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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 | import OAI.Probability.GaussianReplacement.Hybrid
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
namespace CurrentProjection
open InnerProductSpace
open InformationTheory
lemma shannonEntropy_le_log_card {W : Type*} [Fintype W] [MeasurableSpace W]
[MeasurableSingletonClass W] (π : Measure W) [IsProbabilityMeasure π] :
(shannonEntropy π).toReal ≤ Real.log (Fintype.card W) := by
let : Nonempty W := nonempty_of_isProbabilityMeasure π
let Q := (PMF.uniformOfFintype W).toMeasure
have hQ : ∀ w, Q {w} = (Fintype.card W:ℝ≥0∞)⁻¹ := by
intro w
change (PMF.uniformOfFintype W).toMeasure {w} = _
rw [PMF.toMeasure_apply_singleton _ _ (measurableSet_singleton w),PMF.uniformOfFintype_apply]
have hq0 : (Fintype.card W:ℝ≥0∞)⁻¹ ≠ 0 := ENNReal.inv_ne_zero.mpr (ENNReal.natCast_ne_top _)
have hPQ : π ≪ Q := countable_ac_of_atom π Q (by intro w hw; exact (hq0 ((hQ w).symm.trans hw)).elim)
obtain ⟨_,hb⟩ := countable_entropy_le_crossEntropy π Q hPQ (Integrable.of_finite)
have he : (fun w => -Real.log (Q {w}).toReal) = fun _ : W => Real.log (Fintype.card W) := by
funext w
rw [hQ w,ENNReal.toReal_inv,ENNReal.toReal_natCast,Real.log_inv,neg_neg]
simpa [he] using hb
lemma fiber_comparison {d : ℕ} (hd : 100 ≤ d) {W : Type*} [Fintype W]
[MeasurableSpace W] [MeasurableSingletonClass W]
(P : Measure ((Sphere d×W)×Rows (d/10) d)) [IsProbabilityMeasure P]
(hInd : (P.map swapLast).fst = (uniformSphere d).prod (gaussianRows (d/10) d))
(hH : shannonEntropy P.fst.snd ≤ ENNReal.ofReal ((d:ℝ)^2)) :
freshInformation P.fst (d/3) ≤ exposedInformation (alignedExperiment (r:=d/3-d/10) P) +
ENNReal.ofReal (1000*d) := by
let : IsProbabilityMeasure (uniformSphere d) := uniformSphere_probability_of_pos (by omega)
let : Nonempty (Sphere d) := nonempty_of_isProbabilityMeasure (uniformSphere d)
let : Nonempty W := nonempty_of_isProbabilityMeasure P.fst.snd
have hHfin : shannonEntropy P.fst.snd ≠ ∞ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hH
have hHr : (shannonEntropy P.fst.snd).toReal ≤ (d:ℝ)^2 :=
ENNReal.toReal_le_of_le_ofReal (sq_nonneg _) hH
have hdim : AdmissibleDimensions d (d/10) (d/3-d/10) (d/10) := by
unfold AdmissibleDimensions
omega
obtain ⟨hJ0,hJ1,hcomp⟩ := critical_unrefined_comparison hdim P hInd hHfin
have hσ := signal_marginal_of_independence P (uniformSphere d) (gaussianRows (d/10) d) hInd
obtain ⟨hE,hEb,hF,hFb⟩ := criticalSignal_entropy P.fst hσ hdim.2.2.2.1 hHfin
obtain ⟨he,hl⟩ := critical_entropy_scalars (show 1 ≤ d by omega) ENNReal.toReal_nonneg hHr
hEb ENNReal.toReal_nonneg hFb
obtain ⟨hRow,hRowB⟩ := rowInformation_le_entropy P (uniformSphere d) (gaussianRows (d/10) d)
hInd (by rwa [swapLast_snd])
rw [swapLast_snd] at hRowB
have hqr : (0:ℝ) < (d/10:ℕ) := by exact_mod_cast (show 0<d/10 by omega)
have hqrB : (d:ℝ) ≤ 20*(d/10:ℕ) := by exact_mod_cast (show d≤20*(d/10) by omega)
have hℓr : ((d/10+(d/3-d/10):ℕ):ℝ) ≤ d := by exact_mod_cast (show d/10+(d/3-d/10)≤d by omega)
simp only [Nat.cast_add] at hℓr
have hd1 : (1:ℝ) ≤ d := by exact_mod_cast (show 1≤d by omega)
have hRowr : (rowInformation P (gaussianRows (d/10) d)).toReal/(d/10:ℕ) ≤ 20*(d:ℝ) := by
apply (div_le_iff₀ hqr).mpr
nlinarith [hRowB.trans hHr]
have hbound : exposedInformation (P.fst.prod (gaussianRows (d/10+(d/3-d/10)) d)) ≤
exposedInformation (alignedExperiment (r:=d/3-d/10) P) + ENNReal.ofReal (1000*d) := by
apply (ENNReal.toReal_le_toReal hJ0 (ENNReal.add_ne_top.mpr ⟨hJ1,ENNReal.ofReal_ne_top⟩)).mp
rw [ENNReal.toReal_add hJ1 ENNReal.ofReal_ne_top,ENNReal.toReal_ofReal (by positivity)]
nlinarith
have hdimEq : d/10+(d/3-d/10) = d/3 := by omega
change freshInformation P.fst (d/10+(d/3-d/10)) ≤ _ at hbound
have hfresh : freshInformation P.fst (d/10+(d/3-d/10)) = freshInformation P.fst (d/3) :=
congrArg (freshInformation P.fst) hdimEq
rwa [hfresh] at hbound
theorem fiberBlockMain : FiberBlockMain := by
refine ⟨1000,100,by norm_num,?_⟩
intro d hd U V _ _ _ _ _ _ P _ hS κ _ N hN hcard hNlog
let : IsProbabilityMeasure (uniformSphere d) := uniformSphere_probability_of_pos (by omega)
let : Nonempty (Sphere d) := nonempty_of_isProbabilityMeasure (uniformSphere d)
let : Nonempty U := nonempty_of_isProbabilityMeasure P.snd
let : Nonempty V := nonempty_of_isProbabilityMeasure (endingSignalLaw P κ).snd
have hEnt : shannonEntropy (endingSignalLaw P κ).snd ≤ ENNReal.ofReal ((d:ℝ)^2) := by
apply (ENNReal.toReal_le_toReal (shannonEntropy_finite _) ENNReal.ofReal_ne_top).mp
rw [ENNReal.toReal_ofReal (sq_nonneg _)]
apply (shannonEntropy_le_log_card _).trans
exact (Real.log_le_log (by exact_mod_cast Fintype.card_pos) (by exact_mod_cast hcard)).trans hNlog
have hInd : ((endingRowLaw P κ).map swapLast).fst = (uniformSphere d).prod (gaussianRows (d/10) d) := by
rw [endingRowLaw_independence,hS]
have h := fiber_comparison hd (endingRowLaw P κ) hInd (by rwa [endingRowLaw_fst])
simp only [endingRowLaw_fst] at h
have hproc := block_exposed_le_fresh (r:=d/3-d/10) P κ
have he : freshInformation P (d/10+(d/3-d/10)) = freshInformation P (d/3) :=
congrArg (freshInformation P) (by omega)
rw [he] at hproc
exact h.trans (add_le_add hproc le_rfl)
end CurrentProjection
end
end OAI
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