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7d9f027 | 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 | import Mathlib
import OAI.Probability.SKRatio.Gaussian.PlantedEmpirical
import OAI.Probability.SKRatio.Quantization.QuantizerExistence
import OAI.Probability.SKRatio.Quantization.BinPullback
import OAI.Probability.SKRatio.Quantization.CompactTolerance
namespace OAI
noncomputable section
open scoped BigOperators NNReal ENNReal Topology
open MeasureTheory ProbabilityTheory Filter Real
namespace SKRatio.Bins
open Planted Scalar SKRatioClock.Regression
lemma l2Norm_field (β : ℝ) : l2Norm (fieldLaw β) id=sqrt (β^2+β^4) := by
unfold l2Norm
exact congrArg sqrt (fieldLaw_secondMoment β)
lemma quantizer_l2Norm_le {β R : ℝ} (hR : 0<R) {N : ℕ} (hN : 0<N)
{a : ℝ} (ha : 0≤a) (herr : (∫ x, (x-gridQuant R N x)^2 ∂fieldLaw β)≤a^2) :
l2Norm (fieldLaw β) (gridQuant R N)≤ sqrt (β^2+β^4)+a := by
have hs := measurable_intervalBin (Nat.succ_pos N) (gridCut_strictMono hR hN)
have hH := memLp_finite_comp (μ := fieldLaw β) hs (gridRep R N)
have he : l2Norm (fieldLaw β) (fun x => gridQuant R N x-x)≤a := by
unfold l2Norm
apply (sqrt_le_iff).mpr ⟨ha,?_⟩
simpa only [sub_sq_comm] using herr
have ht := l2Norm_add (memLp_field β) (hH.sub (memLp_field β))
change l2Norm (fieldLaw β) (fun x => x+(gridQuant R N x-x))≤
l2Norm (fieldLaw β) id+l2Norm (fieldLaw β) (fun x => gridQuant R N x-x) at ht
have heq : (fun x => x+(gridQuant R N x-x))=gridQuant R N := by ext x; ring
rw [heq] at ht
rw [l2Norm_field] at ht
linarith only [ht,he]
lemma finiteV_le_of_quantizer {β C a : ℝ} (hβ : 0≤β) (ha : 0≤a)
(hscalar : ∀ b r : ℝ → ℝ, MemLp b 2 (fieldLaw β) → MemLp r 2 (fieldLaw β) →
(∫ x, b x^2+r x^2 ∂fieldLaw β)=1 → variational β b r≤C)
{R : ℝ} (hR : 0<R) {N : ℕ} (hN : 0<N) (hβne : β≠0)
(herr : (∫ x, (x-gridQuant R N x)^2 ∂fieldLaw β)≤a^2)
{T : ℝ} (hT : 0≤T) (hsize : sqrt (β^2+β^4)+a≤T)
(hv : ∀ x, |v x-v (gridQuant R N x)|≤a)
(hmv : ∀ x, |m x*v x-m (gridQuant R N x)*v (gridQuant R N x)|≤a)
(hf : ∀ x y, |v y^2/(w x+w y)-v (gridQuant R N y)^2/
(w (gridQuant R N x)+w (gridQuant R N y))|≤a)
(hk : ∀ x y, |kernel x y-kernel (gridQuant R N x) (gridQuant R N y)|≤a)
(z : Parameters (Fin (N+2))) :
finiteV β (massRoot (fieldLaw β) (intervalBin (gridCut R N))) (gridRep R N) z ≤
C+(2*a+2*T*a+5*β^2*a+4*β*sqrt a) := by
let σ := intervalBin (gridCut R N)
have hs : Measurable σ := measurable_intervalBin (Nat.succ_pos N) (gridCut_strictMono hR hN)
have hm : ∀ d, 0<binMass (fieldLaw β) σ d :=
intervalBin_mass_pos (Nat.succ_pos N) (gridCut_strictMono hR hN) β hβne
let b := pullback (fieldLaw β) σ (paramB z)
let r := pullback (fieldLaw β) σ (paramR z)
