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dd14c26 | 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 | import Mathlib
import OAI.Probability.LogConcave.OraclePrograms.CenteringMeanCircuitFull
import OAI.Probability.LogConcave.Sampling.NumericalScalars
namespace OAI
section
noncomputable section
namespace LogConcaveSampling
open Filter Set MeasureTheory Quadrature ProbabilityTheory
open scoped Classical BigOperators NNReal
def CalibratedMeanCircuit {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
(hF : Primitive F lam) (x : Point d) {r T h ψ s : ℝ} (hr : 0<r)
(hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1) (hh : 0<h)
(n N : ℕ) (err : ℝ) : Prop :=
let e := probabilityEndpoint hT0 hT1 hh (n+1)
let μ := gibbs (centeringPotential F x r T)
∃X : StationaryPath μ
(skewLieField (centeringPotential F x r T)
(jointSkew (centeringKernel hF x hr hl hT0.le hT1) s)),
(∀i,Integrable (fun y => ‖stationaryPicard n N
(centeringVelocityJoint F x r T h ψ s n n N e) y i-X.path y (probabilityNodes n i)‖^2) μ) ∧
(∀i,(∫y,‖stationaryPicard n N
(centeringVelocityJoint F x r T h ψ s n n N e) y i-X.path y (probabilityNodes n i)‖^2 ∂μ)≤
(circuitD F x)^2*err) ∧
Measurable (stationaryMeanTarget hF x hr hl hT0.le hT1 X) ∧
μ.map (stationaryMeanTarget hF x hr hl hT0.le hT1 X)=
(stdGaussian (Point d)).map (fun g => primitiveExpectedField F x r+s • g) ∧
Integrable (fun y => ‖centeringMeanCircuit F x r T h ψ s n n N n N e y-
stationaryMeanTarget hF x hr hl hT0.le hT1 X y‖^2) μ ∧
(∫y,‖centeringMeanCircuit F x r T h ψ s n n N n N e y-
stationaryMeanTarget hF x hr hl hT0.le hT1 X y‖^2 ∂μ)≤(s*circuitD F x)^2*err
theorem exists_calibrated_mean_circuit {κ J b t : ℝ}
(hκ : 0<κ) (hJ : 1≤J) (hb : 1/2≤b) (ht : 0<t) (htsmall : t<1/100) :
∀{w : ℝ},0<w → w<t → ∃n : ℕ,
J+10<w*((n:ℝ)-3) ∧ J+10<t*((n:ℝ)-3) ∧ 4*(J+10)<(n:ℝ) ∧
∀{S : ℝ},0≤S → ∀ksize : ℕ,∀ᶠ d : ℕ in atTop,
let T := sourceCorrelation (κ*t) d
let h := (d:ℝ)^(-(κ*w))
let ψ := (d:ℝ)^(-(κ*(4*t)))
∃hT0 : 0<T,∃hT1 : T<1,∃hh : 0<h,
∀{F : Point d → ℝ} {lam : ℝ≥0},∀hF : Primitive F lam,∀x : Point d,
∀{r s : ℝ},∀hr : 0<r,r≤1 → 0<lam → (lam:ℝ)*r≤1 → 0<s →
(lam:ℝ)*r^2≤S*dimensionLog d^ksize*(d:ℝ)^(-(κ*b)) →
(lam:ℝ)*r/s≤S*dimensionLog d^ksize*(d:ℝ)^(-(κ*b)) →
∃hl : (lam:ℝ)*r^2≤1/2,
CalibratedMeanCircuit (ψ:=ψ) (s:=s) hF x hr hl hT0 hT1 hh n (numericalLayerCount J)
((d:ℝ)^(-(2*κ*J))) := by
intro w hw hwt
obtain ⟨n,hnw,hnt,hnn⟩ := exists_numerical_order (J:=J) ht hw
have hn : 0<n := by
have he : (0:ℝ)<n := by linarith
exact_mod_cast he
have hn1 : 1≤n := hn
obtain ⟨A,B,Ap,Ah,Lp,Lh,C,J₀,D,hC,hJ₀,hD,k,hproducer⟩ := centeringMeanCircuit_full_rms n
let U := A+B+Ap+Ah+Lp+Lh
have hU : A≤U ∧ B≤U ∧ Ap≤U ∧ Ah≤U ∧ Lp≤U ∧ Lh≤U := by
dsimp only [U]
have ha : (0:ℝ≥0)≤A := bot_le
have hb : (0:ℝ≥0)≤B := bot_le
have hc : (0:ℝ≥0)≤Ap := bot_le
have hd : (0:ℝ≥0)≤Ah := bot_le
have he : (0:ℝ≥0)≤Lp := bot_le
have hf : (0:ℝ≥0)≤Lh := bot_le
constructor <;> first | nlinarith | skip
constructor <;> first | nlinarith | skip
constructor <;> first | nlinarith | skip
constructor <;> first | nlinarith | skip
constructor <;> nlinarith
refine ⟨n,hnw,hnt,hnn,?_⟩
intro S hS ksize
have hc := centeringScalarCalibration_uniform k n (numericalLayerCount J) ksize A Ap Ah Lp U C J₀ D
hS hκ hJ hb ht htsmall hw hwt hnw hnt hnn (numericalLayerCount_bound J)
have hT := sourceCorrelation_eventually (mul_pos hκ ht)
have hmesh := sourceMesh_eventually (mul_pos hκ ht) (mul_pos hκ hw)
have hangle := sourceAngle_eventually (show 0<κ*(4*t) by positivity) n
filter_upwards [hc,hT,hmesh,hangle,eventually_ge_atTop (1:ℕ)] with d hdc hdT hdmesh hdangle hd
intro T h ψ
let R := (d:ℝ)^(-(κ*t))
have hd0 : (0:ℝ)<d := by exact_mod_cast hd
have hh : 0<h := Real.rpow_pos_of_pos hd0 _
have hR : 0<R := Real.rpow_pos_of_pos hd0 _
have hψ : 0<ψ := Real.rpow_pos_of_pos hd0 _
refine ⟨hdT.1,hdT.2.1,hh,?_⟩
intro F lam hF x r s hr hr1 hlam hL hs hl₀ hα
have hgood := hdc ⟨lam,r,s⟩ ⟨hr,hs,hl₀,hα⟩
rcases hgood with ⟨hl,hcontr,hq,hz,hm⟩
have bound {K : ℝ≥0} (hK : K≤U) : K*probabilityMeanLipschitz lam r≤1/2 :=
(mul_le_mul_of_nonneg_right hK bot_le).trans hcontr
refine ⟨hl,?_⟩
obtain ⟨X,hsi,hsb,hmeas,hlaw,hmi,hmb⟩ := hproducer hF x hr hr1 hlam hl hL hd
hdT.1 hdT.2.1 hh hdmesh.1 hdmesh.2
(bound hU.2.2.1) (bound hU.2.2.2.1) (bound hU.2.2.2.2.1) (bound hU.2.2.2.2.2)
(bound hU.1) (bound hU.2.1) (numericalLayerCount J) R hR hdT.2.2.le n hn1 ψ hψ hdangle
s hs n (numericalLayerCount J) hn hq
refine ⟨X,hsi,?_,hmeas,hlaw,hmi,?_⟩
· intro i
exact (hsb i).trans (mul_le_mul_of_nonneg_left hz (sq_nonneg _))
· exact hmb.trans (mul_le_mul_of_nonneg_left hm (sq_nonneg _))
end LogConcaveSampling
end
end
end OAI
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