File size: 4,101 Bytes
b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | 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
|