have hb : MemLp b 2 (fieldLaw β) := memLp_pullback hs _
have hr : MemLp r 2 (fieldLaw β) := memLp_pullback hs _
have he := pullback_normalized (μ := fieldLaw β) hs hm z
have hH : MemLp (gridQuant R N) 2 (fieldLaw β) := memLp_finite_comp hs (gridRep R N)
have herr' : l2Norm (fieldLaw β) (fun x => x-gridQuant R N x)≤a :=
(sqrt_le_iff).mpr ⟨ha,herr⟩
have hbnd := fieldForm_error (μ := fieldLaw β) β hβ hb hr
(measurable_pullback hs _) (measurable_pullback hs _) measurable_id
(measurable_gridQuant hR hN) (memLp_field β) hH he ha hT herr'
((quantizer_l2Norm_le hR hN ha herr).trans hsize) hv hmv hf hk
have hp := fieldForm_pullback (μ := fieldLaw β) hs hm β (gridRep R N) z
change fieldForm (fieldLaw β) β (gridQuant R N) b r = _ at hp
rw [fieldForm_id,hp] at hbnd
have hu := hscalar b r hb hr he
have hle := neg_le_abs (variational β b r-finiteV β (massRoot (fieldLaw β) σ) (gridRep R N) z)
change finiteV β (massRoot (fieldLaw β) σ) (gridRep R N) z≤_
linarith only [hle,hbnd,hu]
theorem scalar_quantizer_exists
{Ω : ℕ → Type*} [∀ n, MeasurableSpace (Ω n)]
{ρ : ∀ n, Measure (Ω n)} {X : ∀ n, Ω n → Fin n → ℝ}
(hX : ExponentialSquareTails ρ X)
{β K C e : ℝ} (hβ : 0<β) (hK : 0≤K) (he : 0<e)
(hscalar : ∀ b r : ℝ → ℝ, MemLp b 2 (fieldLaw β) → MemLp r 2 (fieldLaw β) →
(∫ x, b x^2+r x^2 ∂fieldLaw β)=1 → variational β b r≤C) :
∃ a R : ℝ, ∃ N : ℕ, 0<a ∧ a<1 ∧ 0<R ∧ 0<N ∧
(2*a+2*(sqrt (β^2+β^4)+2)*a+5*β^2*a+K*a+4*β*sqrt a<e) ∧
(∀ x, |v x-v (gridQuant R N x)|≤a) ∧
(∀ x, |m x*v x-m (gridQuant R N x)*v (gridQuant R N x)|≤a) ∧
(∀ x y, |v y^2/(w x+w y)-v (gridQuant R N y)^2/
(w (gridQuant R N x)+w (gridQuant R N y))|≤a) ∧
(∀ x y, |kernel x y-kernel (gridQuant R N x) (gridQuant R N y)|≤a) ∧
(∀ z, finiteV β (massRoot (fieldLaw β) (intervalBin (gridCut R N))) (gridRep R N) z<C+e) ∧
ExponentiallyRare ρ (fun n => {ω | a≤ siteNorm (fun i => X n ω i-gridQuant R N (X n ω i))}) := by
let T := sqrt (β^2+β^4)+2
have hT : 0≤T := by dsimp [T]; positivity
obtain ⟨a,ha,ha1,hea⟩ := field_error_tolerance β (T+K/2) he
have htotal : 2*a+2*T*a+5*β^2*a+K*a+4*β*sqrt a<e := by linarith only [hea]
obtain ⟨δ,hδ,ht⟩ := compact_coefficients_tolerance ha
obtain ⟨R,N,hR,hN,hcomp,hl2,hemp⟩ := quantizer_exists hX (μ := fieldLaw β)
(memLp_field β).integrable_sq hδ ha
obtain ⟨hv,hmv,hf,hk⟩ := ht (gridQuant R N) hcomp
refine ⟨a,R,N,ha,ha1,hR,hN,htotal,hv,hmv,hf,hk,?_,hemp⟩
intro z
have hp := finiteV_le_of_quantizer hβ.le ha.le hscalar hR hN hβ.ne'
hl2.le hT (by dsimp [T]; linarith only [ha1]) hv hmv hf hk z
have hKa := mul_nonneg hK ha.le
exact hp.trans_lt (by linarith only [htotal,hKa])
end SKRatio.Bins
end
end OAI
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