| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.ScalarHierarchyAverageAlgebra |
| import OAI.Probability.SKBarriers.Hierarchy.HierarchyScoreGeneral |
| import OAI.Probability.SKBarriers.Scalar.ScalarLevelRepresentation |
|
|
| namespace OAI |
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| section |
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| noncomputable section |
| open scoped BigOperators NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
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|
| theorem scalarPath_square_bound_convex (n : ℕ) (m v : Fin n → ℝ) |
| (hm : ∀ i, m i∈Icc (0:ℝ) 1) (hmono : Monotone m) |
| {f : ℝ → ℝ} (hf : BoundedDerivs f) (hc : ScalarSpinConvex f) : |
| (∫ z, (coordinateLinear n v z)^2 ∂hierarchyPathLaw n m |
| (fun z => f (coordinateLinear n v z)) 0) ≤ |
| 2*(∑ i, (v i)^2)+4*(∑ i, (v i)^2)^2 := by |
| have H := hierarchy_square_bound n m hm hmono (fun z => f (coordinateLinear n v z)) |
| (hf.compCLM (coordinateLinear n v)) (fun i => |v i|) (fun i => abs_nonneg _) |
| (fun z i => by |
| rw [(scalar_comp_directional hf (coordinateLinear n v) z (coordinateAxis n i)).1, |
| coordinateLinear_coordinateAxis,abs_mul] |
| exact (mul_le_mul_of_nonneg_right (hc.bounds _).1 (abs_nonneg _)).trans_eq (one_mul _)) 0 v |
| have he : (∑ i, |v i| * |v i|) = ∑ i, (v i)^2 := |
| Finset.sum_congr rfl (fun i _ => by rw [← sq_abs]; ring) |
| rw [he] at H |
| nlinarith |
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| theorem scalarHierarchyAverage_compact_lower (n : ℕ) (m v : Fin n → ℝ) |
| (hm : ∀ i, m i∈Icc (0:ℝ) 1) (hmono : Monotone m) |
| {f g : ℝ → ℝ} (hf : BoundedDerivs f) (hc : ScalarSpinConvex f) |
| (hg : BoundedScalar g) {T c : ℝ} (hT : 0≤T) (hV : ∑ i, (v i)^2 ≤ T) |
| (hc0 : 0≤c) (hg0 : ∀ x,0≤g x) |
| (hgc : ∀ x, |x|≤1+2*T+4*T^2 → c≤g x) : |
| c/2≤ scalarHierarchyAverage n m v f g 0 := by |
| let μ := hierarchyPathLaw n m (fun z => f (coordinateLinear n v z)) 0 |
| let L := coordinateLinear n v |
| let M := 2*T+4*T^2 |
| let R := 1+M |
| have hM : 0≤M := by dsimp [M]; positivity |
| have hR : 0<R := by dsimp [R]; positivity |
| have hR2 : 2*M≤R^2 := by dsimp only [R]; nlinarith [sq_nonneg M] |
| have hRdef : 1+2*T+4*T^2=R := by dsimp [R,M]; ring |
| have he (x : ℝ) : c*(1-x^2/R^2)≤g x := by |
| by_cases hx : |x|≤R |
| · have H : 1-x^2/R^2≤1 := sub_le_self _ (div_nonneg (sq_nonneg _) (sq_nonneg _)) |
| exact (mul_le_of_le_one_right hc0 H).trans (hgc x (hRdef ▸ hx)) |
| · have hsq : R^2≤x^2 := by |
| simpa only [sq_abs] using (sq_le_sq₀ hR.le (abs_nonneg x)).mpr (le_of_not_ge hx) |
| have hdiv : 1≤x^2/R^2 := (one_le_div (sq_pos_of_pos hR)).mpr hsq |
| exact (mul_nonpos_of_nonneg_of_nonpos hc0 (by linarith)).trans (hg0 x) |
| have hF : BoundedDerivs (fun z => f (L z)) := hf.compCLM L |
| let := hierarchyPathLaw_probability n m _ hF 0 |
| have hi : Integrable (fun z => (L z)^2) μ := by |
| rw [show μ=(fiberGaussian n 0).tilted (hierarchyPathLogDensity n m (fun z => f (L z))) from |
| hierarchyPathLaw_eq_tilted n m _ hF 0] |
| exact ((HasExpGrowth.linear L).pow 2).integrable_tilted_fiberGaussian n _ |
| (hierarchyPathLogDensity_regular n m hF) (L.continuous.pow 2) 0 |
| have hig : Integrable (fun z => g (L z)) μ := by |
| obtain ⟨B,hB,hb⟩ := hg.bounded |
| exact hierarchyPathLaw_integrable n m _ _ hF (hg.continuous.comp L.continuous) |
| (C:=B) (fun z => by simpa only [Real.norm_eq_abs,Function.comp_apply] using hb (L z)) 0 |
| have H := integral_mono (((integrable_const (1:ℝ)).sub (hi.div_const (R^2))).const_mul c) hig |
| (fun z => he (L z)) |
| change (∫ z, c*(1-(L z)^2/R^2) ∂μ)≤∫ z,g (L z) ∂μ at H |
| simp only [integral_const_mul,integral_sub (integrable_const _) (hi.div_const _), |
| integral_div,integral_const,probReal_univ,smul_eq_mul,one_mul] at H |
| have hs : 0≤∑ i,(v i)^2 := Finset.sum_nonneg (fun _ _ => sq_nonneg _) |
| have HM : (∫ z, (L z)^2 ∂μ)≤M := (scalarPath_square_bound_convex n m v hm hmono hf hc).trans (by |
| dsimp only [M]; nlinarith [sq_le_sq₀ hs hT |>.mpr hV]) |
| have Hd : (∫ z, (L z)^2 ∂μ)/R^2≤1/2 := (div_le_iff₀ (sq_pos_of_pos hR)).mpr (by linarith) |
| rw [scalarHierarchyAverage_integral hf hg] |
| simp only [zero_add] |
| change c/2≤∫ z,g (L z) ∂μ |
| nlinarith [mul_le_mul_of_nonneg_left Hd hc0] |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
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| end OAI |
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|
