File size: 6,098 Bytes
67aee1d | 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 | import OAI.Analysis.FreeEntropy.MatrixFlow
namespace OAI
universe u1 u2
noncomputable section
open MeasureTheory Set
namespace FiniteEntropySeparation
variable {ι : Type u1} [Fintype ι]
theorem smoothedSecond_integrable (μ : Measure (EuclideanSpace ℝ ι)) [IsProbabilityMeasure μ]
{s K : ℝ} (hs : 0<s) (hK : 0≤K) (hμ : ∀ᵐy ∂μ,‖y‖≤K) :
Integrable (fun x => smoothedDensity μ s x*‖x‖^2) := by
have hi := (smoothedPolynomial_integrable μ hs hK hμ 1).sub (smoothedDensity_integrable μ hs)
convert hi using 1
funext x
simp only [pow_one,Pi.sub_apply]
ring
theorem normalCross_integrable (μ : Measure (EuclideanSpace ℝ ι)) [IsProbabilityMeasure μ]
{s K : ℝ} (hs : 0<s) (hK : 0≤K) (hμ : ∀ᵐy ∂μ,‖y‖≤K) (v : ℝ) :
Integrable (fun x => smoothedDensity μ s x*Real.log (normalKernel v x)) := by
have he x : smoothedDensity μ s x*Real.log (normalKernel v x)=
smoothedDensity μ s x*(-(Fintype.card ι:ℝ)/2*Real.log (2*Real.pi*v))-
(smoothedDensity μ s x*‖x‖^2)/(2*v) := by rw [normalKernel,Real.log_exp]; ring
simp only [he]
exact ((smoothedDensity_integrable μ hs).mul_const _).sub ((smoothedSecond_integrable μ hs hK hμ).div_const _)
theorem normalCross_integral (μ : Measure (EuclideanSpace ℝ ι)) [IsProbabilityMeasure μ]
{s K : ℝ} (hs : 0<s) (hK : 0≤K) (hμ : ∀ᵐy ∂μ,‖y‖≤K) (v : ℝ) :
-(∫x,smoothedDensity μ s x*Real.log (normalKernel v x))=
(Fintype.card ι:ℝ)/2*Real.log (2*Real.pi*v)+(∫x,smoothedDensity μ s x*‖x‖^2)/(2*v) := by
have he x : smoothedDensity μ s x*Real.log (normalKernel v x)=
smoothedDensity μ s x*(-(Fintype.card ι:ℝ)/2*Real.log (2*Real.pi*v))-
(smoothedDensity μ s x*‖x‖^2)/(2*v) := by rw [normalKernel,Real.log_exp]; ring
simp only [he]
rw [integral_sub ((smoothedDensity_integrable μ hs).mul_const _)
((smoothedSecond_integrable μ hs hK hμ).div_const _),integral_mul_const,integral_div,smoothedDensity_integral μ hs]
ring
theorem density_event_toReal {Ω : Type u2} [MeasurableSpace Ω] (ν : Measure Ω)
(f : Ω → ℝ) (hf : Integrable f ν) (hf0 : ∀x,0≤f x) (E : Set Ω) (hE : MeasurableSet E) :
((ν.withDensity (fun x => ENNReal.ofReal (f x))) E).toReal=∫x in E,f x ∂ν := by
rw [withDensity_apply _ hE,← ofReal_integral_eq_lintegral_ofReal hf.integrableOn (ae_of_all _ hf0),
ENNReal.toReal_ofReal (integral_nonneg hf0)]
theorem classicalEntropy_event_gaussian (μ : Measure (EuclideanSpace ℝ ι)) [IsProbabilityMeasure μ]
{s K v z : ℝ} (hs : 0<s) (hK : 0≤K) (hμ : ∀ᵐy ∂μ,‖y‖≤K) (hv : 0<v) (hz : 0≤z)
(E : Set (EuclideanSpace ℝ ι)) (hE : MeasurableSet E)
(hα : ENNReal.ofReal (1/2:ℝ)≤(volume.withDensity (fun x => ENNReal.ofReal (smoothedDensity μ s x))) E)
(hβ : normalLaw v E≤ENNReal.ofReal (Real.exp (-z))) :
classicalEntropy μ s≤(Fintype.card ι:ℝ)/2*Real.log (2*Real.pi*v)+
(∫x,smoothedDensity μ s x*‖x‖^2)/(2*v)-z/2+Real.log 2 := by
