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Tensor network representations of discrete maximum entropy distributions via mean polytopes
Alex Goessmann, Martin Eigel
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
This paper introduces Computation-Activation Networks (CompActNets), a tensor network architecture that represents discrete maximum entropy distributions under expectation constraints. By leveraging the geometry of mean polytopes, the authors show that any maximum entropy distribution can be represented within this framework, linking proper faces of the polytope to the boundary closure of exponential families and restricted distribution supports. The work provides explicit tensor network representations for these supports and suggests tensor network ranks as complexity measures for polytope faces, culminating in a case study connecting Boolean statistics and 0/1-polytopes to propositional formulas.
Entities (7)
Relation Signals (5)
Computation-Activation Networks → represents → Maximum Entropy Distributions
confidence 96% · we represent any maximum entropy distribution in the same architecture.
Computation-Activation Networks → subsumes → Exponential Families
confidence 95% · Computation-Activation Networks (CompActNets), a tensor network architecture that subsumes exponential families.
Mean Polytopes → hasfacescorrespondingto → Boundary Closure of Exponential Families
confidence 94% · proper faces of this polytope correspond to the boundary closure of exponential families
Boolean Statistics → links → 0/1-polytopes
confidence 93% · a case study on Boolean statistics links the geometry of 0/1-polytopes directly to propositional formulas.
Tensor Network Ranks → measurescomplexityof → Faces
confidence 91% · The proposed framework suggests tensor network ranks as complexity measures for faces.
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Abstract
Abstract:We present tensor network representations for discrete maximum entropy distributions under expectation constraints. To this end, we introduce Computation-Activation Networks (CompActNets), a tensor network architecture that subsumes exponential families. By leveraging the geometry of the convex polytope of realizable expectation vectors, we represent any maximum entropy distribution in the same architecture. We exploit the fact that proper faces of this polytope correspond to the boundary closure of exponential families, which restricts the distribution's support. We then derive explicit representations for the support within the CompActNet architecture. The proposed framework suggests tensor network ranks as complexity measures for faces. Finally, a case study on Boolean statistics links the geometry of 0/1-polytopes directly to propositional formulas.
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- Source: https://arxiv.org/abs/2609.07184v1
- Canonical: https://arxiv.org/abs/2609.07184v1
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Tensor network representations of discrete maximum entropy distributions via mean polytopes Alex Goessmann* and Martin Eigel Weierstrass Institute of Applied Analysis and Stochastics Anton-Wilhelm-Amo-Straße 39 Berlin, 10117, Germany *alex.goessmann@wias-berlin.de ABSTRACT We present tensor network representations for discrete maximum entropy distributions under ex- pectation constraints. To this end, we introduce Computation-Activation Networks (CompActNets), a tensor network architecture that subsumes exponential families. By leveraging the geometry of the convex polytope of realizable expectation vectors, we represent any maximum entropy distri- bution in the same architecture. We exploit the fact that proper faces of this polytope correspond to the boundary closure of exponential families, which restricts the distribution’s support. We then derive explicit representations for the support within the CompActNet architecture. The proposed framework suggests tensor network ranks as complexity measures for faces. Finally, a case study on Boolean statistics links the geometry of 0/1-polytopes directly to propositional formulas. 1 Introduction The Shannon entropy of a distribution quantifies the amount of structure, or information, and therefore appears as an important concept across many scientific areas. By Shannon’s source code theorem Shannon (1948), entropy describes fundamental limits of information transfer. In statistical physics, founded on the ergodic assumptions, maximum entropy distributions such as Maxwell-Boltzmann distributions are the gold standard to model equilibrium distributions Uffink (2007); Pressé et al. (2013). The principle of entropy maximization also appears naturally in learning problems Murphy (2023). Extracting characteristics observed in data does not determine an estimated model yet. Choosing the distribution with the least structure among those reflecting the observed characteristics is a regularization principle. Here one can choose the Shannon entropy to quantify the negative amount of structure in the distribution and maximize the quantity with respect to the constraints. These maximum entropy problems are the dual problems to convex maximum likelihood problems Koller and Friedman (2009). Maximum entropy principles appear further in reinforcement learning Yoon et al. (2024); Song et al. (2025), AGI Miotto and Monacelli (2025) and Bayesian N Rathnakumar et al. (2026). A classical result states that maximum entropy distributions belong to exponential families, if and only if the corresponding mean parameter is in the relative interior of the mean polytope Barndorff-Nielsen (1978). A natural extension is therefore the closure of the exponential families as investigated in Csiszar and Matus (2005); Malagò and Pistone (2010). Alternative methods use toric ideals to allow for infinite weight vectors Sottile (2008). Numerical stability of the maximum entropy distributions near the boundary is studied in Straszak and Vishnoi (2017, 2019). Less obvious, studying general maximum entropy distributions might be a key step towards a mathematically sound unification of logical and probabilistic models. Combining logical and probabilistic AI is a necessity in statistical relational AI, neuro-symbolic AI and explainable AI De Raedt et al. (2008); Getoor and Taskar (2019); Sarker et al. (2022). Probability theory with exponential families typically focuses on distributions with maximal support Koller and Friedman (2009). In contrast, logic can be understood as a study of the support of models, marking the worlds consistent with a knowledge base. Logical and probabilistic models can be treated in the tensor network formalism, which thus serves as a unifying representation language Goessmann et al. (2026). arXiv:2609.07184v1 [math.ST] 7 Sep 2026 We in this work derive tensor network representations of all discrete maximum entropy distributions, based on the framework presented in Goessmann (2025); Goessmann et al. (2026). Tensor network formats originate from quantum many-body physics White (1993) as a mitigation scheme for the curse of dimensionality Bellman (1961). From a numerical perspective Hackbusch and Kühn (2009); Hackbusch (2012) formats such as the CP format Hitchcock (1927); Beylkin and Mohlenkamp (2005) and the T format Oseledets (2011) have been studied. They have been applied to various areas such as in high-dimensional PDEs Eigel et al. (2019, 2020), or quantum simulation Sander et al. (2025a). Lastly, tensor network schemes are applied to artificial intelligence Badreddine et al. (2022); Domingos (2025); Ali (2025). 