File size: 2,017 Bytes
e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators Topology
open Filter
universe uV uE uIndex uι
namespace MatchingEntropyBounds
structure LooplessGraph (V : Type uV) (E : Type uE) where
left : E → V
right : E → V
loopless : ∀ e, left e ≠ right e
variable {V : Type uV} {E : Type uE}
[Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
namespace LooplessGraph
def Incident (G : LooplessGraph V E) (v : V) (e : E) : Prop :=
G.left e = v ∨ G.right e = v
instance incidentDecidable (G : LooplessGraph V E) (v : V) (e : E) :
Decidable (G.Incident v e) := inferInstanceAs (Decidable (G.left e = v ∨ G.right e = v))
def IsPerfectMatching (G : LooplessGraph V E) (M : Finset E) : Prop :=
∀ v, ∃! e, e ∈ M ∧ G.Incident v e
abbrev Matching (G : LooplessGraph V E) := {M : Finset E // G.IsPerfectMatching M}
instance matchingFintype (G : LooplessGraph V E) : Fintype G.Matching :=
Fintype.ofFinite _
def indicator (G : LooplessGraph V E) (M : G.Matching) (e : E) : ℝ :=
if e ∈ M.val then 1 else 0
def polytope (G : LooplessGraph V E) : Set (E → ℝ) :=
convexHull ℝ (Set.range G.indicator)
def mean (G : LooplessGraph V E) (p : G.Matching → ℝ) (e : E) : ℝ :=
∑ M, p M * G.indicator M e
end LooplessGraph
def probabilitySimplex (Index : Type uIndex) [Fintype Index] : Set (Index → ℝ) :=
{weights | (∀ index, 0 ≤ weights index) ∧ ∑ index, weights index = 1}
namespace LooplessGraph
def feasibleLaws (G : LooplessGraph V E) (y : E → ℝ) : Set (G.Matching → ℝ) :=
{p | p ∈ probabilitySimplex G.Matching ∧ G.mean p = y}
end LooplessGraph
def entropy {ι : Type uι} [Fintype ι] (p : ι → ℝ) : ℝ :=
∑ i, Real.negMulLog (p i)
def maxMatchingEntropy (G : LooplessGraph V E) (y : E → ℝ) : ℝ :=
sSup (entropy '' G.feasibleLaws y)
namespace Refined
def complementEntropy (x : E → ℝ) : ℝ := entropy (fun e => 1 - x e)
end Refined
end MatchingEntropyBounds
end
end OAI
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