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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import Mathlib
import OAI.Probability.GaussianInformation.Kernel.DriftCutoff
namespace OAI
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
open MeasureTheory ProbabilityTheory
open MeasureTheory
open scoped ENNReal
namespace KernelPotential.NoisyReference
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
open Projection Projection.Support Noise ReferenceGaussian GaussianTools RowConcat Scale
lemma sample_density {k d L : ℕ} (ν : Measure (Ambient d)) (r : Fin L → ℝ)
(s : Ambient d) (t : Observation k d) :
PairReference.density ν r (sampleEquiv s t).1.1 (sampleEquiv s t).1.2
(sampleEquiv s t).2.1 (sampleEquiv s t).2.2 =
mixture ν r (join k t.1) (applyRows (join k t.1) s+join k t.2) := by
change mixture ν r (join k t.1) (join k (applyRows t.1.1 s+t.2.1,applyRows t.1.2 s+t.2.2))=_
rw [join_add,show join k (applyRows t.1.1 s,applyRows t.1.2 s)=applyRows (join k t.1) s from join_rows k d _ _ s]
lemma pair_noise_support (k : ℕ) {η : ℝ} (hη : 0 < η) :
∀ᵐ e ∂(noiseLaw k η).prod (noiseLaw k η),∀ i,|join k e i| ≤ η := by
have := noise_probability k hη
have h1 : ∀ᵐ e ∂(noiseLaw k η).prod (noiseLaw k η),e.1 ∈ closedBox 0 η :=
measurePreserving_fst.quasiMeasurePreserving.ae (noise_support k η)
have h2 : ∀ᵐ e ∂(noiseLaw k η).prod (noiseLaw k η),e.2 ∈ closedBox 0 η :=
measurePreserving_snd.quasiMeasurePreserving.ae (noise_support k η)
filter_upwards [h1,h2] with e h1 h2
apply join_bound k e.1 e.2 (fun x : ℝ => |x| ≤ η)
· simpa only [closedBox,Set.mem_ofPred_eq,Pi.zero_apply,sub_zero] using h1
· simpa only [closedBox,Set.mem_ofPred_eq,Pi.zero_apply,sub_zero] using h2
lemma sampling_noise_support (k d : ℕ) {η : ℝ} (hη : 0 < η) :
∀ᵐ t ∂samplingLaw k d η,∀ i,|join k t.2 i| ≤ η := by
have := noise_probability k hη
have : IsProbabilityMeasure (gaussianRows k d) := by unfold gaussianRows; infer_instance
exact measurePreserving_snd.quasiMeasurePreserving.ae (pair_noise_support k hη)
lemma integrable_pairEnergy (k d : ℕ) :
Integrable (fun t : (Mat k d) × (Mat k d) => matEnergy (join k t))
((gaussianRows k d).prod (gaussianRows k d)) := by
have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance
exact (join_measurePreserving k (scalarRow d)).integrable_comp_of_integrable integrable_matEnergy
lemma sampling_log_integrable {k d L : ℕ} (hL : 0 < L)
(ν : Measure (Ambient d)) [IsProbabilityMeasure ν] (r : Fin L → ℝ) (hr : ∀ j,0 < r j)
(s : Ambient d) {R η : ℝ} (hR : 0 ≤ R) (hη : 0 < η)
(hs : ∀ᵐ u ∂ν,u ∈ closedBall s R) :
Integrable (fun t : Observation k d => Real.log (PairReference.density ν r
(sampleEquiv s t).1.1 (sampleEquiv s t).1.2 (sampleEquiv s t).2.1 (sampleEquiv s t).2.2))
(samplingLaw k d η) := by
have := noise_probability k hη
have : IsProbabilityMeasure (gaussianRows k d) := by unfold gaussianRows; infer_instance
obtain ⟨M,_,hb⟩ := log_mixture_uniform_bound (p := 2*k) hL ν r hr hR hη.le
have hm : Measurable (fun t : Observation k d => Real.log (PairReference.density ν r
(sampleEquiv s t).1.1 (sampleEquiv s t).1.2 (sampleEquiv s t).2.1 (sampleEquiv s t).2.2)) := by
have hD := PairReference.density_measurable (k := k) ν r