1.1 Tensor notation We introduce tensors based on index variables X k to each of the k ∈ [d] =0,...,d− 1 axes, each taking values in [m k ] =0,...,m k − 1, where m k ∈ N is the dimension of the variable. Tensors are then maps τ X [d] : × k∈[d] [m k ]→ R from indices x [d] = (x 0 ,...,x d−1 ) ∈ × k∈[d] [m k ] to their coordinates τ X [d] = x [d] ∈ R. One-hot encodings of index tuples x [d] are specific tensors ε x [d] X [d] such that ε x [d] X [d] = ̃x [d] = 1 if ̃x [d] = x [d] and 0 else. Given a hypergraphG = V,E of an arbitrary node setV and hyperedges e ∈ E as arbitrary subsets ofV , we define decorating tensor network as follows. Each node v ∈ V is decorated by a variable X v of dimension m v and each hyperedge e ∈ E by a tensor τ e [X e ] with the index variables to the contained nodes. The set of decorating tensors then builds the tensor network τ G on the hypergraph. A contraction of the tensor network keeping the nodesU ⊂ V open is then the tensor τ G [X U ] with variables X U and coordinates to the indices x U by τ G [X U =x U ] = X x V Y e∈E τ e [X e = x e ] . We in this work always assume that to each node in the hypergraph there is at least one hyperedge containing it. 1.2 Maximum entropy problem Given a non-negative and non-vanishing base measure ν X [d] , a probability distribution is a non-negative tensor P X [d] such that P X [d] ,ν X [d] [∅] = 1. We denote the set of such distributions by Λ ν = P X [d] : P X [d] ≥ 0 X [d] , P X [d] ,ν X [d] [∅] = 1, ∀ x [d] ∈ × k∈[d] [m k ] : (ν X [d] = x [d] = 0)⇒ (P X [d] = x [d] = 0) . Note that we here restrict the support of the tensor P X [d] to be included in the support of the base measure, to avoid ambiguity in the representation. The entropy of a distribution relative to the base measure ν is H ν P X [d] =− P X [d] , ln P X [d] ,ν X [d] [∅] . Note that this quantity depends on the chosen base measure. A statistic is a function t : × k∈[d] [m k ]→ R p assigning a real vector of dimension p to each state x [d] ∈ × k∈[d] [m k ]. We denote by t ℓ the restriction to the ℓ-th coordinate of t. The mean parameter of a distribution P X [d] to a statistic t is the vector μ [L]∈ R p with the coordinates μ [L = ℓ] = E [f ℓ ] = t ℓ X [d] , P X [d] ,ν X [d] [∅] . To express the computation of the mean parameter as a single contraction, we introduce a coordinate selection variable L of dimension p and the selection encoding σ t X [d] ,L with coordinates by σ t X [d] = x [d] ,L = ℓ = t ℓ X [d] = x [d] . 2 The mean parameter is then μ [L] = P X [d] ,σ t X [d] ,L [L] . The maximum entropy problem given a mean parameter μ ∗ [L] is stated by argmax P [ X [d] ] ∈Λ ν H ν P X [d] subject to P X [d] ,σ t X [d] ,L ,ν X [d] [L] = μ ∗ [L](P t,μ,ν ) 1.3 Computation-Activation Networks We now introduce Computation-Activation Networks, a family of tensor networks representing, as we will show in this work, the solution of any instance of Problem P t,μ,ν . Given a statistic t : × k∈[d] [m k ] → R p we define an index enumeration variable Y ℓ of dimension|im (t ℓ )| and injective index enumeration vectors I ℓ [Y ℓ ] : [n ℓ ]→ R. Based on the enumeration of the image we define the basis encoding tensor of the statistic as β t Y [p] ,X [d] = X x [d] ∈ × k∈[d] [m k ] O ℓ∈[p] ε I −1 ℓ (t ℓ [ x [d] ] ) [Y ℓ ] ⊗ ε x [d] X [d] . A computation network is any representation of β t Y [p] ,X [d] as a tensor network. These can be constructed in the case where the statistics are a composition of connective functions, see Goessmann et al. (2026). An activation tensor is τ Y [p] and the corresponding Computation-Activation Network with respect to the statistic t and the base measure ν X [d] the tensor P X [d] = β t Y [p] ,X [d] ,τ Y [p] [ X [d] ] β t Y [p] ,X [d] ,τ Y [p] ,ν X [d] [ X [d] ] . We are interested in decomposition formats of τ Y [p] , where we use sets of tensor networksT G on a hypergraphG. Definition 1. The Computation-Activation Networks to a statistic t and a hypergraphG is the family Λ t,G,ν = τ Y [p] ,β t Y [p] ,X [d] [ X [d] ] τ Y [p] ,β t Y [p] ,X [d] ,ν X [d] [∅] : τ Y [p] ∈T G . 1.4 Contributions We exploit the novel tensor network formalism developed in Goessmann et al. (2026) to represent generic maximum entropy distributions based on the position of their mean parameters in the mean polytope, see Thm. 4. The connection to logical formalisms of artificial intelligence is exploited in the study of Boolean statistics, where we interpret mean polytopes based on satisfiability of formulas (see Thm. 7) and construct statistics to arbitrary polytopes (see Thm. 8). 1.5 Structure of the paper To prepare for the presentation of our main results we introduce in section Sect. 2 the mean polytope, its faces, and their tensor network representations. In Sect. 3 we study exponential families as classes of elementary Computation- Activation Networks and show that they are the maximum entropy distributions to mean parameters in the relative interior of the mean polytope. This result is generalized in Sect. 4 where we characterize generic maximum entropy distributions based on Computation-Activation Networks. In Sect. 5 we investigate consequences of these results in case of Boolean statistics and connect them to propositional logics. 2 The mean polytope Given a statistic function t : × k∈[d] [m k ]→ R p 3 on the states × k∈[d] [m k ] and a base measure ν X [d] the mean polytope is the set M t,ν : = n P X [d] ,σ t X [d] ,L ,ν X [d] [L] : P X [d] ∈ Λ ν o containing all mean parameters for the distributions in Λ ν . By σ t X [d] ,L we denote the selection encoding of the statistic (see introduction). Since Λ ν is the convex hull of the vertices 1 ν [ X [d] =x [d] ] ·ε x [d] X [d] , and any mean parameter is a linear transformation of the distribution, the mean polytope is the convex hull of the vectors N t,ν : =σ t X [d] = x [d] ,L : x [d] ∈ × k∈[d] [m k ], ν X [d] = x [d] ̸= 0. 