| import OAI.Analysis.FreeEntropy.GaussianDecay |
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| namespace OAI |
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| universe u1 |
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| 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 |
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| end |
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| end OAI |
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