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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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Analysis.FreeEntropy.GaussianDecay
namespace OAI
universe u1
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped Topology
namespace FiniteEntropySeparation
variable {ι : Type u1} [Fintype ι]
theorem normal_prefactor_bounded {a b : ℝ} (ha : 0<a) :
∃ D : ℝ,0 ≤ D ∧ ∀s∈Icc a b, |-(Fintype.card ι:ℝ)/2*Real.log (2*Real.pi*s)| ≤ D := by
have hc : ContinuousOn (fun s : ℝ => -(Fintype.card ι:ℝ)/2*Real.log (2*Real.pi*s)) (Icc a b) := by
apply continuousOn_const.mul
apply ContinuousOn.log (by fun_prop)
intro s hs; exact ne_of_gt (by have : 0<s := ha.trans_le hs.1; positivity)
obtain ⟨D,hD⟩ := isCompact_Icc.exists_bound_of_continuousOn hc
refine ⟨max D 0,le_max_right _ _,?_⟩
intro s hs
exact (show |-(Fintype.card ι:ℝ)/2*Real.log (2*Real.pi*s)| ≤ D by simpa only [Real.norm_eq_abs] using hD s hs).trans (le_max_left _ _)
theorem normalKernel_shift_bounds {a b s D K : ℝ} (ha : 0<a) (hs : s∈Icc a b)
(hD : |-(Fintype.card ι:ℝ)/2*Real.log (2*Real.pi*s)| ≤ D) (_hK : 0 ≤ K)
(x y : EuclideanSpace ℝ ι) (hy : ‖y‖ ≤ K) :
Real.exp (-D-(‖x‖^2+K^2)/a) ≤ normalKernel s (x-y) ∧
normalKernel s (x-y) ≤ Real.exp (D+K^2/(2*b))*Real.exp (-‖x‖^2/(4*b)) := by
have hs0 : 0<s := ha.trans_le hs.1
have hb : 0<b := hs0.trans_le hs.2
have hnorm : ‖x-y‖ ≤ ‖x‖+K := (norm_sub_le x y).trans (add_le_add le_rfl hy)
have hnorm' : ‖x‖ ≤ ‖x-y‖+K := by
calc _ = ‖(x-y)+y‖ := by rw [sub_add_cancel]
_ ≤ ‖x-y‖+‖y‖ := norm_add_le _ _
_ ≤ _ := add_le_add_right hy _
have hu : ‖x-y‖^2 ≤ 2*(‖x‖^2+K^2) := by
nlinarith [sq_nonneg (‖x‖-K),pow_le_pow_left₀ (norm_nonneg _) hnorm 2]
have hl : ‖x‖^2 ≤ 2*(‖x-y‖^2+K^2) := by
nlinarith [sq_nonneg (‖x-y‖-K),pow_le_pow_left₀ (norm_nonneg _) hnorm' 2]
have hdivu : ‖x-y‖^2/(2*s) ≤ (‖x‖^2+K^2)/a := by
calc _ ≤ ‖x-y‖^2/(2*a) := div_le_div_of_nonneg_left (sq_nonneg _) (by positivity) (by linarith [hs.1])
_ ≤ _ := by apply (div_le_iff₀ (show 0<2*a by positivity)).mpr; field_simp; nlinarith
have hdivl : ‖x‖^2/(4*b)-K^2/(2*b) ≤ ‖x-y‖^2/(2*s) := by
calc _ ≤ ‖x-y‖^2/(2*b) := by apply (le_div_iff₀ (show 0<2*b by positivity)).mpr; field_simp; nlinarith
_ ≤ _ := div_le_div_of_nonneg_left (sq_nonneg _) (by positivity) (by linarith [hs.2])
constructor
· apply Real.exp_le_exp.mpr
have := (abs_le.mp hD).1
linarith
· rw [← Real.exp_add]
apply Real.exp_le_exp.mpr
have := (abs_le.mp hD).2
simp only [neg_div] at *
linarith
theorem smoothedDensity_bounds (μ : Measure (EuclideanSpace ℝ ι)) [IsProbabilityMeasure μ]
{a b s D K : ℝ} (ha : 0<a) (hs : s∈Icc a b)
(hD : |-(Fintype.card ι:ℝ)/2*Real.log (2*Real.pi*s)| ≤ D) (hK : 0 ≤ K)
(hμ : ∀ᵐ y ∂μ,‖y‖ ≤ K) (x : EuclideanSpace ℝ ι) :
Real.exp (-D-(‖x‖^2+K^2)/a) ≤ smoothedDensity μ s x ∧
smoothedDensity μ s x ≤ Real.exp (D+K^2/(2*b))*Real.exp (-‖x‖^2/(4*b)) := by
have hi := normalKernel_integrable_finite μ (ha.trans_le hs.1) x
constructor
· have h := integral_mono_ae (integrable_const (Real.exp (-D-(‖x‖^2+K^2)/a))) hi
(hμ.mono (fun y hy => (normalKernel_shift_bounds ha hs hD hK x y hy).1))