2.1 Faces When studying maximum entropy distributions, we will pay special attention to the faces of a polytope, which form the boundary. Here, we define them based on convex hulls and introduce tensor representations by refined base measures. Definition 2. We say that the convex hull of a subsetN F t,ν ⊂N t,ν is a face of a polytopeM t,ν , if and only if there is a vector θ F [L]∈ R p such that N F t,ν = argmax μ[L]∈N t,ν ⟨μ [L],θ F [L]⟩ [∅] . We denote the face asF = conv N F t,ν . The set of all faces of the mean polytopeM t,ν is denoted by L (M t,ν ). The set L (M t,ν ), partially ordered by the inclusion operation, is a lattice (see Ziegler (2013)). We now show that each face itself is a convex polytope corresponding to a refined base measure. Definition 3. The refined base measure with respect to a faceF ∈ L (M t,ν ) is the tensor ν t,F X [d] with coordinates to x [d] ∈ × k∈[d] [m k ] by ν t,F X [d] = ν X [d] = x [d] if t x [d] ∈N F t,ν 0else . The refined base measure has by definition a smaller support than the base measure. We now show that restricting the distributions in Λ ν to those in Λ ν t,F amounts to a restriction of the polytopeM t,ν to the corresponding face. Lemma 1. For any faceF ∈ L (M t,ν ) we have F =M t,ν t,F . Proof. For anyx [d] ∈ × k∈[d] [m k ] we haveν t,F X [d] = x [d] ̸= 0 if and only ift x [d] ∈N F t,ν andν X [d] = x [d] ̸= 0. Thus we have F = conv N F t,ν = conv σ t X [d] = x [d] ,L : t x [d] ∈N F t,ν , ν X [d] = x [d] ̸= 0 = conv σ t X [d] = x [d] ,L : ν t,F X [d] = x [d] ̸= 0 =M t,ν t,F . Representability of a distribution with respect to face measures is an equivalent condition for the mean parameter of a distribution to be on a face, as we show next. Lemma 2. If and only if for a distribution P X [d] ∈ Λ ν and a faceF ∈ L (M t,ν ) we have σ t X [d] ,L , P X [d] ,ν X [d] [L] ∈F , then P X [d] ∈ Λ ν t,F . 4 Proof. For a distribution P X [d] ∈ Λ ν to be in Λ ν t,F , it is enough to show that P X [d] is only supported at ν t,F X [d] . We have μ [L] = X x [d] P X [d] = x [d] · ν X [d] = x [d] · γ t X [d] = x [d] ,L . Now, let θ F [L] be a face normal to the faceF . We then have ⟨μ [L],θ F [L]⟩ [∅] = X x [d] ∈ × k∈[d] [m k ] P X [d] = x [d] · ν X [d] = x [d] · γ t X [d] = x [d] ,L θ F [L] [∅] . Now if and only if P X [d] = x [d] · ν X [d] = x [d] is supported only for x [d] with t x [d] ∈N F t,ν we have that ⟨μ [L],θ F [L]⟩ [∅] =max μ[L]∈M t,ν ⟨μ [L],θ F [L]⟩ [∅] which is equal to μ [L] ∈ F . Thus, if and only if μ [L] ∈ F then P X [d] is only supported at x [d] in the support of ν t,F X [d] . Let us now investigate tensor network representations of refined base measures, based on the basis encoding β t of a statistic. Theorem 1. For any faceF ofM we have ν t,F X [d] = β t Y [p] ,X [d] ,κ F Y [p] ,ν X [d] [ X [d] ] where κ F Y [p] = X v[L]∈N F t,ν ε v[L] Y [p] . Proof. For any v [L]∈N F t,ν the tensor τ v X [d] = β t Y [p] ,X [d] ,ε v[L] Y [p] [ X [d] ] is the indicator of the preimage of μ under σ t . Since preimages of the elements in N F t,ν are disjoint, the support of τ v X [d] is disjoint and their sum X v[L]∈N F t,ν τ v X [d] is the indicator of the preimage ofF under σ t . The face measure obeys thus ν t,F X [d] = * X v[L]∈N F t,ν τ v X [d] ,ν X [d] + [ X [d] ] = X v[L]∈N F t,ν β t Y [p] ,X [d] ,ε v[L] Y [p] ,ν X [d] [ X [d] ] = β t Y [p] ,X [d] ,κ F Y [p] ,ν X [d] [ X [d] ] We now investigate the representation of refined base measures by Computation-Activation Networks. Definition 4. LetG be a hypergraph whose nodes include [p]. We say that a faceF is representable byG if and only if there is a setU of basis vectors withU∩N t,ν =N F t,ν and there is a tensor network κ G with respect to the hypergraph G such that κ G [ Y [p] ] = X v∈U ε v Y [p] . We call any such tensor network a face activating tensor network. 5 Lemma 3. If and only if a face is representable by a hypergraphG, we have 1 ν t,F X [d] [∅] · ν t,F X [d] ∈ Λ t,G,ν . Proof. For anyU withU : U ∩N t,ν =N F t,ν and tensor network κ G respecting the assumptions of Def. 4 we have D κ G [ Y [p] ] ,β t Y [p] ,X [d] E [ X [d] ] = * X v∈U ε v Y [p] ,β t Y [p] ,X [d] + [ X [d] ] = * X v∈N F t,ν ε v Y [p] ,β t Y [p] ,X [d] + [ X [d] ] . Here we used in the second equation that only the vertices inN t,ν are in the image of t. It follows, that 1 ν t,F X [d] [∅] · ν t,F X [d] = D κ G [ Y [p] ] ,β t Y [p] ,X [d] E [ X [d] ] D ⟨κ G ⟩ [ Y [p] ] ,β t Y [p] ,X [d] ,ν X [d] E [∅] and thus ν t,F X [d] |∅ ∈ Λ t,G,ν . Example 1 (Vertices). VerticesF =v [L] are proper faces of affine dimension 0, that is they consist of single vectors v [L]. Since each vertex is in the imageN t,ν , there exists an index tuple x [d] ∈X such that ν X [d] = x [d] ̸= 1 and v [L] = σ t X [d] = x [d] ,L . Then κ F Y [p] is the one-hot encoding of the vector σ t X [d] = x [d] ,L , that is κ F Y [p] = ε σ t [ X [d] =x [d] ,L ] Y [p] = D ε σ t [ X [d] =x [d] ,L=ℓ ] [Y ℓ ] : ℓ∈ [p] E [ Y [p] ] . Such tensors are examples of elementary tensors corresponding to the elementary hypergraph (see Figure 1) EL = [p],ℓ : ℓ∈ [p] . In particular, the activation core is elementary and the corresponding refined base measure is in Λ t,EL,ν . While vertices are the minimal non-vanishing faces in the face-lattice (see Ziegler (2013)), we now show that the maximal face, namely the polytope itself, is also representable with respect to the elementary hypergraph EL (see Figure 1). Example 2 (Maximal face). The maximal faceF ∅ = M t,ν coincides with the mean polytope itself and is given by the choice θ ∅ [L] = 0 [L]. In this case the corresponding activation tensor to the face measure is κ ∅ Y [p] = I Y [p] = O ℓ∈[p] I [Y ℓ ] , where by I Y [p] we denote tensors with all coordinates 1. κ ∅ Y [p] is elementary and the maximal face is always representable by an elementary activation. 2.2 Representation in the CP format Extending Example 1 and Example 2, we can provide a coarse estimation of the hypergraphG required to decompose κ F for generic faces F . To this end, we use the CP format describing tensor networks on the CP hypergraph (see Figure 2) CP = [p]∪I,ℓ,I : ℓ∈ [p] . The dimension of the variable I is called the hidden rank of the format. 6 a) Y 0 Y 1 · Y p−1 b) τ 0 Y 0 τ 1 Y 1 · τ p −1 Y p−1 Figure 1: Elementary tensor format consisting of the elementary hypergraph EL a), and corresponding tensor networks b) being the tensor products of vectors. a) Y 0 Y 1 · Y p−1 I b) τ 0 Y 0 τ 1 Y 1 · τ p −1 Y p −1 I Figure 2: The CP tensor format consist of the CP hypergraph a) and decorating tensor networks b). Tensors in this format constructed as in Lem. 4 represent arbitrary faces. Lemma 4. Any faceF is representable by a CP graph with hidden rank r ≤ min N F t,ν , N t,ν − N F t,ν + 1 . Proof. We show the claim by constructing two face activating tensor networks toF in a CP hypergraph with hidden rank N F t,ν and in a CP hypergraph with hidden rank N t,ν − N F t,ν +1. To show the first representation we enumerate the vertices by a variable I with dimension r = N F t,ν , i.e. N F t,ν =v i [L] : i∈ [n]. Then we define for ℓ∈ [p] core tensors τ ℓ [Y ℓ ,I] τ ℓ [Y ℓ ,I = i] = ε v i [L=ℓ] [Y ℓ ] . Then we have τ ℓ [Y ℓ ,I] : ℓ∈ [p] [ Y [p] ] = X v∈N F t,ν ε v Y [p] and thus have found an activation tensor network in a CP graph with hidden rank N F t,ν representing the faceF . We continue with the second representation, for which we enumerate the set N t,ν /N F t,ν by v i [L] where i ∈ [ N t,ν − N F t,ν ]. We define variable I with dimension r = N t,ν − N F t,ν + 1 and define for ℓ∈ [p] core tensors τ ℓ [Y ℓ ,I = i] = −ε v i [L=ℓ] [Y ℓ ] if i < N t,ν − N F t,ν I [Y ℓ ]if i = N t,ν − N F t,ν . We then have τ ℓ [Y ℓ ,I] : ℓ∈ [p] [ Y [p] ] = I Y [p] − X v∈N t,ν /N F t,ν ε v Y [p] = X v∈N F t,ν ε v Y [p] + X v∈ (× ℓ∈[p] [p ℓ ] ) /N t,ν ε v Y [p] . We have thus found a face activating tensor network forF in a CP format with hidden rank r = N t,ν − N F t,ν +1. We notice that the CP rank bound of Lem. 4 is tight in Example 1 and Example 2, since the elementary format is a restriction of the CP format to rank 1. 