| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.Compensation |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace KernelPotential.Noise |
| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| open Projection.Support Projection Reweight EntropySupport DriftTools Sphere |
|
|
| lemma mixedCube_integrand_measurable {k d : ℕ} (η : ℝ) : |
| Measurable (fun t : ((Fin k → Fin d → ℝ) × (Fin k → Fin d → ℝ)) × Ambient d => |
| boxKernel η (projectVector t.1.1 t.2) 0 * boxKernel η (projectVector t.1.2 t.2) 0) := by |
| have hA := measurable_boxKernel (k := k) η |
| have hP := continuous_projectVector (k := k) (d := d) |
| fun_prop |
|
|
| lemma mixedCube_measurable {k d : ℕ} (μ : Measure (Ambient d)) [SFinite μ] |
| (η : ℝ) : Measurable (fun t : (Fin k → Fin d → ℝ) × (Fin k → Fin d → ℝ) => |
| mixedCubeAverage μ η t.1 t.2) := |
| Measurable.lintegral_prod_right (mixedCube_integrand_measurable η) |
|
|
| lemma mixedCube_le {k d : ℕ} (μ : Measure (Ambient d)) (η : ℝ) |
| (X Z : Fin k → Fin d → ℝ) : |
| mixedCubeAverage μ η X Z ≤ boxScale k η^2*μ univ := by |
| calc |
| _ ≤ ∫⁻ _u,boxScale k η*boxScale k η ∂μ := lintegral_mono |
| (fun _ => mul_le_mul' (boxKernel_le η _ _) (boxKernel_le η _ _)) |
| _ = _ := by simp [pow_two] |
|
|
| lemma centered_rho_mass {d : ℕ} (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] |
| (hν : ∀ᵐ s ∂ν, ‖s‖=1) (s : Ambient d) : |
| centeredRho ν s univ ≤ ENNReal.ofReal ((2:ℝ)^(2*(d/16))) := by |
| rw [centeredRho,Measure.map_apply (by fun_prop) MeasurableSet.univ] |
| simp only [preimage_univ] |
| exact (rho_le ν hν univ).trans_eq (by simp) |
|
|
| noncomputable def momentConstant (d L : ℕ) : ℝ := |
| (9^(d/16-1)*3^(d-1)*2^((d-1)+4*(d/16)+2):ℝ)*((L:ℝ)+2) |
|
|
| lemma momentConstant_pos (d L : ℕ) : 0 < momentConstant d L := by |
| unfold momentConstant; positivity |
|
|
| lemma log_momentConstant (d L : ℕ) (hd : 16 ≤ d) : |
| Real.log (4*(1+2^(2*(d/16))*momentConstant d L)) ≤ |
| 10*(d:ℝ)+Real.log ((L:ℝ)+2) := by |
| have hg : 1 ≤ momentConstant d L := by |
| unfold momentConstant |
| have h9 : (1:ℝ) ≤ 9^(d/16-1) := one_le_pow₀ (by norm_num) |
| have h3 : (1:ℝ) ≤ 3^(d-1) := one_le_pow₀ (by norm_num) |
| have h2 : (1:ℝ) ≤ 2^((d-1)+4*(d/16)+2) := one_le_pow₀ (by norm_num) |
| have hL : (1:ℝ) ≤ (L:ℝ)+2 := by have := Nat.cast_nonneg (α := ℝ) L; linarith |
| exact one_le_mul_of_one_le_of_one_le (one_le_mul_of_one_le_of_one_le (one_le_mul_of_one_le_of_one_le h9 h3) h2) hL |
| have hc : (1:ℝ) ≤ 2^(2*(d/16)) := one_le_pow₀ (by norm_num) |
| have hbig : 4*(1+2^(2*(d/16))*momentConstant d L) ≤ |
| 8*2^(2*(d/16))*momentConstant d L := by nlinarith [one_le_mul_of_one_le_of_one_le hc hg] |
| have hlog := Real.log_le_log (by have := momentConstant_pos d L; positivity) hbig |
| have he : Real.log (8*2^(2*(d/16))*momentConstant d L) = |
| Real.log 8+(2*(d/16):ℕ)*Real.log 2+(d/16-1:ℕ)*Real.log 9+ |
| (d-1:ℕ)*Real.log 3+((d-1)+4*(d/16)+2:ℕ)*Real.log 2+ |
| Real.log ((L:ℝ)+2) := by |
| rw [momentConstant] |
| repeat' rw [Real.log_mul (by positivity) (by positivity)] |
