| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.Entropy |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| namespace KernelPotential |
| open Filter |
| open scoped Topology NNReal |
| namespace Coercivity |
| open _root_.OAI.KernelPotential.Geometry Sphere |
|
|
| lemma inverse_kernel_bound {d n q : ℕ} (μ : Measure (Ambient d)) [IsProbabilityMeasure μ] |
| {C : ℝ} (hC : 0 ≤ C) (hq : 0 < q) (hqn : q+1 ≤ n) |
| (hball : ∀ s r, 0 < r → μ (ball s r) ≤ ENNReal.ofReal (C*r^n)) (s : Ambient d) : |
| (∫⁻ u, (ENNReal.ofReal ‖u-s‖)⁻¹^q ∂μ) ≤ ENNReal.ofReal (1+C*2^(n+q+1)) := by |
| have ht : ∀ j : ℕ, μ {u | ‖u-s‖ ≤ (1/2:ℝ)^j} ≤ |
| ENNReal.ofReal (C*2^n)*ENNReal.ofReal ((1/2:ℝ)^((q+1)*j)) := by |
| intro j |
| calc |
| _ ≤ μ (ball s (2*(1/2:ℝ)^j)) := measure_mono (by |
| intro u hu |
| change dist u s < _ |
| rw [dist_eq_norm] |
| have hp : 0 < (1/2:ℝ)^j := by positivity |
| change ‖u-s‖ ≤ (1/2:ℝ)^j at hu |
| linarith) |
| _ ≤ ENNReal.ofReal (C*(2*(1/2:ℝ)^j)^n) := hball _ _ (by positivity) |
| _ ≤ _ := by |
| rw [← ENNReal.ofReal_mul (by positivity)] |
| apply ENNReal.ofReal_le_ofReal |
| rw [mul_pow, ← mul_assoc, Nat.mul_comm (q+1) j, pow_mul] |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| exact pow_le_pow_of_le_one (by positivity : 0 ≤ (1/2:ℝ)^j) (pow_le_one₀ (by norm_num) (by norm_num)) hqn |
| have h := inverseMoment_extended_of_tails μ (fun u => ‖u-s‖) |
| (measurable_id.sub_const s).norm (fun u => norm_nonneg _) q hq |
| (ENNReal.ofReal (C*2^n)) ENNReal.ofReal_ne_top ht |
| apply h.trans_eq |
| rw [measure_univ, ← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_one, |
| ← ENNReal.ofReal_add (by norm_num) (by positivity)] |
| congr 1 |
| rw [mul_assoc, ← pow_add] |
| congr 3 |
|
|
| lemma integral_square_le {α : Type*} [MeasurableSpace α] (μ : Measure α) |
| [IsProbabilityMeasure μ] (f : α → ℝ≥0∞) (hf : Measurable f) : |
| (∫⁻ x,f x ∂μ)^2 ≤ ∫⁻ x,(f x)^2 ∂μ := by |
| have h := ENNReal.lintegral_mul_le_Lp_mul_Lq μ |
| (p := (2:ℝ)) (q := (2:ℝ)) (Real.holderConjugate_iff.mpr ⟨by norm_num,by norm_num⟩) |
| hf.aemeasurable (measurable_const (a := (1:ℝ≥0∞))).aemeasurable |
| simp only [Pi.mul_apply,mul_one,ENNReal.one_rpow,lintegral_one,measure_univ] at h |
| have H := pow_le_pow_left₀ zero_le h 2 |
| simpa only [← ENNReal.rpow_natCast, Nat.cast_ofNat, ← ENNReal.rpow_mul, |
| show (1/(2:ℝ))*(2:ℝ)=1 by norm_num,ENNReal.rpow_one] using H |
|
|
| lemma measurable_kernel {d q : ℕ} : |
| Measurable (fun x : Ambient d × Ambient d => (ENNReal.ofReal ‖x.1-x.2‖)⁻¹^q) := |
| ((measurable_fst.sub measurable_snd).norm.ennreal_ofReal.inv).pow_const q |
|
|
| lemma measurable_J {d : ℕ} (ν : Measure (Ambient d)) [SFinite ν] : Measurable (J ν) := by |