simpa only [smoothedDensity,integral_const,probReal_univ,one_smul] using h
· have h := integral_mono_ae hi (integrable_const (Real.exp (D+K^2/(2*b))*Real.exp (-‖x‖^2/(4*b))))
(hμ.mono (fun y hy => (normalKernel_shift_bounds ha hs hD hK x y hy).2))
simpa only [smoothedDensity,integral_const,probReal_univ,one_smul] using h
theorem smoothedDensity_log_bound (μ : Measure (EuclideanSpace ℝ ι)) [IsProbabilityMeasure μ]
{a b s D K : ℝ} (ha : 0<a) (hs : s∈Icc a b) (_hD0 : 0 ≤ D)
(hD : |-(Fintype.card ι:ℝ)/2*Real.log (2*Real.pi*s)| ≤ D) (hK : 0 ≤ K)
(hμ : ∀ᵐ y ∂μ,‖y‖ ≤ K) (x : EuclideanSpace ℝ ι) :
|Real.log (smoothedDensity μ s x)| ≤ D+(‖x‖^2+K^2)/a := by
have hs0 := ha.trans_le hs.1
have hb : 0<b := hs0.trans_le hs.2
have hp := smoothedDensity_pos μ hs0 x
obtain ⟨hl,hu⟩ := smoothedDensity_bounds μ ha hs hD hK hμ x
rw [abs_le]
constructor
· have h := Real.log_le_log (Real.exp_pos _) hl
rw [Real.log_exp] at h
linarith
· have hu' : smoothedDensity μ s x ≤ Real.exp D := by
unfold smoothedDensity
have h := integral_mono_ae (normalKernel_integrable_finite μ hs0 x) (integrable_const (Real.exp D))
(ae_of_all _ (fun y => (normalKernel_le s hs0 (x-y)).trans (Real.exp_le_exp.mpr (le_trans (le_abs_self _) hD))))
simpa only [smoothedDensity,integral_const,probReal_univ,one_smul] using h
have h := Real.log_le_log hp hu'
rw [Real.log_exp] at h
exact h.trans (le_add_of_nonneg_right (by positivity))
theorem integrable_of_normBound (μ : Measure (EuclideanSpace ℝ ι)) [IsFiniteMeasure μ]
{K : ℝ} (hμ : ∀ᵐ y ∂μ,‖y‖ ≤ K) {f : EuclideanSpace ℝ ι → ℝ} (hf : Continuous f) : Integrable f μ := by
apply integrable_of_compact_ae μ (isCompact_closedBall (0 : EuclideanSpace ℝ ι) K) _ hf.aestronglyMeasurable hf.continuousOn
simpa only [Metric.mem_closedBall,dist_zero_right] using hμ
theorem smoothedGradient_bound (μ : Measure (EuclideanSpace ℝ ι)) [IsFiniteMeasure μ]
{a s K : ℝ} (ha : 0<a) (hs : a ≤ s) (hK : 0 ≤ K) (hμ : ∀ᵐ y ∂μ,‖y‖ ≤ K)
(x : EuclideanSpace ℝ ι) (i : ι) :
|smoothedGradient μ s x i| ≤ ((‖x‖+K)/a)*smoothedDensity μ s x := by
have hs0 := ha.trans_le hs
have hi : Integrable (fun y => normalGradient s (x-y) i) μ := integrable_of_normBound μ hμ ((normalGradient_continuous s i).comp (continuous_const.sub continuous_id))
calc
_ ≤ ∫y,|normalGradient s (x-y) i| ∂μ := by simpa only [smoothedGradient,Real.norm_eq_abs] using norm_integral_le_integral_norm (fun y => normalGradient s (x-y) i)
_ ≤ ∫y,((‖x‖+K)/a)*normalKernel s (x-y) ∂μ := by
apply integral_mono_ae hi.norm ((normalKernel_integrable_finite μ hs0 x).const_mul _)
filter_upwards [hμ] with y hy
unfold normalGradient
rw [Real.norm_eq_abs,abs_mul,abs_div,abs_neg,abs_of_pos hs0,abs_of_pos (normalKernel_pos _ _)]
apply mul_le_mul_of_nonneg_right _ (normalKernel_pos _ _).le
have hn0 : |(x-y) i| ≤ ‖x-y‖ := by simpa only [Real.norm_eq_abs] using PiLp.norm_apply_le (x-y) i
have hn : |(x-y) i| ≤ ‖x‖+K := hn0.trans ((norm_sub_le x y).trans (add_le_add le_rfl hy))
exact (div_le_div_of_nonneg_right hn hs0.le).trans (div_le_div_of_nonneg_left (by positivity) ha hs)
_ = _ := integral_const_mul _ _
end FiniteEntropySeparation
end
end OAI
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