7 2.3 Examples of elementarily representable faces We show in the following example that all faces of a hypercube and a simplex are representable by elementary activation tensors. Example 3 (Hypercube). In cases where X [d] are Boolean and we have ℓ∈ [p] features f ℓ X [d] = x [d] = x ℓ the mean polytope is the hypercube M f ℓ :ℓ∈[p],I = [0, 1] p . The non-empty faces in the face lattice L ([0, 1] p ) can be enumerated by subsetsU ⊂ [p] and indices y U ∈ × ℓ∈U [2] and represented by the cartesian products F (U,y U ) = × ℓ∈[p] I l,(U,y U ) where I l,(U,y U ) = [0, 1] if ℓ /∈U y ℓ if ℓ∈U . Each of these faces can be represented with respect to the elementary graph EL, namely by the tensor product of leg vectors κ ℓ,(U,y U ) [Y ℓ ] = I [Y ℓ ]if ℓ /∈U ε y ℓ [Y ℓ ] if ℓ∈U . Example 4 (Simplices). The (p− 1)-dimensional standard simplexM △,p−1 in [0, 1] p is the convex hull of the vertex sets N =ε ℓ [L] : ℓ∈ [p]. The simplex is the mean polytope of the p-dimensional statistic with t ℓ : [p]→ [2] , t ℓ h ̃ ℓ i = 1 if ̃ ℓ = ℓ 0 if ̃ ℓ̸= ℓ . The simplex represents all distributions with respect to the trivial base measure I which coordinates are all 1. When flattened to a vector, any tensor in Λ I is an element ofM △,p−1 where p = Q k∈[d] m k . The face lattice of the simplex is enumerated by subsets of [p] as L (M △,p−1 ) =F U : U ⊂ [p] where the faces are F U = conv (ε ℓ [L] : ℓ∈U ) and are therefore themselves (|U|− 1)-dimensional standard simplices. The partial order of the faces coincides with the inclusion order of subsetsU . 2.4 Partition into relative interiors of faces Let us now introduce relative interiors, which enable us to find disjoint partitions of the mean polytope. Definition 5 (Relative Interior). The relative interior (U ) ◦ of an arbitrary setU ⊂ R p is the interior ofU in the affine hull ofU . Lemma 5. To each μ∈M t,ν there is a unique faceF , which we denote byF (μ), such that μ∈ (F ) ◦ . Proof. For any mean vector μ on a faceF we either have μ∈ (F ) ◦ or there is a face ̃ F of lower affine dimension with ̃ F ⊂ F and μ ∈ ̃ F . Starting from the faceM t,ν (for which by assumption μ ∈ M t,ν ) we iteratively choose a face F of lower dimension containing μ until we reach a face with μ ∈ (F ) ◦ . This procedure is guaranteed to terminate, since the affine dimension is decreasing and any face of affine dimension 0 coincides with its relative interior. 8 3 Exponential families It is well known that exponential families are specific maximum entropy distributions. However, not every maximum entropy distribution is an exponential family member. To prepare for the characterization of generic maximum entropy distributions, we will in this section represent exponential family members as Computation-Activation Networks. We further show that a maximum entropy distribution is an exponential family member if and only if its mean parameter is in the relative interior of the mean polytope. 3.1 Representation as elementary Computation-Activation Networks Let us first define exponential families as classes of probability distributions and then derive their representation by elementary Computation-Activation Networks. Definition 6 (Exponential Family). Given a statistic function t : × k∈[d] [m k ]→ R p and a base measure ν X [d] with⟨ν⟩ [∅] ̸= 0, the set Γ t,ν =P (t,θ,ν) : θ [L]∈ R p ⊂ Λ ν of probability distributions P (t,θ,ν) X [d] = exp h σ t X [d] ,L ,θ [L] [ X [d] ] i D exp h σ t X [d] ,L ,θ [L] [ X [d] ] i ,ν X [d] E [∅] is called the exponential family with respect to t. To connect with standard notation of exponential families we notice that the coordinate of any distribution P (t,θ,ν) X [d] at some state x [d] ∈ × k∈[d] [m k ] is P (t,θ,ν) X [d] = x [d] = exp X ℓ∈[p] θ [L = ℓ]· t ℓ x [d] − ψ t,ν (θ) where ψ t,ν (θ) = ln D exp h σ t X [d] ,L ,θ [L] [ X [d] ] i ,ν X [d] E [∅] is the cumulant generating function. The function ψ t,ν (θ) and its Legendre transform ψ ∗ t,ν (μ) define the dually flat Riemannian structure on the exponential family manifold Amari (2016). We now continue with the notation of Def. 6 to find representations of exponential families by elementary Computation-Activation Networks. Theorem 2 (Exponential Families are elementary Computation-Activation Networks). Given any base measure ν and a sufficient statistic t we enumerate for each coordinate ℓ ∈ [p] the image im (t ℓ ) by a variable Y ℓ taking values in [|im (t ℓ )|], given an interpretation map I ℓ : [|im (t ℓ )|]→ im (t ℓ ) . For any canonical parameter vector θ [L] ∈ R p we build the activation cores α ℓ,θ [Y ℓ ] for each coordinate y ℓ ∈ [|im (t ℓ )|] by α ℓ,θ [Y ℓ = y ℓ ] = exp [θ [L = ℓ]· I ℓ [y ℓ ]] and have (see Figure 3) P (t,θ,ν) X [d] = β t Y [p] ,X [d] ∪α ℓ,θ [Y ℓ ] : ℓ∈ [p] [ X [d] ] β t Y [p] ,X [d] ∪α ℓ,θ [Y ℓ ] : ℓ∈ [p]∪ν X [d] [∅] . 9 P (t,θ,ν) X 0 X 1 · X d−1 = 1 Z · α 0,θ α p−1,θ Y 0 · Y p−1 β t X 0 X 1 · X d−1 Figure 3: Representation of a member of an exponential family by a Computation-Activation Network with elementary activation. HereZ ∈ R is a normalization constant. Proof. For each x [d] ∈ × k∈[d] [m k ] we have β t Y [p] ,X [d] ∪α ℓ,θ [Y ℓ ] : ℓ∈ [p] [ X [d] =x [d] ] = Y ℓ∈[p] β t ℓ Y ℓ ,X [d] = x [d] ,α ℓ,θ [Y ℓ ] [∅] = Y ℓ∈[p] exp θ [L = ℓ]· I ℓ t ℓ x [d] = exp X ℓ∈[p] θ [L = ℓ]· I ℓ t ℓ x [d] = exp h σ t X [d] ,L ,θ [L] [ X [d] =x [d] ] i . Therefore we have β t Y [p] ,X [d] ∪α ℓ,θ [Y ℓ ] : ℓ∈ [p] [ X [d] ] = exp h σ t X [d] ,L ,θ [L] [ X [d] ] i . The claim follows, since this implies that also the contraction of both sides with ν X [d] is equivalent. We will use the following well known property (see e.g. Brown (1987)), that the mean parameters of the members of an exponential family are exactly the relative interior of the mean polytope. Lemma 6. The set of mean parameters of the members of an exponential family is the relative interior of the mean polytope. Proof. See The 3.3 in Wainwright and Jordan (2008). Based on this property we define the forward and backward mappings to an exponential family. Definition 7. The forward map of an exponential family is the map F (t,ν) : R p → (M t,ν ) ◦ defined as F (t,ν) (θ) = P t,θ,ν [X [d] ],σ t X [d] ,L ,ν X [d] [L] Any map B (t,ν) : (M t,ν ) ◦ → R p with F (t,ν) (B (t,ν) (μ)) = μ for all μ∈ (M t,ν ) ◦ is called a backward map. 3.2 Maximum entropy distributions One universal property of exponential families amongst others (see Murphy (2023)) is that they maximize the entropy among those with the same mean vector. Lemma 7. Any member of an exponential family Γ t,ν is a maximum entropy distribution with respect to the moment constraint on t and the base measure ν. 10 Proof. Let P t,θ,ν be an exponential family member with canonical parameter θ and let μ [L] be the corresponding mean parameter. Let ̃ P[X [d] ] be any feasible distribution for the maximum entropy problem with respect to the moment constraint on t by μ [L] and the base measure ν. We also have D ̃ P[X [d] ],σ t X [d] ,L ,ν X [d] E [L] = μ [L] and thus H ν h ̃ P, P t,θ,ν i =− D ̃ P[X [d] ], ln P t,θ,ν [X [d] ] ,ν X [d] E [∅] =− D ̃ P[X [d] ],σ t X [d] ,L ,θ [L],ν X [d] E [∅] + A (t,ν) (θ) =−⟨θ [L],μ [L]⟩ [∅] + A (t,ν) (θ) = H ν P t,θ,ν . With the Gibbs inequality we have if ̃ P̸= P t,θ,ν H ν P t,θ,ν − H ν h ̃ P i = H ν h ̃ P, P t,θ,ν i − H ν h ̃ P i > 0 and thus H ν h ̃ P i < H ν P t,θ,ν . Therefore, any feasible ̃ P different from P t,θ,ν has less entropy than P t,θ,ν and P t,θ,ν is a maximum entropy distribution. Together with Lem. 6 stating the existence of an exponential family member for any mean parameter in the relative interior of the mean polytope we arrive at the following theorem. Theorem 3. If and only if μ [L] is in the relative interior ofM t,ν , then the unique solution of the maximum entropy problem is the distribution P t,θ,ν [X [d] ]∈ Γ t,ν where for some backward map B t,ν θ = B t,ν (μ). Proof. By Lem. 6 if μ [L]∈ (M t,ν ) ◦ , there is at least one weight parameter θ [L]∈ R p with P t,θ,ν [X [d] ],σ t X [d] ,L ,ν X [d] [L] = μ [L] and B t,ν (μ) with respect to any backward map B t,ν is such one. With Lem. 7 we conclude that P (t,B t,ν (μ),ν) is the maximum entropy distribution with respect to the moment constraint on t by μ [L] and the base measure ν. While the maximum entropy distribution is unique, the weight vector θ representing the maximum entropy distribution is not always unique. This is reflected by multiple possible choices of backward mappings. Only if the exponential family is minimal (see Wainwright and Jordan (2008)) then the weight vector is unique. However, we did not restrict to such situations, since exponential families on refined base measures are not minimal. 4 Generic maximum entropy distributions In the previous section we have shown that the members of an exponential family are maximum entropy distributions. However, while their mean parameters constitute the relative interior of the mean polytope, they never lie on faces of lower dimension. We now exploit the tensor representation study of such faces (see Sect. 2) to characterize the maximum entropy distributions in the general case. 4.1 Main result Our main result generalizes the maximum entropy characterization of Thm. 3 to arbitrary mean parameters. 11 Theorem 4 (Generic characterization of Maximum Entropy Solutions). Let t be a statistic and ν a base measure. For any θ [L] andF ∈ L (M t,ν ) the distribution P t,θ,ν t,F [X [d] ] is a maximum entropy distribution. For any μ [L] the maximum entropy problem has a feasible distribution, if and only if μ [L] ∈ M t,ν . If μ [L] ∈ M t,ν the unique maximum entropy distribution is P t,θ,ν t,F whereF =F (μ) (see Lem. 5) and θ = B t,ν t,F (μ) where ν t,F is the refined base measure (see Def. 3) and B t,ν t,F is any backward map to the exponential family Γ t,ν t,F (see Def. 7). For any hypergraphG and any faceF ∈ L (M t,ν ) we have that all maximum entropy distributions to mean parameters onF ◦ are in Λ t,G,ν ifF is representable with respect toG (see Def. 4). To prepare for the proof of this theorem we first show in an auxiliary lemma that we can reduce the set of feasible distributions in Problem P t,μ,ν . Lemma 8. For any μ [L] ∈ M t,ν and a faceF ∈ L (M t,ν ) with μ [L] ∈ (F ) ◦ the solutions of P t,μ,ν and P t,μ,ν t,F coincide. Proof. By Lem. 2 all feasible distributions are representable by the base measure refined to the faceF . We have that any for P t,μ,ν feasible distribution P X [d] satisfies P X [d] ,ν t,F [∅] = 1 and thus P X [d] ∈ Λ ν t,F . Conversely, any P X [d] ∈ Λ ν t,F satisfies P X [d] ,ν t,F [∅] = P X [d] ,ν [∅] = 1 and thus P X [d] ∈ Λ ν . Problem P t,μ,ν is thus equal to argmax P [ X [d] ] ∈Λ ν t,F H ν P X [d] subject to P X [d] ,σ t X [d] ,L ,ν X [d] [L] = μ ∗ [L] We further have that any P X [d] ∈ Λ ν t,F H ν P X [d] = H ν t,F P X [d] and arrive together with the above equivalence at the claim. Proof of Thm. 4. To show that for any θ [L] and F ∈ L (M t,ν ) the distribution P t,θ,ν t,F is a maximum entropy distribution, we apply Lem. 8 and Thm. 3. If μ [L] /∈ M t,ν then Problem P t,μ,ν is not feasible by definition. In case μ [L] ∈ M t,ν we findF ∈ L (M t,ν ) with F = F (μ) and by Lem. 8 the solution of maximum entropy problem Problem P t,μ,ν coincides with P t,μ,ν t,F . Since by Lem. 1 the face is the polytope F =M t,ν t,F we find a weight parameter θ = B t,ν t,F (μ). and by Thm. 3 P t,θ,ν t,F is the corresponding maximum entropy distribution. To show the last claim, it is enough to show that for any faceF representable byG and for any θ ∈ R p we have P t,θ,ν t,F ∈ Λ t,G,ν . To this end, we construct activation cores as follows. By Def. 4 and Lem. 3 there is a tensor network κ G such that ν t,F X [d] = κ G ∪ν X [d] [ X [d] ] . 12 P X 0 X 1 X d−1 · = 1 Z · κ F Y 0 · α 0,θ Y p−1 α p −1,θ β t X 0 X 1 X d−1 · Figure 4:Tensor network decomposition of maximum entropy distributions to the constraint μ [L] = P X [d] ,σ t X [d] ,L [L] . Blue: Boolean tensor κ F Y [p] dependent on the face F (μ), which represents the refinement of the base measure to the face, and is possibly non-elementary. Red: Elementary activation tensor α θ Y [p] = N s∈[n] α ℓ,θ [Y ℓ ] to the weight θ [L] = B t,ν t,F (μ). For each ℓ∈ [p] we find an edge e(ℓ)∈E with ℓ∈ e(ℓ). Now we define a tensor network τ G by the cores to e∈E as τ e [X e ] = κ e [X e ]∪α ℓ,θ [Y ℓ ] : e = e(ℓ) [X e ] . Then, we have D β t Y [p] ,X [d] , τ G [ Y [p] ] E [ X [d] ] = D β t Y [p] ,X [d] ,α θ Y [p] , κ G [ Y [p] ] E [ X [d] ] and therefore P t,θ,ν t,F = β t Y [p] ,X [d] ∪ τ G [ X [d] ] β t Y [p] ,X [d] ,ν X [d] ∪ τ G [∅] . We conclude P t,θ,ν t,F ∈ Λ t,G,ν . 