{k},|\cdot|_{1})$:

[TABLE]

In particular, the algebraic structure and the pseudo-distance are preserved under the isomorphism.

In case of single probability spaces, Theorem LABEL:p:spaces-chains is a direct consequence of the asymptotic equipartition property and the following lemma. For chains, a similar argument works.

{lemma}

p:aikd-uniform Denote by $U_{n}$ a finite uniform probability space of cardinality $n$, then

[TABLE]

and

[TABLE]

Proof: We will construct a specific two-fan $U_{n}\stackrel{{\scriptstyle f}}{{\leftarrow}}U_{nm}\stackrel{{\scriptstyle g}}{{\rightarrow}}U_{m}$. Identify $U_{\ell}$ with $\{0,\dots,\ell-1\}$. Let $k\in U_{nm}$. Then $k$ can be written uniquely as

[TABLE]

and we set $f(k):=i_{0}$ and $g(k):=i_{1}$.

Now that we have constructed a two-fan $U_{n}\stackrel{{\scriptstyle f}}{{\leftarrow}}U_{nm}\stackrel{{\scriptstyle g}}{{\rightarrow}}U_{m}$, let $U_{n}\stackrel{{\scriptstyle}}{{\leftarrow}}Z\stackrel{{\scriptstyle}}{{\rightarrow}}U_{m}$ be its minimal reduction. We estimate $|Z|\leq n+m$, which implies that

[TABLE]

thus establishing inequality (6).

To show equality (6), recall that the entropy as a map is $\operatorname{\mathbf{k}}$-Lipschitz with Lipschitz constant $1$. Therefore, we have

[TABLE]

and

[TABLE]

$\boxtimes$