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b317718 | 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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.Probability.GaussianReplacement.Comparison
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
namespace CurrentProjection
open InnerProductSpace
open InformationTheory
def blockInput {d m : ℕ} {U : Type*} (z : (Sphere d × U) × Rows m d) :
U × (Rows m d × (Fin m → ℝ)) :=
(z.1.2, z.2, label z.2 (z.1.1 : Vector d))
lemma measurable_blockInput {d m : ℕ} {U : Type*} [MeasurableSpace U] :
Measurable (blockInput (d:=d) (m:=m) (U:=U)) := by
exact measurable_fst.snd.prodMk (measurable_snd.prodMk
((measurable_own_label (d:=d) (k:=m)).comp
(measurable_snd.prodMk measurable_fst.fst)))
def blockLaw {d m : ℕ} {U V : Type*} [MeasurableSpace U] [MeasurableSpace V]
(P : Measure (Sphere d × U))
(κ : Kernel (U × (Rows m d × (Fin m → ℝ))) V) :
Measure (((Sphere d × U) × Rows m d) × V) :=
(P.prod (gaussianRows m d)) ⊗ₘ κ.comap blockInput measurable_blockInput
instance blockLaw_probability {d m : ℕ} {U V : Type*}
[MeasurableSpace U] [MeasurableSpace V]
(P : Measure (Sphere d × U)) [IsProbabilityMeasure P]
(κ : Kernel (U × (Rows m d × (Fin m → ℝ))) V) [IsMarkovKernel κ] :
IsProbabilityMeasure (blockLaw P κ) := by
unfold blockLaw
infer_instance
def endingSignalLaw {d m : ℕ} {U V : Type*}
[MeasurableSpace U] [MeasurableSpace V]
(P : Measure (Sphere d × U))
(κ : Kernel (U × (Rows m d × (Fin m → ℝ))) V) : Measure (Sphere d × V) :=
(blockLaw P κ).map (fun z => (z.1.1.1, z.2))
instance endingSignalLaw_probability {d m : ℕ} {U V : Type*}
[MeasurableSpace U] [MeasurableSpace V]
(P : Measure (Sphere d × U)) [IsProbabilityMeasure P]
(κ : Kernel (U × (Rows m d × (Fin m → ℝ))) V) [IsMarkovKernel κ] :
IsProbabilityMeasure (endingSignalLaw P κ) := by
unfold endingSignalLaw
infer_instance
def freshInformation {d : ℕ} {U : Type*} [MeasurableSpace U] [StandardBorelSpace U]
(P : Measure (Sphere d × U)) [IsFiniteMeasure P] (k : ℕ) : ℝ≥0∞ :=
exposedInformation (P.prod (gaussianRows k d))
def hybridFirst (d : ℕ) : ℕ := (d+3)/4
def hybridLast (d : ℕ) : ℕ := d/2
def hybridCount (d : ℕ) : ℕ := hybridLast d - hybridFirst d + 1
def averagedFreshInformation {d : ℕ} {U : Type*}
[MeasurableSpace U] [StandardBorelSpace U]
(P : Measure (Sphere d × U)) [IsFiniteMeasure P] : ℝ :=
(∑ k ∈ Finset.Icc (hybridFirst d) (hybridLast d), (freshInformation P k).toReal) /
hybridCount d
def HybridBlockMain : Prop :=
∃ (C C' : ℝ) (d₀ : ℕ), 0 < C ∧ 0 < C' ∧ ∀ d, d₀ ≤ d →
∀ m, 1 ≤ m → m ≤ d/8 →
∀ (U V : Type) [Fintype U] [Fintype V]
[MeasurableSpace U] [MeasurableSpace V]
[MeasurableSingletonClass U] [MeasurableSingletonClass V],
∀ (P : Measure (Sphere d × U)) [IsProbabilityMeasure P],
P.fst = uniformSphere d →
∀ (κ : Kernel (U × (Rows m d × (Fin m → ℝ))) V) [IsMarkovKernel κ],
∀ (B : ℝ), 0 ≤ B →
shannonEntropy (endingSignalLaw P κ).snd ≤ ENNReal.ofReal B →
averagedFreshInformation (endingSignalLaw P κ) - averagedFreshInformation P ≤
C*m + (m:ℝ) / hybridCount d * (C + B / (d/8:ℕ)) + B / (d/32:ℕ) ∧
(B ≤ (d:ℝ)^2 →
averagedFreshInformation (endingSignalLaw P κ) - averagedFreshInformation P ≤ C'*d)
def FiberBlockMain : Prop :=
∃ (K : ℝ) (d₀ : ℕ), 0 < K ∧ ∀ d, d₀ ≤ d →
∀ (U V : Type) [Fintype U] [Fintype V]
[MeasurableSpace U] [MeasurableSpace V]
[MeasurableSingletonClass U] [MeasurableSingletonClass V],
∀ (P : Measure (Sphere d×U)) [IsProbabilityMeasure P],
P.fst = uniformSphere d →
∀ (κ : Kernel (U×(Rows (d/10) d×(Fin (d/10)→ℝ))) V) [IsMarkovKernel κ],
∀ N : ℕ, 1 ≤ N → Fintype.card V ≤ N → Real.log N ≤ (d:ℝ)^2 →
freshInformation (endingSignalLaw P κ) (d/3) ≤ freshInformation P (d/3) + ENNReal.ofReal (K*d)
end CurrentProjection
end
end OAI
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