have hfi := smoothedDensity_integrable μ hs
have hgi := normalKernel_integrable (ι := ι) hv
have hfm : (volume.withDensity (fun x => ENNReal.ofReal (smoothedDensity μ s x))) univ=1 := by
rw [withDensity_apply _ MeasurableSet.univ,setLIntegral_univ,
← ofReal_integral_eq_lintegral_ofReal hfi (ae_of_all _ (fun x => (smoothedDensity_pos μ hs x).le)),smoothedDensity_integral μ hs]
simp
have hfE : (volume.withDensity (fun x => ENNReal.ofReal (smoothedDensity μ s x))) E≠⊤ :=
ne_top_of_le_ne_top (by simp) ((measure_mono (subset_univ E)).trans_eq hfm)
have ha := (ENNReal.ofReal_le_iff_le_toReal hfE).mp hα
rw [density_event_toReal volume _ hfi (fun x => (smoothedDensity_pos μ hs x).le) E hE] at ha
have hb := ENNReal.toReal_mono (ENNReal.ofReal_ne_top) hβ
rw [normalLaw,density_event_toReal volume _ hgi (fun x => (normalKernel_pos v x).le) E hE,
ENNReal.toReal_ofReal (Real.exp_pos _).le] at hb
have he := entropy_event_bound volume (smoothedDensity μ s) (normalKernel v)
(smoothedDensity_pos μ hs) (normalKernel_pos v) hfi hgi (smoothedEntropy_integrable μ hs hK hμ)
(normalCross_integrable μ hs hK hμ v) (smoothedDensity_integral μ hs) (normalKernel_integral hv) E hE hz ha hb
rwa [normalCross_integral μ hs hK hμ v] at he
variable {n d : ℕ}
theorem integral_heatLaw_totalSecond (hd : 0<d) (μ : Measure (MatrixCoordinates n d)) [IsFiniteMeasure μ]
{s : ℝ} (hs : 0<s) :
(∫x,totalSecond (matrixMoment x) ∂matrixHeatLaw μ s)=(d:ℝ)⁻¹*(∫x,smoothedDensity μ (s/d) x*‖x‖^2) := by
rw [integral_matrixHeatLaw μ hs hd]
simp only [totalSecond_matrix,mul_left_comm _ (d:ℝ)⁻¹,integral_const_mul]
theorem normalizedEntropy_event_gaussian (hd : 0<d) (μ : Measure (MatrixCoordinates n d)) [IsProbabilityMeasure μ]
{s K v : ℝ} (hs : 0<s) (hK : 0≤K) (hμ : ∀ᵐy ∂μ,‖y‖≤K) (hv : 0<v)
(E : Set (MatrixCoordinates n d)) (hE : MeasurableSet E)
(hα : ENNReal.ofReal (1/2:ℝ)≤ matrixHeatLaw μ s E)
(hβ : normalLaw (v/d) E≤ENNReal.ofReal (Real.exp (-4*(d:ℝ)^2))) :
normalizedMatrixEntropy μ s≤(n:ℝ)/2*Real.log (2*Real.pi*v)+
(∫x,totalSecond (matrixMoment x) ∂matrixHeatLaw μ s)/(2*v)-2+Real.log 2/(d:ℝ)^2 := by
have hd' : (0:ℝ)<d := by exact_mod_cast hd
have he := classicalEntropy_event_gaussian μ (div_pos hs hd') hK hμ (div_pos hv hd')
(show 0≤4*(d:ℝ)^2 by positivity) E hE hα (by simpa only [neg_mul] using hβ)
have hlog : Real.log (2*Real.pi*(v/d))=Real.log (2*Real.pi*v)-Real.log d := by
rw [show 2*Real.pi*(v/d)=(2*Real.pi*v)/d by ring,Real.log_div (by positivity) hd'.ne']
rw [hlog] at he
rw [integral_heatLaw_totalSecond hd μ hs]
unfold normalizedMatrixEntropy
have hh := div_le_div_of_nonneg_right he (sq_nonneg (d:ℝ))
simp only [Fintype.card_prod,Fintype.card_fin,Nat.cast_mul] at hh
convert add_le_add hh (le_refl ((n:ℝ)/2*Real.log d)) using 1 ; first | rfl | (field_simp; ring)
end FiniteEntropySeparation
end
end OAI
|