| unfold J |
| apply Measurable.lintegral_prod_right |
| exact ((measurable_snd.sub measurable_fst).norm.ennreal_ofReal.inv).pow_const _ |
|
|
| lemma J_square_integral {d : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] : |
| (∫⁻ s,(J ν s)^2 ∂sphereProbability d) ≤ |
| ENNReal.ofReal (1+(9*3^d:ℝ)*2^((d-1)+4*(d/16)+1)) := by |
| have hk : 0 < d/16 := Nat.div_pos (by omega) (by norm_num) |
| calc |
| _ ≤ ∫⁻ s,∫⁻ u, ((ENNReal.ofReal ‖u-s‖)⁻¹^(2*(d/16)))^2 ∂ν ∂sphereProbability d := by |
| apply lintegral_mono |
| intro s |
| exact integral_square_le ν _ (((measurable_id.sub_const s).norm.ennreal_ofReal.inv).pow_const _) |
| _ = ∫⁻ u,∫⁻ s, (ENNReal.ofReal ‖u-s‖)⁻¹^(4*(d/16)) ∂sphereProbability d ∂ν := by |
| simp_rw [← pow_mul, show 2*(d/16)*2=4*(d/16) by omega] |
| exact lintegral_lintegral_swap (measurable_kernel.comp measurable_swap).aemeasurable |
| _ ≤ ∫⁻ u, ENNReal.ofReal (1+(9*3^d:ℝ)*2^((d-1)+4*(d/16)+1)) ∂ν := by |
| apply lintegral_mono |
| intro u |
| simp_rw [norm_sub_rev u] |
| exact inverse_kernel_bound (sphereProbability d) (by positivity) (by omega) |
| (by omega) (fun s r hr => cap_bound (by omega) s hr) u |
| _ = _ := by simp |
|
|
| lemma bounded_density_representation {α : Type*} [MeasurableSpace α] |
| (μ ν : Measure α) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| {H : ℝ} (hH : 0 < H) (hν : ν ≤ ENNReal.ofReal H • μ) : |
| ∃ f : α → ℝ, Measurable f ∧ (∀ x,0 ≤ f x) ∧ (∀ x,f x ≤ H) ∧ |
| ν = μ.withDensity (fun x => ENNReal.ofReal (f x)) := by |
| have hac : ν ≪ μ := Measure.absolutelyContinuous_of_le_smul hν |
| have hr : ν.rnDeriv μ ≤ᵐ[μ] fun _ => ENNReal.ofReal H := by |
| apply ae_le_of_forall_setLIntegral_le_of_sigmaFinite (Measure.measurable_rnDeriv ν μ) |
| intro s hs _ |
| calc |
| _ ≤ ν s := Measure.setLIntegral_rnDeriv_le s |
| _ ≤ (ENNReal.ofReal H • μ) s := hν s |
| _ = _ := by simp [Measure.smul_apply] |
| let f : α → ℝ := fun x => min H (ν.rnDeriv μ x).toReal |
| refine ⟨f,measurable_const.min (Measure.measurable_rnDeriv ν μ).ennreal_toReal, |
| fun x => le_min hH.le ENNReal.toReal_nonneg,fun x => min_le_left _ _,?_⟩ |
| rw [← Measure.withDensity_rnDeriv_eq ν μ hac] |
| apply withDensity_congr_ae |
| filter_upwards [hr] with x hx |
| dsimp [f] |
| have hxt : ν.rnDeriv μ x ≠ ∞ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hx |
| have hxH : (ν.rnDeriv μ x).toReal ≤ H := by |
| exact (ENNReal.toReal_le_toReal hxt ENNReal.ofReal_ne_top).mpr hx |>.trans_eq (ENNReal.toReal_ofReal hH.le) |
| rw [min_eq_right hxH, ENNReal.ofReal_toReal hxt] |
|
|
| lemma bounded_J {d : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] {H : ℝ} (hH : 0 < H) |
