| import Mathlib | |
| import OAI.Probability.GaussianInformation.Kernel.RowConcatenation | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory Metric Set | |
| open scoped ENNReal | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal | |
| open MeasureTheory ProbabilityTheory | |
| namespace KernelPotential.PairReference | |
| open MeasureTheory ProbabilityTheory Metric Set | |
| open scoped ENNReal | |
| open Sphere Coercivity Scale ReferenceGaussian GaussianTools RowConcat | |
| noncomputable def density {k d L : ℕ} (ν : Measure (Ambient d)) (r : Fin L → ℝ) | |
| (X Z : Mat k d) (y z : Fin k → ℝ) : ℝ := | |
| mixture ν r (join k (X,Z)) (join k (y,z)) | |
| lemma density_pos {k d L : ℕ} (hL : 0 < L) (ν : Measure (Ambient d)) | |
| [IsProbabilityMeasure ν] (r : Fin L → ℝ) (hr : ∀ j,0 < r j) | |
| (X Z : Mat k d) (y z : Fin k → ℝ) : 0 < density ν r X Z y z := | |
| mixture_pos hL ν r hr _ _ | |
| lemma density_measurable {k d L : ℕ} (ν : Measure (Ambient d)) [IsFiniteMeasure ν] | |
| (r : Fin L → ℝ) : Measurable (fun t : (Mat k d × Mat k d) × ((Fin k → ℝ) × (Fin k → ℝ)) => | |
| density ν r t.1.1 t.1.2 t.2.1 t.2.2) := by | |
| have hM := measurable_mixture (p := 2*k) ν r | |
| have hX := (join (E := Fin d → ℝ) k).measurable | |
| have hY := (join (E := ℝ) k).measurable | |
| unfold density | |
| fun_prop | |
| lemma density_probability {k d L : ℕ} (hL : 0 < L) (ν : Measure (Ambient d)) | |
| [IsProbabilityMeasure ν] (r : Fin L → ℝ) (hr : ∀ j,0 < r j) (X Z : Mat k d) : | |
| (∫⁻ yz : (Fin k → ℝ) × (Fin k → ℝ),ENNReal.ofReal (density ν r X Z yz.1 yz.2) | |
| ∂(volume : Measure (Fin k → ℝ)).prod volume)=1 := by | |
| have hm : Measurable (fun y : Fin (2*k) → ℝ => ENNReal.ofReal (mixture ν r (join k (X,Z)) y)) := | |
| ((measurable_mixture ν r).comp (measurable_const.prodMk measurable_id)).ennreal_ofReal | |
| have h := (join_volume_measurePreserving k).lintegral_comp hm | |
| change (∫⁻ yz,ENNReal.ofReal (mixture ν r (join k (X,Z)) (join k yz)) ∂volume.prod volume)=1 | |
| rw [h] | |
| exact lintegral_mixture hL ν r hr _ | |
| lemma rows_integral {k d L : ℕ} (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] | |
| (r : Fin L → ℝ) (s : Ambient d) (e e' : Fin k → ℝ) : | |
| (∫ XZ : Mat k d × Mat k d,Real.log (density ν r XZ.1 XZ.2 | |
| (applyRows XZ.1 s+e) (applyRows XZ.2 s+e')) | |
| ∂(Measure.pi (fun _ : Fin k => scalarRow d)).prod (Measure.pi (fun _ : Fin k => scalarRow d)))= | |
| ∫ X : Mat (2*k) d,Real.log (mixture ν r X (applyRows X s+join k (e,e'))) | |
| ∂Measure.pi (fun _ : Fin (2*k) => scalarRow d) := by | |
| have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance | |
| have he (X Z : Mat k d) : join k (applyRows X s+e,applyRows Z s+e')= | |
| applyRows (join k (X,Z)) s+join k (e,e') := by | |
| rw [join_add,show join k (applyRows X s,applyRows Z s)=applyRows (join k (X,Z)) s from join_rows k d X Z s] | |
| simp only [density,he] | |
| exact (join_measurePreserving k (scalarRow d)).integral_comp (join k).measurableEmbedding | |
| (fun X => Real.log (mixture ν r X (applyRows X s+join k (e,e')))) | |
| lemma reference_J_lower_pair {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 : 0 < L) | |
| (hcut : ENNReal.ofReal (H*(9*3^d)*(4:ℝ)^(d-1))* | |
| ENNReal.ofReal ((2:ℝ)^(2*(d/16)+1))*ENNReal.ofReal ((1/2:ℝ)^L) ≤ 1) | |
| (e e' : Fin (d/16) → ℝ) (he : ∀ i,|e i| ≤ radius L) (he' : ∀ i,|e' i| ≤ radius L) : | |
| Real.log (J ν s).toReal-10*(d:ℝ)-2*Real.log ((L:ℝ)+2) ≤ | |
| (∫ XZ : Mat (d/16) d × Mat (d/16) d, | |
| Real.log (density ν (fun j : Fin L => radius j) XZ.1 XZ.2 | |
| (applyRows XZ.1 s+e) (applyRows XZ.2 s+e')) | |
| ∂(Measure.pi (fun _ : Fin (d/16) => scalarRow d)).prod | |
| (Measure.pi (fun _ : Fin (d/16) => scalarRow d))) := by | |
| rw [rows_integral] | |
| exact reference_J_lower hd ν hH hν s hs L hL hcut _ | |
| (join_bound _ _ _ (fun x : ℝ => |x| ≤ radius L) he he') | |
| end KernelPotential.PairReference | |
| end OAI | |