4.2 Family of maximum entropy distributions The family of maximum entropy distributions is the set Γ t,ν =P X [d] : ∃μ∈M t,ν : P X [d] is a solution of Problem P t,μ,ν . Thm. 4 suggests a parametrization of this family as tuples of corresponding faces F and weight vectors θ [L]. As sketched in Figure 4 the activation tensor of each maximum entropy distribution is decomposed into a possibly non- elementary Boolean tensor corresponding to the base measure refinement on a face F , and an elementary tensor dependent on the weight vector θ ∈ R p . We thus define the canonical parameter space P t,ν : = L (M t,ν )× R p and have Γ t,ν = α θ Y [p] ,κ F Y [p] ,β t Y [p] ,X [d] [ X [d] ] α θ Y [p] ,κ F Y [p] ,β t Y [p] ,X [d] ,ν X [d] [∅] : θ [L]∈ R p ,F ∈P t,ν . Note that these families are disjoint, since each member of an exponential family has support by the support of the face measure. 5 Boolean statistics We say that a statistic is Boolean, if im (t ℓ )⊂0, 1 for all ℓ∈ [p]. The mean polytope is then a 0/1-polytope Ziegler (2000), which vertices are exactly the contained Boolean vectors. The special case of Boolean statistics is especially interesting for two reasons. First, we can provide additional intuition on those faces representable by elementary face activating tensors, namely based on intersections with cube faces. Based on these properties we also show that any elementary Computation-Activation Network to a Boolean statistic is a maximum entropy distribution. Second, the statistic itself can be interpreted in terms of logical formulas. We exploit this property to provide examples and to characterize each 0/1-polytope as a mean polytope to a collection of propositional formulas. 13 5.1 Characterization of elementary representable faces Orienting on Example 3 we now introduce cube-likeness of faces and polytopes. Definition 8. We say that a faceF ofM t,ν is cube-like, if it is empty or there isU ⊂ [p] and y U ∈ × ℓ∈U [2] such that F =M t,ν ∩F (U,y U ) . Here we denote byF (U,y U ) the faces of the hypercube [0, 1] p (see Example 3). We further say that a polytopeM t,ν is cube-like, if all facesF ∈ L (M t,ν ) are cube-like. We now show that a face is cube-like if and only if it is representable by an elementary tensor. Theorem 5. Let t be a Boolean statistic, ν a base measure andF be a face ofM t,ν . Then the following are equivalent: (i) F is cube-like (see Def. 8). (i) F is representable by an elementary tensor (see Def. 4). Proof. If the face is empty, i.e. F = ∅, it is by definition cube-like and has 0 Y [p] as an elementary activation tensor. We therefore assume in the followingF ̸= ∅. (i)⇒(i): Let us assume thatF is cube-like, that is there isU ⊂ [p] and y U ∈ × ℓ∈U [2] such thatF =M t,ν ∩F (U,y U ) . We use thatF,M t,ν andF (U,y U ) are the convex hulls of the cube vertex setsN F t,M ,N t,M and that for the set N U,y U =v [L] : ∀ℓ∈U : v [L = ℓ] = y U ,(1) we have N F t,M =N U,y U ∩N t,M . Thus, we can chooseN U,y U for the representation of the faceF (see Def. 4) and further have that X v[L]∈N (U,y U ) ε v[L] Y [p] = O ℓ∈[p] κ ℓ,(U,y U ) [Y ℓ ] , where κ ℓ,(U,y U ) [Y ℓ ] = ε y ℓ [Y ℓ ] if ℓ∈U I [Y ℓ ]if ℓ /∈U . We have thus found an elementary activation tensor for the faceF . (i)→(i) Conversely, let ξ Y [p] be an elementary activation tensor of the face F in M t,ν . Since ξ Y [p] is the sum of different one-hot encodings it is Boolean and we find an elementary decomposition ξ Y [p] = N ℓ∈[p] ξ ℓ [Y ℓ ] such that the leg vectors ξ Y [p] are Boolean. Since F ̸= ∅ we further have for ℓ ∈ [p] that ξ ℓ [Y ℓ ] ̸= 0 [Y ℓ ], and thus ξ ℓ [Y ℓ ]∈ε 0 [Y ℓ ],ε 1 [Y ℓ ], I [Y ℓ ] We construct a setU ⊂ [p] U = ℓ : ξ ℓ [Y ℓ ]̸= I [Y ℓ ] and a tuple y ℓ = 1 if ξ ℓ [Y ℓ ] = ε 1 [Y ℓ ] 0 if ξ ℓ [Y ℓ ] = ε 0 [Y ℓ ] . By construction ξ ℓ [Y ℓ ] = κ ℓ,(U,y U ) [Y ℓ ] (see Example 3) follows for all ℓ∈ [p] and therefore ξ Y [p] = κ (U,y U ) Y [p] . We therefore have for the set (1) ξ Y [p] = X v∈N U,y U ε v Y [p] and since ξ Y [p] is an activation tensor forF , that N F t,M =N U,y U ∩N t,M . Taking convex hulls on both sides, this is equivalent to F =F (U,y U ) ∩M t,ν . We conclude thatF is cube-like. 14 Thm. 5 implies that the maximum entropy distributions to hypercubes (see Example 3) and simplices (see Example 4) are in Λ t,EL,ν , since these polytopes are cube-like. We can further show, that in case of Boolean statistics any distribution in Λ t,EL,ν is a maximum entropy distribution. Theorem 6. Let t be a Boolean statistic. Any distribution in Λ t,EL,ν is a maximum entropy distribution with respect to (t,μ,ν) where μ is its mean parameter. Proof. Let P∈ Λ t,EL,ν be arbitrary. We then find an elementary tensor N s∈[n] τ ℓ [Y ℓ ] andZ ∈ R such that P X [d] = 1 Z * β t Y [p] ,X [d] , O s∈[n] τ ℓ [Y ℓ ] + [ X [d] ] . We then define for each ℓ∈ [p] the vector κ ℓ [Y ℓ ] as the support of τ ℓ [Y ℓ ] and θ [L = ℓ] = ( ln h τ ℓ [Y ℓ =1] τ ℓ [Y ℓ =0] i if τ ℓ [Y ℓ = 0],τ ℓ [Y ℓ = 1]̸= 0 0else . Since t is a Boolean statistic the N ℓ∈[p] κ ℓ [Y ℓ ] is a face activation tensor to the face F : =M t,I ∩F (U,y U ) whereU =ℓ : κ ℓ [Y ℓ ]̸= I [Y ℓ ] and for ℓ∈U y ℓ = 0 if κ ℓ [Y ℓ = 1] = 0 1 else . Therefore, P is the maximum entropy distribution to the canonical parametersF and θ. 5.2 Interpretation by propositional formulas Let us now assume that m k = 2 for any k ∈ [d]. Following the formalism in Goessmann et al. (2026) we understand each coordinate t ℓ of the statistic as a propositional formula and the variables X [d] as atoms of a propositional theory. We use propositional connectives for syntactical representations of the formulas and denote by¬ 1 the logical negation and by¬ 0 the identity. A Boolean tensor f X [d] is called satisfiable, if there is at least one x [d] ∈ × k∈[d] [m k ] with f X [d] = x [d] = 1. Theorem 7. Let t be a Boolean statistic. Any Boolean vector v [L]∈0, 1 p is a vertex ofM t,I if and only if f v X [d] : = ℓ∈[p] ¬ 1−v[L=ℓ] f ℓ X [d] is satisfiable. Proof. Let v [L] ∈ 0, 1 p be arbitrary. If and only if v [L] is a vertex of M t,I we find x [d] ∈ × k∈[d] [m k ] with v [L] = σ t X [d] = x [d] ,L. This is further equivalent to f ℓ X [d] = x [d] = v [L = ℓ] for any ℓ ∈ [p], ¬ 1−v[L=ℓ] f ℓ X [d] = x [d] = 1 for any ℓ∈ [p], and f v X [d] = x [d] = 1. Since the presence of Boolean vectors inM t,I is related by Thm. 7 to the satisfiability of the conjunction f v X [d] , the absence is related to entailment statements. To be more precise, for any v [L] ∈ 0, 1 p and arbitraryU ⊂ [p] we have v [L] /∈M t,I , if and only if ℓ∈U ¬ 1−v[L=ℓ] f ℓ X [d] ! |= _ ℓ/∈U ¬ v[L=ℓ] f ℓ X [d] ! . By definition (see Russell and Norvig (2021)) the latter holds if and only if ℓ∈U ¬ 1−v[L=ℓ] f ℓ X [d] ! ∧¬ _ ℓ/∈U ¬ v[L=ℓ] f ℓ X [d] ! = f v X [d] is not satisfiable. Let us now relate two well-studied sets of formulas, namely atomic formulas and minterm formulas, with two well- studied polytopes, namely hypercubes and simplices. 