| (hν : ν ≤ ENNReal.ofReal H • sphereProbability d) (s : Ambient d) : |
| J ν s ≤ ENNReal.ofReal (H*(1+(9*3^d:ℝ)*2^((d-1)+2*(d/16)+1))) := by |
| have hk : 0 < d/16 := Nat.div_pos (by omega) (by norm_num) |
| calc |
| J ν s ≤ ∫⁻ u,(ENNReal.ofReal ‖u-s‖)⁻¹^(2*(d/16)) ∂(ENNReal.ofReal H • sphereProbability d) := |
| lintegral_mono' hν (le_refl _) |
| _ = ENNReal.ofReal H * J (sphereProbability d) s := by rw [J,lintegral_smul_measure,smul_eq_mul] |
| _ ≤ ENNReal.ofReal H * ENNReal.ofReal (1+(9*3^d:ℝ)*2^((d-1)+2*(d/16)+1)) := by |
| gcongr |
| exact inverse_kernel_bound (sphereProbability d) (by positivity) (by omega) |
| (by omega) (fun s r hr => cap_bound (by omega) s hr) s |
| _ = _ := (ENNReal.ofReal_mul hH.le).symm |
| |
| lemma lower_J_on_sphere {d : ℕ} (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] |
| (hν : ∀ᵐ u ∂ν, ‖u‖=1) (s : Ambient d) (hs : ‖s‖=1) : |
| ENNReal.ofReal ((1/2:ℝ)^(2*(d/16))) ≤ J ν s := by |
| calc |
| _ = ∫⁻ u, ENNReal.ofReal ((1/2:ℝ)^(2*(d/16))) ∂ν := by simp |
| _ ≤ J ν s := by |
| apply lintegral_mono_ae |
| filter_upwards [hν] with u hu |
| have hn : ‖u-s‖ ≤ 2 := by calc |
| ‖u-s‖ ≤ ‖u‖+‖s‖ := norm_sub_le _ _ |
| _ = 2 := by rw [hu,hs]; norm_num |
| rw [ENNReal.ofReal_pow (by norm_num),show ENNReal.ofReal (1/2:ℝ)=(ENNReal.ofReal (2:ℝ))⁻¹ by rw [one_div,ENNReal.ofReal_inv_of_pos (by norm_num)]] |
| exact pow_le_pow_left₀ zero_le (ENNReal.inv_le_inv.mpr (ENNReal.ofReal_le_ofReal hn)) _ |
| |
| lemma integrable_log_of_bounds {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ) (hf : Measurable f) |
| {a b : ℝ} (ha : 0 < a) |
| (hbound : ∀ᵐ x ∂μ, a ≤ f x ∧ f x ≤ b) : Integrable (fun x => Real.log (f x)) μ := by |
| apply Integrable.mono' (integrable_const (|Real.log a|+|Real.log b|)) hf.log.aestronglyMeasurable |
| filter_upwards [hbound] with x hx |
| have hl := Real.log_le_log ha hx.1 |
| have hu := Real.log_le_log (ha.trans_le hx.1) hx.2 |
| simp only [Real.norm_eq_abs] |
| apply abs_le.mpr |
| constructor |
| · have hh := neg_abs_le (Real.log a) |
| have hb0 := abs_nonneg (Real.log b) |
| linarith |
| · have hh := le_abs_self (Real.log b) |
| have ha0 := abs_nonneg (Real.log a) |
| linarith |
|
|
| lemma J_real_bounds {d : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] {H : ℝ} (hH : 0 < H) |
| (hν : ν ≤ ENNReal.ofReal H • sphereProbability d) (s : Ambient d) (hs : ‖s‖=1) : |
| (1/2:ℝ)^(2*(d/16)) ≤ (J ν s).toReal ∧ |
| (J ν s).toReal ≤ H*(1+(9*3^d:ℝ)*2^((d-1)+2*(d/16)+1)) := by |
| have hu := bounded_J hd ν hH hν s |
| have ht := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hu |
| have hnorm : ∀ᵐ u ∂ν, ‖u‖=1 := (Measure.absolutelyContinuous_of_le_smul hν).ae_le (ae_unit_norm (by omega)) |
| have hl := lower_J_on_sphere ν hnorm s hs |
