I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives Variations on information theory: categories, cohomology, entropy. Juan Pablo Vigneaux IMJ-PRG - Universit´ e Paris 7 May 17, 2016
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives I NTRODUCTION (Co)homology Information I NFORMATION STRUCTURES Observables Probabilities Functions C OHOMOLOGY De Rham cohomology Definition Perspectives Perspectives
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives (C O ) HOMOLOGY In geometry, homology and cohomology are related to the notion of “shape”. Define H 1 = { 1-dimensional cycles } / { 1-dimensional boundaries } . The fact that dim H 1 ( sphere ) = 0 and dim H 1 ( torus ) = 2 is stable under continuous deformations.
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives I NFORMATION THEORY Shannon (1948) defined the information content of a random variable X : Ω → { x 1 , ..., x n } as n � H ( X ) = − P ( X = x i ) log 2 P ( X = x i ) , (1) k = 0 where P denotes a probability law on the space Ω . The function H is called entropy.
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives I NFORMATION THEORY Shannon (1948) defined the information content of a random variable X : Ω → { x 1 , ..., x n } as n � H ( X ) = − P ( X = x i ) log 2 P ( X = x i ) , (1) k = 0 where P denotes a probability law on the space Ω . The function H is called entropy. Information is related to uncertainty . 1. Uniform distribution on { x 1 , ..., x n } implies H ( X ) maximal. 2. If P ( X = x i ) = 1 for certain i , then H ( X ) = 0.
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives I NFORMATION THEORY Shannon (1948) defined the information content of a random variable X : Ω → { x 1 , ..., x n } as n � H ( X ) = − P ( X = x i ) log 2 P ( X = x i ) , (1) k = 0 where P denotes a probability law on the space Ω . The function H is called entropy. Information is related to uncertainty . 1. Uniform distribution on { x 1 , ..., x n } implies H ( X ) maximal. 2. If P ( X = x i ) = 1 for certain i , then H ( X ) = 0. Shannon recognized an important relation, H ( X , Y ) = H ( X ) + H ( Y | X ) .
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives O BSERVABLES Consider a set of observables 1 , X 1 , X 2 , X 3 , ... (where 1 corresponds to a certitude/a constant variable). We are just interested in the algebras of events defined by each variables... (we consider X ∼ = Y if σ ( X ) = σ ( Y ) ). We can write an arrow X → Y if σ ( Y ) ⊂ σ ( X ) (if “ X refines Y ”).
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives I NFORMATION STRUCTURES : EXAMPLES Example 1. Set Ω = { 1 , 2 , 3 } and define X i := {{ i } , Ω \ { i }} . M is the atomic partition. 1 Ω X 1 X 2 X 3 M
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives I NFORMATION STRUCTURES : EXAMPLES Example 2. As before, but the observable X 2 is not available. 1 Ω X 1 X 3 M
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives I NFORMATION STRUCTURES : EXAMPLES Example 3. From quantum physics. Here, L x , L y , L z are the quantum observables that correspond to angular momentum and L 2 = L 2 x + L 2 y + L 2 z . 1 L 2 L x L y L z L x L 2 L y L 2 L z L 2 We cannot measure simultaneously two components of the angular momentum since the operators do not commute.
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives I NFORMATION STRUCTURE : GENERAL DEFINITION An information structure is a category, whose objects are observables (seen as partitions/ σ -algebras) and whose arrows are refinements (they form a poset for this relation). We suppose that: ◮ given any three observables X , Y and Z in S , such that X refines Y and Z , then the joint observable YZ := ( Y , Z ) , ω �→ ( Y ( ω ) , Z ( ω )) also belongs to S . ◮ S has a final object (a constant variable/ a certitude).
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives P ROBABILITIES Each observable X defines an algebra of sets σ ( X ) . Fix a set Q X of allowed laws on (Ω , σ ( X )) , parametrized in some way. To each arrow of refinement X → Y , we want a surjective Y ∗ application Q X → Q Y , called marginalization.
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives Example. Set Ω = { 1 , 2 , 3 } , X i := {{ i } , Ω \ { i }} , M atomic. ∆ k := { ( x 0 , . . . , x k ) ∈ R 2 ≥ 0 : x 0 + . . . + x k = 1 } , the k -simplex. 1 Ω X 1 X 3 M { 1 } ∆ 1 ∆ 1 ( p 1 , p 2 + p 3 ) ( X 1 ) ∗ ∆ 2 ( p 1 , p 2 , p 3 )
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives F UNCTIONAL MODULE Similarly, for each observable X , consider the real vector space F X of measurable functions on Q X (the entropy H [ X ] lives here!). If X → Y , a function f ∈ Q X can be mapped naturally to F X : just set f X | Y ( P ) = f ( Y ∗ P ) . The set F X accepts a natural action of S X (these are the variables refined by X ): for an observable Y (call the possible values { y 1 , ..., y k } ) in S X and f ∈ F ( Q X ) , the new function Y . f ∈ F X is given by k � ( Y . f )( P ) = P ( Y = y i ) f ( P | Y = y i ) . i = 1
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives Example. Set Ω = { 1 , 2 , 3 } , X i := {{ i } , Ω \ { i }} , M atomic, ∆ k the k -simplex. F M = { f : ∆ 2 → R } , etc. 1 Ω X 1 X 3 M F 1 f ( x , y ) F X 1 F X 2 f X | Y ( x , y , z ) = f ( x , y + z ) F M
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives F INITE QUANTUM CASE ◮ The role of Ω is played by a fixed finite dimensional, complex vector space E with a distinguished basis (or a non-degenerate hermitian form). ◮ Observables are self-adjoint operators, they induce decompositions of E as direct-sum of subspaces (Spectral theorem). ◮ We can measure simultaneously two quantities only if the corresponding observables commute as operators. In this case the joint ( X , Y ) determines a refined decomposition. ◮ We obtain a category S of observables. ◮ Quantum laws are positive hermitian forms. ◮ Etc.