have hS := (sampleEquiv (k := k) s).measurable
exact Real.measurable_log.comp (hD.comp hS)
refine ((integrable_const 1).add ((integrable_pairEnergy k d).comp_fst _)).const_mul M |>.mono' hm.aestronglyMeasurable ?_
filter_upwards [sampling_noise_support k d hη] with t ht
rw [Real.norm_eq_abs,sample_density]
exact hb s hs (join k t.1) (join k t.2) ht
lemma q_probability {k d L : ℕ} (hL : 0 < L) (ν : Measure (Ambient d))
[IsProbabilityMeasure ν] (r : Fin L → ℝ) (hr : ∀ j,0 < r j) :
(∫⁻ z : Observation k d,ENNReal.ofReal (PairReference.density ν r z.1.1 z.1.2 z.2.1 z.2.2)
∂observationBase k d)=1 := by
have : IsProbabilityMeasure (gaussianRows k d) := by unfold gaussianRows; infer_instance
rw [observationBase,lintegral_prod _ (PairReference.density_measurable ν r).ennreal_ofReal.aemeasurable]
simp only [PairReference.density_probability hL ν r hr,lintegral_one,measure_univ]
lemma noisy_reference_J_lower {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)
{η : ℝ} (hη : 0 < η) (hηL : η ≤ radius L) :
Real.log (J ν s).toReal-10*(d:ℝ)-2*Real.log ((L:ℝ)+2) ≤
(∫ z : Observation (d/16) d,Real.log (PairReference.density ν (fun j : Fin L => radius j)
z.1.1 z.1.2 z.2.1 z.2.2)
∂(observationBase (d/16) d).withDensity (fun z => observationKernel η z s)) := by
have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance
have := noise_probability (d/16) hη
have : IsProbabilityMeasure (gaussianRows (d/16) d) := by unfold gaussianRows; infer_instance
have hiSupport : ∀ᵐ u ∂ν,u ∈ closedBall s 2 := by
filter_upwards [(Measure.absolutelyContinuous_of_le_smul hν).ae_le (Sphere.ae_unit_norm (by omega : 0 < d))] with u hu
rw [mem_closedBall,dist_eq_norm]
exact (norm_sub_le _ _).trans (by rw [hu,hs]; norm_num)
have hi := sampling_log_integrable (k := d/16) hL ν (fun j : Fin L => radius j) (fun j => radius_pos j)
s (by norm_num : (0:ℝ) ≤ 2) hη hiSupport
rw [sample_integral hη s,samplingLaw,integral_prod _ hi,integral_integral_swap hi]
have hir := hi.integral_prod_right
calc
_ = ∫ _e : (Fin (d/16) → ℝ) × (Fin (d/16) → ℝ),
Real.log (J ν s).toReal-10*(d:ℝ)-2*Real.log ((L:ℝ)+2)
∂(noiseLaw (d/16) η).prod (noiseLaw (d/16) η) := by simp
_ ≤ _ := by
apply integral_mono_ae (integrable_const _) hir
filter_upwards [pair_noise_support (d/16) hη] with e he
have he' : ∀ i,|join (d/16) e i| ≤ radius L := fun i => (he i).trans hηL
have h := reference_J_lower hd ν hH hν s hs L hL hcut _ he'
have hrw : (∫ XZ : Mat (d/16) d × Mat (d/16) d,
Real.log (PairReference.density ν (fun j : Fin L => radius j)
(sampleEquiv s (XZ,e)).1.1 (sampleEquiv s (XZ,e)).1.2
(sampleEquiv s (XZ,e)).2.1 (sampleEquiv s (XZ,e)).2.2)
∂(gaussianRows (d/16) d).prod (gaussianRows (d/16) d))=
∫ X : Mat (2*(d/16)) d,Real.log (mixture ν (fun j : Fin L => radius j) X
(applyRows X s+join (d/16) e)) ∂Measure.pi (fun _ : Fin (2*(d/16)) => scalarRow d) := by
simp only [sample_density]
exact (join_measurePreserving (d/16) (scalarRow d)).integral_comp'
(fun X : Mat (2*(d/16)) d => Real.log (mixture ν (fun j : Fin L => radius j) X
(applyRows X s+join (d/16) e)))
rw [hrw]
exact h
end KernelPotential.NoisyReference
end OAI
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