| constructor |
| · convert ENNReal.toReal_mono ht hl using 1 |
| rw [ENNReal.toReal_ofReal (by positivity)] |
| · convert ENNReal.toReal_mono ENNReal.ofReal_ne_top hu using 1 |
| rw [ENNReal.toReal_ofReal (by positivity)] |
|
|
| lemma integrable_log_J {d : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] (hν : BoundedDensity ν) (μ : Measure (Ambient d)) |
| [IsFiniteMeasure μ] (hμ : ∀ᵐ s ∂μ, ‖s‖=1) : |
| Integrable (fun s => Real.log (J ν s).toReal) μ := by |
| obtain ⟨H,hH,hνH⟩ := hν |
| apply integrable_log_of_bounds μ _ (measurable_J ν).ennreal_toReal (by positivity : 0 < (1/2:ℝ)^(2*(d/16))) |
| filter_upwards [hμ] with s hs |
| exact J_real_bounds hd ν hH hνH s hs |
|
|
| open EntropySupport |
|
|
| lemma density_integral_one {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) [IsProbabilityMeasure μ] (f : α → ℝ) (hf : Measurable f) |
| (hf0 : ∀ x,0 ≤ f x) {H : ℝ} (hfH : ∀ x,f x ≤ H) |
| [IsProbabilityMeasure (μ.withDensity (fun x => ENNReal.ofReal (f x)))] : |
| ∫ x,f x ∂μ = 1 := by |
| have hif := integrable_bounded_measurable μ f hf (fun x => by rw [abs_of_nonneg (hf0 x)]; exact hfH x) |
| have h := measure_univ (μ := μ.withDensity (fun x => ENNReal.ofReal (f x))) |
| rw [withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ, |
| ← ofReal_integral_eq_lintegral_ofReal hif (ae_of_all _ hf0)] at h |
| exact ENNReal.ofReal_eq_one.mp h |
|
|
| lemma density_entropy_nonneg {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) [IsProbabilityMeasure μ] (f : α → ℝ) (hf : Measurable f) |
| (hf0 : ∀ x,0 ≤ f x) {H : ℝ} (hfH : ∀ x,f x ≤ H) |
| [IsProbabilityMeasure (μ.withDensity (fun x => ENNReal.ofReal (f x)))] : |
| 0 ≤ ∫ x,xlog (f x) ∂μ := by |
| have hif := integrable_bounded_measurable μ f hf (fun x => by rw [abs_of_nonneg (hf0 x)]; exact hfH x) |
| have hix := integrable_xlog_bounded μ f hf hf0 hfH |
| have h := integral_mono (hif.sub (integrable_const 1)) hix |
| (fun x => Real.self_sub_one_le_mul_log (hf0 x)) |
| change (∫ x,f x-1 ∂μ) ≤ _ at h |
| rw [integral_sub hif (integrable_const 1),density_integral_one μ f hf hf0 hfH] at h |
| simpa using h |
|
|
| lemma entropy_young {f j M : ℝ} (hf : 0 ≤ f) (hj : 0 < j) (hM : 0 < M) : |
| 2*f*Real.log j ≤ xlog f+f*(Real.log M-1)+j^2/M := by |
| have h := xlog_lower_tangent hf (div_pos (sq_pos_of_pos hj) hM) |
| have hl : Real.log (j^2/M) = 2*Real.log j-Real.log M := by |
| rw [Real.log_div (pow_ne_zero _ hj.ne') hM.ne',Real.log_pow]; norm_num |
| dsimp [xlog] at h ⊢ |
| rw [hl] at h |
| nlinarith |
|
|
| noncomputable def sphereMoment (d : ℕ) : ℝ := 1+(9*3^d:ℝ)*2^((d-1)+4*(d/16)+1) |
| lemma sphereMoment_pos (d : ℕ) : 0 < sphereMoment d := by dsimp [sphereMoment]; positivity |