15 Example 5 (Atomic formulas). Let us consider the case of atomic formulas f ℓ X [d] defined with the coordinates at x [d] ∈ × k∈[d] [m k ] by f ℓ X [d] = x [d] = x ℓ . The mean polytope in this case is the d-dimensional hypercube (see Example 3) M F [d] ,I = [0, 1] p which is called a simple polytope, since each vertex is contained in the minimal number of d facets. Each face is characterized by the projections onto each variable, which is either0, 1 or [0, 1]. The projections are represented by the tuple (U,y U ) defined in the following way: • We define the setU ⊂ [d] of variables, such that the projection onto the variable is0 or1 • We define to each ℓ∈U an index y ℓ = 0 if the projection is0 and y ℓ = 1 if the projection is1. Trivially, each face of the hypercube is a cube face. More generally, the mean polytope to a collection f ℓ X [d] of propositional formulas is the hypercube, if and only if for any Boolean vector v [L] we have * ℓ∈[p] ¬ 1−v[L=ℓ] f ℓ X [d] + [∅] ̸= 0. This means that no conjunction of possibly negated formulas from this statistic entails nor contradicts another formula. Example 6 (Minterm formulas build simplices). The set of minterm formulas is indexed by v [L]∈0, 1 p and given by f v X [d] = ℓ∈[d] ¬ 1−v[L=ℓ] X ℓ = ε v[L] X [d] where by X ℓ we denote the ℓ-th atomic formula (see Example 5). The mean polytope to the minterm statistic and base measure ν is the standard simplex of dimension v [L] : v ∈0, 1 p , ν X [d] = v [L] ̸= 0 − 1, that is the set M F ∧ ,ν = conv ε v[L] Y [p] : v ∈0, 1 p , ν X [d] = v [L] ̸= 0 . We notice that for Boolean base measuresM F ∧ ,ν = Λ ν . The simplex is further cube-like, since each face is indexed by a subsetU ⊂0, 1 p of vertices and the intersection of the simplex with the cube faceF (0,1 p ,1 0,1 p ) . More generally, the mean polytope to a Boolean statistic is a standard simplex, if and only if each target coordinate restriction f ̃ ℓ contradicts all others, that is ∀ℓ̸= ̃ ℓ∈ [p] : f ℓ |=¬f ̃ ℓ . While hypercubes and simplices are cube-like, we now provide examples of statistics, which polytopes are not cube- like. Example 7 (Maximum entropy distribution with non-elementary activation cores). Consider two atomic variables X 0 and X 1 and a statisticF consisting of the formulas f 0 [X 0 ,X 1 ] = (X 0 ∧ X 1 ) , f 1 [X 0 ,X 1 ] = (X 0 ⇒ X 1 ) with the coordinatewise expressions f 0 [X 0 ,X 1 ] = 0 0 0 1 [X 0 ,X 1 ] , f 1 [X 0 ,X 1 ] = 1 1 0 1 [X 0 ,X 1 ] . We can think of X 0 as a feature on an invoice, and X 1 as a feature on the accounting proposal. 16 From this we have β (f 0 ,f 1 ) [Y 0 = 0,Y 1 = 0,X 0 ,X 1 ] = 0 0 1 0 [X 0 ,X 1 ] , β (f 0 ,f 1 ) [Y 0 = 0,Y 1 = 1,X 0 ,X 1 ] = 1 1 0 0 [X 0 ,X 1 ] , β (f 0 ,f 1 ) [Y 0 = 1,Y 1 = 0,X 0 ,X 1 ] = 0 0 0 0 [X 0 ,X 1 ]and β (f 0 ,f 1 ) [Y 0 = 1,Y 1 = 1,X 0 ,X 1 ] = 0 0 0 1 [X 0 ,X 1 ] . Since the only vanishing slice of β F with respect to the head variables is that to y 0,1 = (1, 0), the vertices of the mean polytope are the vectors to the other head indices. The mean polytope is the convex hull of these vertices M (f 0 ,f 1 ) = conv 0 0 [L], 0 1 [L], 1 1 [L] . This polytope has a non cube-like face (sketched blue in Figure 5), which is the convex hull of the vertices (0 0) T , (1 1) T . This face is parametrized by the (CP-rank 2) hard activation core κ (0,0),(1,1) [Y 0 ,Y 1 ] = ε (0,0) [Y 0 ,Y 1 ] + ε (1,1) [Y 0 ,Y 1 ] = 1 0 0 1 [Y 0 ,Y 1 ] and has the refined base measure D κ (0,0),(1,1) [Y 0 ,Y 1 ],β F [Y 0 ,Y 1 ,X 0 ,X 1 ] E [X 0 ,X 1 ] = 0 0 1 1 [X 0 ,X 1 ] . Any mean parameter μ on the interior of that face can be parametrized by a scalar λ∈ (0, 1) μ λ [L] = ( λ λ ) [L] . With the canonical parameters θ [L]∈ R 2 of the maximum entropy distributions on this face by P [X 0 ,X 1 ] = 1 1 + exp [θ [L = 0] + θ [L = 1]] 00 1 exp [θ [L = 0] + θ [L = 1]] [X 0 ,X 1 ] we get the correspondence by the sigmoid λ = 1 1 + exp [−(θ [L = 0] + θ [L = 1])] . Note, that the hard activation core κ (0,0),(1,1) [Y 0 ,Y 1 ] to the blue face is the only non-elementary activation core. While the vertices have always elementary cores, the other non-vertex faces have elementary activation cores κ (0,0),(1,0),(1,1) [Y 0 ,Y 1 ] = 1 1 1 1 [Y 0 ,Y 1 ] = I [Y 0 ]⊗ I [Y 1 ], κ (0,0),(1,0) [Y 0 ,Y 1 ] = 1 0 1 0 [Y 0 ,Y 1 ] = I [Y 0 ]⊗ ε 0 [Y 1 ], κ (1,0),(1,1) [Y 0 ,Y 1 ] = 1 1 0 0 [Y 0 ,Y 1 ] = ε 0 [Y 0 ]⊗ I [Y 1 ] . The maximum entropy distributions to mean parameters on the interior of all other faces than the blue face are represented by Computation-Activation Networks with only elementary activation cores. The corresponding lattice is sketched in Figure 6. 5.3 Faces activated in the T format The T format is a well-studied tensor network format with favorable numerical properties Holtz et al. (2012) and applied across several scientific areas Sander et al. (2025b). This format describes tensor networks on a hypergraph (see Figure 7) T = X [d] ∪ I [d−1] ,I k−1 ,X k ,I k : k ∈1,...,d− 2∪X 0 ,I 0 ,I d−2 ,X d−1 , where I [d−1] are auxiliary hidden variables. We now construct an example of a polytope with a face activatable with T, in which all hidden variables are of dimension 2. 17 a) 0 0 1 0 1 1 0 1 μ λ b) 1 0 0 1 Y 0 1 exp [θ [L = 0]] Y 1 1 exp [θ [L = 1]] β (f 0 ,f 1 ) X 0 X 1 Figure 5: a) Mean polytope of the statistic t = (X 0 ∧ X 1 ,X 0 ⇒ X 1 ) (thick), as a subset of the cube [0, 1] 2 (dashed). The blue line is the face of the polytope, which is not cube-like, that is not an intersection of the polytope with the faces of the hypercube. We further define for λ ∈ (0, 1) a mean parameter μ λ [L] = [λ] T which is on the interior of the blue face. b) Corresponding Computation-Activation Network being the maximum entropy distribution reproducing μ λ [L], when λ is the sigmoid of θ [L = 0] + θ [L = 1]. F ∅ F (0,0) F (1,0) F (1,1) F (0,0),(1,0) F (1,0),(1,1) F (0,0),(1,1) rank (F) = 2rank (F) = 1 F (0,0),(1,0),(1,1) Figure 6: Face lattice L ((X 0 ∧ X 1 ,X 0 ⇒ X 1 )) to Example 7. The directed arrows represent inclusion of the faces, which is a partial order of the faces. The faceF (0,0),(1,1) is the only face, which is not representable by an elementary hypergraph. It is representable in a CP with hidden rank 2 Example 8 (T representation of diagonal faces). Consider the vertex set N : =0, 1 p /v [L] for some Boolean vector v [L]. We are interested in the faceF ◁,1 with the normal θ [L] = 1 2 (2v [L]−I [L]). Its vertices are the p elements of0, 1 p , which differ from v [L] in exactly one coordinate (i.e. those with Hamming distance of 1 from v [L]). We denote these vertices by v ℓ for ℓ∈ [p], which coordinates to ̃ ℓ∈ [p] are v ℓ h L = ̃ ℓ i = v h L = ̃ ℓ i if ̃ ℓ̸= ℓ 1− v h L = ̃ ℓ i if ̃ ℓ = ℓ . We now represent their sum in an T format. The Boolean hidden variables are denoted by I [p−1] and are interpreted as indicators, whether the coordinate flip has happened in [ℓ] coordinates. We construct a T core for ℓ = 0 by τ 0 [Y 0 ,I 0 ] = ε v[L=0] [Y 0 ]⊗ ε 0 [I 0 ] + ε 1−v[L=0] [Y 0 ]⊗ ε 1 [I 0 ] , a) X 0 X 1 X 2 · X d−1 I 0 I 1 · I d−2 e 0 e 1 e 2 e d−2 b) τ 0 X 0 τ 1 X 1 τ 2 X 2 · τ d−1 X d −1 I 0 I 1 I 2 I d −1 Figure 7: T format consisting of the T hypergraph a) and decorating tensor networks b). 