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives D E R HAM COHOMOLOGY Question: U ⊂ R 2 , functions f 1 , f 2 : U → R . Is ∂ f 1 ∂ y − ∂ f 2 ∂ x = 0 a sufficient condition for the existence of F such that ∇ F = ( f 1 , f 2 ) ? 1. If U is star-shaped (radially convex): yes! 2. if U = R 2 \ { 0 } : no.
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives D E R HAM COHOMOLOGY Question: U ⊂ R 2 , functions f 1 , f 2 : U → R . Is ∂ f 1 ∂ y − ∂ f 2 ∂ x = 0 a sufficient condition for the existence of F such that ∇ F = ( f 1 , f 2 ) ? 1. If U is star-shaped (radially convex): yes! 2. if U = R 2 \ { 0 } : no. � � − x 2 x 1 For example, for ( f 1 , f 2 ) = 2 , such F does not exist, x 2 1 + x 2 x 2 1 + x 2 2
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives D E R HAM COHOMOLOGY Question: U ⊂ R 2 , functions f 1 , f 2 : U → R . Is ∂ f 1 ∂ y − ∂ f 2 ∂ x = 0 a sufficient condition for the existence of F such that ∇ F = ( f 1 , f 2 ) ? 1. If U is star-shaped (radially convex): yes! 2. if U = R 2 \ { 0 } : no. � � − x 2 x 1 For example, for ( f 1 , f 2 ) = 2 , such F does not exist, since x 2 1 + x 2 x 2 1 + x 2 2 � 2 π d d d θ F ( cos θ, sin θ ) d θ = F ( 1 , 0 ) − F ( 1 , 0 ) = 0 but d θ F ( cos θ, sin θ ) = 1 0 by the chain rule. The answer depends on the “shape” (the topology) of U .
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives S OME ALGEBRA ... C ∞ ( U , R ) { 1 − forms } { 2 − forms } δ 0 = ∇ δ 1 = curl Ω 0 ( U ) Ω 1 ( U ) Ω 2 ( U ) ∂ x d x + ∂ f ∂ f f ∂ y d y � � ∂ g ∂ y − ∂ h g ( x , y ) d x + h ( x , y ) d y d x ∧ d y . ∂ x Remark that curl ( ∇ f ) = 0... this means that im ∇ ⊂ ker ( curl ) .
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives δ 0 = ∇ δ 1 = curl Ω 0 ( U ) Ω 1 ( U ) Ω 2 ( U ) Define, H 1 ( U ) = ker δ 1 / im δ 0 = ker ( curl ) / im ∇ . Then, 1. H 1 ( U ) ∼ = { 0 } if U is star-shaped.
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives δ 0 = ∇ δ 1 = curl Ω 0 ( U ) Ω 1 ( U ) Ω 2 ( U ) Define, H 1 ( U ) = ker δ 1 / im δ 0 = ker ( curl ) / im ∇ . Then, 1. H 1 ( U ) ∼ = { 0 } if U is star-shaped. 2. H 1 ( R 2 \ { 0 } ) � = { 0 } .
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives δ 0 = ∇ δ 1 = curl Ω 0 ( U ) Ω 1 ( U ) Ω 2 ( U ) Define, H 1 ( U ) = ker δ 1 / im δ 0 = ker ( curl ) / im ∇ . Then, 1. H 1 ( U ) ∼ = { 0 } if U is star-shaped. 2. H 1 ( R 2 \ { 0 } ) � = { 0 } . = R n if U has n holes. 3. In general, H 1 ( U ) ∼
I NTRODUCTION I NFORMATION STRUCTURES C OHOMOLOGY Perspectives T HE TRICKY TECHNICAL POINTS ... 1. Consider your category S . Over each X ∈ S there is monoid S X of variables coarser than X . Denote by A X the algebra generated over R by this monoid. 2. Put the trivial Grothendieck topology on S . The couple ( S , A ) is a ringed site. We work in the category Mod ( A ) : sheaves of groups with an action of A (the sheaf F lives here!). 3. Define the information cohomology as (cf. Bennequin-Baudot, 2015 [1]): H n ( S , Q ) = Ext n ( R S , F ) . 4. The bar resolution construction allows us to construct a complex δ 0 δ 1 δ 2 C 0 C 1 C 2 0 . . . and compute H n ( S , Q ) ∼ = ker δ n / im δ n − 1 .
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