|
|
| lemma square_integrable_bound {d : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] (hν : BoundedDensity ν) : |
| Integrable (fun s => (J ν s).toReal^2) (sphereProbability d) ∧ |
| (∫ s,(J ν s).toReal^2 ∂sphereProbability d) ≤ sphereMoment d := by |
| have hb := J_square_integral hd ν |
| have ht : (∫⁻ s,(J ν s)^2 ∂sphereProbability d) ≠ ∞ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hb |
| have hm := (measurable_J ν).pow_const 2 |
| obtain ⟨H,hH,hνH⟩ := hν |
| have hp : ∀ s,(J ν s)^2 < ∞ := fun s => |
| ENNReal.pow_lt_top (lt_top_iff_ne_top.mpr (ne_top_of_le_ne_top ENNReal.ofReal_ne_top (bounded_J hd ν hH hνH s))) |
| constructor |
| · simpa only [ENNReal.toReal_pow] using integrable_toReal_of_lintegral_ne_top hm.aemeasurable ht |
| · have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top hb |
| change _ ≤ (ENNReal.ofReal (sphereMoment d)).toReal at hh |
| rw [ENNReal.toReal_ofReal (sphereMoment_pos d).le] at hh |
| simpa only [← ENNReal.toReal_pow, integral_toReal hm.aemeasurable (ae_of_all _ hp)] using hh |
|
|
| lemma Phi_density_formula {d : ℕ} (hd : 16 ≤ d) (f : Ambient d → ℝ) |
| (hf : Measurable f) (hf0 : ∀ x,0 ≤ f x) {H : ℝ} (hfH : ∀ x,f x ≤ H) |
| (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] (hν : BoundedDensity ν) |
| (heq : ν = (sphereProbability d).withDensity (fun x => ENNReal.ofReal (f x))) : |
| Phi ν = ∫ s, xlog (f s)-f s*Real.log (J ν s).toReal ∂sphereProbability d := by |
| have : IsProbabilityMeasure ((sphereProbability d).withDensity (fun x => ENNReal.ofReal (f x))) := heq ▸ inferInstance |
| have hix := integrable_xlog_bounded (sphereProbability d) f hf hf0 hfH |
| have hil := integrable_log_J hd ν hν (sphereProbability d) (ae_unit_norm (by omega)) |
| have hib := hil.bdd_mul hf.aestronglyMeasurable (ae_of_all _ (fun x => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (hf0 x)]; exact hfH x)) |
| have hD : (InformationTheory.klDiv ν (sphereProbability d)).toReal = ∫ s,xlog (f s) ∂sphereProbability d := by |
| rw [heq,klDiv_withDensity_bounded _ f hf hf0 hfH, |
| ENNReal.toReal_ofReal (density_entropy_nonneg _ f hf hf0 hfH)] |
| have hB : B ν = ∫ s,f s*Real.log (J ν s).toReal ∂sphereProbability d := by |
| unfold B |
| nth_rw 1 [heq] |
| rw [integral_withDensity_eq_integral_toReal_smul hf.ennreal_ofReal (ae_of_all _ (fun x => ENNReal.ofReal_lt_top))] |
| simp only [ENNReal.toReal_ofReal (hf0 _),smul_eq_mul] |
| rw [Phi,hD,hB,integral_sub hix hib] |
|
|
| lemma B_entropy_bound {d : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] (hν : BoundedDensity ν) : |
| 2*B ν ≤ (InformationTheory.klDiv ν (sphereProbability d)).toReal+Real.log (sphereMoment d) := by |