18 for ℓ /∈0,p− 1 the cores τ ℓ [I ℓ−1 ,Y ℓ ,I ℓ ] =ε 0 [I ℓ−1 ]⊗ ε 1−v[L=ℓ] [Y ℓ ]⊗ ε 1 [I ℓ ] + ε 0 [I ℓ−1 ]⊗ ε v[L=ℓ] [Y ℓ ]⊗ ε 0 [I ℓ ] + ε 1 [I ℓ−1 ]⊗ ε v[L=ℓ] [Y ℓ ]⊗ ε 1 [I ℓ ] and for ℓ = p− 1 τ p−1 [I p−2 ,Y p−1 ] = ε 0 [I p−2 ]⊗ ε 1−v[L=p−1] [Y p−1 ] + ε 1 [I p−2 ]⊗ ε v[L=p−1] [Y p−1 ] . For this tensor network in the T format we then have X ℓ∈[p] ε v ℓ Y [p] = τ 0 [Y 0 ,I 0 ],τ p−1 [I p−2 ,Y p−1 ]∪τ ℓ [I ℓ−1 ,Y ℓ ,I ℓ ] : ℓ∈1,...,p− 2 [ Y [p] ] . The dimension 2 to the hidden variables is furthermore minimal, since each matrification based on a partition of [p] into non-empty sets has a matrix rank of 2 Holtz et al. (2012). 5.4 Construction of Boolean statistics to polytopes We now show that any convex polytope with Boolean vertices in R p is the mean polytope to a Boolean statistic. To construct the corresponding formulas we denote byM| U the restriction of the vectors in the setM to the coordinates inU . Theorem 8. LetM be an arbitrary polytope with Boolean vertices in R p . Then we construct propositional formulas on atoms X [p] by f 0 [X 0 ] = ⊤if M| 0 =1 ⊥if M| 0 =0 X 0 if M| 0 = [0, 1] and iteratively for ℓ∈ [p] with ℓ≥ 1 by f ℓ X [ℓ+1] = v[L]∈(M∩0,1 p )| [ℓ] ̃ ℓ∈[ℓ] ¬ 1−v [ L= ̃ ℓ ] f ̃ ℓ h X [ ̃ ℓ+1] i ⇒ ⊤if (M∩ v× R p−ℓ )| ℓ =1 ⊥if (M∩ v× R p−ℓ )| ℓ =0 X ℓ if (M∩ v× R p−ℓ )| ℓ = [0, 1] . Here we denote byM| V the projections of the vertices inM onto the subspaces V , and by 0 p the zero vector in R p . Proof. We show per induction, that for any ℓ ∈ [p] the family of Hybrid Logic Networks with the statistic f [ℓ+1] by the first ℓ + 1 formulas has the mean polytope M f [ℓ+1] ,I =M| [ℓ] .(2) ”ℓ = 0”: The polytopeM| [ℓ] =1 (respectivelyM| [ℓ] =0) is reproduced by f 0 being a tautology (respectively a contradiction). In the caseM| [ℓ] = [0, 1] the polytope is reproduced by the any formula, which is neither a tautology nor a contradiction, and the atomic formula X 0 is an example of such a contingency. Since the projection of a 0-1 polytope onto the first coordinates is itself a 0-1 polytope, these are the only possible cases and we conclude that in all M f [ℓ] ,I =M| [ℓ] . ”ℓ→ ℓ + 1”: We define for any v ∈M f [ℓ] ,I ∩0, 1 ℓ the formula f ℓ,v X [ℓ+1] : = ̃ ℓ∈[ℓ] ¬ 1−v [ L= ̃ ℓ ] f ̃ ℓ h X [ ̃ ℓ+1] i . and notice that all formulas are satisfiable (since the corresponding vertices are in theM f [ℓ] ,I ) and have disjoint models. We have for any Boolean vector v ∈0, 1 ℓ and λ∈0, 1 that v×λ∈M| [ℓ+1] , if and only if v ∈M| [ℓ] and λ∈M∩ v× R p−ℓ | [ℓ] . With modus ponens on f ℓ+1 X [ℓ+2] this is further equivalent to the satisfaction of f ℓ,v X [ℓ+1] ∧¬ 1−λ f ℓ+1 X [ℓ+2] and thus to v ×λ ∈ M f [ℓ+1] ,I . The vertices of the 0/1 polytopes M f [ℓ+1] ,I and M| [ℓ+1] thus coincide and both polytopes are therefore equivalent. By induction the equation (2) holds for arbitrary ℓ∈ [p]. For ℓ = p− 1 this is the claim. 19 5.5 Examples in statistical physics and coding theory Let us now investigate cube-like faces to standard examples in statistical physics and coding theory. Example 9 (Ising model on 2 nodes). Consider two Boolean variables X 0 ,X 1 and the Ising statistic t = (f 0 ,f 1 ,f 2 ) by three propositional formulas f 0 X [2] = X 0 , f 1 X [2] = X 1 and f 2 X [2] = X 0 ∧ X 1 . In the Ising interpretation, the Boolean variables represent interacting spins at two locations. Their value is then measured by the first two formulas and their interaction by the third. The vertices of the mean polytope to this statistic are σ t X [2] = (0, 0),L = 0 0 0 ! [L] , σ t X [2] = (0, 1),L = 0 1 0 ! [L] , σ t X [2] = (1, 0),L = 1 0 0 ! [L]and σ t X [2] = (1, 1),L = 1 1 1 ! [L] . The mean polytope is the convex hull of these, sketched as: 000 010 100 111 μ [L = 0] μ [L = 1] μ [L = 2] We notice, that the convex hull of any three out of the four vertices builds a facet. The only cube-like facet of these four (sketched in cyan in the above plot) is the convex hull of conv 0 0 0 ! [L], 1 0 0 ! [L], 0 1 0 ! [L] ! to which we have the parametrization by (U,y U ) = (2, 0). As a consequence, there are maximum entropy distributions of mean parameters in the Ising statistics, which do not have a representation by an elementary CompActNet in the Ising statistic. Example 10 (Codeword Polytopes). Consider the codeword polytope with the vertices N = y [3] : X k∈[3] y k mod 2 = 0 = ( 0 0 0 ! [L], 0 1 1 ! [L], 1 0 1 ! [L], 1 1 0 ! [L] ) We interpret the vertices by the strings of length 3, which parity vanishes. Such constraints are common in the construction of codewords using parity-check codes (see Gallager (1963)). 000 011 101 110 μ [L = 0] μ [L = 1] μ [L = 2] 20 All facets are non-cube-like, whereas all other faces are cube-like (the edges are labeled by the 6 facets of [0, 1] 3 ). Each facet is characterized by the Hamming distance of exactly 1 to a single of the 4 cube vertices, which are not codewords (see Example 3.9 in Wainwright and Jordan (2008)). We can construct T activation tensors with rank 2, which calculate the Hamming distance to these vertices via hidden ranks. Each core tensor in that format adds the contribution of a mode to the Hamming distance. In more generality, Low-Density-Parity-Checking Codes (see Gallager (1963)) construct codewords by a collection of parity constraints on subsets of variables. Such polytopes are also known as cycle polytopes, and studied e.g. in Grötschel et al. (1993). 6 Outlook This work characterized all discrete maximum entropy distributions as Computation-Activation Networks, a class of tensor networks. The mean polytope contains those mean parameters, to which entropy maximization problems are feasible. While exponential families are known to be the maximum entropy distributions to the interior of the polytope, we showed that refining the base measure with respect to faces enables the representation of maximum entropy distributions to mean parameters on the particular face. To this end we utilized the tensor network formalism and showed how to represent exponential families and their generalizations to maximum entropy distributions. One important application is estimation of maximum entropy distributions from data, such as learning in exponential families and graphical models Jordan (1998). Maximum likelihood problems are known to be dual to maximum entropy problems Koller and Friedman (2009) and therefore face similar representation challenges. Here the mean parameters of empirical distributions serve as sufficient statistics and are matched by trainable weights in algorithms such as iterative proportional fitting Darroch and Ratcliff (1972); Csiszar (1989). Optimizing parameters within a poor choice of learning architecture will lead to numerical instabilities, resulting from the absence of an optimum. Choosing the right tensor format representing the refined base measure guarantees the existence of an optimum. Further research might derive a relation between tensor format complexity Hackbusch (2012) to notions of facet complexity Grötschel et al. (1993). 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