| simp only [Real.log_pow] |
| ring |
| rw [he] at hlog |
| have hl8 : Real.log 8 ≤ 7 := by linarith [Real.log_le_sub_one_of_pos (show (0:ℝ)<8 by norm_num)] |
| have hl9 : Real.log 9 ≤ 8 := by linarith [Real.log_le_sub_one_of_pos (show (0:ℝ)<9 by norm_num)] |
| have hl3 : Real.log 3 ≤ 2 := by linarith [Real.log_le_sub_one_of_pos (show (0:ℝ)<3 by norm_num)] |
| have hl2 : Real.log 2 ≤ 1 := by linarith [Real.log_le_sub_one_of_pos (show (0:ℝ)<2 by norm_num)] |
| have hk : (d/16:ℕ) ≤ (d:ℝ)/16 := by |
| have hh : (16*((d/16:ℕ):ℝ)) ≤ (d:ℝ) := by exact_mod_cast Nat.mul_div_le d 16 |
| linarith |
| have hk1 : ((d/16-1:ℕ):ℝ) ≤ (d/16:ℕ) := by exact_mod_cast Nat.sub_le (d/16) 1 |
| have hd1 : ((d-1:ℕ):ℝ) ≤ (d:ℝ) := by exact_mod_cast Nat.sub_le d 1 |
| have hdR : (16:ℝ) ≤ d := by exact_mod_cast hd |
| push_cast at hlog |
| nlinarith [mul_le_mul_of_nonneg_left hl9 (show (0:ℝ) ≤ (d/16-1:ℕ) by positivity), |
| mul_le_mul_of_nonneg_left hl3 (show (0:ℝ) ≤ (d-1:ℕ) by positivity), |
| mul_le_mul_of_nonneg_left hl2 (show (0:ℝ) ≤ 2*(d/16:ℕ) by positivity), |
| mul_le_mul_of_nonneg_left hl2 (show (0:ℝ) ≤ (d-1:ℕ)+4*(d/16:ℕ)+2 by positivity)] |
| |
| lemma weighted_centered_projection {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) |
| (L : ℕ) (hL : H*(9*3^d:ℝ) ≤ 2^L) {η : ℝ} (hη : 0 < η) |
| (u : (Fin (d/16) → Fin d → ℝ) → ℝ) (hu : Measurable u) |
| (hu0 : ∀ X,0 ≤ u X) (hu1 : ∀ X,u X ≤ 1) : |
| (∫ X,u X*(∫ Z,Real.log (1+2^(2*(d/16))* |
| (mixedCubeAverage (centeredRho ν s) η X Z).toReal) ∂gaussianRows (d/16) d) |
| ∂gaussianRows (d/16) d) ≤ |
| -(∫ X,u X ∂gaussianRows (d/16) d)*Real.log (∫ X,u X ∂gaussianRows (d/16) d)/(d/16:ℕ)+ |
| (∫ X,u X ∂gaussianRows (d/16) d)*(10*(d:ℝ)+Real.log ((L:ℝ)+2)) := by |
| have hnorm := (Measure.absolutelyContinuous_of_le_smul hν).ae_le (ae_unit_norm (by omega : 0 < d)) |
| have := centered_finite ν hnorm s |
| have : IsProbabilityMeasure (gaussianRows (d/16) d) := by unfold gaussianRows; infer_instance |
| let B : ℝ := (boxScale (d/16) η).toReal^2*2^(2*(d/16)) |
| have hB : 0 ≤ B := by dsimp [B]; positivity |
| have hb (X Z : Fin (d/16) → Fin d → ℝ) : |
| mixedCubeAverage (centeredRho ν s) η X Z ≤ ENNReal.ofReal B := by |
| calc |
| _ ≤ boxScale (d/16) η^2*centeredRho ν s univ := mixedCube_le _ _ _ _ |
| _ ≤ boxScale (d/16) η^2*ENNReal.ofReal ((2:ℝ)^(2*(d/16))) := |
| mul_le_mul_right (centered_rho_mass ν hnorm s) _ |
| _ = _ := by |
| dsimp [B] |
| rw [ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_pow ENNReal.toReal_nonneg, |
| ENNReal.ofReal_toReal (boxScale_ne_top _ hη)] |
| have h := fresh_weighted_log_moment (gaussianRows (d/16) d) (gaussianRows (d/16) d) |
| u (fun t => mixedCubeAverage (centeredRho ν s) η t.1 t.2) hu (mixedCube_measurable _ _) |
| hu0 hu1 hB (momentConstant_pos d L).le (by positivity : (0:ℝ) ≤ 2^(2*(d/16))) hb |
| (d/16) (by omega) (centered_mixed_moment hd ν hH hν s hs L hL η hη) |
| exact h.trans (add_le_add_right (mul_le_mul_of_nonneg_left (log_momentConstant d L hd) |
| (integral_nonneg hu0)) _) |
| |
| end KernelPotential.Noise |
| |
| end OAI |
| |