| obtain ⟨H,hH,hνH⟩ := hν |
| obtain ⟨f,hf,hf0,hfH,heq⟩ := bounded_density_representation (sphereProbability d) ν hH hνH |
| have : IsProbabilityMeasure ((sphereProbability d).withDensity (fun x => ENNReal.ofReal (f x))) := heq ▸ inferInstance |
| have hif := integrable_bounded_measurable (sphereProbability d) f hf |
| (fun x => by rw [abs_of_nonneg (hf0 x)]; exact hfH x) |
| have hix := integrable_xlog_bounded (sphereProbability d) f hf hf0 hfH |
| have hil := integrable_log_J hd ν ⟨H,hH,hνH⟩ (sphereProbability d) (ae_unit_norm (by omega)) |
| have hib := hil.bdd_mul hf.aestronglyMeasurable (ae_of_all _ (fun x => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (hf0 x)]; exact hfH x)) |
| obtain ⟨hij,hijbound⟩ := square_integrable_bound hd ν ⟨H,hH,hνH⟩ |
| have hpoint : ∀ᵐ s ∂sphereProbability d, |
| 2*(f s*Real.log (J ν s).toReal) ≤ |
| xlog (f s)+f s*(Real.log (sphereMoment d)-1)+(J ν s).toReal^2/sphereMoment d := by |
| filter_upwards [ae_unit_norm (by omega : 0 < d)] with s hs |
| have hj := (J_real_bounds hd ν hH hνH s hs).1 |
| have hjpos : 0 < (J ν s).toReal := lt_of_lt_of_le (by positivity) hj |
| simpa only [mul_assoc] using entropy_young (hf0 s) hjpos (sphereMoment_pos d) |
| have h := integral_mono_ae (hib.const_mul 2) |
| ((hix.add (hif.mul_const _)).add (hij.div_const _)) hpoint |
| change (∫ s,2*(f s*Real.log (J ν s).toReal) ∂sphereProbability d) ≤ |
| (∫ s,xlog (f s)+f s*(Real.log (sphereMoment d)-1)+(J ν s).toReal^2/sphereMoment d ∂sphereProbability d) at h |
| rw [integral_const_mul,integral_add (f := fun s => xlog (f s)+f s*(Real.log (sphereMoment d)-1)) |
| (g := fun s => (J ν s).toReal^2/sphereMoment d) (hix.add (hif.mul_const _)) (hij.div_const _), |
| integral_add hix (hif.mul_const (Real.log (sphereMoment d)-1)),integral_mul_const,integral_div, |
| density_integral_one _ f hf hf0 hfH,one_mul] at h |
| have hrat : (∫ s,(J ν s).toReal^2 ∂sphereProbability d)/sphereMoment d ≤ 1 := |
| (div_le_one (sphereMoment_pos d)).mpr hijbound |
| have hD : (InformationTheory.klDiv ν (sphereProbability d)).toReal = ∫ s,xlog (f s) ∂sphereProbability d := by |
| rw [heq,klDiv_withDensity_bounded _ f hf hf0 hfH, |
| ENNReal.toReal_ofReal (density_entropy_nonneg _ f hf hf0 hfH)] |
| have hB : B ν = ∫ s,f s*Real.log (J ν s).toReal ∂sphereProbability d := by |
| unfold B |
| nth_rw 1 [heq] |
| rw [integral_withDensity_eq_integral_toReal_smul hf.ennreal_ofReal (ae_of_all _ (fun x => ENNReal.ofReal_lt_top))] |
| simp only [ENNReal.toReal_ofReal (hf0 _),smul_eq_mul] |
| rw [hD,hB] |
| linarith |
|
|
| lemma log_sphereMoment_le {d : ℕ} (hd : 16 ≤ d) : Real.log (sphereMoment d) ≤ 20*d := by |
| let e := (d-1)+4*(d/16)+1 |
