| import Mathlib |
| import OAI.Probability.GaussianInformation.Dyadic.MixtureBounds |
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|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
| open MeasureTheory Metric Set |
| open scoped ENNReal NNReal |
| open MeasureTheory Set Filter |
| open scoped ENNReal Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Topology |
| open InnerProductSpace Matrix MeasureTheory Set |
| open scoped RealInnerProductSpace ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace RepeatedLocalization |
|
|
| noncomputable def weightedProjectionMoment (n k m : ℕ) (B : ℝ) : ℝ≥0∞ := |
| ENNReal.ofReal ((2:ℝ)^(k*(m-1))) * |
| (1 + tubeWeightConstant n B * ENNReal.ofReal ((2:ℝ)^(k+1)))^(m-1) |
|
|
| lemma weightedProjectionMoment_ne_top (n k m : ℕ) (B : ℝ) : |
| weightedProjectionMoment n k m B ≠ ∞ := by |
| apply ENNReal.mul_ne_top ENNReal.ofReal_ne_top |
| apply ENNReal.pow_ne_top |
| exact ENNReal.add_ne_top.mpr ⟨by simp, ENNReal.mul_ne_top (tubeWeightConstant_ne_top _ _) ENNReal.ofReal_ne_top⟩ |
|
|
| lemma actual_smoothedProjection_moment {n k m : ℕ} (hn : 0 < n) (hk : 0 < k) (hm : 1 < m) |
| (hdim : m-2+2*(k+1) ≤ n/2) (Q : Cell n) |
| (μ : Measure (Coordinate n)) [IsProbabilityMeasure μ] |
| (hsupport : μ (cellSet Q)ᶜ = 0) {B : ℝ} (hB : 0 ≤ B) (hspread : DyadicSpread μ Q B) |
| (h : Coordinate n → ℝ) (hh : Measurable h) (hnorm : eLpNorm h 2 μ ≤ 1) |
| (η : ℝ) (hη : 0 < η) : |
| (∫⁻ x, smoothedProjection (k := k) (μ.withDensity (fun z => ENNReal.ofReal |h z|)) |
| (normalizedChart Q) η x ^ m ∂(gaussianRows k (n+1)).prod volume) ≤ |
| weightedProjectionMoment n k m B := by |
| let ν := μ.withDensity (fun z => ENNReal.ofReal |h z|) |
| have hν : ν univ ≤ 1 := weightedMeasure_mass_le_one μ h hh hnorm |
| have : IsFiniteMeasure ν := ⟨hν.trans_lt (by simp)⟩ |
| have he : m-1+1 = m := by omega |
| unfold weightedProjectionMoment |
| rw [← he] |
| have hm1 : m-1+1-1 = m-1 := by omega |
| simp only [hm1] |
| apply smoothedProjection_moment_le ν hν (normalizedChart Q) (normalizedChart_measurable Q) |
| hk (m-2) _ _ (m-1) (by omega) η hη |
| intro V hV hd |
| exact weighted_affine_inverseMoment_ennreal hn Q μ hsupport hB hspread h hh hnorm V hV k hk |
| (by omega) |
|
|
| lemma actual_projection_borel_test {n k m : ℕ} (hn : 0 < n) (hk : 0 < k) (hm : 1 < m) |
| (hdim : m-2+2*(k+1) ≤ n/2) (Q : Cell n) |
| (μ : Measure (Coordinate n)) [IsProbabilityMeasure μ] |
| (hsupport : μ (cellSet Q)ᶜ = 0) {B : ℝ} (hB : 0 ≤ B) (hspread : DyadicSpread μ Q B) |
| (h : Coordinate n → ℝ) (hh : Measurable h) (hnorm : eLpNorm h 2 μ ≤ 1) |
| (g : (Fin k → Fin (n+1) → ℝ) × (Fin k → ℝ) → ℝ) (hg : Measurable g) |
| (hg0 : ∀ p, 0 ≤ g p) (hg1 : ∀ p, g p ≤ 1) : |
| (∫⁻ q, ENNReal.ofReal (g (q.1,projectVector q.1 (normalizedChart Q q.2))) |
| ∂(gaussianRows k (n+1)).prod (μ.withDensity (fun z => ENNReal.ofReal |h z|))) ≤ |
| (weightedProjectionMoment n k m B)^(1/(m:ℝ)) * |
| (∫⁻ p, ENNReal.ofReal (g p) ∂(gaussianRows k (n+1)).prod volume)^(1-1/(m:ℝ)) := by |
| let ν := μ.withDensity (fun z => ENNReal.ofReal |h z|) |
| have hν : ν univ ≤ 1 := weightedMeasure_mass_le_one μ h hh hnorm |
| have : IsFiniteMeasure ν := ⟨hν.trans_lt (by simp)⟩ |
| have : IsProbabilityMeasure (gaussianRows k (n+1)) := by unfold gaussianRows; infer_instance |
| have hb : Measurable (fun q : (Fin k → Fin (n+1) → ℝ) × Coordinate n => |
| projectVector q.1 (normalizedChart Q q.2)) := |
| continuous_projectVector.measurable.comp (measurable_fst.prodMk |
| ((normalizedChart_measurable Q).comp measurable_snd)) |
| have : IsLocallyFiniteMeasure ((gaussianRows k (n+1)).prod (volume : Measure (Fin k → ℝ))) := |
| Measure.prod.instIsLocallyFiniteMeasure |
| have H := exactMixtureLaw_borel_test (gaussianRows k (n+1)) ν _ hb hm |
| (weightedProjectionMoment n k m B) (weightedProjectionMoment_ne_top _ _ _ _) |
| (actual_smoothedProjection_moment hn hk hm hdim Q μ hsupport hB hspread h hh hnorm) |
| g hg hg0 hg1 |
| simpa only [exactMixtureLaw, lintegral_map hg.ennreal_ofReal (measurable_fst.prodMk hb)] using H |
|
|
| end RepeatedLocalization |
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| end OAI |
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