| import OAI.Analysis.FreeEntropy.MatrixFlow |
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| namespace OAI |
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| universe u1 u2 |
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| 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 |
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