| have he : e ≤ 2*d := by dsimp [e]; omega |
| have heR : (e:ℝ) ≤ 2*d := by exact_mod_cast he |
| have hdR : (16:ℝ) ≤ d := by exact_mod_cast hd |
| have h3 : (1:ℝ) ≤ 3^d := one_le_pow₀ (by norm_num) |
| have h2 : (1:ℝ) ≤ 2^e := one_le_pow₀ (by norm_num) |
| have hQ : (1:ℝ) ≤ 9*3^d*2^e := by nlinarith |
| have hM : sphereMoment d ≤ 18*3^d*2^e := by dsimp [sphereMoment,e] at *; nlinarith |
| have hl := Real.log_le_log (sphereMoment_pos d) hM |
| have heq : Real.log (18*3^d*2^e:ℝ) = Real.log 18+(d:ℝ)*Real.log 3+(e:ℝ)*Real.log 2 := by |
| rw [Real.log_mul (by positivity) (by positivity), Real.log_mul (by norm_num) (by positivity),Real.log_pow,Real.log_pow] |
| rw [heq] at hl |
| have hl18 : Real.log (18:ℝ) ≤ 17 := by linarith [Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ) < 18)] |
| have hl3 : Real.log (3:ℝ) ≤ 2 := by linarith [Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ) < 3)] |
| have hl2 : Real.log (2:ℝ) ≤ 1 := by linarith [Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ) < 2)] |
| have he0 : (0:ℝ) ≤ e := Nat.cast_nonneg e |
| nlinarith [mul_le_mul_of_nonneg_left hl3 (Nat.cast_nonneg d),mul_le_mul_of_nonneg_left hl2 he0] |
|
|
| lemma coercivity {d : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] (hν : BoundedDensity ν) : |
| (1/2:ℝ)*(InformationTheory.klDiv ν (sphereProbability d)).toReal-10*d ≤ Phi ν := by |
| have h := B_entropy_bound hd ν hν |
| have hl := log_sphereMoment_le hd |
| unfold Phi |
| linarith |
|
|
| lemma Phi_uniform_le {d : ℕ} (hd : 16 ≤ d) : |
| Phi (sphereProbability d) ≤ (2*(d/16):ℕ)*Real.log 2 := by |
| have hσ : BoundedDensity (sphereProbability d) := ⟨1,by norm_num,by simp⟩ |
| have hil := integrable_log_J hd (sphereProbability d) hσ (sphereProbability d) (ae_unit_norm (by omega)) |
| have hl : ∀ᵐ s ∂sphereProbability d, |
| -((2*(d/16):ℕ):ℝ)*Real.log 2 ≤ Real.log (J (sphereProbability d) s).toReal := by |
| filter_upwards [ae_unit_norm (by omega : 0 < d)] with s hs |
| have hlow := (J_real_bounds hd (sphereProbability d) (by norm_num : (0:ℝ) < 1) (by simp) s hs).1 |
| have h := Real.log_le_log (by positivity : 0 < (1/2:ℝ)^(2*(d/16))) hlow |
| rw [Real.log_pow,Real.log_div one_ne_zero (by norm_num),Real.log_one,zero_sub] at h |
| simpa only [mul_neg,neg_mul] using h |
| have h := integral_mono_ae (integrable_const (-((2*(d/16):ℕ):ℝ)*Real.log 2)) hil hl |
| simp only [integral_const,measureReal_def,measure_univ,ENNReal.toReal_one,one_smul] at h |
| rw [Phi,InformationTheory.klDiv_self,ENNReal.toReal_zero,zero_sub] |
| change -B (sphereProbability d) ≤ _ |
| change _ ≤ B (sphereProbability d) at h |
| linarith |
|
|
| end Coercivity |
| end KernelPotential |
|
|
| end OAI |
|
|
